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 THE ELEMENTS OF LOGIC
 
 THE ELEMENTS 
 
 OF 
 
 LOGIC 
 
 THEORETICAL AND PRACTICAL 
 
 BY 
 
 JAMES H. HYSLOP PH.D. 
 
 INSTRUCTOR IN LOGIC PSYCHOLOGY AND ETHICS COLUMBIA COLLEGE 
 NEW YORK 
 
 THIRD EDITION 
 
 NEW YORK 
 
 CHAELES SCEIBNEE'S SONS 
 1894
 
 COPYRIGHT, 1892, BY 
 CHARLES SCRIBNERS SONS 
 
 mow oiRECTanr 
 
 i AND BOOKBINDING COMPANY 
 NEW YORK
 
 PEEFACE 
 
 ALL who have had anything to do with Logic will recognize, 
 without being told it, the extent to which I am indebted to 
 Jevons for both matter and method in the treatment of this 
 subject. But they will quite as readily perceive the deviations 
 from him, and the additions which I have made with the hope 
 of improving upon his work. Jevons designed his " Element- 
 ary Lessons " to direct the student in practical reasoning and 
 correct thinking in professional vocations. I have intended 
 the present work to serve the same end, and, if possible, more 
 completely than Jevons. For this reason I have been deter- 
 mined in the development of the subject by the questions con- 
 stantly put me by students. As far as possible I have endeav- 
 ored to answer all questions likely to be raised when framing 
 rules about the logical treatment of conceptions and proposi- 
 tions. It will be apparent, therefore, that the plan has neces- 
 sitated many judicious omissions both of irrelevant matter in 
 Jevons and others, and of certain theoretical discussions pecul- 
 iar to the scientific rather than the practical aspect of Logic. 
 With a view to the student's guidance and mental discipline I 
 have added a large number of practical questions and exer- 
 cises at the close of the book. 
 
 It is important to remark, however, that I have aimed to 
 satisfy two wants at the same time. I have tried to produce 
 a work that could be used both for beginners and for advanced 
 students of the subject, but not for those who care to go into 
 it exhaustively. Beginners can be directed to the definitions 
 and illustrations, and the heavier matter omitted, while the 
 special questions developed at length and in a more technical 
 manner can be taken up by the advanced student, or by those 
 who are interested sufficiently to press further inquiries. But 
 
 2223781
 
 vi PREFACE 
 
 only one or two chapters are inserted which do not have ref- 
 erence to the practical advantages of Logic. The full treat- 
 ment of all subjects is designed to afford students a better 
 guide than Jevons can possibly be. With this in view I have 
 laid considerable stress upon the nature of Conceptions, Prop- 
 ositions, and the Classification of Fallacies. The last subject 
 always gives students a great deal of trouble, because they 
 find no means of distinguishing one fallacy from another as 
 clearly as is desirable. By the manner in which I have tried 
 to treat the subject I hope I have rendered their work easier 
 in this important part of Logic. De Morgan is the only 
 author I know who has approximated what the subject of fal- 
 lacies deserves ; but he leaves very much to be desired. He 
 has not attempted either a classification or a discussion of the 
 way in which several fallacies may coincide at the same time, 
 and differ only in the point of view from which they are re- 
 garded. 
 
 In the treatment of certain questions it has seemed fit to 
 venture upon some distinctions which may be regarded as 
 innovations. I have distinguished between two kinds of 
 " General Terms," which I have called " Mathematical " and 
 " Logical Generals," for reasons that the text must explain. In 
 pursuance of this distinction and the double signification of 
 the term " Genus," I have also coined the term " Conferentia," 
 as denoting the " logical " as opposed to the " mathematical 
 genus." A term is required to contrast with " differentia," as 
 " genus " contrasts with " species." Whether I am justified in 
 the invention must be a matter between me and my critics 
 when they have examined the work. 
 
 If I mistake not, I have somewhat modified the ordinary 
 treatment of Induction. First, I have carefully distinguished 
 between " inductive reasoning " or " inference," and what is 
 ordinarily called "Induction " or " Inductive Method." Then, 
 to complete the thought involved in this distinction, I have 
 discussed " Scientific Method " as somewhat extra-logical, and 
 included therein the process of verification as involving, in 
 some instances at least, more than purely inductive inference.
 
 PREFACE vii 
 
 The importance of this will be apparent when we observe the 
 constant confusion by logicians of formal Induction as a pro- 
 cess of reasoning with the methods of discovering and verify- 
 ing new knowledge. Bacon and Mill have not sufficiently dis- 
 tinguished them, and on that account have done much to mis- 
 represent and to disparage the importance of Deduction. I 
 have endeavored, so far as I am able, to remove this defect 
 from the discussion. But of my success I am probably not a 
 competent judge. Besides, the limits to which I have been 
 compelled to reduce the discussion may appear to make this 
 part of the work unsatisfactory. I am content, however, if I 
 have given a hint in the right direction. 
 
 Special acknowledgments are due to Mr. Fowler and Mr. 
 Venn for the valuable help which I have received from their 
 works on Logic. Mr. Venn's recent volume on " Empirical 
 Logic " is unusually sagacious and suggestive in all the most 
 fundamental matters. Jevons receives due acknowledgment 
 in the extent to which I have modelled my work after his. 
 
 JAMES H. HYSLOP. 
 COLUMBIA COLLEGE, April 23, 1892.
 
 PREFACE TO THE SECOND EDITION. 
 
 THE favorable reception and rapid exhaustion of the first 
 edition offer an opportunity to correct a number of typographical 
 errors and to introduce a few minor changes. The most impor- 
 tant of these were made on pages 120 and 170. In the former 
 case the doctrine about negative exclusive propositions has been 
 modified, and in the latter a few changes were made to correct 
 the consequences of an error previously overlooked. Rules for the 
 sorites have been added at the end of Chapter XIV., and a note 
 to Chapter XIX. has also been added on page 274, in order to 
 prevent any possible misunderstanding of what was said about 
 exclusive propositions in the main part of the chapter. No other 
 important changes have been made. 
 
 JAMES H. HYSLOP. 
 COLUMBIA COLLEGE, 
 January 11, 1883. 
 
 PREFACE TO THE THIRD EDITION. 
 
 THE third edition of this work offers an opportunity to make 
 some further corrections of statement, and of typographical errors 
 overlooked in the previous editions. Thus page 80 has been 
 wholly rewritten in deference to criticisms in Mind, which gained 
 their point wholly from my lack of clear statement. This diffi- 
 culty, I hope, is now overcome. The note also on page 274 has 
 been recast and lengthened, and a few changes of words or sen-* 
 tences here and there have been made, in order to avoid wrong 
 or ambiguous statement. These are all of the important altera- 
 tions affecting the book. 
 
 JAMES H. HYSLOP. 
 COLUMBIA COLLEGE, 
 October 2T, 1894.
 
 CONTENTS 
 
 CHAPTER I. PAGE 
 
 INTRODUCTION, DEFINITION AND DIVISIONS OP LOGIC, . . 1 
 
 CHAPTER II. 
 ELEMENTS OF LOGICAL DOCTRINE, 16 
 
 CHAPTER III. 
 TERMS OR CONCEPTS AND THEIR KINDS, . . . . .31 
 
 CHAPTER IV. 
 THE AMBIGUITY OF TERMS, 50 
 
 CHAPTER V. 
 THE INTENSION AND THE EXTENSION OF CONCEPTS, ... 68 
 
 CHAPTER VI. 
 DEFINITION AND DIVISION, 82 
 
 CHAPTER VII. 
 PROPOSITIONS OR JUDGMENTS, ...... 105 
 
 CHAPTER VIII. 
 THE RELATION BETWEEN SUBJECT AND PREDICATE, . . . 122
 
 X CONTENTS 
 
 CHAPTER IX. PAGE 
 
 OPPOSITION, . . . . 141 
 
 CHAPTER X. 
 IMMEDIATE INFERENCE, 154 
 
 CHAPTER XL 
 PRINCIPLES OF MEDIATE REASONING, 171 
 
 CHAPTER XII. 
 MOODS AND FIGURES OF THE SYLLOGISM, . ... 181 
 
 CHAPTER XIII. 
 REDUCTION OF MOODS AND FIGURES, 190 
 
 CHAPTER XIV. 
 FORMS OF SYLLOGISTIC REASONING, 19? 
 
 CHAPTER XV. 
 HYPOTHETICAL REASONING 204 
 
 CHAPTER XVL 
 DISJUNCTIVE SYLLOGISMS, 212 
 
 CHAPTER XVII. 
 CLASSIFICATION OF FALLACIES, 219 
 
 CHAPTER XVIII. 
 MATERIAL FALLACIES 228
 
 CONTENTS xi 
 
 CHAPTER XIX. PA GB 
 
 QUANTIFICATION OF THE PREDICATE 262 
 
 CHAPTER XX. 
 
 MATHEMATICAL AND OTHER REASONING, 275 
 
 CHAPTER XXI. 
 
 THE LAWS OF THOUGHT, 290 
 
 CHAPTER XXII. 
 
 INDUCTIVE REASONING, 295 
 
 CHAPTER XXIII. 
 
 SCIENTIFIC METHOD, . 336 
 
 PRACTICAL QUESTIONS AND PROBLEMS, 367 
 
 PRACTICAL EXERCISES, 383 
 
 INDEX, 399
 
 CHAPTER I. 
 INTRODUCTION DEFINITION AND DIVISIONS OF LOGIC 
 
 I. DEFINITION. Logic has sometimes been defined as a 
 science, sometimes as an art, and sometimes as both a science 
 and an art. Dr. Watts calls it "the Art of Thinking ;" Thomp- 
 son, " the Science of the Laws of Thought ; " and Whately, 
 " the Science and also the Art of Reasoning." More elaborate 
 and technical definitions are such as Hamilton's, that " Logic 
 is the Science of the Formal and Necessary Laws of Thought 
 as Thought ; " or Ueberweg's, that it is " the Science of the 
 Regulative Laws of Human Knowledge." Most writers define 
 it as a science instead of an art, and in so far as they so regard 
 it their views of it are substantially the same. They begin to 
 diverge from each other only when they speak of it as an art. 
 Yet a careful examination of the various usages of the term 
 " art " will show that the difference between it and a science 
 is less than general disputes would imply. This is apparent if 
 we but reflect that science and art usually have the same sub- 
 ject-matter, although they have different ends in view. A 
 science teaches us to know ; an art, to do. Science endeavors 
 to discover truth or knowledge, art to apply it, or to formu- 
 late rules for applying it, to the realization of some other end. 
 But it is evident in such a view that art assumes knowledge as 
 a condition of itself, and hence science and art may go together 
 as complementary of each other. This is true of Logic to the 
 extent that it may be regarded both as a science and as an art, 
 according as it aims at certain truths, or at the application of
 
 2 ELEMENTS OF LOGIC 
 
 them in practice. Logic as a science aims to ascertain what 
 are the laws of thought ; as an art it aims to apply these laws 
 to the detection of fallacies or for the determination of correct 
 reasoning. As a science Logic will be concerned chiefly, if not 
 wholly, with the general principles of thinking, and only indi- 
 rectly with truths of any other kind. The laws of thought, the 
 object to which they can be applied, or the kind of phenomena 
 in which they are embodied, are all that Logic as a science 
 need occupy itself with. Truths or knowledge in other fields 
 of mental interest may be useful for illustration, but they are 
 not to be investigated by it. Only the truth, extent, nature, 
 and validity of the laws of rational thinking come under its 
 scientific aspect. But when we wish to know whether other 
 bodies of knowledge, in which reasoning is involved, have been 
 correctly obtained or proved, we subject them to the test of 
 logical laws, and to do this we may be obliged to adopt a large 
 system of rules for practice. The formulation of such rules 
 and the testing of the various material truths of other sciences 
 are left to Logic as an art-. The truth and validity of its gen- 
 eral laws will be assumed and admitted. The problem will be 
 to find whether individual processes of reasoning have been 
 conducted in conformity with those laws or not. Hence, as an 
 art, it is concerned with a larger system of truths than it is as 
 a science. In both it is concerned with the laws of thought ; 
 but as a science it treats those laws as an end for its own sake, 
 and as an art it treats them as a means to a remoter end. 
 Hence there is no necessity of deciding whether it is one or 
 the other. The old controversy concerning that question imay 
 thus be settled by defining the subject as both a science and 
 an art under limitations.* Other aspects of the definition 
 
 * For a discussion of the nature of Logic, and of the relation between 
 science and art, see Whately : Elements of Logic, Introduction ; Fowler : 
 Element of Deductive Logic, Introduction, Chapter II. ; Thompson : Laws 
 of Thought, Introduction, Section 6 ; Hamilton : Lectures on Metaphys- 
 ics, Lecture VII. ; Lectures in Logic. Lecture I. ; J. S. Mill: Logic, In- 
 troduction ; and Examination of the Philosophy of Sir William Hamilton, 
 Chapter XX.
 
 DEFINITION AND DIVISIONS OF LOGIC 3 
 
 will require separate consideration. But to indicate what 
 these are we shall adopt as the most complete definition for 
 our purposes that which makes Logic "the Science of the 
 Formal Laws of Thought." Three terms of this definition re- 
 quire special examination. 
 
 1st. Thought. "Thought," in common usage, is a very 
 comprehensive term. It is even coextensive with consciousness 
 or mind. We use the expression, " I have such a thing in my 
 thoughts," whereby we mean merely that attention perhaps is 
 occupied with a particular idea. It is not in any such sense 
 that Logic must use the term. To have an idea in conscious- 
 ness does not necessarily imply that any logical processes are 
 going on, or that the mind is " thinking " logically about the 
 fact. It may denote no more than an act of perception or at- 
 tention. In the loose sense of the term, therefore, " thought " 
 might denote any conscious act of the mind. But to regard 
 Logic as the science of such activities and their laws would 
 identify it with Psychology. Hence "thought" must denote 
 a more specific act of the mind, and this is supposed to be the 
 act which compares and reasons. The term may denote both 
 the act and the product of the rational faculty. 
 
 The various acts of the mind may be denominated sensation, 
 perception, apprehension or cognition, memory, association, 
 attention, etc. But all of these are comparatively simple acts. 
 They do not require any act of comparison by the mind. 
 " Thought," in the logical sense, does require such compari- 
 son. The simple perceptive acts of the mind have but one 
 thing as the object of consciousness, and hence denote either 
 presentations or individual states of mind without taking into 
 account any relations that might be connected with them. 
 " Thought," on the other hand, does explicitly express the 
 consciousness of some relation between two or more objects 
 held together in consciousness at the same time. Thus I may 
 perceive a tree, a house, a man, without performing any men- 
 tal act which thinks them in relation to other objects of a like 
 or different kind, or without apprehending their meaning. 
 But if I think of a tree as a vegetable, of a house as a useful
 
 4 ELEMENTS OF LOGIC 
 
 structure, or of a man as rational, I am apprehending the 
 meaning or relation of the several objects, holding two con- 
 ceptions in the mind at the same time and pronouncing upon 
 their connection or disconnection, as the case may be. Thus 
 "thought," as the subject-matter of logical science is an act 
 connecting two distinct ideas, or is the product of such an act. 
 Verbally it is the act, nominally it is the product. Logic does 
 not need specially to distinguish between them for its pur- 
 poses. But it does require to consider " thought " in its nar- 
 rower signification as an act of comparison between concep- 
 tions in order to distinguish more clearly its own laws from 
 the laws with which Psychology is concerned. 
 
 " The term thought" says Sir William Hamilton, " is used 
 in two significations of different extent. In the wider mean- 
 ing, it denotes every cognitive act whatever ; by some philos- 
 ophers, as Descartes and his disciples, it is used for every 
 mental modification of which we are conscious, and thus in- 
 cludes the Feelings, Volitions, and the Desires. In the more 
 limited meaning, it denotes only the acts of the Understand- 
 ing properly so called, that is, of the Faculty of Comparison, 
 or that which is distinguished as the Elaborative or Discur- 
 sive Faculty. It is in this more restricted signification that 
 thought is said to be the object-matter of Logic. Thus Logic 
 does not consider the laws which regulate the other powers 
 of mind. It takes no immediate account of the faculties by 
 which we acquire the rude materials of knowledge ; it supposes 
 these materials in possession, and considers only the manner 
 of their elaboration. It takes no account, at least in the de- 
 partment of Pure Logic, of Memory and Imagination, or of 
 the blind laws of Association, but confines its attention to con- 
 nections regulated by the laws of intelligence. Finally, it 
 does not consider the laws themselves of Intelligence as given 
 in the Regulative Faculty ;" namely, the Intuitions of pure in- 
 telligence, or the ultimate data, facts, and principles which are 
 involved in the primary experiences of mind. But such are 
 the functions with which Logic is not conversant. It remains 
 to determine positively what the nature of its object-matter is.
 
 DEFINITION AND DIVISIONS OF LOGIC 5 
 
 " The contemplation of the world presents to our subsidiary 
 faculties a multitude of objects. These objects are the rude 
 materials submitted to elaboration by a higher and self-active 
 faculty, which operates upon them in obedience to certain 
 laws, and in conformity to certain ends. The operation of 
 this faculty is Thought. All thought is a comparison, a recog- 
 nition of similarity or difference ; a conjunction or disjunc- 
 tion ; in other words, a synthesis or analysis of its objects. 
 In Conception, that is, in the formation of concepts (or gen- 
 eral notions), it compares, disjoins, or conjoins attributes ; in 
 an act of Judgment, it compares, disjoins, or conjoins con- 
 cepts ; in Reasoning, it compares, disjoins, or conjoins judg- 
 ments. In each step of this process there is one essential 
 element ; to think, to compare, to conjoin, or disjoin, it is nec- 
 essary to recognize one thing through or under another ; and 
 therefore in defining Thought proper, we may either define it 
 as an act of Comparison, or as a recognition of one notion as 
 in or under another. It is in performing this act of thinking 
 a thing under a general notion, that we are said to understand 
 or comprehend it. For example, an object is presented, say a 
 book ; this object determines an impression, and I am even 
 conscious of the impression, but without recognizing to myself 
 what the thing is ; in that case there is only perception, and 
 not properly a thought. But suppose I do recognize it for 
 what it is, in other words, compare it with, and reduce it 
 under a certain concept, class, or complement of attributes, 
 which I call book ; in that case, there is more than a percep- 
 tion there is a thought." 
 
 Thought is, therefore, the act or product of the Under- 
 standing or Reason as distinct from the various processes of 
 simple Apprehension or Cognition, and consequently Logic is 
 conversant with the laws affecting or regulating this act of 
 comparative knowledge rather than with the laws of percep- 
 tion. It is the science of thought as an act of conception, 
 judgment, and reasoning, or as the cognition of relations be- 
 tween conceptions. This act has its own laws distinct from 
 those of other mental acts. But the limitations of the science
 
 6 ELEMENTS OF LOGIC 
 
 and its laws can be determined only by comparing its field 
 and functions with those of other sciences. The nature of its 
 general object-matter suffices to define and distinguish it from 
 cognate sciences, on the one hand, and from the physical sci- 
 ences on the other. " Thought " is, first, a fact quite distinct 
 from physical events, and is, second, in its technical sense, as 
 logically defined, distinct from mental events which are not 
 acts of comparison : hence it implies a double limitation of the 
 subject of Logic ; one its distinction from sciences which in- 
 vestigate the physical laws and causes of events, and the other 
 its distinction from the philosophical sciences which are occu- 
 pied either with the efficient causes of mental phenomena or 
 with the nature of the being of which they are phenomena. 
 Logic is occupied with the relations between those phenomena, 
 as objects of reason or rational thought, or with the laws 
 which attest the validity of thought, and which serve, at least, 
 as negative criteria of the truth in so far as it is determined 
 by processes of reasoning. This, however, will be rendered 
 clearer when we examine the relation of Logic to the various 
 sciences. 
 
 2d. The Nature of the Laws of Thought. The laws of 
 thought will be best understood by defining the several uses of 
 the term " law." That there are " laws " of thought we may 
 at present take for granted, unless we are to assume that 
 there are no regulative principles or conditions which deter- 
 mine the uniformity of the mental processes involved in the 
 various acts of thought. But the assumption that there are 
 "laws" of thought neither defines their specific nature nor 
 affords any indication of what they are. This desirable end 
 can be achieved only by an examination of what we mean by 
 "law." 
 
 The first of the two general meanings of the term is of little 
 importance to a discussion in Logic, except that it may be 
 serviceable for bringing the true meaning of it into proper 
 relief. But this first conception is its moral and political 
 sense, in which it denotes a command or prohibition in regard 
 to the doing of certain actions. This idea does not imply any
 
 DEFINITION AND DIVISIONS OF LOGIC 7 
 
 absolute necessity of the event commanded, nor does it imply 
 any regularity of such events. It denotes only an injunction 
 to act or not to act. But such a conception is quite the con- 
 trary of the idea of " law " as employed by the sciences, where 
 it denotes the regularity with which events occur, or the uni- 
 formity of their dependence upon certain conditions which ne- 
 cessitate them. Hence the notion of " law," as applied in the 
 physical sciences, is but an abbreviation for the uniformity of 
 coexistence and sequence, or the uniformity of causation. It 
 describes or implies those conditions which make events regu- 
 lar and inevitable, or which explain why they follow a given 
 order of occurrence. It is thus opposed to chance or caprice. 
 In so far as the idea is synonymous with the notion of cause, 
 it is a convenient term for referring to the explanation of 
 phenomena. In so far as it denotes merely uniformity of 
 events, it is convenient for indicating the general principles 
 of unity that are exhibited or expressed thereby in the multi- 
 plicity of phenomenal events. 
 
 But this form of discussion is perhaps not quite so clear as 
 illustration. Each of the sciences endeavors to ascertain the 
 laws which prevail in a special class of phenomena. Chem- 
 istry tries to find out the laws of affinity and combination 
 regulating the relations between atomic bodies. Thus oxy- 
 gen and hydrogen must combine in a certain proportion to 
 produce water, and they will combine in no other proportion 
 to produce the same effect. Tbis represents a certain " law " 
 of affinity between the elements. A similar law operates 
 among all elements which combine only in certain definite 
 proportions which can be expressed with perfect mathematical 
 accuracy. In Astronomy we speak of the law of gravitation 
 which expresses the uniform tendency of material bodies to 
 seek the centre of gravity, or to move toward it, when free, 
 with a determinate velocity. In Physics we have the laws 
 of the expansion and diffusion of gases ; of contraction and 
 expansion under changes of temperature ; of the conservation 
 of energy, and of the transmission of motion. In all these 
 cases there are certain fixed ways of acting which bodies must
 
 8 ELEMENTS OF LOGIC 
 
 follow or obey. All facts must be regulated or determined by 
 tlie conditions which such laws impose. 
 
 But there is a science of human reason, and the processes 
 of thought are as much under law as any other phenomena. 
 These laws, like all other laws or conditions of events, are uni- 
 formities of nature ; in this case, of the nature of the mind. 
 They are embodied in principles or propositions which show 
 how we must think and reason, if we think and reason at all. 
 They are necessary laws of thought, because the mind has no 
 power to evade them. Thus conception, judgment, and rea- 
 soning are according to a principle which conditions and vali- 
 dates them, and so is known as their law, or the uniformity 
 of mental action in them which enables us to accept their 
 results. Such laws may be illustrated in the following man- 
 ner : 
 
 If I take the judgment that " All men are mortal," and infer 
 from it, by a process which is yet to be considered, that 
 " those who are not mortal are not men," or, " The immortals 
 are not men," and if I can apply a similar process to all such 
 propositions, it is because of the law that what is excluded from 
 the conception "mortal" must be excluded from the conception 
 " men." The assumption of this law is necessary to the mak- 
 ing of this inference, and unless it always be true I have no 
 means of guaranteeing the legitimacy of the results. Again, I 
 may take three conceptions which are capable of agreeing 
 with each other when conjoined in the form of a judgment. 
 
 Metals. Iron. Useful 
 
 By connecting these as subject and predicate I obtain three 
 propositions, one of which is a conclusion or inference from 
 the other two. Thus : 
 
 Metals are useful. Iron is a metal. . . Iron is useful. 
 Illustrations could be multiplied indefinitely showing the 
 same form and process of thinking, but the one suffices to call 
 attention to the principle upon which the mental act is based. 
 We observe that in the first two propositions there is a com- 
 mon term compared with two others. The first proposition 
 states an agreement between " metals" and " useful," and the
 
 DEFINITION AND DIVISIONS OF LOGIC 9 
 
 second between " iron " and " metals." The common term 
 with which "iron "and "useful" agree is " metals," and this 
 fact affords a reason to suppose that " iron " and " useful " 
 also agree to the same extent and in the same sense. If we 
 find that all such cases of reasoning exemplify the same form 
 of process and conditions we may formulate the law which 
 regulates and legitimates it. It is a logical axiom. Thus, if 
 two things are identical or agree with a third common thing, they 
 are identical or agree with each other. This is a fundamental 
 law of thought and sustains the same relation to the validity 
 of reasoning that the law of gravitation sustains to the phe- 
 nomena of falling bodies or the motion of the planets about 
 the sun. It is a law, because it expresses the uniform way in 
 which the mind acts, and must act, when comparing concep- 
 tions and judgments. There are many other such laws, but 
 they do not require examination at this stage of the discus- 
 sion. One illustration suffices to show what it is that Logic 
 endeavors to investigate and establish. Its laws are generally 
 expressed as principles or assumptions, while those of the 
 physical sciences are usually conceived as causes, conditions, 
 or uniformities of coexistence and sequence. But in both, the 
 essential idea is uniformity of some kind. In the physical 
 sciences it is the regularity with which certain events occur, 
 and may be expected to occur, under given conditions. In 
 logical science, the uniformity is that of mental action when 
 thinking and reasoning, as the fixed modes of comparison and 
 inference which the mind must obey when its action is healthy 
 and rational. In that respect Logic is a science like all other 
 forms of inquiry into the nature and principles of phenomena ; 
 only its laws are the laws of reason, and not of physical events. 
 3d. Form and Matter. The distinction between "form" 
 and "matter," and the definition of the term "formal" as 
 employed in Logic, are two of the most important things to 
 be accomplished, as their peculiar signification appears to 
 influence many of the doctrines of the science and to explain 
 many of the perplexities which are incident to logical pro- 
 cesses. The term " formal " we shall find often to be equiva-
 
 10 ELEMENTS OF LOGIC 
 
 lent in many respects to the idea of "law." The "forms " of 
 thought are often called the laws of thought, and they are 
 such in their nature. But why they are so can be more dis- 
 tinctly understood when we have made clear the difference be- 
 tween the "form " and the " matter" of thought, or between 
 the uses of those terms in reference to any subject-matter 
 whatever. 
 
 What we mean by " form " when applied to material or 
 physical objects is generally clear enough. It is synonymous 
 with shape, the geometrical outline of a body, or the spacial 
 limitations under which a physical object is conceived. The 
 " form " of a body may be regular or irregular, long or short, 
 round, square, rectangular, thick, thin, flat, etc. But no such 
 predicates can be applied to thought, or to states of con- 
 sciousness. Mental states can neither be said to have " form " 
 in the sense of having the quality of extension, nor to exist in 
 any assignable relation to space. Hence we cannot speak of 
 their "form" as we would of material objects. But inas- 
 much as the "form" of physical objects does not necessarily 
 depend upon the stuff or matter of which they consist, but 
 may remain the same amid all changes of the latter, or the 
 same when the matter may be of different kinds, because ex- 
 tension is independent of material substances, this conception 
 of the case may be chosen to describe, at least by analogy, 
 certain relations between the processes and the objects of 
 thought or consciousness. The fact first suggests, however, 
 the definition of "matter," which in its relation to "form," as 
 applied to physical objects, is merely the physical elements, 
 apart from extension, which make a body or object what it is 
 other than its " form." It is, as already said, the stuff of which 
 it consists. But by the same analogy which transfers the use 
 of the term " form " to other than physical objects the term 
 " matter " is also transferred, and the two terms are chosen to 
 express certain relations of uniformity and variation existing 
 between all kinds of phenomena, physical or mental Hence, 
 as two things may be alike in their '"form," while they are 
 of different " matter " or substance, two mental processes may
 
 DEFINITION AND DIVISIONS OF LOGIC 11 
 
 be the same in kind, although occupied about different ob- 
 jects of thought. Thus the judgments, " All men are mortal," 
 and "All metals are elements," are alike in the one aspect 
 of grammatical structure, but are different in respect to the 
 conceptions of which they are composed. Two or more syllo- 
 gisms consisting of different propositions will illustrate the 
 same truth. The one aspect in which they resemble each 
 other may be called their " form," and the conceptions which 
 compose them, and which may vary indefinitely without affect- 
 ing the " form," grammatical or logical, may be called their 
 " matter." Hence we may define the " form," at least provi- 
 sionally, that which remains constant when the " matter" changes, 
 and, correlatively, the " matter " may be denned as that which 
 remains constant when the "form" changes. 
 
 Objections can be made to these definitions, the force of 
 which can be readily admitted. But they are intended to be 
 merely tentative and approximate, and for the purpose of 
 making them general enough to cover all conceivable objects 
 of consciousness. It is a misfortune to be compelled to define 
 the terms reciprocally. But their simplicity and the fact that 
 they are purely relative to each other is an excuse for such a 
 course, and may even make it necessary. All objects, real or 
 ideal, must have both their "matter" and their "form" at 
 the same time. But the fact that either quality may be con- 
 stant while the other is variable proves the value of defining 
 them in a way to recognize both this variable relation and the 
 permanent coexistence of the two qualities in one way or an- 
 other at the same time. 
 
 But the " form" is usually regarded as what is essential to a 
 thing as well as what is constant, This means that a certain 
 quality cannot be dispensed with in reference to a given end, 
 although others may be immaterial or accidental in this re- 
 spect. Thus a house may serve the same purpose whether 
 made of brick or wood, while the " form " may not only be the 
 same, but must be more or less essential. A judgment must 
 have a subject and predicate, whatever the conceptions con- 
 stituting it, and so the "form" is essential to its being a judg-
 
 12 ELEMENTS OF LOGIC 
 
 ment, while its matter is not. This idea of what is essential 
 must be added to that of constancy in order to complete the 
 notion of " form " and its distinction from " matter." 
 
 But it is well to remark again the purely relative character 
 of the two conceptions in order to anticipate and remove a 
 possible objection. The two qualities are so variable that 
 what is the " form " in one relation may become the " matter" 
 in another. It depends wholly upon the nature of the relation 
 in which an object is conceived. The fact, however, does not 
 require development in an elementary treatise of the science. 
 It is noticed only to indicate that the circumstance has not 
 been overlooked in our account of the subject under consid- 
 eration. 
 
 The relation between "form " and "matter," as we have de- 
 nned it, expresses very clearly how Logic is a science of the for- 
 mal laws of thought. They are the laws which are not only 
 essential to it, but which are the same whatever the subject- 
 matter involved in our reasoning. The laws of thought remain 
 the same in the reasonings of Astronomy, Physics, Politics, or 
 Ethics, but the " matter " changes and does not affect the va- 
 lidity of the process. The " form " of our reasoning in all 
 these cases is essential to its being such a process. Hence 
 Logic, as a science, is " formal," and deals only with the " for- 
 mal " principles of thought in distinction from the material 
 objects of reason. Logic thus becomes the most general of all 
 the sciences, and its principles are the " formal " conditions of 
 the truth attained in them. What these "forms" of thought 
 are will appear with the development of the science. It is only 
 important at present to know that we do not require to take 
 any account of the particular " matter " of knowledge in order 
 to ascertain these "formal " laws. 
 
 //. THE RELATION OF LOGIC TO THE SEVERAL SCI- 
 ENCES. Logic we have found to be the science of the formal 
 laws of thought. This fact has led to its definition as the Sci- 
 ence of Sciences (scientia scientiarum). This is true if consid- 
 ered only in its formal functions. Any other conception of it 
 would imply that it was concerned with their material truths.
 
 DEFINITION AND DIVISIONS OF LOGIC 13 
 
 But it can employ itself with the material knowledge of other 
 sciences only in its function as an Art (ars artium). As a sci' 
 ence, however, it has a " formal " relation to all other sciences 
 inasmuch as it determines the laws of thought everywhere 
 affecting the acquisition and legitimation of truth. This is 
 all that it is necessary to recognize in a general way. But in- 
 asmuch as it sustains a more intimate and complex relation to 
 the mental sciences, often dealing with precisely the same sub- 
 ject-matter, we require to distinguish between its relation, in 
 some particulars, to the physical sciences, and its relation to 
 the mental. While it formally conditions the mental processes 
 of the physical sciences, it is materially occupied with the 
 laws of thought, and they, the physical sciences, with the laws 
 of things. The other mental sciences, however, to which it 
 has a more peculiar relation beside the general formal one, 
 have quite a distinct object in view as compared with Logic. 
 The two sciences are Psychology and Ethics. With Logic 
 these constitute, properly, the mental sciences. 
 
 Psychology deals with all mental phenomena ; Logic deals 
 with only a part of them. They therefore partly coincide. In 
 so far as they both deal with the thought processes of judg- 
 ment and reasoning they are occupied with the same field. 
 But they do not deal with these phenomena in the same way, 
 nor with the same object in view. - " Psychology deals with 
 them as laws in the sense of uniformities, that is, as laws in ac- 
 cordance with which men are found by experience normally to 
 think and reason ; Psychology investigates also their genesis 
 and origin. Logic, on the other hand, deals with them purely 
 as regulative and authoritative, as affording criteria by the aid 
 of which valid and invalid- reasonings may be discriminated, 
 and as determining the formal relations in which the different 
 products of thought stand to one another." These observa- 
 tions of Keynes may perhaps be rendered a little more com- 
 plete if we remark that Psychology investigates all mental 
 phenomena, including the rational, with a view to ascertaining, 
 first, their uniformities as actual events, and second, their 
 causes, but does not require to distinguish between normal and
 
 14 ELEMENTS OF LOGIC 
 
 abnormal states, valid or invalid ideas. On the other hand, 
 Logic, in addition to being conversant with a more limited 
 field than Psychology, does not deal with the phenomena in 
 common to the two sciences with the same object in view 
 throughout. It deals with the uniformities of rational pro- 
 cesses, but neither with their causes, nor with any matters re- 
 garding the origin of those uniformities. It endeavors to as- 
 certain the uniformities of thought which are the grounds of 
 truth or valid thought, and which exclude all other ideas from 
 recognition, except as facts. Psychology is thus concerned 
 with the origin and nature of mental phenomena in general, 
 and Logic with the conditions and the validity of the rational 
 processes. The latter, therefore, has to do with the formal in 
 contradistinction to the efficient principles of thought. Ethics, 
 of course, is the science of the ends of conduct, and although 
 it is formally conditioned by Logic, the relation between it 
 and Logic is not so close as between Logic and Psychology. 
 But their respective spheres can be very well denned by the 
 Aristotelian and scholastic formulas expressing the various 
 kinds of causes. This will characterize the difference between 
 them as sciences. Consequently Psychology is the science of 
 the efficient causes (causa efficient, ratio fiendi) of mental phe- 
 nomena, including the rational processes : Logic is the sci- 
 ence of the formal causes (causa formalis, ratio essendi) of 
 thought, of valid rational processes : Ethics is the science of 
 the final causes (causa finalis, ratio agendi) of human conduct, 
 of the mental phenomena of desire and volition. 
 
 It will thus be seen how Logic and Psychology may to some 
 extent overlap each other. This, however, is mainly in regard 
 to the phenomena with which they deal, and not with regard 
 to the objects which they aim to accomplish. They both deal 
 with uniformities of mental phenomena. But Psychology as 
 such does not distinguish between the true and the false ; it 
 explains mental processes ; Logic validates those of reasoning 
 and ascertains the principles which condition them. Psychol- 
 ogy deals with causes, Logic with principles. This will per- 
 haps show the intimate relation subsisting between the two
 
 DEFINITION AND DIVISIONS OF LOGIC 15 
 
 sciences, and at the same time indicates the difference of 
 method and object pursued by them in their investigations. 
 
 III. THE DIVISIONS OF LOGIC. Ijogic may be divided 
 according to the particular object which it has in view, or ac- 
 cording to the kind of reasoning with which it deals. The 
 first principle of division gives us Pure or Formal Logic, and 
 Material or Applied Logic. The same distinction is observed 
 in the divisions, Theoretical and Practical Logic. These divis- 
 ions are based upon the distinction between Logic as a sci- 
 ence and Logic as an art. Pure or Formal Logic is a science ; 
 Material or Applied Logic is an art. The former is conversant 
 only with the pure or formal laws of thought, and does not 
 concern itself with the material truth of any particular propo- 
 sition ; the latter is conversant with the material conceptions 
 of various sciences and endeavors to apply formal laws to the 
 attestation of truth or knowledge. Pure Logic is abstract and 
 theoretical ; applied Logic is concrete and practical. 
 
 The second principle of division gives us Deductive and In- 
 ductive Logic. Deductive Logic is usually defined as occu- 
 pied with the laws of a priori reasoning, and Inductive Logic 
 with the laws of a posteriori reasoning. The one assumes gen- 
 eral principles or facts in order to elicit into consciousness 
 something which is not explicitly there in the premises, or 
 which, when it is in consciousness, requires a general assump- 
 tion to validate it as truth ; the other assumes facts and en- 
 deavors to arrive at general truths beyond them, or ideas not 
 directly deducible from the premises. But this division of the 
 subject can be adjusted to the former, as the following scheme 
 will illustrate : 
 
 (Pure or Forma. 
 '
 
 CHAPTER IL 
 
 /. GENERAL PRINCIPLES. The definition of logic 
 teaches us that the science is employed about the laws of 
 thought. Thought, we also found to be an act of comparison, 
 and as the object of Logic and logical doctrine it is usually 
 identified with the processes of reasoning. But all the higher 
 and more complex mental processes presuppose the lower, and 
 the material which they furnish. Thus reasoning deals with 
 ideas, notions, conceptions, etc., which may be called its ele- 
 ments. We cannot understand the nature of that process un- 
 til we know the nature of the elements involved in it. It may 
 be the function of Psychology to tell us how these elements 
 have originated, but it may not be its function to tell us what 
 they are, or what their relation to reasoning is. Logic must 
 know what the qualities of its elements are, in order to formu- 
 late its laws about their relations in the processes of thought. 
 Hence we proceed to inquire what the elements of logical 
 doctrine are. 
 
 There are two distinct ways of regarding this question, ac- 
 cording as we consider the objects of logical science or the 
 mental processes concerned in it. Hence we may divide the 
 elements of logical doctrine into two classes : (a) The matter 
 of logical science which consists of Terms, Propositions, and 
 Syllogisms ; and (6) the form or process of thought which is 
 the principal object of the science, and which is usually rep- 
 resented as consisting of three subordinate mental modifica- 
 tions ; namely, Conception, Judgment, and Reasoning. The 
 former looks at the question from the point of view of lan- 
 guage, and makes no special examination, introspectively or 
 otherwise, into the mental activities of which terms, proposi-
 
 ELEMENTS OF LOGICAL DOCTRINE 17 
 
 tions, and syllogisms are the object. It simply relies upon 
 the normal accuracy of mental processes and develops Logic 
 as a practical doctrine concerning the proper mode of con- 
 ducting arguments and avoiding error. But from the point 
 of view of the process of thought there is a desire to get a 
 deeper knowledge of the way in which the mind conceives and 
 uses its material, and of the actions involved in understanding 
 the nature and relations of its material objects. Thus we may 
 produce a Logic without entering into any analysis of terms, 
 propositions, etc., and without considering the nature of the 
 mental processes involved. On the other hand, a complete 
 conception of the problem is not possible until we have formed 
 some definite idea of the process as well as the matter of 
 thought. The two aspects may be treated together, as they 
 imply each other. But their peculiarities may be best exhib- 
 ited by viewing them apart. 
 
 II. DEFINITION OF THE MATERIAL ELEMENTS OF 
 LOGICAL DOCTRINE. 1st. Terms. A term may be vari- 
 ously defined. Grammatically considered it is a word which 
 is the name of an idea, or conception. Logically considered 
 it is any word or group of words which is capable of constitut- 
 ing a distinct part of a sentence or a proposition. A gram- 
 matical term is single; such as "man," "tree," "with," 
 " from," " walk," etc. A logical term may be either a single 
 word, or a phrase ; such as " man," " house," or " the wife of 
 Socrates," "the Queen of England." The singleness of the 
 idea is all that is required to make it a logical term. But 
 terms are the elements, or atoms, as it were, which combine 
 to form a proposition. Logic may deal with them to some ex- 
 tent without considering the processes implied in apprehend- 
 ing their meaning. This fact has given rise to the belief in 
 some schools of thought that the science is exclusively occu- 
 pied about language. But very little observation is required 
 to perceive that it can deal with language only as a symbol of 
 thought. 
 
 2d. Propositions. A proposition, grammatically consid- 
 ered, is a sentence consisting of words or terms arranged to
 
 18 ELEMENTS OF LOGIC 
 
 make sense. Logically considered it is a connection of terms, 
 as subject and predicate, in a way to express their agreement 
 or disagreement with each other. Thus " Iron is an element," 
 and " The security of life is one of the primary objects of gov- 
 ernment," are propositions respectively of the simpler and the 
 more complex form. They express a relation between the sub- 
 ject and predicate, which cannot be expressed by a term. 
 
 3d. Syllogisms. A syllogism is purely an object of Logic. 
 It is the combination of three or more propositions in such a 
 relation to each other that the last one is an inference or con- 
 clusion from the others. It is a simple syllogism if it consists 
 of three propositions, and a complex one when it consists of 
 more than three. It is the material form which all our rea- 
 soning assumes, and various characteristics of it which are yet 
 to be examined make it the principal object of the investiga- 
 tions conducted by logical science. 
 
 ///. EXAMINATION OF THE FORMAL ELEMENTS OF 
 LOGICAL DOCTRINE. As we have already ascertained, the 
 formal elements are Conception, Judgment, and Reason- 
 ing. The last two are more conspicuous objects of logical sci- 
 ence inasmuch as their laws are more readily determined 
 and the processes themselves can be more easily regulated. 
 The process of Conception is so spontaneous and follows so 
 closely the psychological laws of mental phenomena that con- 
 cepts are always found and completed before logical science 
 can take any account of them. There are, therefore, no well- 
 defined laws of Conception which we can regard beforehand 
 in the making of the primary elements with which Judgment 
 and Reasoning have to deal. The qualities of its products, 
 namely, concepts, can be observed and their influence upon 
 different mental processes considered. But it is more difficult 
 to formulate any laws by which Conception is governed. Yet 
 it has its own laws, although they are not usually formulated 
 or discussed, except in Psychology. Logicians are usually 
 content with a statement of the characteristics of various no- 
 tions, ideas, or conceptions, and hence they omit any detailed 
 account of the processes involved in their formation. But we
 
 ELEMENTS OF LOGICAL DOCTRINE 19 
 
 shall give the subject a little more attention in the present 
 treatise. We proceed, therefore, to a careful examination of 
 each process. 
 
 1st. Conception. The terra " conception " is an ambigu- 
 ous one. It is used sometimes verbally, and sometimes nom- 
 inally or substantively. Hence it may denote either a. process 
 or a product. At present we are concerned with it only as a 
 process, and its object or product will appear for more partic- 
 ular consideration in a moment. But the product of concep- 
 tion as a process is called a concept. For the purposes of 
 Logic a concept is the same as a term. In itself it is to be 
 viewed from the mental side, and represents the ideal rather 
 than the symbolical element of thought. A word or term is 
 only symbolical of ideas, an index of thought ; a concept ex- 
 presses the content of the mental act as conceived apart from 
 language, and so is, to some extent at least, ideal as opposed 
 to real. It expresses more clearly than " term " what is con- 
 noted or denoted by a word. The distinction between it as a 
 product, and conception as a process, must be kept clear if we 
 are to understand the unity and simplicity of the latter as 
 compared with the diversity and complexity of the former. 
 Hence we must define them as accurately as possible. 
 
 1. DEFINITION OF CONCEPTION AND CONCEPT. Conception is 
 the mental act of connecting percepts or individual ideas into a 
 whole. This whole may be of several kinds, as we shall presently 
 see. But the act is illustrated in the process of conceiving 
 "man," for example, as uniting the properties of animality and 
 rationality. These terms may denote a whole group of quali- 
 ties. For instance, " animality " may denote material proper- 
 ties, such as solidity, color, form, etc., organic properties, such 
 as peculiar physiological structure, assimilative, circulatory, 
 respiratory organs, etc., and " rationality " may denote sensi- 
 bility, consciousness, reasoning capacity, etc. All these prop- 
 erties are grouped together in the notion of man. The act 
 of mind which conceives them, or thinks them as combined in 
 a single object is Conception. The act is the same for all 
 objects comprehending a number of attributes, or for all terms
 
 20 ELEMENTS OF LOGIC 
 
 comprehending a number of individual objects. Thus man 
 besides representing a whole which is a combination of quali- 
 ties, may denote a whole which is a common concept for a 
 class made up of individual men. It requires, however, the 
 same act of synthesis or comprehension for both kinds of con- 
 cepts. The nature and properties of these various concepts 
 must be examined later. We can now only take account of 
 the comparing and combining act which groups different qual- 
 ities or different individuals under the same idea or term. 
 The laws by which the process is governed are the laws of as- 
 sociation, of identity and difference, and of necessary connec- 
 tion. 
 
 Concepts, as already intimated, are the products of Concep- 
 tion. They denote the ideas themselves instead of their 
 names. Technically a concept may be defined as a synthesis 
 of percepts, or a synthesis of individual wholes to form a gen- 
 eral notion. If the former could be called a simple concept, 
 the latter could be denned as the synthesis of simple concepts, 
 making a complex and general concept of greater extent. A 
 simple concept in this view of the case would be such as 
 " man," " tree," "lion," "institute," "President," etc., in so far 
 as these terms or ideas represented an individual group of 
 qualities. But in so far as they were general names for a large 
 number of individuals of the same kind, the concept would be 
 cdmplex as involving the idea of a group of qualities compre- 
 hended in each member of a larger group of individual ob- 
 jects. But the definition of it as a synthesis is perhaps too 
 technical for common use. Hence it may be best to define a 
 concept, in its broadest sense, as the notion of a group of qual- 
 ities or individuals which are capable of being thought of at the 
 same time, the former of which belongs to the same thing and 
 the latter of which represents the same general idea, or is ap- 
 plicable to a number of the same kind. But this definition 
 will not throw much light on the subject because it undertakes 
 to do so much. It is an endeavor to define an idea which is 
 supposed to apply equally to an individual group of qualities 
 and a general group of individuals. The difference between
 
 ELEMENTS OF LOGICAL DOCTRINE 21 
 
 these two things is so great that some logicians use the 
 term " concept " as if it could apply but to one of them and 
 that to the latter, where it expresses a general abstract idea. 
 Some have endeavored to distinguish between "idea," "no- 
 tion," and " concept." For certain purposes this may be 
 proper and necessary. A complete and accurate logical doc- 
 trine may be greatly helped thereby inasmuch as such a dis- 
 tinction is intended to evade the ambiguities incident to the 
 use of the term " concept." But practical purposes may not 
 require any refined niceties in the matter, and as we are at 
 present interested only in throwing light upon the fundamen- 
 tal nature of Conception as a process, we may postpone the 
 immediate consideration of that question. It will come up 
 again when studying the formation of concepts and their sev- 
 eral divisions. For the moment we need only to know that 
 there is a mental process by which more than one object of 
 consciousness is perceived as constituting a single whole. 
 This process is one of comparison and synthesis. It is ele- 
 mentary, and conditions all the higher and more complex acts 
 of comparison and synthesis. The complete investigation of 
 its nature and functions maybe left either to Psychology or to 
 more elaborate treatises of Logic than the present one pro- 
 fesses to be. It is sufficient at present to know enough of it 
 for appreciating its relation to concepts and the kind of 
 knowledge with which Logic deals. 
 
 2. THE FORMATION OF CONCEPTS. I have stated my intention 
 not to draw any important distinctions between "idea," "no- 
 tion," and " concept." For practical purposes I shall assume 
 them as identical and as denoting objects of knowledge or 
 products of mental action other than the primary experiences 
 of sense and consciousness. They may require to be distin- 
 guished from representative states which are the objects of 
 memory. But in a broader sense the products of memory are 
 non-compared data, and may be characterized as a form of 
 ideal percepts. I cannot, however, go into any special ques- 
 tions of Psychology. We must be content to assume that con- 
 cepts are the result of some form of comparison and of group-
 
 22 ELEMENTS OF LOGIC 
 
 ing either qualities or objects. With this once accepted, at 
 least tentatively, we may proceed to see how they have been 
 formed. 
 
 I said that a concept might be a synthesis of percepts. But 
 the term " percept " requires definition and suggests the ne- 
 cessity of considering briefly the elementary processes of 
 knowledge. 
 
 A percept is an individual object of cognition, such as a color, 
 a sound, a taste, solidity, weight, extension. We may include 
 any state of self-consciousness as such, and as a single object 
 of inner cognition. The consciousness even of a "thought," 
 in the sense of a logical act, so far as it is an individual ob- 
 ject of inner perception, may be regarded as a percept. But 
 those illustrations taken from the phenomena of sense percep- 
 tion are the clearest instances, and they suffice to show how a 
 concept is formed. We must be careful, however, in thinking of 
 " a percept as an individual object of cognition," that we do not 
 understand by it any object of sense perception in the ordinary 
 sense of the term, such as a tree, a house, a cloud, matter, etc. 
 We may say practically that we see such objects, but in reality 
 we do not. We perceive certain colors and relations of form, 
 and other experiences of the use and nature of objects having 
 these qualities enable us to interpret the meaning of what we 
 see. Thus what appears to be a very simple idea turns out to be 
 very complex. Hence a percept strictly considered can only be 
 a single presentation of sense or of consciousness. Sound is 
 given by hearing only, savor by taste, solidity and resistance 
 by touch, color and form by -vision. Certain signs also in 
 other senses may be called the percept by which extension 
 or space is derived. But not to go into special problems, we 
 find these illustrations sufficient to show what is meant by " an 
 individual object of cognition." They are the elements out of 
 which concepts are formed. 
 
 The process of knowledge which gives us percepts is Per- 
 ception or Apprehension, which we may denominate internal 
 or external, according as it gives us material or mental per- 
 cepts. Into its nature Logic does not require us to enter. It
 
 ELEMENTS OF LOGICAL DOCTRINE 23 
 
 is the process which combines them that concerns the prob- 
 lems of Logic. We call that process Conception, as an act, 
 because it involves some form of comparison and discrimina- 
 tion ; that is, the holding of two or more elements in con- 
 sciousness at the same time, and combining or separating them 
 in its own peculiar way. But the process is conditioned by 
 the Laws of Association. By repeated experiences in the con- 
 joint perception of a number of qualities, physical or mental, 
 or physical and mental combined, we come to form a concept 
 of an individual object or whole. By conjoint perception we 
 mean the activity of different senses at the same time, or un- 
 der such circumstances as enables the mind to think that the 
 qualities perceived belong to the same thing. Thus certain ex- 
 periences of taste, color, and touch enable me to form an idea 
 of an orange as uniting the three sets of qualities, so that when 
 I see one of them I may think of the others as associated with 
 it, and requiring only the proper experience to verify them. 
 Again, certain experiences of form, color, solidity, etc., and per- 
 ceptions of usefulness enable me to form the notion of a house. 
 I think of these qualities as cohering in the same object. The 
 same is true of all objects or wholes which we know. The in- 
 dividual percepts, through the agency of association, and per- 
 haps other mental acts, are conceived as constituting one 
 object or mental totality, which is a concept of the first order. 
 All such objects representing a simple synthesis of qualities 
 or percepts I shall call individual or attribute-wholes. The 
 simplest illustrations of them is that of proper names, such 
 as Plato, Bucephalus, etc. They represent objects which are 
 merely a combination of qualities or attributes, and whose 
 name is not applicable to any other individual. They are only 
 attribute-wholes. Other and general concepts may be more 
 than attribute- wholes. Thus "man," "quadruped," "tree," 
 " circle," " country," " institution," " state," etc., are attribute- 
 wholes, inasmuch as they represent certain combinations of 
 qualities. But they are also more than these at the same time 
 because they may denote a class of individuals. This fact 
 makes it necessary to examine into the formation of concepts
 
 24 ELEMENTS OF LOGIC 
 
 of the second order. In comparison with this, those of the 
 first order may be best denominated individual-vjholes. Those 
 of the second order shall be called class-wholes. 
 
 The process of formation is that of association and com- 
 parison. If we perceive objects having like qualities there is 
 a tendency to associate them and to give them a common name. 
 Thus I may see an oak having roots, trunk, branches, leaves, 
 color, form, etc., of a particular kind, and then an elm with the 
 same properties slightly modified. But there may be a suffi- 
 cient number of qualities alike for me to think that the objects 
 bearing them are entitled to be called by a common name. 
 Hence, instead of calling them both "oaks," or both "elms," I 
 may use the term tree, which will apply to the common quali- 
 ties, while " oak " and " elm " may denote the differences. We 
 may even compare different " oak " trees and use that term as 
 a general one to denote all trees having the essential qualities 
 of " oak," but with slight differences ; or to denote all indivi- 
 dual oak-trees of exactly the same kind, whether there be any 
 differences or not. The comparison is one based upon the 
 similarity or resemblance between individual objects, not upon 
 the coexistence of two or more qualities in the same thing, but 
 rather the coexistence or succession of the same quality or 
 qualities in different things. The concept is thus a class con- 
 cept because it applies to more than one individual. It may 
 denote at the same time an individual-whole, which is perhaps 
 what we represent to ourselves in imagination when we think 
 of it. But it also denotes a class of individuals to each of which 
 it is equally applicable. The terms "man," "quadruped," 
 "tree," "circle," "institution," "state," etc., to which ref- 
 erence was made as representing attribute-wholes, are also 
 class-wholes, or general concepts, as opposed to individual con- 
 cepts, although they denote at the same time a totality of co- 
 hering attributes or qualities which make them representative 
 of individual concepts. But there are general concepts which 
 represent a class of individuals without, perhaps, indicating 
 that the individuals composing it are complex attribute wholes. 
 These will be such as represent different percepts, qualities, or
 
 ELEMENTS OF LOGICAL DOCTRINE 25 
 
 objects of the same kind, or perhaps of mutually exclusive 
 qualities not belonging to the same object and the same space. 
 Thus "color," "sound," "hardness," may be general or class 
 concepts without denoting individual-wholes. Take the idea 
 of " red," for example. It is a single percept. But this percept 
 may be found in connection with various objects, and so we 
 may form a general idea of "redness" applicable to a number 
 of percepts without denoting their synthesis or combination 
 in the same object or the same space. These will afford the 
 clearest illustration of class- wholes independently of individual 
 wholes, while most concepts may be understood in both rela- 
 tions at the same time. But in all, the same process of com- 
 parison, discrimination, and association is necessary to their 
 formation. The diffei-ence between the two classes of concepts, 
 and hence between the processes forming them, is, that indi- 
 vidual wholes, as a synthesis of percepts, are a combination of 
 different qualities, and class-wholes are a combination of similar 
 qualities in different objects under a common name. To illus- 
 trate again, "Lincoln," "Socrates," "Italy," "God," etc., are 
 pure individual wholes ; "color," "sound," "redness," "stature," 
 "number," etc., are pure class-wholes, inasmuch as they de- 
 note more than one thing without implying a union of various 
 attributes in it ; while " man," "vegetable," "oak," "nation," 
 etc., may denote both individual and class concepts. 
 
 This distinction between the two kinds of class concepts 
 is the basis of the distinction between the Mathematical or 
 Quantitative Sciences, and the Metaphysical or Qualitative 
 Sciences. The element of the distinction to which I refer is 
 that in which a class term or concept may be used to denote a 
 number of individuals, perceptive or conceptive, absolutely iden- 
 tical in kind, or it may denote a number of individuals essen- 
 tially different in kind, but with common properties of a char- 
 acter to justify the use of a common name. The comparison 
 in one case is between individual-wholes, or properties to which 
 the common name applies in the same sense and without a 
 difference. In the other the comparison is between individual- 
 wholes, or properties to which the name applies with a differ-
 
 26 ELEMENTS. OF LOGIC 
 
 ence. Thus "two," "red," "length," etc., will apply to a class 
 of objects without reference to a difference in kind. These 
 may be called mathematical concepts, because each object de- 
 noted by them may be representative of the whole class irre- 
 spective of the distinction between common and accidental, or 
 what are called essential and differential qualities. But there 
 are other concepts, such as "man," "tree," "biped," "Greek," 
 "country," etc., which may be called logical or metaphysical 
 concepts, inasmuch as they denote individuals with such differ- 
 ences that no one of the individuals is wholly representative of 
 the class. They are concepts connoting only the common or 
 essential qualities and not taking any special account of the 
 accidental or differential properties. It may not be proper to 
 call them " logical " or " metaphysical," and I shall not contend 
 for that name longer than to indicate the distinction which I 
 have defined. The importance of the distinction itself mil 
 appear when we discuss the matter of genus or essentia, and 
 differentia or accident. We require at present only to recog- 
 nize the two acts of comparison, one of which involves an act 
 of abstraction, and the other does not. The formation of math- 
 ematical concepts does not require any abstraction of special 
 properties, and the bringing of them together to the neglect of 
 certain others, unless it be in a manner which we hardly need 
 to take account of. On the contrary, the formation of what I 
 have called logical or metaphysical concepts, such as "man," 
 "tribe," "race," "animal," etc., requires that we abstract cer- 
 tain common qualities and ignore the accidental ones, so as to 
 denote the former by a common name. " Abstraction " is here 
 used merely to denote the thinking away of certain properties 
 from their exclusive application to any particular object. A 
 farther conception of it is not required at present. It is suffi- 
 cient to remark a process producing two kinds of class con- 
 cepts with which later problems in logic will have to reckon. 
 
 3. THE DENOMINATION OF CONCEPTS. The naming of con- 
 cepts has always been considered an important matter in 
 Logic. It is not especially a process of thought, although it is 
 necessary to make thought an effective instrument in the com-
 
 ELEMENTS OF LOGICAL DOCTRINE 27 
 
 munication of knowledge, and to give stability and fixity to the 
 various products of mental activity. Some have even consid- 
 ered names and naming the most important aspects of Logic. 
 Into this controversy it is not necessary to enter either on one 
 side or the other. We may be content with one or two re- 
 marks about the nature of language and its service to thought. 
 Language is the symbol of ideas, or consists of the signs by 
 which we can indicate the resemblances and differences be- 
 tween concepts. By it the infinite number of individuals and 
 classes can, to some extent, be tabulated or indexed for use. 
 Without it, perhaps, we should not be able to develop our 
 thinking processes above the grade of animal intelligence. 
 With it we can name an idea so as to keep it by itself, if re- 
 quired, or conceive and speak of a class of ideas, if need be. 
 The same word even may have a double denotation, as we have 
 seen in general concepts denoting either individual or class- 
 wholes. In this way a word may indicate the quality that sep- 
 arates the concept or object named by it from others, or it 
 may apply equally to all members of the class. " Man " may 
 imply or represent the quality or qualities which separate him 
 from a " lion," an " eagle," or all other objects. At the same 
 time it will denote the qualities by which the term may be 
 employed to indicate all individual men. Thus economy of 
 language is obtained, on the one hand, and clearness of con- 
 ception on the other. But the general service of language is il- 
 lustrated in the simple fact that where any ambiguity is asso- 
 ciated with a single term, our intellectual confusion is com- 
 pletely overcome by the use of two or more terms which may 
 specify the distinct qualities confounded under a single term. 
 The denomination of concepts, therefore, is an important pro- 
 cess either in completing the act of thought, or in making it 
 useful when it is completed.* 
 
 * The student may consult the following references for a discussion 
 of Language and Denomination in their relation to Logic : Bosanquet : 
 Logic, vol. i., Introduction, Sections, 4, 5, pp. 8-30; Thomson: Laws 
 of Thought, Introduction, Chapter on Language ; Hamilton : Lectures on 
 Logic, Lecture VIII.
 
 28 ELEMENTS OF LOGIC 
 
 2d. Judgment. Like Conception the term Judgment may 
 denote both a process and a product. Its product is a propo- 
 sition. At present we are concerned only with the act, which 
 can be briefly denned, and which, with a slight modification, 
 will apply to the product. As an act a judgment is the per- 
 ception of the relation between two concepts. It is an act of 
 comparison still more distinct than any involved in Concep- 
 tion. The relation which it expresses is that of agreement or 
 difference between subject and predicate, or of inhesion or 
 non-inhesion between subject and attribute. Much discussion 
 would be required to give a satisfactory theory of that rela- 
 tion, and to decide whether it is wholly one of agreement or 
 difference, or partly of some other character. But larger 
 treatises may be consulted upon this problem. It is sufficient 
 to know that the act of connecting two concepts is the funda- 
 mental characteristic of Judgment. The laws which determine 
 it may be ascertained without any particular theory of the 
 process, even if such a theory be helpful in the final solution 
 of problems centring about it. 
 
 The principal matter of importance to be observed in con- 
 nection with the nature of Judgment is that the act of compar- 
 ison involved in it is in its essential elements the same as the 
 act of Conception. The difference between them is only the 
 way in which the result is expressed, or the object-matter with 
 which they deal. Conception is the connecting of percepts ; 
 Judgment is the connecting of concepts. The connecting of 
 percepts can be expressed in terms or words : the connecting 
 of concepts must be expressed in propositions. The relation 
 between subject and predicate in Judgment may be variously 
 expressed. But the process will be identical with that which 
 is involved in the formation of concepts. This is perhaps ap- 
 parent in the fact that the two kinds of judgments correspond 
 to the two general kinds of concepts. There are judgments 
 which express the relation between substance and attribute, 
 corresponding to concepts representing individual wholes ; for 
 example, " Iron is hard." Then there are judgments which 
 express the relation of resemblance or difference between sub-
 
 ELEMENTS OF LOGICAL DOCTRINE 29 
 
 ject and predicate, or the relation of classes, corresponding to 
 concepts representing class-wholes ; for example, " Iron is a 
 metal." We shall examine the importance of this distinction 
 again. The nature of the mental process of Judgment and its 
 relation to that of Conception is all we require to know at this 
 stage of the discussion. 
 
 3d. Reasoning. Reasoning is a process only a little 
 more complex than Judgment. It may be briefly defined as 
 inference. But this will require further explanation. Hence 
 we may adopt Jevons's definition of reasoning as adequate for 
 all practical purposes. It is " the progress of the mind from 
 one or more given propositions to a proposition different from 
 those given. Those propositions from which we argue are 
 called the Premises, and that which is drawn from them is 
 called the Conclusion" The definition here covers what is 
 known as Immediate Inference, and Mediate Inference. If 
 from the proposition that "All metals are elements," I infer 
 that "All that are not elements are not metals," I am mak- 
 ing an immediate inference ; if from the two propositions 
 that "All metals are elements," and "Iron is a metal," I in- 
 fer that " Iron is an element," I am making a mediate infer- 
 ence. But in both cases my reasoning consists in the percep- 
 tion of a relation between propositions, through the medium 
 or agency of concepts whose relation is known, implied, or ex- 
 pressed. 
 
 This last definition of Reasoning identifies the process in its 
 nature with that of Conception and Judgment. The difference 
 between them is not in the form of the mental act, but in the 
 matter to which it is applied. In reasoning it is the resem- 
 blance or difference of relations between propositions that con- 
 stitutes the peculiar nature of the matter dealt with, while in 
 Judgment it is the resemblance or difference of relations be- 
 tween concepts. The distinction is thus only a matter of 
 complexity, Reasoning representing in a more complex form 
 only what is found substantially in the earlier process of Con- 
 ception. At any rate, this way of regarding the question helps 
 to give unity to the mental processes, while it justifies our
 
 30 ELEMENTS OF LOGIC 
 
 turning to the differences of matter to which the various men- 
 tal acts are applied, in order that we may determine the differ- 
 ent laws of thought regulating the process according to the 
 changes of matter apparent in the development of the subject 
 of Logic.
 
 CHAPTEK HI. 
 
 TERMS OR CONCEPTS, AND THEIR KINDS 
 
 * 
 
 TERMS and concepts have already been denned as denoting 
 ideas. We come next to their divisions, which can be made 
 on several different principles of distinction. 
 
 1st. Categorematic and Syncategorematic Terms. 
 A categorematic word is one which can stand as the subject 
 or predicate of a proposition. Such are " animal " " nation," 
 "excellence," "wise," "beautiful," "perfect." These show 
 that categorematic terms are limited to nouns and adjectives. 
 Verbs ought to be included. 
 
 A Syncategorematic word is one which cannot stand alone 
 as the subject or predicate of a proposition. Such are " with," 
 " and," " through," " nobly," " very," " indeed," etc. From this 
 we perceive that Syncategorematic terms are either modal or 
 relational : modal, if they are adverbs, relational, if they are 
 conjunctions and prepositions. 
 
 A term in the logical sense, as the subject or predicate 
 of a proposition, may consist of one or more categorematic 
 words, or of categorematic and Syncategorematic words. In 
 the latter case it must consist of a grammatical phrase or 
 clause. 
 
 2d. Singular and General Terms. This division of 
 terms is limited to categorematic words, and perhaps to the 
 class of substantives. But this fact is less important than that 
 the division is a new one and not to be confused with the one 
 already made. It is based upon the differences between the 
 number of individuals denoted by various concepts, and at 
 least partly coincides with that between individual-wholes and 
 class-wholes. 
 
 1. SINGULAR TERMS. A singular term is one which can be
 
 32 ELEMENTS OF LOGIC 
 
 affirmed, in the same sense, only of a single object, real or im- 
 aginary. Proper names are the best illustration. Thus, Xa- 
 poleon, Paris, Greece, St. Paul, etc., are singular terms because 
 they can apply, in the same sense, only to one object. Ex- 
 pressions like " the Secretary of State," " the Prime Minister 
 of I'liy-laml," " the Kin-' of Spain," \vill he singular when they 
 refer to an individual or particular person. But they are also 
 capable of being general* terms. This will be the case when 
 they arc used to denote the class of officers by those respec- 
 tive names. Thus when we refer indefinitely to " the Secretary 
 of State " as any man or officer in that station, we use it in its 
 general sense ; if we refer definitely to a particular man occu- 
 pying the place, it is singular. Keynes also indicates how the 
 same expressions may become singular, if an individualizing 
 prefix is added to them : Thus, " the present Prime Minister," 
 "the present Secretary of State," "the reigning Queen of 
 England," etc. Likewise, he thinks such expressions as " the 
 first man," " the pole star," are singular. Perhaps we could 
 add such terms as " the highest good," " the supreme or ulti- 
 mate end." These are simple cases where the addition to a 
 general name of an adequate prefix transforms it into a sin- 
 gular one. But great caution must be exercised in our judg- 
 ment of such cases. For example, " the eldest child," is an ex- 
 pression which is applicable only to a particular child, but it 
 is an indefinite particular, and so is general in its import. But 
 the illustrations preceding show that there may be singular 
 terms other than proper names. Some of them, however, are 
 capable of both a general and a singular use. For example, 
 sjoace, when it refers to the totality by that name, is singu- 
 lar, but when it is a name for a number of definite spaces, 
 or the different portions of aggregate space, it has all the 
 distinctive characters of a general term. The same is true 
 of the term universe, and Professor Bain thinks it true of 
 all aggregate substances which are divided into parts, not 
 kinds; as "water," "stone," "salt," "mercury," "flame." 
 But Keynes's remark that we can say "some water," "some 
 salt," "some mercury," which cannot be said of proper names
 
 TERMS OR CONCEPTS, AND THEIR KINDS 33 
 
 or concepts denoting only one object, or an individual whole, 
 seems to me decisive in favor of considering them general 
 terms. Keynes ought, perhaps, to have remarked the reason 
 that such terms are liable to be mistaken for singular ones. 
 It is that general terms apply either to objects which are dif- 
 ferent in kind, or to objects which have an independent exist- 
 ence, and never unite to form one continuous mass of homo- 
 geneous matter, as do "water," "air," "space," and "stone" 
 when denoting rock of the same kind. Oneness of kind is a 
 characteristic of singular terms, as well as individuality, and 
 hence it is easy to confuse such terms as "water," "air," 
 "mercury," etc., with the singularity of proper names. But 
 they represent objects which can be divided into individual 
 parts without modifying the qualities of the parts which re- 
 ceive the same names as the aggregates. This is not true of 
 proper names, or of any singular term which is singular in its 
 absolute sense. It would have been more apt if Professor 
 Bain had classified the terms as collective, which are still to 
 be examined, for their analogy to such terms is closer than to 
 singular terms. But, nevertheless, I think they can be shown 
 not to ba collective. I regard them as abstract. 
 
 Proper names may become general terms when used to de- 
 note a class of individuals having a given characteristic. Thus 
 "the Napoleons," "the Csesars," "the Platos," "the Washing- 
 tons," are general terms because they denote a class of persons 
 with certain characteristics which belonged to the original 
 person by that name. Again, the name " God " will be singu- 
 lar to a monotheist aud general to a polytheist : uncapitalized 
 it is general to a monotheist. 
 
 There is, then, no absolute rule by which the mere form of a 
 word may be taken to indicate its character. Some general 
 terms by adding a prefix may become singular ; some singular 
 terms, used in the plural, or to denote, on certain emergencies, 
 more than one of a kind, become general. It is a case where 
 a change of the matter, when the form of the term remains 
 absolutely or virtually the same, affects the character of the 
 concept. Hence the rule can be absolute only when the two 
 3
 
 34 ELEMENTS OF LOGIC 
 
 aspects of the concept, form and matter, remain constant and 
 according to definition. 
 
 2. GENERAL TEBMS. A general term or concept is one which 
 can be applied, in the same sense, to an indefinite number of 
 objects, real or imaginary. It is a name applied to class- 
 wholes, such as "man," "vertebrate," " quadruped," "genera- 
 tion," "triangle," etc. Also such terms as "army," "forest," 
 "crowd," "nation," etc., are general terms. But there is so 
 marked a difference between the first class of illustrations and 
 the second that a subdivision of general terms into distributive 
 and collective has to be recognized. A distributive term is 
 one which applies in the same sense to~"each individual in the 
 class, such as in the first set of illustrations given ; namely, 
 " man," " vertebrate," etc. A collective term is one which 
 applies to an aggregate of individuals composing a whole, and 
 which will not apply to any of the individuals alone. Thus 
 " army," "flock," " bevy," " family," "tribe," etc., are collective 
 terms because they denote a composite or aggregate whole. 
 They are at the same time general terms because they are 
 applicable in the same sense to an indefinite number of such 
 aggregates. 
 
 It is important to remark that a collective term may be 
 singular instead of general. Thus the Vatican Library, the 
 American people, the Greek nation, the Seventy -second Regi- 
 ment, Company B, etc., are singular aggregates because the 
 name will apply to no other objects of the same general kind. 
 Some few terms may be used either distributively or collective- 
 ly according to the emergency. Thus "people," "the Greeks," 
 " the English," and often the plural of ordinary general terms, 
 as "the trees," " the houses," etc., may be used in either sense, 
 as illustration will presently show. The following diagram 
 will show the relation subsisting between singular and general 
 terms, and their subdivision : 
 
 ( Singular \ Individual. Singular \ ,, vu 
 
 J ( Or ( Collective.
 
 TERMS OR CONCEPTS, AND THEIR KINDS 35 
 
 Perhaps a doubt about the accuracy of this representation 
 is possible, since a collective term when general may at the 
 same time have the characteristics of a distributive term. 
 This is true of such terms as "army," "herd," "regiment," 
 etc. They are collective in so far as they denote an aggregate 
 whole of individuals ; they are distributive in so far as they 
 denote or can apply to a class of such aggregates. But since 
 the last is true only when the aggregates are treated as indi- 
 vidual-wholes, the general distinction between them may still 
 be observed. It is merely a case again where the difference is 
 due to some variation between the form and matter of the 
 concept. Thus in the sentence, " Mobs are crowds of enraged 
 men," there can be no distinction between the form and mat- 
 ter of the collective term " mobs." But in the sentence, " Mobs 
 are dangerous," the assertion is made of all such aggregates, 
 and hence while the form of the term is collective its matter 
 may be both collective and distributive, or distributive alone. 
 
 But the distinction between distributive and collective 
 terms is more important for Logic than the distinction be- 
 tween singular and general terms. This is apparent for the 
 reason that the nature of propositions is less affected by the 
 latter than by the former distinctions. We shall learn later 
 that the same logical laws apply to "singular propositions" 
 as apply to "universal propositions," and the distinction be- 
 tween these is parallel with, and determined by, that between 
 singular and general terms. They are not liable to easy con- 
 fusion. But the collective and distributive uses of terms are 
 often confused and give rise to corresponding fallacies in rea- 
 soning. This liability to confusion is illustrated in such 
 propositions as the following : " All the angles of a triangle 
 are equal to two right-angles," and " All the angles of a tri- 
 angle are less than two right-angles." The two propositions 
 seem contradictory, because the same thing cannot be equal 
 to and less than another at the same time and taken in the 
 same seiise. But the first proposition taken collectively in 
 its subject is true, and the second taken distributively is true. 
 Again, "All the trees in the forest produce a thick shade,"
 
 36 ELEMENTS OF LOGIC 
 
 may be taken in the same double sense. Similarly, " The peo- 
 ple filled the hall," and " The people are honest," or, "The 
 Greeks are a nation," and "The Greeks are Caucasian." A 
 case of the plural of an ordinary distributive term becoming 
 collective is the following : " The trees make a forest ; " but in 
 the following it is distributive : " The trees are beautiful." 
 
 3d. Concrete and Abstract Terms. It is difficult to 
 give a satisfactory definition of concrete and abstract terms. 
 Scarcely any two authors agree upon the subject, and even if 
 they did agree, observation would teach us that any attempt 
 to apply the definition to an actual classification of concepts 
 would meet with the serious obstacle that the two classes seem 
 to shade off into each other by insensible degrees. It is only in 
 certain cases that the distinction can be made clear, and often, 
 in spite of this clearness, a term may be concrete in one of its 
 applications and abstract in another. The difficulty is largely 
 caused by the unfixed and varied use of the term " abstract," 
 which is sometimes used as the equivalent of " general," and 
 again as denoting the conception of a quality apart from its 
 subject. The confusion occasioned by this usage wiD appear 
 as we proceed. 
 
 The definition of " concrete " and " abstract " terms can be 
 made very simple. The only difficulty we have to encounter 
 after that, is the determination of the particular terms which 
 fall under the one class or the other. But we can at least 
 begin with a definition of them in their purest form, and if 
 subsequent facts require us to qualify it we can do so. 
 
 A concrete term is a name which stands for a thing, or for 
 an attribute of a thing conceived as an attribute ; e.g., "Web- 
 ster," "Bucephalus," "Parthenon," "white," "clear," "gener- 
 ous," etc. A concrete concept is the same in its meaning, but 
 we do not think of it as a word. It denotes a thing or a qual- 
 ity as the object of consciousness ; while spoken of as a term 
 it is the object of Grammar. With the same qualification we 
 may define abstract terms or concepts. 
 
 An abstract term is a name which stands for an attribute or 
 quality considered apart from the thing possessing it, or con-
 
 TERMS OR CONCEPTS, AND THEIR KINDS 37 
 
 ceived and used as a thing, e.g., "redness," "cheapness," 
 "purity," "perfection," "righteousness," "justice," "ability," 
 etc. 
 
 Mill's definition is as follows : " A concrete name is a name 
 which stands for a thing ; an abstract name is a name which 
 stands for an attribute of a thing." It is not clear what such 
 a definition will do with adjectives. They are names of attri- 
 butes and yet Mill includes them among concrete names. On 
 the other hand, they can hardly be included among concrete 
 terms because they do not denote " things " in the strict sense. 
 We must therefore either make a class for them apart from 
 the concrete and abstract, or modify the definition. I have 
 attempted to overcome this difficulty in the definition which 
 I have given. It provides for two kinds of concrete terms, 
 the substantive and the attributive. I shall also divide abstract 
 terms into two classes, static and dynamic, or adjectival and 
 verbal nouns. To complete the classification I shall set aside 
 a third class of mixed concrete and abstract terms. Thus it 
 appears in the following table : 
 
 / p , ^ Substantive = Singular Nouns, e.g., Homer, 
 
 p , ) | Attributive = Adjectives, e.g., Pure. 
 
 Terms 4 "* / Abstract ] Static = Ad J ectival Nouns, e.g., Generosity. 
 
 ' l } Dynamic = Verbal Nouns, e.g., Acceleration. 
 Mixed = Concrete and Abstract, e.g., Government, Institu- 
 tion, etc. 
 
 In this classification no mention of such terms as " man," 
 " animal," "race," " vegetable," " triangle," etc., has been made. 
 Keynes considers them as concrete. The illustrations of con- 
 crete substantives were chosen from singular terms, and it is 
 now a question whether general terms are concrete or abstract. 
 The mere fact of being general terms does not make them 
 concrete, as is shown in verbal abstracts, which may be general, 
 and also in the adjectival abstracts, if they can be consider- 
 ed as general. Jevons thinks them singular. This is ex- 
 tremely doubtful, to say the least. But "man," "animal," 
 "race," etc., denote objects which we are accustomed to speak 
 and think of as " concrete," and so they seem most naturally
 
 38 ELEMENTS OF LOGIC 
 
 entitled to be called concrete names. On the other hand, 
 some writers call them abstract, and so regard all general con- 
 cepts as abstract because the process of forming them is one 
 of abstraction. Keynes disputes the legitimacy of this treat- 
 ment of them. He seems to hold that the process has nothing 
 to do with the nature of the product. This seems to my mind 
 doubtful. But in so far as such terms as "man," "animal," 
 " tree/' etc., denote real objects which are definite things, they 
 may with, at least, tolerable propriety, be called concrete. 
 But in so far as they are indefinite and do not seem to repre- 
 sent any clearly conceived individual object, they closely re- 
 semble abstract terms in this respect. In fact, as terms be- 
 come more general, that is, as their extension increases they 
 approximate in indefiniteness the abstract concepts which are 
 chai'acterized by this indifference to particular things, and 
 hence they may with some propriety be spoken of as abstract. 
 Perhaps their ambiguity in this respect would justify us in 
 regarding them as abstract in one relation and concrete in 
 another, their abstractness depending upon the proportion of 
 indefiniteness, and their concreteness upon the proportion of 
 definiteness expressed by them. There are other terms which 
 give still greater trouble than these. They are such examples 
 as "color," "sound," "pleasure," "thought," etc. It may be 
 a problem to determine whether they are concrete or abstract. 
 On the one hand, " color " and " sound," as denoting quali- 
 ties apart from a definite conception of their subject, might be 
 regarded as abstract. On the other hand, as general terms 
 for attributes which are concrete they might be considered as 
 concrete. So "pleasure," "thought," and all names of states 
 of consciousness, as verbal nouns, dynamic concepts, might be 
 abstract. But as names of individual facts clearly represent- 
 able in some way, they might be considered by many writers 
 as concrete. In other words, as general tenns denoting facts 
 or individual events, they will be usually conceived as con- 
 crete, but as terms denoting attributes, but not definitely de- 
 noting their subject, they will appear usually as abstract. 
 This may be true of a large number of terms. If so, they may
 
 TERMS OR CONCEPTS, AND THEIR KINDS 39 
 
 be treated in two relations at the same time, according to the 
 degree of definiteness and indefiniteness with which they are 
 conceived. 
 
 But this manner of speaking about such terms suggests 
 the common usage of the words "concrete" and "abstract," 
 which logical discussion cannot wholly ignore. Ordinarily 
 " concrete " denotes any sensible or real object, which may 
 represent the meaning of a term, singular or general. Thus 
 " man," " tree," " biped," denote sensible objects. Even 
 " color," " sound," " odors," denote individual percepts, and so 
 there is always some distinct or definite reality expressed by 
 them, which is supposed to be identical with the concrete. 
 But such terms as "emotion," "reasoning," "thought," "pleas- 
 ure," " figure," " form," are either not sensible objects or are 
 so vague and indefinite, being often called the " higher ab- 
 stractions " of the mind, that the common consciousness thinks 
 and speaks of them, perhaps loosely, as "abstract" concep- 
 tions. In the same way general terms are often conceived as 
 abstract in proportion to their generalized character, or their 
 remoteness from the individuals which they comprehend. 
 This distinction, then, is mainly that between what is presenta- 
 live or representative, and what is merely thought in conscious- 
 ness. Conceptions which call up distinctly to the mind the 
 individual objects which they denote are thus commonly taken 
 for concrete, and those which do not indicate clearly the char- 
 acteristics named, and which might be called symbolical con- 
 ceptions, after the manner of Leibnitz, are taken as abstract. 
 For this reason it might serve the purposes of Logic much 
 better if the distinction were made between definite and indefi- 
 nite concepts, the former taken to denote all terms which 
 clearly denote or connote certain marks, such as Peter, man, 
 ivhite, and the latter taken to denote such as do not indicate 
 distinctly any communicable mark of an object or a fact, as 
 "life," "organic," "humanity," " government," " institution," 
 etc. It is this distinction rather than that between the con- 
 crete and the abstract, as I have defined them, that is of im- 
 portance in Logic, because fallacious reasoning is occasioned
 
 40 ELEMENTS OF LOGIC 
 
 more by ambiguities of meaning due to indefiniteness than by 
 the question whether a term is abstract or not, unless " ab- 
 tract " is taken as synonymous with indefinite. As this mdef- 
 initeness increases with the extension of a concept, general 
 terms will partake of this character as they recede from the 
 individual and concrete, and so will often be called "abstract" 
 in the same degree, although they may not wholly lose a con- 
 crete reference. It is possible, therefore, to consider them 
 either as mixed concrete and abstract terms, or as combining 
 definite and indefinite characteristics in an inverse ratio to 
 each other. If the former, we simply adapt them to our defi- 
 nitions ; if the latter, we use "concrete" and "abstract" to 
 denote the distinction between the representable and the non- 
 representable concepts, which is the most important for Logic, 
 as later chapters will show. Wundt has some excellent obser- 
 vations upon this question, which sustain the position I have 
 taken, and should be quoted in this connection : 
 
 " It is a necessary consequence of the formation of concepts 
 from the connection of percepts that certain conceptions 
 should stand much nearer than others to the presentations of 
 sense. We have generally expressed this fact by the distinc- 
 tion between concrete and abstract ideas, but have described the 
 logical process, by which abstract notions have been formed 
 from concrete, as the process of abstraction. The influence of 
 this latter procedure, moreover, has essentially changed the 
 meaning of the terms ' concrete ' and ' abstract ' in the 
 course of history. Scholastic Nominalism, which introduced 
 it into Logic, used the terms for the mere purpose of distin- 
 guishing between words. Every substantive noun which de- 
 noted an individual object, or a class of objects, was concrete, 
 whereas, on the other hand, a word formed from a concrete 
 term and used to denote a universal property was called ab- 
 stract. Words like man, white, etc., therefore were regarded as 
 concrete, and those like humanity, whiteness, etc., were regarded 
 as abstract. In modern Logic this distinction gradually became 
 confused with that between individual and general concepts, 
 since, in depending upon the use of the word ' to abstract,'
 
 TERMS OR CONCEPTS, AND THEIR KINDS 41 
 
 we became accustomed to characterize as abstract all concep- 
 tions whose formation was marked by a distinct process of 
 abstraction. But as this was peculiar to all generic concep- 
 tions, there remained finally nothing but singular or individual 
 terms which could represent the territory of the concrete. 
 This confusion is, in fact, logically quite excusable ; for, in 
 whatever senses the terms ' concrete ' and ' abstract ' are 
 applied, it is evident that it is neither a difference in the pro- 
 cess of forming, nor the essential constitution of concepts, but 
 a much more external circumstance which is expressed by the 
 terms. Even Mill's proposition to restore scholastic usage of 
 them may be opposed to the practical consideration that, in 
 the modified sense in which they refer to the degree of apply- 
 ing the process of abstraction, they have already obtained, 
 through common usage, a right to a place in the language of 
 science, as well as in practical life, which at the same time in- 
 dicates a demand for a corresponding logical distinction. If 
 we regard only the present practice of language, it will not 
 appear doubtful that we have here to do chiefly with the rela- 
 tion of a conception to its representative percept. So long as 
 the latter consists in a sensible presentation in which the essen- 
 tial elements of the concept are realized, and not merely in 
 the word denoting it, we name it concrete. But, on the other 
 hand, so soon as the written or spoken word becomes a single 
 sign or symbol for the concept, it is abstract. In other words, 
 those concepts are abstract to which no adequate representa- 
 tive percept corresponds, and for which, in thought, we can 
 only choose an external and apparently arbitrary symbol. In 
 this sense we should doubtlessly describe such a concept as 
 ' man ' or ' animal ' as concrete, and such a concept as 
 ' humanity ' as abstract. But in opposition to scholastic 
 usage we could call 'the righteous' as well as ' righteous- 
 ness' by the name of abstract. And further, an individual 
 concept would most frequently be concrete at the same time, 
 while a concrete term would very frequently be general. Also 
 it would certainly happen that, in individual cases, the distinc- 
 tion would remain indeterminate or doubtful. Though words
 
 42 ELEMENTS OF LOGIC 
 
 have gradually developed into signs of abstract conceptions, 
 and though, as the history of the change of meaning every- 
 where shows, abstract conceptions have been developed from 
 concrete, why should we not at times meet a conception which 
 remains in the intermediate stage of development ? Concep- 
 tions like 'machine,' 'weight,' etc., may, in fact, be com- 
 pletely abstract to on person, and in another be attached to a 
 sense picture, and consequently concrete. From this a single 
 conclusion is evident ; namely, that this distinction has little 
 importance for the nature of a conception, but that it has just 
 as great an importance for the development of abstract no- 
 tions. This development depends upon the constant ap- 
 plication of the same process which we find is active in the 
 origin of all conceptions, even the most individual and con- 
 crete." * 
 
 A pure abstract concept, therefore, I shall consider as de- 
 fined to be a quality or attribute conceived and treated as a 
 substantive or thing. A mixed abstract and concrete term 
 will then be capable of either reference according to circum- 
 stances and the degree of its definiteness. It is general terms 
 that are so frequently of this mixed character and that are the 
 source of confusion in Logic. The pure forms also give rise to 
 a certain order of errors. In the first place, there is the danger 
 of treating abstract conceptions as if they represented inde- 
 pendent realities. Thus such ideas as "truth," "beauty," 
 "excellence," "virtue," "nature," "law," etc., are often spoken 
 of by writers as if they represented existences independent 
 of particular objects or persons, of which they are in reality 
 only qualities. The error here is first in the conception, and 
 reasoning is subsequently affected by it as the conclusion is 
 always vitiated or validated by the character of the premises. 
 But modern thought is better provided against the confusion 
 of the abstract and the concrete than ancient and mediaeval 
 speculation, when it concerns the pure concrete and the pure 
 abstract concepts. This is not the case with the second form of 
 error which arises from the confusion of the concrete and the 
 * Wundt: Logik, Bd. I., Zweiter Abschiiitt, Cap. I., 3, p. 97.
 
 TERMS OR CONCEPTS, AND THEIR KINDS 43 
 
 abstract in the same terms and propositions. One person may 
 have the abstract and another the concrete aspect in conscious- 
 ness when using the concept, and unless what is affirmed of it 
 be true of both aspects there will be a difference of opinion 
 between the two, or one will be guilty of error in thought and 
 reasoning. This is less apparent in conceptions than in prop- 
 ositions. Hence it is more frequent that abstract thoughts, 
 propositions, or principles should produce error than the mere 
 concept alone, or the confusion of the abstract and concrete 
 aspects of it, although the error in the use of principles must 
 begin with an error in the use of the concept. But as pos- 
 sible sources of fallacy in the conception of mixed terms 
 we have such general concepts as " religion," " institution," 
 "home," "history," "organism," "socialism," where we may 
 not be assured whether it is the concrete or the abstract form 
 which is prominent in consciousness. But all this is brought 
 out more clearly in propositions than in concepts. Thus if I 
 take the proposition, "Keligion is useful," I may mean "relig- 
 ion " in the abstract ideal form, or I may mean in its particular 
 concrete form ; that is, all religions and denominations. Or I 
 may say, " Governments are good institutions," and mean either 
 all particular governments, or government in the abstract. In 
 the concrete, governments might be very bad, while in the ab- 
 stract they might be very good. " Government," as an abstract 
 concept, is only an ideal quality of the aggregate of men or of 
 those in power, and so, in the concrete, can never be better 
 than the men composing it. But in speaking and writing of it, 
 it is often convenient to treat the concept independently of its 
 reference to the men who made it, and then it is considered 
 in the abstract. As long as the men who compose government 
 are bad, I can saj; that " government is a good institution " 
 only when I consider it abstractly. Concretely government 
 could only be what the men are who compose it. The same 
 observations apply to any other mixed concepts and proposi- 
 tions. This will also be true, but perhaps in a less degree, of 
 such terms as "man," "animal," "vegetable," etc., for we 
 occasionally give them an abstract import, although their con-
 
 44 ELEMENTS OF LOGIC 
 
 crete reference may be the most frequent. But the signifi- 
 cance and importance of these distinctions will be more ap- 
 parent when we come to consider the essence and accident* of 
 concepts. For the difference between the abstract and the 
 concrete is closely related or connected with that between es- 
 sence and accident. 
 
 4th. Positive, Negative, Privative, and Nego-positive 
 Terms. The usual division of terms in this respect is into 
 Positive, Negative, and Privative, but I add the fourth for 
 reasons which will appear in the sequel. I shall first define 
 and illustrate them. 
 
 Positive terms or concepts are those which signify the exist- 
 ence, presence, or possession of certain qualities ; for example, 
 " good," " pure," " excellence," " metal," " organic," " human," 
 etc. A positive term is thus one which is so both grammati- 
 cally and logically, or both in form and matter. 
 
 Negative terms or concepts are those which signify the 
 absence of certain qualities, as " impure," " inorganic," " un- 
 human," " non-metal," " ingratitude," " insipid," etc. They are 
 thus negative in both form and matter. 
 
 Privative terms or concepts are those w T hich denote the de- 
 privation of certain qualities once possessed or the normal 
 characteristic of the subject. Thus " deaf," " dumb," " blind," 
 " dead," are privative terms. In so far as they denote the 
 absence of qualities they are negative terms. But they differ 
 in their form from negative terms, although materially con- 
 sidered they are only modifications of them. They may be 
 more strictly defined as terms which are positive in their 
 form, but negative in their matter. This will be apparent 
 from the illustrations. 
 
 Nego-positive terms or concepts are those which denote the 
 presence of a positive quality, although they appear to be neg- 
 ative. Thus " disagreeable," " inconvenience," " infamous," 
 "ignorant," "displeasure," " immediate," "undone," etc. They 
 are thus negative in form and positive in matter. They can in 
 many cases be most easily distinguished by comparing them 
 with their corresponding positive conceptions. Thus "un-
 
 TEEMS OR CONCEPTS, AND THEIR KINDS 45 
 
 happiness " and " invaluable " have their equivalents in " mis- 
 ery " and "costly," both of which are positive. 
 
 Some terms may be taken in either a negative or a nego-posi- 
 tive sense. Thus "uncertain," "unhealthy," "unpleasant," 
 " indistinct," may be conceived as the negatives of " certain," 
 "healthy," "pleasant," " distinct," or as the nego-positive 
 equivalents of " doubtful," " sickly," " painful," " obscure." 
 They are, however, the same modification of positive concep- 
 tions that privative terms are of negatives. In fact, privative 
 and nego-positive terms are simply mixed concepts, having an 
 element from each of the other two, but combining them in a 
 reversed relation. In the broader sense, therefore, we can 
 divide all terms into positive and negative. In this broader 
 sense the symbols or indices of negative terms are in, un, 
 non, less, dis, a or an, anti, and sometimes de, and perhaps 
 mis, 
 
 The following tabular outlines will give the divisions and 
 indicate their nature and relations to each other. 
 
 f Positive = Positive in both form and matter. 
 Terms j Negative = Negative in both form and matter. 
 or Concepts. | Privative = Positive in form and negative in matter. 
 
 [ Nego-positive = Negative in form and positive in matter. 
 
 Terms 
 or 
 
 (Positive. m ( ( Pure = Simple Positive. 
 
 Negative. Terms Positive 1 ( p . 
 
 Privative. p or \ Mixed rlvatlv %. 
 
 Concepts I Mixed \ Nego- Co ": 1 Negative. \ ' 1 Nego-positive. 
 
 ( positive. [ I Pure = Simple Negative. 
 
 The reason for distinguishing a separate class of nego- 
 positive tei'ins is their liability at times to be mistaken for 
 negatives, and the confusion often incident to the transition, 
 or immediate inference from a purely negative conception to 
 one which is really positive in its meaning, although expressed 
 by the same term. Thus if we were to argue that a thing or an 
 act is bad because it is not-good, we should be committing 
 a fallacy, considering that "not-good" is a purely negative 
 conception. The same remark applies to the passage from 
 the iwt-just to the unjust, and from the non-moral or not-moral
 
 46 ELEMENTS OF LOGIC 
 
 to the immoral. For instance, phj^sical acts are " not-moral," 
 but they are also not immoral. In ethics there are three 
 classes of acts the moral, the non-moral, and the immoral or 
 in stoical parlance good, indifferent, and bad. Or we might 
 divide them first into moral and not-moral, and subdivide the 
 latter into non-moral and immoral. Similarly we may recog- 
 nize just, not- just, and unjust acts. The distinction is here 
 the same. Also in many other departments of thought this 
 triple division is the proper one, although the usual division is 
 dichotomous instead of trichotomous. Where it occurs the 
 negative and nego-positive conceptions do not necessarily 
 coincide. It is true, however, that a negative term is often 
 used as the equivalent of the nego-positive. Thus we some- 
 times describe an act as " not-just " when we think of it as 
 " unjust." This is when we do not think of the conception as 
 infinitated. By an infinitated conception I mean one which 
 comprehends all other possible objects in the world than those 
 denoted by the contradictory positive term. Thus in its 
 strict meaning and extension the negative "not-just" will in- 
 clude all other things in the universe than those expressed by 
 "just." Hence among " not-just " might be found material 
 objects, such as trees, stones, houses, etc. Again, " not-house " 
 would include all other things in the universe than the term 
 " house ; " and so on with all similar conceptions. The infini- 
 tated concept is simply all else than what is expressed by the 
 positive. But often a negative concept is used either as a 
 euphemism, or as an equivalent of the nego-positive. In such 
 cases they are clearly convertible. Thus wherever " unpleas- 
 ant " is thought of as " painful," the two concepts can be sub- 
 stituted for each other. Illustrations of this will appear in 
 Conversion. But when the negative term is infinitated this 
 substitution cannot be made, and it is the business of the stu- 
 dent and the logician to be on the alert for the confusion inci- 
 dent to such a procedure. 
 
 The terms " greater," " less," and " equal " deserve a brief 
 notice. They are all positive concepts, but taken in rela- 
 tion to each other, as they must be, they are relative terms,
 
 TERMS OR CONCEPTS, AND THEIR KINDS 47 
 
 and must be considered in the next section. But being rela- 
 tive concepts we may say that any two of them are the 
 negatives of the third. But their pure negatives are the 
 infinitated forms of their positives, and it is only as rela- 
 tives that any of them can be considered the negative of the 
 others. 
 
 What has been said of the terms " greater," " less," and 
 " equal " can be said of a large number of terms in the lan- 
 guage. Indeed, every term may be said to be negative in rela- 
 tion to all other terms, except its own equivalents or synonyms. 
 It will not need to be so in its form, but only in its meaning a9 
 related to those terms which do not express the same concept. 
 Hence, while we should call " horse " a positive term, in relation 
 to man it would be negative, because it would be negatively 
 conceived as "not man ; " " tree," as not a lion ; " external in- 
 fluences " as " not internal influences," etc. Very often, from 
 the mere habit of conceiving different concepts as excluding 
 each other, we assume that particular cases represent contrary 
 concepts, one the negative or contradictory of each other, 
 when in reality they are not so. Many errors of opinion and 
 of reasoning are incident to this mistake. We require always 
 to examine how far the meaning of concepts excludes that of 
 other terms, and not to assume a negative relation from the 
 mere fact that it generally so exists. A very large number of 
 concepts have this relation ; but the form of the term will be 
 no adequate criterion of the fact, and we must examine the 
 matter of thought in given cases, in order to decide the ques- 
 tion for practical instances. 
 
 5th. Absolute and Relative Terms or Concepts. 
 This distinction, as usually applied, does not have much im- 
 portance in Logic. Hence it may be dismissed very briefly. 
 There is, perhaps, a more comprehensive sense in which the 
 distinction is a valuable one. But it does not apply in any 
 special way to the few terms known particularly as relative. 
 Most terms are absolute concepts in the ordinary sense of the 
 word. 
 
 " Absolute " means literally that which is severed from all
 
 48 ELEMENTS OF LOGIC 
 
 dependence on another. Hence an absolute term is one which 
 expresses what can be thought of by itself, and which does not 
 have its meaning determined by comparison with some corre- 
 late object of consciousness. Thus " man," " tree," " earth," 
 " star," " book," etc., are absolute concepts. They have no 
 correlatives implied in them. They may denote things which 
 exist, both in thought and reality, in relation to something 
 else. But this related object is merely associated with the 
 conception and may not be necessary to the meaning of the 
 term, or to its presentation in consciousness. Quite the con- 
 trary is time of relative terms. 
 
 A " relative " term or concept is one which denotes an object 
 that cannot be thought of without reference to some other 
 object, which is its con-elate. Thus "father" and "son," 
 "parent" and " child " are correlatives. Each has its meaning 
 determined in relation to the other. Again, the terms " mon- 
 arch," "shepherd," "master," "teacher," "subject," "slave," 
 " eldest," etc., are relative. Their correlates can easily be re- 
 marked. It will be apparent also how " greater " and " less " 
 may be regarded as relative. But as little confusion is incident 
 to logical processes connected with the use of relative terms 
 as here defined, the subject does not require further consider- 
 ation. 
 
 In a broader sense, as I have remarked, every term is relative. 
 It is first relative to its negative concept, and we often find this 
 means a convenient one for defining a term. But it is never 
 completely satisfactory, because a true definition demands a 
 statement of the positive content of a concept. The negative 
 term, however, is a very convenient one for representing the 
 actual relation sustained by every concept. Further than this, 
 every object may be said to exist in a relation to some other, 
 or all other objects, and the relation between two or more ob- 
 jects may be so close as to mutually affect each other's meaning. 
 Thus " day " and " night," " joy " and " sorrow," " pleasure " 
 and "pain," " true " and "false," are conceived as relatives by 
 contrast. But while they may be so related in thought, the 
 material or real existence of one of the correlates is not iru-
 
 TERMS OR CONCEPTS, AND THEIR KINDS 49 
 
 plied by the other. Such relation as may exist between other 
 classes of conceptions is too remote and unimportant for logic 
 to take any special notice of it.* 
 
 * For discussion of technical problems connected with the qualities of 
 terms the student may consult the following works: Mill: Logic, Bk. I., 
 Chap. II. ; Venn : Empirical Logic, Chap. VII. ; Keyues : Formal Logic, 
 Part L, Chaps. I.-IIL, pp. 7-50; Jevons: Principles of Science, Bk. I., 
 Chap. II.; Wundt: Logik, Zweiter Absclmitt, Chaps. I.-IV., pp. 86-134. 
 4
 
 CHAPTER IV. 
 
 THE AMBIGUITY OF TERMS 
 
 IN this chapter I shall do little more than transcribe the 
 language of Jevons upon the subject. He has said about all 
 that a practical Logic requires to have said upon it. The im- 
 portance of considering it ought to be apparent to everyone, 
 and will be so to those who know or suspect that the majority 
 of fallacies in our reasoning turn upon the ambiguous use of 
 words. Most of our controversies are logomachies on the same 
 account. We think we are employing the same conceptions 
 when we are only using the same terms ; the difference be- 
 tween our conceptions, in spite of the identity in language, 
 may be as great as between different words. Different terms 
 also may be used for the same concepts, and thus give rise to a 
 converse error. But the most frequent source of error is am- 
 biguity. A syllogism will illustrate it : 
 
 No designing person ought to be trusted. 
 Engravers are by profession designers. 
 .'. They ought not to be trusted. 
 
 It is easy enough to defect in such cases the source of the 
 fallacy. But there are instances where the ambiguity is more 
 subtle, and requires a keener logical insight for its detection. 
 In profounder subjects of speculation it requires a wide knowl- 
 edge of the use of language and a thorough acquaintance with 
 the laws and processes of the mind. In fact, a reasoner should 
 always be on the alert for the ambiguous use of terms, and he 
 cannot have the possibility of such a source of confusion better 
 indicated than by a few observations upon the simplest words 
 of the language.
 
 THE AMBIGUITY OF TERMS 51 
 
 We mny divide terms into uni vocal and equivocal. A uni- 
 vocal term is one with but a single meaning, which is exposed 
 to no mistake of interpretation. An equivocal term is one with 
 more than a single meaning. Very little observation is neces- 
 sary to show that very few terms come under the class univo- 
 caL Proper names, and therefore singular terms, are almost 
 the only conceptions with an unmistakable import, and even 
 some of these are equivocal. A general concept applying only 
 to a single class of individuals of exactly the same kind may 
 be univocal. Thus President Lincoln, St. Paul's Cathedral, 
 Berlin, are univocal. But Washington may be equivocal as 
 applying to a person, a city, or a state. Of common terms 
 that are univocal, Jevons thinks that they are chiefly found in 
 technical and scientific language. He enumerates " steam- 
 engine," "gasometer," and " railway train," and in common 
 life such terms as " penny," " mantelpiece," " tea-cup," " bread," 
 and "butter." "Cathedral" is probably univocal, or of one 
 logical meaning only. But " church" is equivocal, as referring 
 sometimes to a building, and sometimes to a corporate body of 
 men. Compared with the equivocal terms, however, the uni- 
 vocal are very few. 
 
 From this point we may simply quote Jevons. He begins 
 with a division of equivocal terms. " We may distinguish," 
 he says, " three classes of equivocal words, according as they 
 are 
 
 " 1. Equivocal in sound only. 
 
 " 2. Equivocal in spelling only. 
 
 " 3. Equivocal both in sound and spelling. 
 
 " The first two classes are, comparatively speaking, of very 
 slight importance, and do not often give rise to serious error. 
 They produce what we cah 1 trivial mistakes. Thus we may 
 confuse, when spoken only, the words right, wright, and rite 
 (ceremony) ; also the words rein, rain, and reign.; might, mite, 
 etc. Owing partly to defects of pronunciation, mistakes are 
 not unknown between the four words air, hair, hare, and heir. 
 
 " Words equivocal in spelling, but not in sound, are such as 
 tear (drop), and tear, pronounced tare, meaning a rent in
 
 52 ELEMENTS OF LOGIC 
 
 cloth ; or lead, the metal, and lead, as in following the lead of 
 another person. As little more than momentary misapprehen- 
 sion, however, can arise from such resemblance of words, we 
 shall pass at once to the class of words equivocal both in sound 
 and spelling. These I shall separate into three groups, accord- 
 ing as the equivocation arises. 
 
 " (a) From the accidental confusion of different words. 
 
 " (b) From the transfer of meaning by the association of 
 ideas. 
 
 " (c) From the logical transfer of meaning to analogous ob- 
 jects. 
 
 " (a) Under the first class we place a certain number of curi- 
 ous but hardly important cases in which ambiguity has arisen 
 from the confusion of entirely different words, derived from 
 different languages or from different roots of the same lan- 
 guage, but which have in the course of time assumed the 
 same sound and spelling. Thus the word mean denotes either 
 that which is medium or mediocre, from the French moyen 
 and the Latin medius, connected with the Anglo-Saxon mid or 
 middle; or it denotes what is low-minded and base, being then 
 derived from the Anglo-Saxon gemoene, which means 'that 
 belonging to the moene or many,' whatever, in short, is vulgar. 
 The verb to mean can hardly be confused with the adjective 
 mean, but it comes from a third distinct root, probably con- 
 nected with the Sanscrit verb to think. 
 
 " As other instances of this casual ambiguity I may mention 
 rent, a money payment, from the French rente (rendre, to re- 
 turn), or a tear, the result of the action of rending, this word 
 being of Anglo-Saxon origin and one of the numerous class 
 beginning in r or wr, which imitate more or less perfectly the 
 sound of the action which they denote. Pound, from the 
 Latin pondus, a weight, is confused with pound, in the sense of 
 a village pinfold for cattle, derived from the Saxon pydan, to 
 pen up. Fell, a mountain, is a perfectly distinct word from 
 fell, a skin or hide ; and pulse, a throb or beating, and pulse, 
 peas, beans, or potage, though both derived from the Greek 
 or Latin, are probably quite unconnected words. It is curi-
 
 THE AMBIGUITY OF TERMS 58 
 
 ous that gin, in the meaning of trap or machine, is a con- 
 tracted form of engine, and when denoting the spirituous 
 liquor is a corruption of Geneva, the place where the spirit 
 was first made. 
 
 " Certain important cases of confusion have been detected in 
 grammar, as between the numeral one, derived from an Aryan 
 root, through the Latin unus, and the indeterminate pronoun 
 one (as in ' one ought to do one's duty '), which is really a cor- 
 rupt form of the French word homme or man. The Germans 
 to the present day use man in this sense, as in 'man sagt,' i.e., 
 one says. 
 
 " (6) By far the largest part of equivocal words have become 
 so by a transfer of the meaning from the thing originally de- 
 noted by the word to some other thing habitually connected 
 with it so as to become closely associated in thought. Thus 
 in Parliamentary language the House means either the cham- 
 ber in which the members meet, or it means the body of mem- 
 bers who happen to be assembled in it at any time. Similarly 
 the word church originally denoted the building (KV/XOJCOV, 
 the Lord's House) in which religious worshippers assemble, 
 but it has thence derived a vai'iety of meanings ; it may mean 
 a particular body of worshippers accustomed to assemble in 
 any one place, in which sense it is used in Acts xiv. 23 ; or 
 it means any body of persons holding the same religious 
 opinions, and connected in one organization, as in the Angli- 
 can, or Greek, or Roman Catholic Church ; it is also some- 
 times used so as to include the laity as well as the clergy ; but 
 more generally perhaps the clergy and religious authori- 
 ties of any sect or country are so strongly associated with the 
 act of worship as to often be called the church par excellence. 
 It is quite evident, moreover, that the word entirely differs in 
 meaning according as it is used by a member of the Anglican, 
 Greek, Roman Catholic, Scotch Presbyterian, or any other ex- 
 isting church. 
 
 " The word foot has suffered several curious but very evi- 
 dent transfers of meaning. Originally it denoted the foot of 
 a man or an animal, and is probably connected in a remote
 
 54 ELEMENTS OF LOGIC 
 
 manner with the Latin pes, pedis, and the Greek Trot'?, 770809 ; 
 but since the length of the foot is naturally employed as a 
 rude measure of length it came to be applied to a 
 fixed measure of length ; and as the foot is at the bottom of 
 the body the name was extended by analog} 7 to the foot of a 
 mountain, or the feet of a table ; by a further extension, any 
 position, plan, reason, or argument on which we place our- 
 selves and rely, is called the foot or footing. The same word 
 also denotes soldiers who fight upon their feet, or infantry, 
 and the measured part of a verse having a definite length. 
 That these very different meanings are naturally connected 
 with the original meaning is evident from the fact that the 
 Latin and Greek words for foot are subject to exactly similar 
 series of ambiguities. 
 
 " It would be a long task to trace out completely the various 
 and often contradictory meanings of the word fellow. Origi- 
 nally a fellow Avas what follows another, that is, a companion ; 
 thus it came to mean the other of a pair, as one shoe is the 
 fellow of the other, or simply an equal, as when we say that 
 Shakespeare 'hath not a fellow.' From the simple meaning 
 of companion, again, it comes to denote vaguely a person, as in 
 the question, ' Wb&t fellow is that ? ' but then there is a curious 
 confusion of depreciatory and endearing power in the word ; 
 when a man is called a mere fellow, or simply a, fellow in a par- 
 ticular tone of voice, the name is one of severe contempt ; alter 
 the tone of voice of the connected words in the least degree, 
 and it becomes one of the most sweet and endearing appella- 
 tions, as when we speak of a dear or good fellow. "We may 
 still add the technical meanings of the name as applied in the 
 case of a Fellow of a College or of a learned society. 
 
 " Another good instance of the growth of a number of differ- 
 ent meanings from a single root is found in the word i<i.-t. 
 Originally a post was something posited, or placed firmly in 
 the ground, such as an upright piece of wood or stone ; such 
 meaning still remains in the cases of a lamp-post, a gate-post, 
 signal-post, etc. As a post would often be used to mark a 
 fixed spot of ground, as in a mile-post, it came to mean the
 
 THE AMBIGUITY OF TERM8 55 
 
 fixed or appointed place where the post was placed, as in a 
 military post, the post of danger, or honor, etc. The fixed 
 places where horses were kept in readiness to facilitate rapid 
 travelling during the times of the Roman empire were thus 
 called posts, and thence the whole system of arrangement for 
 the conveyance of persons or news came to be called the posts. 
 The name has retained an exactly similar meaning to the pres- 
 ent day in most parts of Europe, and we still use it in post- 
 chaise, post-boy, post-horse, and postilion. A system of post 
 conveyance for letters having been organized for about two 
 centuries in England and other countries, this is perhaps the 
 meaning most closely associated with the word post at present, 
 and a number of expressions have arisen, such as post-office, 
 postage, postal-guide, postman, postmaster, postal-telegraph, 
 etc. Curiously enough, we now have iron letter-posts, in 
 which the word post is restored exactly to its original mean- 
 ing. 
 
 " Although the words described above were selected on ac- 
 count of the curious variety of their meanings, I do not hesi- 
 tate to assert that the majority of common nouns possess vari- 
 ous meanings in greater or less number. Dr. Watts, in his 
 ' Logic,' suggests that the words book, Bible, fish, house, and 
 elephant are univocal terms, but the reader would easily 
 detect ambiguities in each of them. Thus fish bears a very 
 different meaning in natural history from what it does in the 
 mouth of unscientific persons, who include under it not only 
 true fishes, but shell-fish or mollusca, and the cetacea, such as 
 whales and seals, in short, all swimming animals, whether they 
 have the character of true fish or not. Elephant, in a station- 
 er's or bookseller's shop, means a large kind of paper instead 
 of a large animal. Bible sometimes means any particular copy 
 of the Bible, sometimes the collection of works constituting 
 the Holy Scriptures. The word man is singularly ambiguous ; 
 sometimes it denotes man as distinguished from woman ; at 
 other times it is certainly used to include both sexes ; and in 
 certain recent election cases lawyers were unable to decide 
 whether the word man, as used in the Reform Act of 1867,
 
 56 ELEMENTS OF LOGIC 
 
 ought or ought not to be interpreted so as to include women. 
 On other occasions man is used to denote an adult male as 
 distinguished from a boy, and it also often denotes one who 
 is emphatically a man as possessing a masculine character 
 (heroic). Occasionally it is used in the same way as groom, 
 for a servant, as in the proverb, "Like master, like man." At 
 other times it stands specially for husband. 
 
 " (e) Among ambiguous words we must thirdly distinguish 
 those which derive their various meanings in a somewhat dif- 
 ferent manner, namely, by analogy or real resemblance. When 
 we speak of a sweet taste, a sweet flower, a sweet tune, a sweet 
 landscape, a sweet face, a sweet poem, it is evident that we 
 apply one and the same word to very different things ; such a 
 concrete thing as lump-sugar can hardly be compared directly 
 with such an intellectual existence as Tennyson's May Queen. 
 Nevertheless, if the word sweet is to be considered ambiguous, 
 it is in a different way from those we have before considered, 
 because all the things are called sweet on account of a peculiar 
 pleasure which they yield, which cannot be described otherwise 
 than by comparison with sugar. In a similar way we describe 
 a pain as sharp, a disappointment as bitter, a person's temper 
 as sour, the future as bright or gloomy, an achievement as 
 brilliant ; all these adjectives implying comparison with bodily 
 sensations of the simplest kind. The adjective brilliant is de- 
 rived from the French briller, to glitter or sparkle ; and this 
 meaning it fully retains when we speak of a brilliant diamond, 
 a brilliant star, etc. But by what subtle analogy is it that we 
 speak of a brilliant position, a brilliant achievement, brilliant 
 talents, brilh'ant style ! We cannot speak of a clear explana- 
 tion, indefatigable perseverance, perspicuous style, or sore 
 calamity, without employing in each of these expressions a 
 double analogy to physical impressions, actions, or events." 
 
 Continuing the discussion in a later chapter on " The Growth 
 of Language," the same author goes on to show how these 
 ambiguities have originated. It is a matter of considerable 
 importance to the logician to understand the process, and we 
 reproduce, at length, Jevons's treatment of it :
 
 THE AMBIGUITY OF TERMS 57 
 
 " There are two great and contrary processes," he says, 
 " which modify language, as follows : 
 
 " 1. GENERALIZATION, by which a name comes to be applied to 
 a wider class of objects than before, so that the extension of 
 its meaning is increased, and the intension diminished. 
 
 " 2. SPECIALIZATION, by which a name comes to be restricted to 
 a narrower class, the extension being decreased and the inten- 
 sion increased.* 
 
 " The first change arises in the most obvious manner from our 
 detecting a resemblance between a new object which is without 
 a name and some well-known object. To express the resem- 
 blance we are instinctively led to apply the old name to the 
 new object. Thus we are well acquainted with glass, and if we 
 meet any substance having the same glassy nature and appear- 
 ance we shall be apt at once to call it a kind of glass. The 
 word coal has undergone a change of this kind ; originally it 
 was the name of charked or charred wood, which was the prin- 
 cipal kind of fuel used five hundred years ago. As mineral 
 coal came into use it took the name from the former fuel, 
 which it resembled more nearly than anything else, but was at 
 first distinguished as sea-coal or pit-coal. Being now the far 
 more common of the two, it has taken the simple name, and we 
 distinguish charred wood as charcoal. Paper has undergone a 
 like change : originally denoting i\ie papyrus used in the Ko- 
 inan empire, it was transferred to the new writing material, 
 made of cotton or linen rags, which was introduced at a quite 
 uncertain period. The word character is interesting on ac- 
 count of its logical employment ; the Greek ^apa/cr^p denoted 
 strictly a tool for engraving, but it was transferred by asso- 
 ciation to the marks or letters engraved with it, and this 
 meaning is still retained by the word when we speak of Greek 
 characters, Arabic characters, i.e., figures or letters. But inas- 
 much as objects often have natural marks, signs, or tokens 
 which may indicate them as well as artificial characters, the 
 name was generalized and now means any peculiar or distinc- 
 tive mark or quality by which an object is easily recognized. 
 
 * For explanation of the terms intension and extension see Chapter V.
 
 58 ELEMENTS OF LOGIC 
 
 " Changes of this kind are usually effected by no particular 
 person and with no distinct purpose, but by a sort of uncon- 
 scious instinct in a number of persons using the name. In 
 the language of science, however, changes are often made pur- 
 posely, and with a clear apprehension of the generalization 
 implied. Thus soap in ordinary life is applied only to a com- 
 pound of soda or potash with fat, but chemists have purposely 
 extended the name so as to include any compound of a metal- 
 lic salt with a fatty substance. Accordingly there are such 
 things as lime-soap and lead-soap, which latter is employed in 
 making common diachylon plaster. Alcohol at first denoted 
 the product of ordinary fermentation commonly called spir- 
 its of wine, but chemists having discovered that many other 
 substances had a theoretical composition closely resembling 
 spirits of wine, the name was adopted for the whole class and 
 a "long enumeration of different kinds of alcohol will be found 
 in Dr. Roscoe's lessons on chemistry. The number of known 
 alcohols is likewise subject to indefinite increase by the prog- 
 ress of discovery. Every one of the chemical terms, acid, 
 alkali, metal, alloy, earth, ether, oil, gas, salt, may be shown to 
 have undergone great generalizations. 
 
 " In other sciences there is hardly a less supply of instances. 
 A lens originally meant a lenticular-shaped or double convex 
 piece of glass, that being the kind of glass most frequently used 
 by opticians. But as glasses of other shapes came to be used 
 along with lenses, the name was extended to concave or even 
 to perfectly flat pieces of glass. The words lever, plane, cone, 
 cylinder, arc, conic section, curve, prism, magnet, pendulum, 
 ray, light, and many others, have been similarly generalized. 
 
 " In common language we may observe that even proper or 
 singular names are often generalized, as when in the time of 
 Cicero a good actor was called a Roscius, after an actor of pre- 
 eminent talent. The name Caesar was adopted by the succes- 
 sor of Julius Csesar as an official name of the Emperor, with 
 which it gradually became synonymous, so that in the present 
 day the kaisers of Austria and the czars of Russia both 
 take their title from Csesar. The celebrated tower built by
 
 THE AMBIGUITY OF TERMS 59 
 
 the king of Egypt on the island of Pharos, at the entrance of 
 the harbor of Alexandria, has caused light-houses to be called 
 phares in French, and pharos in obsolete English. From the 
 celebrated Roman general, Quintus Fabius Maximus, any one 
 who avoids bringing a contest to a crisis is said to pursue a 
 Fabian policy. 
 
 "In science also singular names are often extended, as when 
 fixed stars are called distant suns, or the companions of Jupiter 
 are called his moons. It is, indeed, one theory, and a probable 
 one, that all general names were created by the process of 
 generalization going on in the early ages of human progress. 
 As the comprehension of general notions requires higher intel- 
 lect than the apprehension of singular and concrete things, 
 it seems natural that names should at first denote individual 
 objects and should afterward be extended to classes. We 
 have a glimpse of this process in the case of the Australian 
 natives, who had been accustomed to call a large dog cadli, but 
 when horses were first introduced into the country they adopted 
 this name as the nearest description of a horse. A very sim- 
 ilar incident is related by Captain Cook of the natives of 
 Otaheite. It may be objected, however, that a certain process 
 of judgment rmist have been exerted before the suitability of a 
 name to a particular thing could have been perceived, and it 
 may be considered probable that specialization as well as 
 generalization must have acted in the earliest origin of lan- 
 guage much as it does at present. 
 
 "Specialization is an exactly opposite process to generaliza- 
 tion, and is almost equally important. It consists in narrow- 
 ing the extension of meaning of a general name, so that it 
 comes to be a name only of an individual or a minor part of 
 the original class. It is thus we are fm*nished with the requi- 
 site names for a multitude of new implements, occupations, 
 and ideas with which we deal in advancing civilization. The 
 name physician is derived from the Greek <f>vo-iK6<;, natural, 
 and </>vo-i9, nature, so that it properly means one who has stud- 
 ied nature, especially the nature of the human body. It has 
 become restricted, however, to those who use this knowledge
 
 60 ELEMENTS OF LOGIC 
 
 for medical purposes, and the investigators of natural science 
 have been obliged to adopt the term physicist. The name nat- 
 uralist has been similarly restricted to those who study ani- 
 mated nature. The name surgeon originally meant handi- 
 craftsman, being a corruption of chirurgeon, derived from the 
 Greek xapoupvds, handworker. It has long been specialized, 
 however, to those who perform the mechanical parts of the 
 sanatory art. 
 
 "Language abounds with equally good examples. Minister 
 originally meant a servant, or one who acted as a minor of 
 another. Now it often means, specially, the most important 
 man in the kingdom. A chancellor was a clerk, or even a 
 doorkeeper, who sat in a place separated by bars or cancelli 
 in the offices of the Roman Emperor's palace ; now it is always 
 the name of a high or even the highest dignitary. Peer was an 
 equal (Latin par], and we still speak of being tried by our 
 peers ; but now, by the strange accidents of language, it means 
 the few who are superior to the rest of the Queen's subjects in 
 rank. Deacon, bishop, clerk, queen, captain, general, are all 
 words which have undergone a like process of specialization. 
 In such words as telegraph, rail, signal, station, and many 
 words relating to new inventions, we may trace the progress 
 of change in a lifetime. 
 
 "One effect of this process of specialization is very soon to 
 create a difference between any two words which happen from 
 some reason to be synonymous. Two or more words are said 
 to be synonymous (from the Greek cruv, with, and wo/za, name) 
 when they have the same meaning, as in the case, perhaps, of 
 teacher and instructor, similarity and resemblance, beginning 
 and commencement, sameness and identity, hypothesis and 
 supposition, intension and comprehension. But the fact is 
 that words commonly called synonymous are seldom perfect- 
 ly so, and there are almost always shades of difference in 
 meaning or use, which are explained in such works as Crabb's 
 ' English Synonyms.' A process called by Coleridge DESYXO- 
 KTMIZATION, and by Herbert Spencer, DIFFERENTIATIOX, is always 
 going on, which tends to specialize one of a pair of synonymous
 
 THE AMBIGUITY OF TERMS 61 
 
 words to one meaning and the other to another. Thus wave 
 and billow originally meant exactly the same physical effect, 
 but poets have now appropriated the word ' billow,' whereas 
 wave is used chiefly in practical and scientific matters. Un- 
 dulation is a third synonym, which will probably become the 
 sole scientific term for wave in course of time. Cab was origi- 
 nally a mere abbreviation of cabriolet, and therefore of similar 
 meaning, but it is now specialized to mean almost exclusively 
 a hackney cab. In America car is becoming restricted to 
 the meaning of a railway car. 
 
 "It may be remarked that it is a logical defect in a language 
 to possess a great number of synonymous terms, since we ac- 
 quire the habit of using them indifferently, without being sure 
 that they are not subject to ambiguities and obscure differ- 
 ences of meaning. The English language is especially subject 
 to the inconvenience of having a complete series of words 
 derived from Greek or Latin roots nearly synonymous with 
 other words of Saxon or French origin. The same statement 
 may, in fact, be put into Saxon or classical English ; and we 
 often, as Whately has well remarked, seem to prove a state- 
 ment by merely reproducing it in altered language. The rhe- 
 torical power of the language may be increased by the copious- 
 ness and variety of diction, but pitfalls are thus prepared for 
 all kinds of fallacies. 
 
 "In addition to the effects of generalization and specializa- 
 tion, vast additions and changes are made in language by the 
 process of analogous or metaphorical extension of the meaning 
 of words. This change may be said, no doubt, to consist in 
 generalization, since there must always be a resemblance be- 
 tween the new and old applications of the term. But the re- 
 semblance is often one of a most distant and obscure kind, 
 such as we should call analogy rather than identity. All words 
 used metaphorically, or as sijnilitudes, are cases of this process 
 of extension. Thus the old similitude of a ruler to the pilot 
 of the vessel gives rise to many metaphors, as in speaking of 
 the Prime Minister being at the helm of the state. The word 
 governor, and all its derivatives, is, in fact, one result of this
 
 62 ELEMENTS OF LOGIC 
 
 metaphor, being merely a corrupt form of gubernator, steers- 
 man. The words compass, pole-star, ensign, anchor, and many 
 others connected with navigation, are constantly used in a met- 
 aphorical manner. From the use of horses and hunting we de- 
 rive another set of metaphors ; as, in taking the reins of gov- 
 ernment, overturning the government, taking the bit between 
 the teeth, the government whip being heavily weighted, etc. 
 No doubt it might be shown that every other important occupa- 
 tion of life has furnished its corresponding stock of metaphors. 
 
 "It is easy to show, however, that this process, besides going 
 on unconsciously at the present day, must have acted through- 
 out the history of language, and that we owe to it almost all, 
 or probably all, the words expressive of refined mental or 
 spiritual ideas. The veiy word spirit, now the most refined 
 and immaterial of ideas, is but the Latin spirit us, a gentle 
 breeze or breathing ; ancl inspiration, esprit, or wit, and many 
 other words, are due to this metaphor. It is truly curious, 
 however, that almost all the words in different languages de- 
 noting mind or soul imply the same analogy to breath. Thus, 
 soul is from the Gothic root denoting a strong wind or storm ; 
 the Latin words animus and anima are supposed to be con- 
 nected with the Greek, ave^os, wind ; ^vxn is certainly derived 
 from $i>x<a, to blow ; irvtvpa, air, or breath, is used in the New 
 Testament for Spiritual Being ; and our word ghost has been 
 asserted to have a similar origin. 
 
 "Almost all the terms employed in mental philosophy or 
 metaphysics, to denote actions or phenomena of mind, are ul- 
 timately derived from metaphors. Apprehension is the putting 
 forward of the hand to take anything ; comprehension is the 
 taking of things together in a handful ; extension is the spread- 
 ing out ; intention, the bending to ; explication, the unfolding ; 
 application, the folding to ; proposition, the placing before ; 
 intuition, the seeing into ; and they might be almost indefi- 
 nitely extended. Our English name for reason, the understand- 
 ing, obviously contains some physical metaphor which has not 
 been fully explained ; with the Latin intellect there is also a 
 metaphor.
 
 THE AMBIGUITY OF TERMS 63 
 
 " Every sense gives rise to words of refined meaning ; sapi- 
 ence, taste, insipidity, gofit, are derived from the sense of taste ; 
 sagacity, from the dog's extraordinary power of smell ; but as 
 the sense of sight is by far the most acute and intellectual, it 
 gives rise to the larger part of language ; clearness, lucidity, 
 obscurity, haziness, perspicuity, and innumerable other expres- 
 sions, are derived from this sense. 
 
 "It is truly astonishing to notice the power which language 
 possesses by the processes of generalization, specialization, and 
 metaphor, to create many words from one single root. Pro- 
 fessor Max Miiller has given a remarkable instance of this in 
 the case of the root spec, which means sight, and appears in the 
 Aryan languages, as in the Sanscrit spas, the Greek o-KeVro/xai, 
 with transposition of consonants in the Latin specio, and even 
 in the English spy. The following is an incomplete list of the 
 words developed from this root : Species, special, especial, 
 specimen, spice, spicy, specious, specialty, specific, specializa- 
 tion, specie (gold or silver), spectre, specification, spectacle, 
 spectator, spectral, spectrum, speculum, specular, speculations. 
 The same root also enters into composition with various pre- 
 fixes ; and we thus obtain a series of words, suspect, aspect, 
 circumspect, expect, inspect, prospect, respect, retrospect, in- 
 trospection, conspicuous, perspicuous, perspective ; with each 
 of which, again, a number of derivatives is connected. Thus 
 from suspect we derive suspicion, suspicable, suspicious, sus- 
 piciousness. I have estimated that there are in all at least 
 two hundred and forty-six words employed at some period or 
 another in the English language which undoubtedly come from 
 the root spec." * 
 
 Jevons's discussion suffices to illustrate quite fully the fact 
 of ambiguous conceptions and the laws of their development, 
 but it does not indicate those instances which are likely to 
 
 * Fora more complete study of the ambiguity of terms and the origin 
 of it, the student may consult the following works: Locke: Essay on 
 Human Understanding, Book III., Chapters IX. and X. ; Mill: Logic, 
 Book IV., Chapter V. ; Trench : On the Study of Words Max Miiller : 
 Lectures on the Science of Language.
 
 64 ELEMENTS OF LOGIC 
 
 give trouble in the study of Logic. It is perhaps just as well 
 that we should first understand the wide extent to which words 
 vary in meaning, and to which they are modifiable by general- 
 ization and specialization, before considering more particular 
 classes affecting logical problems. We can in that way have 
 the general laws regulating or causing their ambiguity most 
 distinctly impressed upon our mind, inasmuch as they affect 
 the meaning of terms even where logical confusion may not be 
 the consequence of the change. Most of the conceptions 
 chosen by Jevons in illustration would very seldom give rise 
 to any serious fallacy in reasoning, as, in spite of a certain 
 kind of ambiguity, they are well enough understood to prevent 
 serious logical mishaps. But this is not the case with a very- 
 large set of conceptions current in science and philosophy, 
 general conceptions whose technical meaning varies with the 
 schools arguing for or against certain doctrines involved in 
 them. They belong largely to the class of terms which are 
 either very abstract in their import, or are liable to the con- 
 fusion of their abstract with their concrete conception. This 
 is a source of ambiguity already touched upon, and which 
 Jevons does not treat of. It is perhaps a source of more logi- 
 cal errors than all the ambiguities his discussion illustrates, 
 although it comes under the same laws as those which he does 
 notice. Pure abstracts and pure concretes, as adjectival nouns 
 and proper or singular names, are not likely to give much 
 difficulty. Fallacies are much more incident to mixed abstract 
 and concrete conceptions, or those concrete terms, usually so- 
 called, whose different meanings are either closely allied, or con- 
 nected with closely resembling objects having nevertheless im- 
 portant differences between them. Where the separate concrete 
 meanings of the same term have no natural affiliation, they will 
 not easily give rise to error. Thus "spirit," denoting mind on 
 the one hand, and alcohol on the other, will not be used in 
 connections where fallacy will be the consequence of such a 
 possible ambiguity. But "spirit," denoting mind or intelli- 
 gence in one case, and something immaterial in another, may 
 be the source of error, because the two conceptions are closely
 
 THE AMBIGUITY OF TERMS 65 
 
 related. The process of generalization gives rise to a tendency 
 away from the concrete and may produce confusion as long as 
 the new and old conceptions are not clearly defined. An op- 
 posite source of error is that of the process of specialization, 
 which is from the more abstract to the concrete ; in so far as 
 the term " abstract " is taken to denote the common qualities 
 denoted by the general concept apart from the particular, con- 
 crete, and differentiated individual to which it may also apply. 
 The generalization and specialization indicated have to do, 
 almost exclusively, with the terms that may be mixed abstracts 
 and concretes, and the source of confusion increases as the 
 indefiniteness of a conception increases with the process of 
 generalization. 
 
 A highly illustrative set of ambiguous terms, more impor- 
 tant to scientific and philosophic stud}' than any which Jevons 
 has stated, and which yet come under the laws which he 
 enunciates, are such as the following : Motive, intuition, expe- 
 rience, idea, cause, God, religion, faith, feeling, knowledge, sen- 
 sation, reason, first principles, a priori, government, law, nature, 
 moral, right, justice, nation, church, authority, origin, freedom, 
 etc. Thus motive may denote in mechanics the cause or force 
 producing motion, and by analogy the cause of volition ; then 
 it may denote the idea of an end, which, in so far as it is the 
 object of the mind, is rather the effect than the cause of voli- 
 tion. Intuition may denote immediate perception, or again 
 it may denote universal perception, the first indicating only 
 a direct process of knowledge without reference to time or 
 the number of persons involved, and the second indicating 
 that all individuals experience it. The term has, further, 
 four or five other meanings, but these two suffice to illustrate 
 an ambiguity which might well give rise to all the contro- 
 versies waged about certain doctrines in philosophy. Add 
 to this the still greater ambiguity of the terms " experience " 
 and "idea," and we can well imagine why so much dispute 
 has centred about the theory of "innate ideas." On the 
 one hand, experience has meant, first, simple sensations ; sec- 
 ond, any realized state of consciousness ; third, a series of men- 
 5 '
 
 66 ELEMENTS OF LOGIC 
 
 tal states giving as a resultant a new component not found 
 in the primary state ; on the other hand, idea has meant, first, 
 simple presentations of sense (Locke) ; second, general concep- 
 tions which are the product of the higher intellectual faculties ; 
 and third, mere opinion. These are 'sufficient to make a per- 
 fect labyrinth of complexities in ai-gument. Again, cause is 
 sometimes merely the condition of anything, and at others the 
 active agent producing a phenomenon. In this latter sense it 
 may be either some influence external to the thing affected, or, 
 as in the case of the mind and its volition, or of any agent 
 which contributes of its own energy to the effect, it may be 
 internal. God sometimes denotes the first cause without in- 
 dicating whether it is more than force ; then it may denote 
 the agent effecting the order of the world, whether regarded 
 or not as the creator of matter ; again the term may denote a 
 supreme intelligence, with various attributes of perfection and 
 power, not necessarily implied in the former conceptions, but 
 not excluded by them. Keligion may denote certain beliefs 
 about God and the world, or it may denote a certain attitude 
 of mind toward these things, or it may further denote the be- 
 lief and practice of certain moral doctrines, with various sub- 
 ordinate meanings. Faith has at least four distinct meanings : 
 First, intellectual assent to propositions above the attestation 
 of reason, and thus equivalent to intuition, if it gives immedi- 
 ate assured knowledge, or to mere probability, if it can only 
 produce less than absolute conviction or certitude ; second, it 
 may denote the acceptance of truth on authority ; third, fidel- 
 ity of disposition in living up to a promise, treaty, or a law ; 
 and fourth, trust in a person. Feeling has a similar applica- 
 tion, now denoting a tactual sensation, now a firm and ineradi- 
 cable conviction = intuition ; again, a variable mental state of 
 excitement which can easily be eradicated from an influence 
 upon knowledge, and so is equivalent to emotion, and lastly, a 
 general conception for the primary elements of knowledge. 
 The remaining terms and many others possess similar multi- 
 plications of meaning, which are the source of all the contro- 
 versies and their incident fallacies in the world of speculative
 
 THE AMBIGUITY OF TERMS 67 
 
 knowledge. The few that we have specified may serve to illus- 
 trate the importance of being on the alert for them, and of 
 first giving a term that analysis or definition of its meaning 
 which is at least a partial provision against the contingency of 
 error. 
 
 To recapitulate. First, words are ambiguous when they 
 are capable of more than a single meaning. Second, they be- 
 come ambiguous through the process of transition from a 
 generalized to a specialized form, or the reverse. Third, this 
 process causes confusion in Logic mainly when it results in 
 the liability to mistake the abstract for the concrete, or one 
 concrete conception for another closely allied to it.
 
 . CHAPTER V. 
 
 THE INTENSION AND THE EXTENSION OF CONCEPTS 
 
 1st. Nature of Intension and Extension. Allusion has 
 been made to the intension and the extension of conceptions 
 without explaining the meaning of those terms. We come 
 now to determine this meaning, which is a very important 
 matter in Logic. Various terms have been employed to ex- 
 press the same fact and relation as are expressed by extension 
 and intension. Thus comprehension, depth, connotation are 
 frequently taken as synonymous with intension, and extent, 
 breadth, denotation for the extension of terms. Much contro- 
 versy exists about the true use of the terms denotation and 
 connotation, which I shall consider later in the chapter. The 
 only matter of real importance is the meaning of the concepts 
 which they are supposed to describe, and this can be deter- 
 mined, in a large measure at least, without complicating our- 
 selves with this controversy. 
 
 Nearly all, if not absolutely all terms have a twofold mean- 
 ing or application, which is expressed by their intension and 
 extension. A simple example will make this apparent. The 
 term " man " may apply to all the individual men represented 
 by the word. It is thus a name for the individuals in the 
 class, and we should call each one by that name when asked to 
 define what he is. On the other hand, the term also expresses 
 a certain number of qualities or marks which make up the in- 
 dividual. " Man " is thus not only a name for the individuals 
 of the class, but for a certain conjunction of qualities, which 
 may be thought of without regard to the range of application 
 possessed by the term. Tree has a similar application. It 
 may be a name for a class of objects, or it may denote a cer- 
 tain number of vegetable qualities. Metal, quadruped, biped,
 
 INTENSION AND EXTENSION OF CONCEPTS 69 
 
 vertebrate, triangle, figure, nation, city, custom, etc., are only 
 other instances of the same fact. There are some terms, how- 
 evei*, which are not class concepts ; for example, singular or 
 proper names. They seem only to denote a combination of 
 qualities, and it is true that they cannot denote more than one 
 individual. But this does not hinder it from having a numer- 
 ical application to that extent, and this is all that is necessary 
 to justify the application of extension to it. In all these terms 
 the intension refers to the quality or qualities possessed by an 
 object having a given name ; the extension refers to the number 
 of objects included under the name. These will at least serve 
 as approximate definitions of the two terms until completed. 
 We require at present merely to know that the extension indi- 
 cates the objects to which a name applies, and the intension, 
 the attributes which it implies. 
 
 The clearest illustration of terms with extension will be 
 class concepts, and especially all concrete general concepts, as 
 " man," " quadruped," " Caucasian," " European," " tree," 
 "book," "animal," "house," etc., and perhaps the clearest il- 
 lustration of intension will be pure abstract terms and singular 
 or proper names, which latter, although they denote an indi- 
 vidual in a class, more particularly indicate a certain quality 
 or union of qualities that are thought of rather than the range 
 of the term, as "Lincoln," "Berlin," "Rome," "Europe," 
 "Plato," "Declaration of Independence," "Magna Charta," 
 etc. But all terms have both references at the same time, and 
 differ only in the degree of their intension and extension in 
 relation to each other. That is, every term has both intension 
 and extension. The extension of " man " is the number of in- 
 dividuals to which it is applicable as a name ; the intension is 
 the number of qualities which it implies, and so with other 
 terms. 
 
 The application of extension and intension to concrete con- 
 ceptions, singular or general, affords no difficulty. But is it 
 possible to apply them to abstract terms? The answer to 
 this question involves the previous questions whether abstract 
 terms are singular or general, and whether singular terms can
 
 70 ELEMENTS OF LOGIC 
 
 be said to have extension. Inasmuch as I have claimed sin- 
 gular concepts for extension, merely indicating that the ex- 
 tension is at its minimum in them, there remains only to settle 
 whether abstract terms are singular or general Jevons thinks 
 them singular. This may possibly be the case in some in- 
 stances. But it is certainly not the case in such terms as 
 ' sweetness," " justice," " ability," " color," etc. For there are 
 several kinds of " sweetness," several divisions of " justice," 
 and various forms of "ability" and "color." Perhaps some 
 writers would make "color " a concrete concept. I hold that 
 it may be either concrete or abstract, according to its refer- 
 ence. But it is not important to settle this question, whether 
 it may be both, or is only one of them. If it be concrete it 
 certainly has extension and intension together, and if it be 
 abstract and general, as it must be, both extension and in- 
 tension are characteristic of it ; so that the two qualities can 
 be denied of it only on the supposition that it is abstract and 
 singular, and that all singulars have only intension. But with 
 the qualification already mentioned these properties are pos- 
 sessed by both concrete and abstract terms, so that the re- 
 duction of abstract concepts to singulars would be no obstacle 
 to their simultaneous possession of intension and extension. 
 
 It remains to consider whether adjectives can possess in- 
 tension and extension. In so far as they are the names of 
 qualities they indicate some degree of intension. But are 
 they class terms? There are two ways of answering this 
 question. The first is Mill's view that attributive terms of 
 this kind always imply a subject. Thus " white " implies all 
 white objects, and so must represent a class. The second 
 is the view that, even if they do not necessarily imply anything 
 definite about things, they may connote different kinds or 
 degrees of the quality expressed by them. Thus there are 
 many shades of "white," "blue," "green;" many kinds or de- 
 grees of " pure," " noble," " benevolent," etc. In this sense 
 they will also have extension. And also they would possess 
 this quality if they were singular, according to the principle I 
 have asserted for the minimum extension of a term, and it is
 
 INTENSION AND EXTENSION OF CONCEPTS 71 
 
 not necessary, perhaps, to take account of any other degree 
 of it, 
 
 It may also be a question whether any great importance 
 attaches to either the affirmation or the denial of extension to 
 
 abstract terms and attributives. The chief interest of the 
 
 
 
 logician, probably, is in the possibility of representing them 
 by the usual symbols, the logical circles, in illustrating the 
 relation between the subject and predicate of a proposition- 
 But as this symbolization is only one of analogy, and as 
 the legitimacy of it even is disputed by some, we may disre- 
 gard the question whether it be strictly applicable to abstract 
 nouns and to adjectives, and, if we desire, leave the whole 
 matter undecided, so far as practical Logic is concerned, be- 
 cause fallacies do not, to any extent, turn upon the question 
 whether abstract terms and adjectives are capable of extension 
 or not. In purely theoretical Logic it may be somewhat 
 different. There, we may be interested merely in the truth 
 about this special case, and undoubtedly we shall find the 
 matter of extension less clear in its application to abstract 
 terms and adjectives, even if it be granted as possible, than 
 to concrete substantives. For, being terms which merely 
 imply a subject, but do not express it definitely, they will most 
 likely represent either a single quality with its various degrees, 
 or only such as the substantive from which they are taken 
 represents. Of course some adjectives are not taken from 
 substantives. But many of those that are so taken, as 
 " manly," " human," " animal," " personal," " heavenly," un- 
 doubtedly connote as many qualities as the substantive, and so 
 equal it in its intension at least, and it is possible to say that 
 their extension is also equal to that of the terms from which 
 they are derived ; although it might be proper to regard it as a 
 sort of relative or derivative extension. Certainly they do not as 
 terms indicate individuals so distinctly as substantive concretes. 
 But even if they do not, Logic does not require us to give them 
 more than the minimum extension, and this will apply to all 
 concepts representing any idea whatever. This is to say that 
 extension must not be confused with the notion of plurality.
 
 72 ELEMENTS OF LOGIC 
 
 After this discussion we may conclude the section with a 
 more accurate definition of extension and intension. The for- 
 mer is usually described as referring to the individuals of a 
 class, and this conception of it leaves the impression that in 
 order to have extension at all a term must be a class concept. 
 But this is only an incident of the disproportion that may ex- 
 ist between the two properties. But to cover the case more 
 distinctly they may be defined as follows : 
 
 Extension is the quantitative power of terms or concepts, and 
 so indicates their numerical application. It may refer to in- 
 dividual- or class-wholes, whether substantive or attributive, 
 real or conceptual. 
 
 Intension is the qualitative power of terms or concepts, and so 
 indicates their denotation of qualities. It may refer to a 
 single quality, a group of qualities in an individual- whole, or 
 the common quality or qualities of a class-whole, whether sub- 
 stantive or attributive, real or conceptual. 
 
 Conceptual-wholes are such as denote thought products 
 which are not conceived properly as either substantive or at- 
 tributive. Illustrations are such concepts as "proposition," 
 " word," " syllogism," " science," " botany," etc. Some terms 
 describing mental states may be regarded as attributive ; e.g., 
 " sensation," " memory," etc. 
 
 2d. The Relation between Extension and Intension. 
 The relation between the intension and the extension of con- 
 cepts is determined by comparing the broader with the narrower 
 term. Thus, if we take the term " metal " and compare it with 
 the term " iron " we shall find that " metal " is a name for a 
 larger number of objects than the term " iron," because it in- 
 cludes all that is denoted by " iron/' and all other metals be- 
 sides. The extension of metal is, therefore, said to be greater 
 than the extension of " iron." But the intension of " iron " is 
 greater than that of " metal," because it contains all the quali- 
 ties necessary to regard it as a "metal," and in addition the 
 quality or qualities necessary to make it " iron " and to dis- 
 tinguish it from other metals, such as gold, silver, lead, etc. 
 Again, "matter" will have greater extension than "metal," and
 
 INTENSION AND EXTENSION OF CONCEPTS 73 
 
 " steel " less than " iron." But the intension of " steel " will 
 be greater than that of " iron," and the intension of " matter " 
 less than that of " metal." The same comparison can be in- 
 stituted between any set of related terms, such as biped, 
 man, European, Frenchman, Louis XIV., or vertebrate, quad- 
 ruped, horse, Bucephalus ; or figure, quadrilateral, square. 
 
 From this we deduce the general law that as the extension in- 
 creases the intension decreases, and vice versa. The same law is 
 sometimes expressed in a different way. Thus, extension and 
 intension vary in an inverse ratio to each other, or they are in- 
 versely related to each other. Hamilton represented the rela- 
 tion by a cone or pyramid, in which the apex indicated the 
 least intension or extension, as the case might be, and the base 
 the greatest. We may thus symbolize the relation for a series 
 of terms, and we may indicate both the order of greatest and 
 least extension, the greatest and least intension, and their re- 
 ciprocal relation inversely considered. The following figures 
 will represent them : 
 
 EXTENSION. INTENSION. INTENSION AND EXTENSION. 
 
 V /& 7 
 
 \ /Greei\ / 
 
 _ 
 
 / Man \ / Greek \ / \Man / \ 
 
 / Vertebrate \ / Plato \ / Ver^/bnte \ 
 
 FIG. 1 Fia. 2. FIG. 3. 
 
 Fig. 1 represents the order of least and greatest extension, 
 beginning with the apex of the triangle or cone. " Plato " has 
 the least extension of the series and applies to only one indi- 
 vidual. "Vertebrate " has the greatest extension of the series, 
 including all the others and all other beings having a certain 
 anatomical structure. In Fig. 2 the order is reversed. " Ver- 
 tebrate " has the least intension because it stands for the 
 fewest qualities, and " Plato " the greatest because it denotes 
 the largest number of qualities. Fig. 3 represents the recip- 
 rocal relation of the two properties. " Plato " has at the 
 same time the greatest intension and the least extension, and
 
 74: ELEMENTS OF LOGIC 
 
 "vertebrate" the greatest extension and the least intension. 
 The intermediate terms vary between these extremes by an in- 
 determinate ratio, but are presumably in the same relation. 
 
 We must bear in mind, however, that the formula for this 
 relation is not strictly accurate. It is not true that the ex- 
 tension always increases as the intension decreases. Some 
 logicians, in their objections to the law thus enunciated, go so 
 far as to say that the relation in some instances may be re- 
 versed, so that the intension would increase as the extension 
 increases. This, however, is an exceptional state of the case 
 and may be dismissed from consideration of the general rule. 
 The law is, perhaps, not meant by any logician to be absolute- 
 ly and universally true in the strict sense in which it is some- 
 times expressed. The formula is a convenient one for indi- 
 cating a relation sometimes strictly true, and sometimes true 
 with qualifications. It is enunciated in mathematical terms 
 for the sake of clearness rather than because it is literally true 
 in all instances of comparison, although in the ideal logical 
 world it might be so. Nevertheless, the absolute and univer- 
 sal application of the law, as we have formulated it, is subject 
 to the following limitations : 
 
 1. The law cannot be interpreted in any strict mathematical 
 sense. It is, for instance, not true that when the intension is 
 doubled the extension is halved. The number of individuals 
 may .even be increased without decreasing the intension of 
 either the class term or of the individual under it. Thus I 
 may increase the number of persons to whom the term " man " 
 is applicable, and still not alter the quantity of intension repre- 
 sented by the term. 
 
 Some logicians might reply that, in fact, this never takes 
 place ; that every individual added to a class differs by some 
 mark from all others, and so may decrease the intension of the 
 general term in the same proportion, at least, that its exten- 
 sion is increased. This is conceivably the case, and, if a fact, 
 would sustain the law in its general sense, although it might 
 not prove its strict or definite mathematical interpretation in 
 terms of any specified ratio. Besides, the law is perhaps de-
 
 INTENSION AND EXTENSION OF CONCEPTS 75 
 
 fensible as a general formula in the same sense in which psy- ' 
 chologists speak of the inverse ratio between sensation and 
 perception, or between the consciousness of feeling and the 
 consciousness of an object, where we wish merely to express 
 the fact that the two do not vary together in the same way, 
 but that as one becomes more distinct the other becomes less 
 so. This mode of expression may be applicable to the varia- 
 tions, at least as a general rule, between extension and inten- 
 sion, and be relatively true when one increases without the 
 other decreasing, and absolutely true, although not by any 
 assignable ratio, when the increase of the one is accompanied 
 by a corresponding decrease of the other. Nevertheless, the 
 variation is so irregular that the law has only a conventional 
 value in the form in which it is usually enunciated. 
 
 Perhaps a distinction between two kinds of general terms 
 would enable us to formulate the law in different ways, one of 
 them to suit its simple mathematical conception, and the other 
 to suit a less definite conception of it. It is a fact that general 
 terms or concepts are of two distinct kinds, although the dis- 
 tinction is not explicitly recognized by logicians. The two 
 divisions I shall call mathematical generals and logical generals. 
 The latter term is perhaps an unfortunate one, because all 
 generals are logical in the broadest sense. But I defend its 
 use in the technical sense to be defined as the only resort 
 at my command for expressing the notion I have of the 
 terms described by it. By mathematical general concepts I 
 mean those which are absolutely alike in their content or in- 
 tension, or such as are grouped together under the same name 
 solely on account of their numerical value. In such cases the 
 extension may be increased either without altering the in- 
 tension, or in connection even with an increase of the inten- 
 sion, supposing, of course, that the increase of the intension 
 was the same in all individuals of the class. Thus if I assume 
 a number of gold coins exactly alike, I may add to the number 
 denoted by a particular name any number of like coins with 
 exactly the same qualities, and here I increase the extension 
 without increasing the intension. And again, if in adding a
 
 76 ELEMENTS OF LOGIC 
 
 new individual to the class, I discover a quality not known be- 
 fore, but which is found to belong to all members of the class, 
 I have increased the intension of the concept (not of the thing) 
 in the same proportion as I have increased the extension. If 
 the new individual has a new property not in the others of the 
 same class, the intension of the general term or genus may 
 remain the same, or fixed, while the extension is increased. 
 Hence wherever the addition to the general concept is a purely 
 numerical or mathematical one the law of inverse ratio be- 
 tween extension and intension does not hold in its strict sense, 
 and can only be taken as a general statement of the indepen- 
 dence of each other in their variations. 
 
 But in the case of the second class, namely, logical concepts, 
 the ratio of variation between extension and intension may 
 more definitely accord with the statement of the law. By a 
 logical general concept I mean one which does not apply in 
 exactly the same sense to the individuals or species which it 
 comprehends, or which strictly connotes only the common 
 qualities of a class, and does not denote the differences. The 
 difficulty, however, in strictly defining them is that they are 
 usually applied in a mathematical sense at the same time, and 
 hence we may have no concepts which denote only the com- 
 mon qualities of a class without enumerating the individuals 
 mathematically at the same time. This is the case with the 
 mixed concrete and abstract terms. But such a general con- 
 cept, if it actually exist in its purity, will be such as compre- 
 hends individuals and species not taken merely in an additive 
 sense, but as denoting certain common qualities allowing great 
 variations and differences in all other characteristics. Thus 
 "vertebrate " is a general concept comprehending individuals 
 and species, with such differences as do not distinguish one of 
 them from another as comprehended in the general concept. 
 It is the same with the general term " animal." It denotes a cer- 
 tain common quality or qualities which permit all sorts of other 
 differences without disturbing or confusing the application of 
 the term. In all such cases the increase of the extension of the 
 concept may be either mathematical, or both mathematical and
 
 INTENSION AND EXTENSION OF CONCEPTS 77 
 
 logical. In case it is mathematical, or the mere addition of an 
 individual exactly like those already denoted by the term, the 
 law is subject to the limitations mentioned in the instances of 
 mathematical generals. But if the increase is due to the ad- 
 dition of a new species, presenting new characteristics com- 
 pared with those denoted before, the term is generalized logi- 
 cally as well as mathematically, and its intension is decreased 
 with its increase of extension. Thus the term " crow " for a 
 long time denoted a certain class of birds with a black, glossy 
 plumage. The color of their feathers was a common quality, 
 being thus involved in the intension of the class. But as soon 
 as " crows " with a white or gray color were discovered, this 
 intension was decreased by throwing the quality of color out 
 of regard as a distinctive characteristic, and the extension was 
 increased in proportion to the new additions to the class. The 
 generalization is accomplished by taking into account fewer 
 common qualities, and hence the ratio between the intension 
 and the extension varies inversely in all such cases. Again, 
 the term " Frenchman " ordinarily denotes a person with cer- 
 tain race characteristics, and born in France. But we often 
 find it used to denote persons having only the race qualities 
 and born in any other locality. " Hebrew " once denoted the 
 inhabitants of Palestine. It now denotes a race without any 
 implications as to country. "Book" was once synonymous 
 with "beechen boards, "having printed or written matter upon 
 them. It is now broadened to denote any mass of paper bound 
 in a particular way, whether containing printed and written 
 matter or not. The matter, indeed, is an entirely unessential 
 characteristic, as it was once the essential. Here again the law 
 is strictly illustrated, as the intension decreases with an increase 
 of the extension. The same fact is clearly illustrated in all 
 cases of generalization and specialization already discussed. 
 Here the intension and extension vary inversely. 
 
 But it is important to remark the limitation with which this 
 is true. The addition to the general term numerically may be 
 manifold as great as the deduction from its intension. A hun- 
 dred, more or less, individuals may be added to the extension
 
 78 ELEMENTS OF LOGIC 
 
 and only one or two qualities subtracted from the intension. 
 And even if only one individual with a difference is added to 
 the class, and two or more qualities subtracted from the inten- 
 sion, the law is still true in the sense it was intended, although 
 not true in the mathematical sense that the ratio of variation is 
 the same for both intension and extension. In its proper logi- 
 cal meaning the law simply indicates, without any numerical 
 implications, that as you extend the application of a term to 
 new species you decrease the number of generic qualities com- 
 prehended by it. This will be quite uniformly true of logical 
 extension, but variable and subject to modifications in mathe- 
 matical extension. 
 
 The importance of this distinction and discussion will appear 
 when we come to consider the question of genus and species, 
 or essentia and differentia. At present it suffices merely to 
 specify the qualifications under which the law is true. 
 
 2. The second limitation of the formula regarding the rela- 
 tion between extension and intension is that it is not applicable 
 to all conceptions independently of the relation between genus 
 and species. Thus the extension of " man " cannot be com- 
 pared with that of "bear ; " of "horse " with "lion ; " of "gov- 
 ernment" with "science." Nor can any measure of the rela- 
 tion between the extension and the intension of such terms be 
 determined. Hence the law cannot apply to terms taken pro- 
 miscuously. It can apply only to conceptions which represent 
 a superordinate and a subordinate notion, or a genus and a 
 species. Neither genera nor species can be compared in this 
 relation with their own kind, but only with each other. The 
 comparision must be limited to terms representing different 
 degrees of generalization. All concepts may thus be brought 
 under one comprehensive head, but apart from such a relation 
 their extension and intension will not be an object of deter- 
 mination at all. Hence this limitation of the law. But it 
 should be remarked that this does not set aside the formula. 
 It only qualifies it and its application. In its proper concep- 
 tion and under appropriate conditions it remains valid, al- 
 though it may be possible to exaggerate its importance.
 
 INTENSION AND EXTENSION OF CONCEPTS 79 
 
 But even those who criticise the doctrine admit considerable 
 importance for the meaning which they assume is latent at the 
 basis of it. Bosanquet, who subjects the formula to a some- 
 what searching criticism, virtually admits all that the law ever 
 meant to express by saying that conceptions may vary their 
 import and range of application in a way much as the doctrine 
 of an inverse ratio asserts. He says: "It is certain that to 
 abstract and to distinguish to know what belongs to one re- 
 lation, and what, again, though conjoined with that relation, 
 yet does not arise out of it, but out of some other condition 
 or cause is the first duty of the scientific intelligence. In 
 consequence of this activity arrangements of individual objects 
 under a series of abstractions, each applying to a wider aggre- 
 gate than the last, meet us on every hand and most obviously 
 of all in family relationships as estimated among civilized na- 
 tions."* When we take into account, therefore, that the 
 diminution of intension, in any case, does not involve, neces- 
 sarily, a decrease in the number of qualities in the individual 
 or sub-class, but only the number of common qualities consti- 
 tuting the genus, as before determined, and so with the 
 increase of the same, there will be nothing to seriously object 
 to in the law, except its precise mathematical and promiscuous 
 application. Its importance is therefore vindicated. 
 
 3d. The Denotation and Connotation of Terms. 
 Some logicians use this distinction as identical with that be- 
 tween the extension and the intension of terms. Others em- 
 ploy it in a somewhat different sense. Indeed, it is only this 
 variation of usage and its frequent coincidence, or close con- 
 nection with the application of extension and intension, that 
 makes it necessary to consider the matter at all. I shall notice 
 briefly the doctrines of Mill, Fowler, and Keynes. 
 
 Mill's distinction is between connotative and non-connotative 
 terms. " A non-connotative term," he says, "is one which signi- 
 fies a subject only, or an attribute only. A connotative term is 
 one which denotes a subject and implies an attribute. By a sub- 
 ject is here meant anything which possesses attributes. Thus 
 * See Bosanquet's Logic, Vol. I., Introduction, 8, pp. 46-71.
 
 80 ELEMENTS OF LOGIC 
 
 John, or London, or England, are names which signify a sub- 
 ject only. Whiteness, length, virtue, signify an attribute only. 
 None of these terms, therefore, are connotative. But white, 
 long, virtuous, are connotative." The distinction here made 
 Mill holds to be a restoration of scholastic usage. As far as 
 can be determined from his language, "denotative" merely 
 means " significative ; " for he does not define it. He practi- 
 cally identifies it with "non-connotative." "Connotative" he 
 defines, according to scholastic usage, as equal to "connotare, 
 to mark along with ; to mark one thing with, or in addition to 
 another." This identifies it with imply, or the implicative 
 power of a term. 
 
 Objections can be produced against his classification of con- 
 notative and non-connotative terms on the basis of these defini- 
 tions themselves. For instance, he says that proper names " do 
 not indicate or imply attributes as belonging to individuals." 
 This can be directly challenged. Bucephalus, The Secretary ot 
 State, Mont Blanc, quite distinctly imply attributes. It does 
 not suffice to make a proper name that we merely capitalize a 
 word. It is such only by virtue of its application to a single 
 individual that is a group of attributes. But if, besides denot- 
 ing a subject, it implies an attribute or attributes, it becomes 
 connotative, according to the definition, though classified as 
 non-connotative. On the other hand, adjectives, though classed 
 as connotative, are as much the names of attributes alone as ab- 
 stract nouns, and. therefore should be non-connotative. Besides 
 they do not denote a subject or imply an attribute. On the 
 contrary, they denote an attribute and imply a subject. This 
 would prevent them from being connotative, according to the 
 definition. But if " connotative " be regarded as synonymous 
 with implication, as seems to be the case with Mill, adjectives 
 would be connotative, but so would abstract nouns for the same 
 reason. Hence Mill must either classify these together, or 
 change the definition of connotative. There is certainly, in this 
 respect, an inconsistency between his definition and his classifi- 
 cation, while at the same time the definition given does not de- 
 scribe the nature of adjectives.
 
 INTENSION AND EXTENSION OF CONCEPTS 81 
 
 It will be apparent in Mill's usage that the terms cannot be 
 identical with that of extension and intension, and hence his 
 doctrine either has a diminished importance or is too confus- 
 ing in its implications to be of much service in the problem 
 under consideration. Fowler and Keynes take the liberty to 
 modify their application or meaning so as to coincide with 
 extension and intension, " denotation " being identical with ex- 
 tension and " connotation " with intension. In this applica- 
 tion they require no further discussion. If I might be allowed 
 to introduce an innovation, I vould prefer, since we already 
 have the terms extension and intension, to employ "denota- 
 tive " to indicate the application of a name to individuals with- 
 out any reference to their qualities, and hence only to denote 
 them as concrete wholes, numerically or mathematically con- 
 sidered. " Connotative " I would employ to connote or com- 
 prehend a union of attributes, or a class of individuals. It 
 would thus apply to individual and general wholes, one as an 
 aggregate of qualities and the other as an aggregate of indi- 
 vidual objects. Denotative might apply to the names of singu- 
 lar attributes or singular individuals as such. In this way I 
 should remain by the strict import of the word connote, and 
 gain an economical term for a logical synthesis of any kind, 
 and retain a separate one for all concepts representing an in- 
 dividual unit of any kind. In this sense I have used the terms 
 where I found it necessary to employ them technically at all. 
 But as no particular logical doctrine is dependent upon this 
 usage, I do not urge either the adoption or the importance of 
 it.* 
 
 * For general discussion of this question and the problem relating to 
 the extension and intension of terms the student may consult the follow- 
 ing references : Mill : Logic, Book I., Chap. II., 5 ; Fowler : Deductive 
 Logic, Chap. II.; Keynes : Formal Logic, Chap. II. ; Hamilton : Lectures 
 on Logic, Lects. XI. and XII. ; Venn : Empirical Logic, Chap. VII. ; 
 Bosanquet : Logic, Vol. I., Introduction, 8, pp. 46-71; De Morgan: 
 Formal Logic, Chap. XII. 
 
 6
 
 CHAPTER VI 
 
 DEFINITION AND DIVISION 
 
 1st. The Predicables. Definition and Division are com- 
 plicated processes, the former especially, since there are several 
 kinds of definition. But to understand them and their place 
 in logical science and disquisition we must examine into the 
 nature and meaning of the so-called predicables. They are 
 usually stated as five in number, as follows : 
 
 Oe-nus (yevos) . = Genus. 
 Species (eTSos) = Species. 
 
 Differentia (Sio^opek) = Difference. 
 Proprium (I$i6v) = Property. 
 
 Accident (crv/*&f0r)K6s) = Accident. 
 
 The most natural order of considering the predicables would 
 be that in which they are stated. But as the meaning of the 
 first three is dependent upon understanding the fourth, and 
 perhaps the fifth, I shall begin with the fourth and return to 
 the others. 
 
 1. PKOPEBTT. This term is, for all practical purposes, synony- 
 mous with quality and attribute. Some writers endeavor to 
 distinguish between them, but the distinction serves no im- 
 portant logical purpose. Hence by a property of a thing I 
 mean any quality, mark, characteristic, or attribute of it which 
 goes to make it what it is. Thus whiteness is a property of 
 snow ; hardness, of iron ; yellowness, of gold ; brilliancy, of a 
 diamond ; instability, of a liquid, etc. It is the same with any 
 namable quality of an object. But qualities or properties are 
 not all of the same value or importance to the existence of a 
 thing. Hence they are usually divided into essential and non- 
 essential. Essential properties are those which are necessary 
 to the particular nature of an object, which would not be what
 
 DEFINITION AND DIVISION 83 
 
 it is except for these qualities. For example, the essential 
 property of a pen is that it be fit for writing ; of man, that he 
 have life and consciousness ; of a lamp, that it be able to give 
 light ; of a tree, that it be of wood, have a trunk and branches, 
 etc. ; of iron, that it have a certain density, metallic lustre, and 
 molecular cohesion. In a great many cases it is difficult to 
 assure ourselves that a given quality is essential rather than 
 non-essential. This is because of the indeterminate extension 
 of ihe term under notice. Thus we might consider sweetness 
 as an essential property of sugar, and so it is in the common 
 use of the term. But to the student of chemistry this is not 
 necessarily the case, as we have seen that science will often 
 generalize a term without carrying along with the increased 
 extension the essential property of its narrower import. An 
 instance of this is the term " metal." Originally it was sup- 
 posed that a specific gravity greater than w#ter was a neces- 
 sary property of metals. But on the discovery that potassium, 
 sodium, and lithium were metals because of their metallic 
 lustre and structure, this property becatne one of their essen- 
 tial characters, and their specific gravity was disi^egarded by the 
 term metal obtaining a more general application. We have, 
 therefore, always first to determine the extension of a concept 
 before indicating its essential properties. Fluidity will be an 
 essential property of water, if we do not include the term ice 
 in it. Animal fibre is an essential property of " meat," if we 
 do not use the term synonymously with the term " food." But 
 all this merely indicates how difficult it is to name absolutely 
 th,e essential property denoted by a given term. We have first 
 to settle what extension is given it, and this will be determined 
 by the limits assigned to the presence of a given quality, and 
 this quality will be an essential one so far as it is identified 
 with the extension of the term, or made to determine that ex- 
 tension. This last modifying clause has to be added because 
 some universal properties are regarded as non-essential. But 
 the essential properties will determine the limits of extension 
 for any given meaning, but will not stand in the way of a 
 higher generalization eliminating that quality as essential, and
 
 84: ELEMENTS OF LOGIC 
 
 substituting another in its place. But in this case the concept 
 is materially changed, and is not the same as before. In par- 
 ticular cases, therefore, before naming the essential quality we 
 must see that the term is clearly understood or that its exten- 
 sion is not greater and cannot be greater than the property to 
 be specified. 
 
 A non-essential property is one which is not necessary to the 
 concept or existence of a thing. For instance, whiteness is 
 not an essential property of man ; redness, of an apple j a 
 specific length, of the sides of a triangle ; iron material, of a 
 ship, etc. The non-essential properties are called accide/ita, 
 which are admirably defined and illustrated by Jevons. " An 
 accident," he says, " is any quality which may indifferently be- 
 long or not belong to a class, as the case may be, without 
 affecting the other qualities of the class. The word means 
 that which falls or happens by chance, and has no necessary 
 connection with the nature of the thing. Thus the absolute 
 size of a triangle is a pure accident as regards its geometrical 
 properties ; for whether the side of a triangle be one-tenth of 
 an inch or a million miles, whatever Euclid proves to be true 
 of one is true of the other. The birthplace of a man is an 
 accident concerning him, as are also the clothes in which he is 
 dressed, the position in which he rests, and so on. Some 
 writers distinguish between separable and inseparable acci- 
 dents. Thus the clothes in which a man is dressed is a separ- 
 able accident, because they can be changed, as can also his 
 position, and many other circumstances ; but his birthplace, 
 his height, his Christian name, etc., are inseparable accidents, 
 because they can never be changed, although they have no 
 necessary or important relation to his general character." 
 
 Accidental properties are almost as indeterminate in par- 
 ticular cases as the essential. This is because the term may, 
 like that, be relative in its import. That is, what is accidental 
 in one relation may be essential in another. Thus muddiness 
 may be an accidental quality of rivers, but an essential quality 
 of a certain river. A better illustration is that of red hair. It 
 is a purely accidental property of a man, as a man, but it may
 
 DEFINITION AND DIVISION 85 
 
 be essential to him as an individual. There are instances, 
 however, where the accident will hardly be considered as in 
 any way essential even to the individual. But this may be 
 controverted, and it is not necessary to sustain the position 
 for our purposes. All we require to remember is that, usually 
 at least, the accidents are relative to certain qualities regarded 
 as essential, and that by narrowing the extension of a concept, 
 we may make them essential in that sphere. The only place 
 where an accident will be absolutely such is in the case of the 
 individual or singular concept, and even here it may be dis- 
 puted whether the distinction between essence and accident 
 can be drawn. It may require for its existence a comparison 
 with some other individual having common attributes and 
 certain differences, in which case the common attributes would 
 be the essentia of a common name and the difference would 
 be the accidents of it. 
 
 If this be the case, however, it only shows the ambiguity of 
 the term " accident." In fact, it has more than one significa- 
 tion. Sometimes it is used synonymously with any quality of 
 an object, and in this case it is equivalent to property or attri- 
 bute. Again, it is used synonymously with differentia, which 
 is the property denning a species. In this sense it is acci- 
 dental in one relation and essential in another. A third mean- 
 ing is that which denotes a property in no way essential to 
 either the genus, the species, or the individual. This is per- 
 haps the sense in which it is usually taken by logicians. But 
 it is sometimes relative in this usage. Thus seven feet stature 
 might be an accident of a giant as a man, but necessary to his 
 being a giant. But a case where this relative import is not 
 apparent, if it exist at all, would be the size of the hand, the 
 presence of a mole on the skin, liability to blush, etc., which 
 would be regarded as accidents of the individual. This is the 
 proper sense of the term, although it may be disputed whether 
 it is ever absolute in its meaning even under the circum- 
 stances just considered. But it is not necessary to decide 
 this point. Its relative import suffices to set aside certain 
 properties which may be disregarded in determining the
 
 86 ELEMENTS OF LOGIC 
 
 essentially important meaning of a concept. It only remains 
 to remark that some accidentice are regarded as universal or 
 coincident with the genus or species, as curly hair of the 
 negro, and others are merely casual or contingent, as sickness, 
 flatness of the nose, shape of the head, etc. 
 
 Some logicians use the distinction peculiar property to de- 
 note what belongs to a whole class and to that class only, as 
 risibility in man. But this comes under the general head of 
 universal accidents, as one kind of them, and needs no further 
 consideration. 
 
 2. DIFFERENTIA. Differentia, or difference, is the name ol 
 that particular property which distinguishes one species from 
 another. For example, bipedality, or two-footedness, is a dif- 
 ferentia of man as compared with quadrupeds ; the possession 
 of feathers, a differentia of birds as compared with horses ; 
 cloven-footedness of cattle is a differentia as compared with 
 horses ; redness of core or pulp is a differentia of " blood- 
 oranges " as compared with ordinary oranges, etc. In all such 
 cases the difference, or differentia, is a quality or property in 
 addition to the generic or common qualities, and merely de- 
 termines the species or individual under the class or genus. In 
 this respect or relation it may even be spoken of as essential, 
 although it is essential only to the species or individual. 
 
 There is no word used by logicians as the counterpart or 
 complement of differentia except the term genus. But this is 
 also contrasted with species, and species is the name for an 
 individual or narrower class rather than of a quality or group 
 of qualities expressed by the term differentia, although it is de- 
 termined by this characteristic. Sometimes the term essence, 
 however, is used as the opposite of difference, and denotes the 
 quality or qualities essential or necessary to the existence of 
 a class or individual. It is identical with essential property. 
 For example, vertebrate anatomy and rationality are essential 
 to the class " man ; " f our-f ootedness to the class quadrupeds ; 
 cloven-footedness to cattle ; a certain pungent taste, color, 
 etc., to oranges; woody structure, trunk, and branches to 
 trees, etc. But here, as in the case of the term differentia, it
 
 DEFINITION AND DIVISION 87 
 
 is relative in its import. Besides, it is somewhat ambiguous 
 in that it is sometimes used to denote the common qualities 
 of the class or genus, and at other times is applicable to the 
 differentia as an essential quality of the species. Usage is not 
 so uniform as is desirable in this matter. Indeed, Whately 
 calls the differentia the formal essence of a thing, and the 
 genus the material part or essence of a concept. 
 
 The confusion incident to the discussion may be avoided if 
 we adopt a new term corresponding to differentia, as genus 
 corresponds to species, and retain essence or essentia as con- 
 trasted only with accident or accidentia. There has been 
 some difficulty in selecting a proper term that would express 
 the characteristic meant to be indicated by the term. But 
 the best I have been able to accomplish in this matter is the 
 selection of conferentia (from con and fero, to bring together, 
 as differentia is from dis and fero, to separate). I had seri- 
 ously thought of using communia (neuter plural of the Latin 
 commune, what is common), because it would etymologically 
 import the common qualities. But the fatal objection was 
 that no singular of it could be used without confusion with 
 the English commune. I have therefore chosen " conferentia," 
 as a good logical and etymological complement of differentia. 
 It has the objection to contend with that the English equiva- 
 lent, " conference," is not associated with any logical usage of 
 the kind here wanted. But we can confine ourselves to the 
 form "conferentia," with the distinctive meaning assigned 
 to it. 
 
 By " conferentia " I shall mean the common quality or 
 qualities which " bring together " like individuals, or consti- 
 tute the application of a term to a class. It is therefore the 
 essence of the genus, as the differentia is the essence of the 
 species. It is also a relative term, as what is conferentia in 
 one class may be the differentia in relation to a co-ordinate 
 species. But in it we have a convenient expression for the 
 common qualities uniting individuals in a class, in contrast 
 with those important qualities that are not common to all 
 under it. Generic and general, as well as " conferential," may
 
 88 ELEMENTS OF LOGIC 
 
 be the corresponding adjectives. In this way we may reserve 
 essence, oressentia, to denote the essential properties of either 
 genus or species, and accidentia for the non-essential. The 
 following table summarizes the discussion and classifies prop- 
 erties : 
 
 { ( Conf erentia = Common properties, or essence of the Genus. 
 
 T. I Essentia < Differentia = Distinctive properties, or essence of the Spe- 
 
 erty." | ' - cie8 ' 
 
 Accidentia ] n-^nTf f Non-essential to either Genus or Species. 
 
 4. GENUS AND SPECIES. These are important terms to define. 
 The concepts to which they apply play a ver} 7 large part in all 
 logical discourse. They are closely related to the extension 
 and the intension of concepts. They differ from them only in 
 implying a distinction between the general or conferential and 
 differential properties of concepts, which is not necessarily in- 
 volved in the difference between extension and intension, al- 
 though when the variation between thess is in an inverse ratio 
 it coincides with, and perhaps in a measure determines, the dis- 
 tinction between genus and species. But this relation is a matter 
 for more advanced Logic to consider. The question of import- 
 ance in all practical reasoning is the influence exerted upon it 
 by those various conceptions known as genera and species, and 
 the logical relation between them. It will be seen most clearly 
 in the fallacies of Equivocation and Accident, when we come to 
 consider them. It will also be valuable in interpreting the rela- 
 tion between subject and predicate in one class of judgments, 
 the process of Conversion, and the relation subsisting between 
 essential and accidental properties, on the one hand, and con- 
 ferential and differential properties, as concomitants, on the 
 other. We now proceed to define and illustrate the two terms 
 very carefully. 
 
 A conception which applies to a whole class of objects is 
 called a genus. Thus " man " is a genus-concept because it 
 is a name which applies to the various kinds of individuals, 
 tribes, or nations of men. " Substance " is a genus because it 
 includes iron, clay, brass, gold, silver, water, etc. It is a 
 general name for various species and individuals under it.
 
 DEFINITION AND DIVISION 89 
 
 A conception which applies to a narrower class, or an indi- 
 vidual under a genus, is a species. Thus " Caucasian " is a 
 species compared with the genus man ; " iron " is a species 
 compared with the genus substance ; triangle is a species com- 
 pared with the genus figure. A species, therefore, has the less, 
 and a genus the greater, extension. 
 
 But it must be farther observed that the terms " Caucasian " 
 and " iron " may also be genera in relation to a lower order of 
 concepts. Thus " Caucasian " includes Germans, Frenchmen, 
 Englishmen, etc. ; iron includes steel, malleable iron, wrought- 
 iron, cast-iron, etc. On the other hand, " man " is a species 
 in comparison with the higher orders " biped," " vertebrate," 
 " organic being," etc. To take a single term iUustrating both , 
 relations, the concept " metal " is a genus compared with 
 " iron," " gold," " silver," " platinum," etc., but a species com- 
 pared with the term " substance." 
 
 From these illustrations it is apparent that genus and spe- 
 cies are relative terms wherever they are convertibly applicable 
 to the same concept in different relations. It will be noticed 
 that a term is always a genus in relation to a narrower exten- 
 sion, and a species in relation to a wider extension. We may 
 thus proceed in either direction until we reach the limits of 
 farther progress, as Being, organized being, vertebrate, man, 
 American, Liiicoln. Here all intermediate terms are either 
 genera or species according as they are conceived in relation 
 to a higher or a lower order ; according as they include a 
 lower class, or are included in a higher. But the two extreme 
 terms cannot be viewed in this twofold relation. "Being" 
 is a genus, but not a species. On the other hand, the term 
 " Lincoln" is a species, but not a genus, supposing, of course, 
 that we can speak of an individual as a species. The former 
 is not a species, because there is no higher genus under which 
 objects can be brought, and the latter is not a genus because 
 it cannot be divided into individuals or species. All singular 
 terms are species, but not genera. They are called infima 
 species, or lowest species. They are always individuals, so far 
 as Logic is concerned. On the other hand, the highest genus
 
 90 ELEMENTS OF LOGIC 
 
 is called the summum genus, or genus generalissimum. It is 
 represented by some such term as " being," " thing," " some- 
 thing," " ultimate reality," but in all cases must be repre- 
 sented by a single concept. It is thus worthy of remark that 
 there is only one absolute summum genus, while there may 
 be an indefinite number of infima species. All intermediate 
 concepts are sometimes called subalterns, as being either gen- 
 era or species, according to the relation in which they are 
 viewed. 
 
 It is necessary to notice the use of the terms genus and spe- 
 cies as used in natural history. A species is there " a class of 
 plants or animals supposed to have descended from common 
 parents, and to be the narrowest class possessing a fixed form ; 
 a genus is the next higher class." This is Jevons's definition, 
 but he does not illustrate it. Perhaps in natural history the 
 term " tree " would represent a genus, and oak, elm, maple, 
 etc., a species, while the distinct kinds of oak, elm, and 
 maple would be varieties. But the peculiar use of the term 
 species here is that it is supposed to be fixed, and not relative 
 as in Logic. In this conception of the term its meaning is 
 quite distinct from that of Logic, where, as we have seen, any 
 but the summum genus and the infima species may be either a 
 genus or species, according to its relation to a higher or lower 
 order. In natural history it is supposed to represent certain 
 fixed characters and relations to a common progenitor. But 
 the acceptance of the doctrine of evolution prevents any such 
 determinate line of distinction from being drawn. The con- 
 ception of " species " becomes an indistinct one, and does not 
 imply any necessary assumption about a particular common 
 ancestor. It denotes only a certain aggregate of characteris- 
 tics with differences less marked and distinct than between 
 genera. This change of meaning, therefore, makes the term 
 approximate its logical import, if it may not ultimately iden- 
 tify the two. But it is necessary to remark the differences that 
 have hitherto prevailed between the logical use of the term 
 and its use in natural history. 
 
 It is important to notice an ambiguity in the logical use of
 
 DEFINITION AND DIVISION 91 
 
 the term genus. This ambiguity is apparent, as I have already 
 remarked in the contrast between genus and species, on the 
 one hand, and the contrast between genus and differentia, on 
 the other, where species and differentia are not identical. The 
 fact determines a double use of the term genus. Differentia, 
 as we have seen, denotes certain properties determining the 
 species, but it does not determine the whole content or inten- 
 sion of the species. It denotes only the distinctive qualities. 
 The species is therefore a concrete thing, combining the dif- 
 ferential and conferential qualities, or the common qualities 
 expressed or implied by the " genus " and the added differen- 
 tial quality which makes it a species distinct from some other 
 species having less or different qualities. But strictly taken, 
 a specific term is concrete and denotes indifferently an individ- 
 ual or an aggregate of qualities. It implies no special relation 
 between extension and intension. But the genus, or terms that 
 represent genera, are not so distinct in their meaning. 'On 
 the one hand, they denote numerically, that is, extensively, 
 all that is denoted by the various species under them ; logi- 
 cally, or intensively, they denote less than any given species. 
 In relation to species, therefore, genera are greater in exten- 
 sion and less in intension, but in so far as they denote the 
 common qualities they apply equally to individual species and 
 to the whole number of species, while specific terms will not 
 apply to the whole of the genus. The relation between genus 
 and species in this respect can be illustrated as follows : 
 " Man," as a genus, applies equally to Germans, Frenchmen, 
 Engli'shmen, or Caucasians, Mongolians, Negroes, etc. The 
 only difference between them is quantity or quality of inten- 
 sion. Germans contain certain qualities which Frenchmen do 
 not. But the difference is not great enough to prevent the 
 use of the term " man " to denote both species. In this sense 
 the genus, or man, is identical ivith all the species taken together 
 and extensively considered. The contrast, therefore, between 
 genus and species is not a complete one. Indeed, strictly 
 speaking, they cannot be contrasted at all. They differ only 
 in regard to the number of individuals denoted by them, and
 
 92 ELEMENTS OF LOGIC 
 
 in this respect are alike concrete in their signification. As 
 applying to all the species the genus is but a common name 
 for a number of individual-wholes, and applies mathematically 
 to all alike. The fact that genus and species are not wholly 
 distinct is apparent in all common judgments involving purely 
 class terms. Thus, " Man is a biped," or " All Germans are 
 men." I cannot reverse the subject and predicate because the 
 difference of extension between them will not permit of it. 
 There is, however, a connection always subsisting between 
 genus and species which allows a statement in the form of a 
 simple proposition. But no such connection exists between 
 the different species under the same genus. Thus I can never 
 say "the Germans are Frenchmen." Of them I can only form 
 a negative proposition, where no relation of extension exists. 
 The distinction, therefore, between species is absolute. Be- 
 tween genus and species it is not absolute, but is only of quan- 
 tify, whether of extension or intension. 
 
 But when the term genus is contrasted with the term dif- 
 ferentia the matter is quite different. Genus and differentia 
 are contrasted not as larger and smaller classes of individual- 
 wholes, but as different qualities or groups of qualities. In 
 this sense genus, or a generic concept, denotes or connotes 
 certain common qualities which characterize the whole class, 
 but which are quite distinct from the differential qualities 
 which form the differential essence of the species. In this 
 meaning the genus is as distinct from the differentia as one 
 species is from another. It cannot apply as a name to any of 
 the qualities represented by the differential elements. It is 
 only a name for the common qualities of a class. Thus " man," 
 as a genus contrasted with differentia, does not denote individ- 
 ual-wholes at all, but only a certain group of common qualities. 
 It is, therefore, not only an abstract conception, thus con- 
 ceived, but denotes only the intension of the concept, exclusive 
 of the differentia, and so serves as the basis for determining 
 its extension. But the extension is not the matter in thought 
 when conceiving it as contrasted with the differentia. It de- 
 notes or connotes only the common qualities, and anything
 
 DEFINITION AND DIVISION 93 
 
 affirmed of it in this sense will not agree with the species or 
 differential concept. For example, I may say, "Pine wood is 
 good for lumber." This is not true of every specific form of 
 pine wood, but only of its substance or generic qualities. 
 Matches might be made of pine wood and yet not be good for 
 lumber. The affirmation, therefore, can be true only of the 
 genus as contrasted with the differentia, and not of the genus 
 including all the species. The distinction, therefore, between 
 the two kinds of genera, as here drawn, is the same as that be- 
 tween the two kinds of general terms, the mathematical and the 
 logical. Hence I distinguish the mathematical and the logical 
 genus : the first, or mathematical, to denote the genus as ap- 
 plicable numerically to all the species, and the second, or logi- 
 cal, to denote or connote simply the conferentia or common 
 essence, and not affirmable of specific characters. 
 
 The importance and meaning of this distinction must be 
 brought out by further illustration. To effect this requires a 
 brief explanation of what is meant by the connection between 
 subject and predicate. Usually we suppose, or are told, that the 
 predicate is more or less identical with the subject. Thus if I 
 say, "Man is a biped," I mean that two-footedness is a quality 
 of man. But I may mean by a similar judgment that one 
 quality invariably and universally accompanies another. Thus 
 to say, "Man is intelligent," may mean that along with a cer- 
 tain representative quality or qualities standing for man will 
 be found the equally universal and necessary quality intelli- 
 gence, but which is not analytically represented to conscious- 
 ness in the mention of the name " man." In other words, I 
 affirm the agreement or concomitance of certain conferential 
 qualities, and this will be true of their connection, generically 
 considered, wherever found. To take the former illustration, 
 " Pine wood is good for lumber," goodness for lumber is con- 
 nected with the generic qualities of pine wood, but it is not 
 connected with every particular or specific form of it. The 
 statement undoubtedly assumes a particular form and quantity 
 of the pine wood as essential to its making lumber. But this 
 only shows that it is not the mathematical genus of which it is
 
 94 ELEMENTS OF LOGIC 
 
 affirmed, but only of the logical, and hence the difficulty when 
 we come to compare the predicate with the species, or the dif- 
 ferential qualities. 
 
 A negative illustration will bring out the same truth. For 
 instance, I cannot say that " All men are white," but I can say, 
 " The Caucasians are white," because " whiteness " is true of 
 the species, and not true of the genus, taken either logically 
 or mathematically. But now if I turn around and say, " The 
 Caucasian race is the most intelligent," this may not be true 
 of all individual Caucasians numerically considered, but only 
 generally ; that is, their intelligence is an accidental character- 
 istic connected with such essential qualities as make them 
 men, and with which whiteness is found frequently enough to 
 make the statement of the race in general. We therefore 
 speak of the. logical and not the mathematical genus in our 
 proposition. 
 
 The importance of the distinction I have drawn will appear 
 when I come to consider the doctrine of Judgments and Fal- 
 lacies. But it is fully justified in the ambiguity remarked in 
 the use of the term genus, contrasting it, on the one hand, 
 with the species, with which it differs only quantitatively in 
 regard to extension, and on the other with the differentia, to 
 which it is opposed and with which it differs qualitatively in 
 regard to intension. We have, therefore, to keep constantly in 
 mind that all generic concepts have a double capacity, the math- 
 ematical, and the philosophic or logical. If this fact is closely 
 guarded the student will be saved many a misstep in reasoning. 
 
 2d. Analysis of Concepts. By the analysis of concepts 
 I mean here the breaking up of them into their parts and sub- 
 ordinate elements. There are two forms of analysis, Division 
 and Partition, which must be considered in their proper order. 
 
 1. DIVISION. Division is the analysis of the extension of a 
 concept, the separation of a genus into its species. The pro- 
 cess is usually called Logical Division. Thus we are said to 
 divide the genus " tree " when we indicate the species of tree 
 to which the term applies, as, for example, into oak, elm, 
 maple, willow, ash, pine, etc., some of which at least are still
 
 DEFINITION AND DIVISION 95 
 
 farther divisible into subordinate species or individuals, as 
 oak into white, black, and red oaks ; pine into white and yel- 
 low pines ; Avillow into white, weeping, and swamp willows, 
 etc. They may be divided by some other principle if we so 
 desire. Thus we might divide " tree " into those of deciduous 
 leaves, and those of evergreen leaves, etc. It is not necessary 
 in every case to proceed upon the same principle. But what- 
 ever principle is used is called the Fundamentum Divisionis. 
 In order to be a principle of division the quality or circum- 
 stance, taken as such, " must be present with some and absent 
 from others, or must vary with the different species compre- 
 hended in the genus. A generic property, of course, being 
 present in the whole of the genus, cannot serve for the pur- 
 pose of division." The principle of division, therefore, must 
 be some differentia, or differential quality, which is the dis- 
 tinctive feature of the species. Thus if I divide apples into 
 red, green, and yellow, color is the principle of division, and 
 each specific color is the differentia of its class. An accidental 
 property will not suffice for any permanent or scientific division. 
 Hamilton enumerates seven rules governing the process of 
 division. They are : 
 
 (a) " Every division should be governed by some principle." 
 (6) "Every division should be governed by only a single 
 principle." 
 
 (c) " The principle of division should be an actual and essen- 
 tial character of the divided notion, and the division, there- 
 fore, neither complex nor without a purpose." 
 
 (d) "No dividing member of the predicate must by itself 
 exhaust the subject." 
 
 (e) "The dividing members, taken together, must exhaust, 
 but only exhaust, the subject." 
 
 (/) "The divisive members must be reciprocally exclusive." 
 (g) " The divisions must proceed continuously from immedi- 
 ate to mediate differences." 
 
 These are not all of equal importance, and might be reduced 
 to a smaller number. For instance, the first is practically the 
 same as the second. The fourth and fifth (d and e) might be
 
 96 
 
 summarized in one. The third and the seventh (c and g), al- 
 though important in a complete enumeration, are more likely 
 to be observed naturally than some of the others. But they 
 require to be kept in mind. Jevons reduces them to three, 
 which serve for most all practical purposes. 
 
 The importance of the rules is seen in what is called Cross 
 Division, which is the naming of species that interpenetrate 
 or overlap. Thus, if I divided trees into tall trees, green trees, 
 pine trees, and dead trees ; or books into octavos, histories, 
 theoretical books, dictionaries, etc., I should be using more 
 than one principle of division, and indicating species that were 
 not mutually exclusive. An illustration of the proper form of 
 division is the following table, or outline : 
 
 f Trilateral = Triangles. 
 
 ( Parallelograms. 
 
 Rectilinear -j Quadrilateral = < Trapezoids. 
 ( Trapeziums. 
 
 Plane -{ (. Multilateral = Polygons of more than four sides. 
 
 Circular = Circles. 
 
 
 Solid 
 
 
 Rectilinear 
 
 Curvilinear 
 
 Hyperbolic = Hyperbolas. 
 f Tetrahedrons. 
 J Pentahedrons. 
 1 Cubes. 
 [Parallelepipeds, etc. 
 
 Spheres. 
 
 Cones. 
 
 Cylinders. 
 
 Paraboloids. 
 
 In division a genus, in relation to a species, is said to bo 
 superordinate ; a species in relation to a genus is said to be 
 subordinate ; and a species in relation to a species is said to 
 be co-ordinate. Thus, in the division of man into Caucasians, 
 Mongolians, etc., " man " is superordinate in prior relation to 
 Caucasians, Mongolians, etc., and they are subordinate to man 
 in ulterior relation, while Caucasians, Mongolians, etc., in re- 
 lation to each other, are co-ordinate. 
 
 We may adopt, as is apparent in this outline, a new princi- 
 ple for each successive process of division. In the division of 
 plane and solid figures, however, it is the same. In the first 
 division it is the form in general ; in the second division it is 
 the kind of bounding lines ; in the third division it is the dif-
 
 DEFINITION AND DIVISION 
 
 97 
 
 ferent relative positions and relations of lines. An illustration 
 of a completely new principle in each division will be the fol- 
 lowing. I do not pretend that it is perfect, but only that it 
 illustrates the point under consideration : 
 
 f 
 
 Science 
 
 Physical 
 
 Moral 
 
 ( Physics. 
 ] Chemistry. 
 j Biology. 
 < Physiology, 
 j History, 
 j Sociology, 
 j ( Noetics. 
 
 Psychological^ /Esthetics. 
 ( Ethics. 
 
 Mechanical 
 
 Organic 
 
 Political 
 
 The simplest form of logical division is called Dichotomy, 
 which is the continual division of a genus into two species, a 
 positive and a negative. This is the simplest mode of making 
 the division exhaustive. A threefold division is called Tri- 
 chotomy ; but there is no technical name for the forms after 
 that. Dichotomy is very useful in certain kinds of discussion, 
 but in other circumstances is not so convenient. An example 
 of it is found in what is called the Tree of Porphyry, named 
 after the Greek logician who originated it. It may be repre- 
 sented thus : 
 
 Substance. 
 
 Corporeal. Incorporeal. 
 
 Body. 
 Animate. Inanimate. 
 
 Living being. 
 Sensible. Insensible. 
 
 Animal. 
 Rational. Irrational. 
 
 Man. 
 Socrates. Plato. Aristotle and others.
 
 98 ELEMENTS OF LOGIC 
 
 Man could be dichotoinously divided into Caucasian and 
 non-Caucasian, the former into Greeks and not-Greeks, and 
 so on. But the process must be terminated in the last analysis 
 with a mention of the individuals, and there may be several 
 points where this may be legitimately done. 
 
 The usefulness of dichotomy has its limitations. Thus it 
 would be useless to divide Europe into France and net-France, 
 the British Empire into England and not-England, or America 
 into Rhode Island and not-Rhode Island. " Dichotomy is use- 
 less and even seems absurd in these cases, because we can ob- 
 serve the rules of division certainly in a much briefer division. 
 But in less certain branches of knowledge our divisions can 
 never be free from possible oversight unless they proceed by 
 dichotomy. Thus, if we divide the population of the world 
 into Aryan, Semitic, and Turanian, some race might ultimately 
 be discovered which is distinct from any of these, and for 
 which no place has been provided ; but had we proceeded 
 thus: 
 
 Man 
 
 I 
 
 Aryan Not- Aryan 
 
 Semitic Not-Semitic 
 
 Turanian Not-Turanian, 
 
 it is evident that the new race would fall into the last group, 
 which is neither Aryan, Semitic, nor Turanian. All the divis- 
 ions of naturalists are liable to this inconvenience. If we di- 
 vide Vertebrate Animals into Mammalia, Birds, Reptiles, and 
 Fish, it may happen at any time that a new form is discovered 
 which belongs to none of these, and therefore upsets the di- 
 vision." 
 
 Jevons might have remarked that dichotomy can be best ap- 
 plied where our knowledge has not been exhausted, or where 
 changes of boundary are likely to take place. Where there are 
 distinct and known limits to a body of knowledge, the genus
 
 DEFINITION AND DIVISION 99 
 
 can be more clearly exhausted by the usual method of division. 
 Dichotomy is needed where there is an indistinct and uncertain 
 field of ideas. 
 
 2. PARTITION. Partition is the analysis of a concept by a 
 statement of its intension. It is simply a process describing 
 the concept by its qualities, or parts constituting it. The con- 
 cept may be viewed either as an individual or a class-whole, 
 and partition merely defines it by its properties. Partition 
 may be mathematical or quantitative and logical or qualitative. 
 It is mathematical when the division or analysis is into its 
 parts expressed in terms of space or time. Thus the concept 
 tree is partitioned mathematically into roots, trunk, branches, 
 and leaves ; logically, it would be divided into its vegetable 
 properties color, woody fibre, raising of the sap by capillary 
 attraction, etc. The concept " life " (a person's age) would be 
 partitioned mathematically into childhood, maturity, old age, 
 etc. ; logically, it might be divided or partitioned into its 
 length, goodness or badness, mode of spending it, etc. But 
 in all cases we should disregard the questions of genus and 
 species, and merely endeavor to consider the properties, essen- 
 tial or accidental, which constitute a concept. To make the 
 partition exhaustive we should be obliged to state all the 
 properties that make up an object ; but as this is often pre- 
 vented by the limitations of our knowledge of them, we have 
 to be content with such as we know. A more complete illus- 
 tration of the analysis than those given may better show the 
 extent to which it may be carried. Take the case of " gold." 
 The qualities of it are that it is material, metallic, solid, ele- 
 mentary, yellow, malleable, precious, useful, conductor of elec- 
 tricity, etc. " Man " again may be partitioned into animality, 
 rationality, color, weight, sociality, etc. 
 
 The usefulness of the process is not so apparent in common 
 conceptions. But if it were carried out carefully with such 
 conceptions as "virtue," "thought," "mind," "religion," 
 " cause," " intuitive," " law," "nature," etc., many a controversy 
 would be modified in its incidents. Thus the notion " cause " 
 may be partitioned into uniformity of sequence and co-exist-
 
 100 ELEMENTS OF LOGIC 
 
 tence, and efficiency of power, and perhaps other qualities. 
 Controversies about our knowledge of it would be materially 
 affected by the presence or the absence of the second quality. 
 Usually the partition implies that an object has more than 
 one, or that a concept represents more than one quality. But 
 we may regard every simple descriptive or declarative proposi- 
 tion as a case of partition. It is a descriptive definition, which 
 partition aims to give. But it is not necessary to carry the 
 idea of partition so far, except to intimate the broad distinc- 
 tion between it and true logical definition. Partition, where 
 it serves any useful purpose, assumes a multiple of qualities 
 which require recognition as well as the relation of genus and 
 species. It is opposed to division as intension is opposed to 
 the extension of concepts. Hence it is a process complement- 
 ary to division. 
 
 3d. Definition of Concepts. The definition of anything 
 in practice is undertaken in several ways. But Logic has 
 strictly to do with only one of them. This is called Logical 
 Definition. It is necessary, however, to notice the several 
 modes of definition in order to distinguish the logical from 
 them in the proper way. They may be called Etymological 
 Definition, Descriptive Definition, and Logical Definition. 
 
 1. ETYMOLOGICAL DEFINITION. This is the definition of a 
 concept by the word-roots from which the term originated. 
 For example, " inquisition " would be etymologically defined 
 by saying it was from two roots or words denoting " to in- 
 quire into ;" "playfulness," from a root and suffixes denoting 
 the quality of being full of sport, etc. But while this form of 
 definition is very valuable in some circumstances it is of no 
 importance in logical doctrine, or for elucidating any of the 
 laws of thought. 
 
 2. DESCRIPTIVE DEFINITION. This is the definition of a con- 
 cept or a thing merery by describing it, and is essentially the 
 same as partition. It most frequently occurs in the mention 
 of accidental properties of objects when distinguished from 
 complete partition and definition proper. It is, in fact, im- 
 perfect definition in the omission of one or the other of the
 
 DEFINITION AND DIVISION 101 
 
 two essential conditions of the logical form of it. Thus, a 
 descriptive definition of a triangle would be, that it is com- 
 posed of straight lines and a symbol very- frequently used in 
 geometry. But such an account of it might as well apply to 
 a rectangle or a parallelogram. A descriptive definition, how- 
 ever, may be made to approximate very closely to the logical. 
 The above illustration wants but little modification to become 
 that. Indeed the true definition is descriptive, but it is com- 
 pletely descriptive, involving a relation not expressed in this 
 imperfect form. The difference between them can perhaps 
 be technically expressed by saying that the ordinary descrip- 
 tive definition depends upon the accidentia or the conferentia, 
 and logical definition upon the differentia. 
 
 3. LOGICAL DEFINITION. Logical definition is the statement 
 of the genus and differentia of a concept, and is thus occupied 
 with the whole of its intension, as Division is occupied with 
 the whole of its extension. To illustrate, I may define "man " 
 as a "rational animal." In this statement "animal" is the 
 genus to which " man " is supposed to belong as a species, 
 and rationality is the differentia which distinguishes him from 
 other species. Again, I may define a circle as " a curved line 
 everywhere equally distant from a point within called the cen- 
 tre ;" or tree as "a vegetable with woody fibre, root, trunk, 
 branches, leaves, and a certain magnitude ;" or a house as "a 
 building used for a place of residence," etc., and in each case 
 fulfil the requirements of a logical definition. 
 
 It is important to make two remarks in regard to these and 
 all logical definitions. First, they do not state the mathemat- 
 ical genus and species, which would identify the process with 
 division in its principles, but the conferentia or logical genus, 
 and the differentia. Second, it is always the species that is de- 
 fined, never the genus. 
 
 The difference between definition and division, in their 
 treatment of the genus, is, that the division of a concept pro- 
 ceeds progressively to the species under it, and may continue 
 on down to the individual, but definition proceeds regressirely 
 to the proximate genus, or some appropriate genus which may
 
 102 ELEMENTS OF LOGIC 
 
 serve as such, and goes no farther unless called upon to define 
 that in the same way. Its progressive movement, if it can be 
 said to have any, is a statement of the differential property or 
 properties of the species defined, not a development of its ex- 
 tension or subordinate species. Since it is the species, there- 
 fore, which is always defined, and never the genus, unless it 
 be also a species in relation to a higher genus, the ultimate or 
 summum genus can never be defined. A logical definition, as we 
 have shown, must state the genus ; but as thesummun genus is 
 never a species it cannot be logically defined. Hence the impos- 
 sibility of defining ultimate truths or principles, or the simplest 
 concepts. They can be dealt with only descriptively or par- 
 titively, or divisively. Everything below them can be defined. 
 But it is important to observe the meaning of stating the 
 "genus and differentia" in a logical definition. We observe, 
 first, that it is not the mathematical " genus and species," as 
 already remarked. Hence the term genus is used in its mean- 
 ing as contrasted with differentia, and so denotes the logical 
 as opposed to the mathematical genus. It therefore denotes 
 the common qualities, or conferentia of the concept defined, in 
 comparison with co-ordinate species. This is to show that 
 logical definition is a statement of the conferentia and differ- 
 entia of a conception, and not of the genus and species mathe- 
 matically considered. Hence we see how it is exclusively oc- 
 cupied, if not explicitly, then implicitly, with the intension of 
 concepts. The genus, as contrasted with the differentia, de- 
 notes only the common qualities of objects in the same class, 
 and the differentia those which separate the individual or 
 species defined from others in the same class. Thus to de- 
 fine a "bed" as "a piece of furniture for reclining upon," is 
 to intimate that "bed" has qualities in common with other 
 things known as furniture, and it is distinguished from a 
 chair, which is used for sitting, by the property of being used 
 for reclining. Instead of specifying the common qualities par- 
 titively, however, a general name suffices to imply them. But 
 not having any corresponding abstract term for differentia, 
 these properties have to be distinctly indicated. The genus
 
 DEFINITION AND DIVISION 103 
 
 and differentia, therefore, as stated in the definition, are sim- 
 ply all the qualities that make up the species defined. These 
 are the conferentia and the differentia. 
 
 The conclusion from this must be that the predicate of a 
 logical definition is always equal to, identical, and convertible 
 with the subject. The importance of this will be apparent in 
 the doctrine of Conversion and of Reasoning. All that we re- 
 quire to observe here is the difference between the ordinary 
 simple proposition and the proposition which is regarded as a 
 definition. The statements that "man is a biped," or "man is 
 mortal," are not definitions. One states merely the genus of 
 man, and the other a property of him. The first implies a 
 property, that of two-footedness, and the second may be said 
 to imply a genus. But neither of them specifies any differen- 
 tia that would distinguish man, on the one hand, from other 
 bipeds, and, on the other hand, from other mortals. They are 
 not definitions because they do not state the whole intension 
 of the species. Hence a true definition expresses the full 
 meaning of the species defined, and can be used convertibly 
 with it. Thus we can say equally that "man is a rational 
 animal," or " rational animals are men," a process which can- 
 not be performed with the simple proposition unless it is con- 
 sidered a definition. The fact that the two forms of propo- 
 sition are the same to all appearances, and .the fact that the 
 mind uses the subject and predicate of definitions couvertibly 
 with each other, often lead to confusion, by inducing the treat- 
 ment of simple propositions as definitions. This source of er- 
 ror will be treated in its proper place. At present it suffices 
 to call attention to the fact. 
 
 In regard to the rules regulating correct definitions it will 
 suffice to state Jevons's account of them, and it will always be 
 important for the student to keep them in mind. They are five : 
 
 (a) " A definition should state the essential attributes of the 
 species defined. So far as any exact meaning can be given to 
 the expression ' essential attributes,' it means the proximate 
 genus and difference." 
 
 (6) " A definition must not contain the name defined. For
 
 104 ELEMENTS OF LOGIC 
 
 the purpose of the definition is to make the species known, 
 and as long as it is not known it cannot serve to make itself 
 known. When this rule is not observed, there is said to 
 be a ' circulus in definiendo,' or ' circle in definition,' because 
 the definition brings us around again to the very word from 
 which we started. This fault will usually be committed by 
 using a word in the definition which is really a synonym of 
 the name defined, as if I were to define a ' plant 'as 'an or- 
 ganized being, possessing vegetable life,' or ' elements ' as 
 ' simple substances,' vegetable being really equivalent to plant, 
 and simple to elementary. If I were to define ' metals ' as 
 ' substances possessing metallic lustre,' I should either commit 
 this fault or use the term metallic lustre in a sense which 
 would admit other substances and thus break the folio wing rule." 
 
 (c) " The definition must be exactly equivalent to the species 
 defined. That is to say, it must be an expression, the denota- 
 tion of which is neither narrower nor wider than the species, so 
 as to include exactly the same objects. The definition, in short, 
 must denote the species, and nothing but the species, and this 
 may really be considered a description of what a definition is." 
 
 (d) "A definition must not be expressed in obscure, figurative, 
 or ambiguous language. In other words, the terms employed in 
 the definition must be all exactly known, otherwise the pur- 
 pose of the definition, to make us acquainted with the sufficient 
 marks of the species, is obviously defeated. There is no worse 
 logical fault than to define ignotum per ignotius, the unknown 
 by the still more unknown. Aristotle's definition of the soul 
 as ' the entelechy, or first form of an organized body which 
 has potential life,' certainly seems subject to this objection." 
 
 (e) " A definition must not be negative when it can be affirm- 
 ative. This rule, however, is often not applicable, and is by no 
 means always binding." * 
 
 * The following references may be consulted on matters pertaining to 
 this chapter : Mill : Logic, Book I., Chaps. VII. and VIII. ; Venn : Em- 
 pirical Logic. Chaps. XI. and XII.; Hamilton: Lectures on Logic, Lects. 
 XXIV. and XXV. ; De Morgan: Formal Logic, Chap. XII. ; Whately: 
 Elements of Logic, Book II., Chap. V.; Supplement to Chap. I., 2-6.
 
 CHAPTER VH. 
 
 PROPOSITIONS OR JUDGMENTS 
 
 1st. Definition. Words or terms unconnected express 
 only concepts outside of any distinctly affirmed relation. In 
 this way they do not convey truth, but only ideas or concep- 
 tions. Logic has to deal with the connection of concepts and 
 their implications. The manner in which terms and concepts 
 are joined together determines what a proposition shall be. 
 It is not every combination of terms that forms a logical prop- 
 osition. Some combinations may be mere phrases or ejacula- 
 tions. But those combinations expressing a certain kind of 
 relation, namely, a declarative relation between two terms, are 
 the propositions with which Logic is concerned. 
 
 A proposition in Grammar is called a sentence ; in Logic, it 
 is most frequently called a, judgment. A proposition or judg- 
 ment, therefore, in Logic, is the affirmation or denial of agree- 
 ment between two conceptions. It involves a comparison be- 
 tween them and a perception of this relation. Thus the 
 proposition, " Gold is a metal," expresses a certain agreement 
 between the concepts " gold " and " metal," an agreement 
 which implies that the same quality is common to both, or 
 that "gold" is a species of "metal." On the other hand, the 
 proposition, "Man is not a quadruped," expresses a disagree- 
 ment in a certain particular between the tw r o concepts a dis- 
 agreement which implies that "man" is not in the class 
 "quadrupeds," or does not possess the particular quality 
 which distinguishes quadrupeds. This agreement or disagree- 
 ment is not limited to single concepts or terms, but may in- 
 clude the same relation between groups of concepts constitut- 
 ing phrases. Thus, " The City of Washington, in the District 
 of Columbia, is the Capital of the United States of America,"
 
 106 ELEMENTS OF LOGIC 
 
 is a proposition, only a little more complex in its elements 
 than the former illustrations. 
 
 The terms between which the relation is asserted or denied 
 are called the subject and the predicate. The subject is that 
 of which something is affirmed or denied ; the predicate is 
 that which is affirmed or denied of the subject. "Subject" 
 (subjectum, vn-oKei'/xevoi') means underlying thing ; " predicate " 
 (prsedicatum, Karr^yopou/xei/ov) means that which is asserted. 
 The subject and predicate may be either grammatical or logi- 
 cal. The grammatical subject or predicate will be a single 
 term ; the logical subject and predicate will consist of the 
 grammatical subject with all its modifiers. Taken together 
 they express in thought a single idea or conception, and hence 
 Logic may treat them accordingly. All complex propositions 
 are thus reduced to a single form. 
 
 The term expressing the connection between the subject and 
 predicate is called the copula, and is always some form of the 
 verb to be, or its equivalent. In many, perhaps the largest 
 number of propositions, the verb to be is not found, and hence 
 they appear to be wanting in a copula. Thus the proposition, 
 " Napoleon ruled France," contains no expressed copula. In 
 all such propositions, however, the predicate is said to include 
 the verb and its dependent terras, and so to include the 
 suppressed copula. Thus in the illustration given, "ruled 
 France " is called the predicate, and the proposition seems to 
 consist of only subject and predicate. But if we resolve the 
 expression " ruled France " into its exact logical equivalent, 
 " was the ruler of France," we have the copula and the predi- 
 cate in the simple form. Hence the term " France " will not 
 be the predicate alone, but " ruled France " must represent it 
 with the copula implied or included in it. It is necessary to 
 so consider the matter in order to deal logically with all such 
 propositions. This logical treatment of them depends upon 
 such a conception of the relation between subject and predi- 
 cate as can be reduced to a general or universal law. A more 
 complete discussion of the nature of this relation will be ap- 
 propriate after we have considered the divisions of judgments.
 
 PROPOSITIONS OR JUDGMENTS 107 
 
 2d. Divisions. Propositions can be divided in a great many 
 ways. The first division into Indicative, Interrogative, and Im- 
 perative, with perhaps the Optative and Exclamatory, as recog- 
 nized by some, is grammatical, and it is only with the first class 
 that Logic has to do. The essential meaning of the others, so 
 far as the relation of concepts is concerned, can be reduced to 
 the first form, the declarative, or indicative in such emergen- 
 cies as require a logical use of their matter. 
 
 1. LOGICO-GBAMMATICAL PROPOSITIONS. There is a second di- 
 vision which is both grammatical and logical, but which has 
 not been uniformly the same with logicians. Sometimes it has 
 been into Categorical and Conditional, with a subdivision of 
 the second into Hypothetical and Disjunctive. Sometimes Con- 
 ditional and Hypothetical simply interchange places in this 
 division, and in a third form they are made synonymous with 
 each other, giving us a co-ordinate division of three kinds, 
 into Categorical, Conditional or Hypothetical, and Disjunctive. 
 This last division I much prefer to all others, for the reason 
 that their relation to each other in structure and meaning can 
 be more easily determined than in any other classification. 
 
 There are, however, two classifications which may be given 
 and that are of considerable convenience in explaining the 
 meaning and relations of various kinds of propositions. The 
 first one proceeds in the order of increasing complexity, and 
 is intended to mark the nature of the additions made to deter- 
 mine the more complex forms. The following diagram ex- 
 hibits the classification with illustrations : 
 
 Propositions - 
 
 n.,*,, . -^ n i ( Declarative = A is B. 
 
 Categorical j Disjunctive = A is . elther B or c 
 
 n A-f i i Hypothetical = If A is B, C is D. 
 
 )nal | Dilemmatic = If A is B, C is either D or E. 
 
 It is apparent in this division that the simplest form is the 
 declarative proposition, where the assertion is absolute and 
 definite as regards both subject and predicate. The disjunc- 
 tive form is equally assertory in its form of expression, but 
 differs from the first in allowing some doubt or choice about 
 the predicate, there being one alternative which excludes the
 
 108 ELEMENTS OF LOGIC 
 
 connection of the other with the subject. The hypothetical 
 expresses a definite dependence of one proposition, a declara- 
 tive proposition upon a condition. Hence it adds a declara- 
 tive assertion to a conditional one. The dilemnmtic proposi- 
 tion simply adds a disjunctive one to a conditional proposi- 
 tion. 
 
 But a second classification is much preferable to this because 
 it conforms to the three forms of reasoning, and, in a measure, 
 determines them. In this classification we make the disjunc- 
 tive appear as co-ordinate with the other two, although it is in 
 reality a combination of them, and to which we apply the dis- 
 tinction between form and matter. The categorical and con- 
 ditional propositions are regarded logically as pure and un- 
 mixed. The disjunctive we make categorical in its form of 
 expression, but conditional in its meaning. This the following 
 diagram will show : 
 
 ( Categorical = Assertory in form and matter. 
 Propositions < Conditional = Hypothetical in form and matter. 
 
 ( Disjunctive = Categorical in form, but Conditional in matter. 
 
 In regard to the disjunctive proposition, under this conception 
 of it, it need only be said that its form of expression is un- 
 doubtedly assertory. It is positively affirmed that "A is either 
 B or C." But the meaning of the disjunction, or the alterna- 
 tive expressed, can be understood only as implying " if A is B, 
 it is not C," or " if A is C, it is not B." We shall discover 
 later on in the discussion of reasoning that this is the only in- 
 terpretation of the case which will enable us to reduce disjunc- 
 tive reasoning to the regular form, or to understand it as a 
 mode of the usual process of reasoning. 
 
 A Categorical proposition is one in which a statement is un- 
 conditionally made ; as, " A is B," or, " Man is mortal." A Con- 
 ditional or Hypothetical proposition is one in which the asser- 
 tion is conditional or dependent upon a supposition of some 
 kind ; as, "If A is B, C is D," or, "If a stone be released from 
 support it will fall to the ground." The first clause of the 
 conditional proposition is called the antecedent, the second the 
 consequent. The symbols of such propositions are if, even if,
 
 PROPOSITIONS OR JUDGMENTS 109 
 
 provided that, although, sometimes when, or any form of ex- 
 pression denoting a condition. A Disjunctive proposition is 
 one which implies or asserts an alternative in the relation be- 
 tween the subject and predicate ; as, " A is either B or C," or, 
 "Metals are either hard or soft." The symbols of the disjunc- 
 tive proposition are either and or. Some ambiguity is con- 
 nected with their meaning, which will have to be considered 
 when discussing the Disjunctive Syllogism. But as it does 
 not affect the form and general meaning of the proposition by 
 that name, the matter need not be discussed at present. We 
 have only to remark what the disjunction means when it is 
 complete, and that is, that the alternatives expressed by the 
 terms either and or should be exhaustive. It means that the 
 connection between the subject and predicate must be one or 
 the other of two things. In the proposition A is either B or C, 
 the question whether A is B or A is C is indefinite or unde- 
 cided, but it is definitely one or the other, and hence the 
 proposition either means that A is B and not C, or it means 
 that A is C and not B. Hence, although the proposition stands 
 as a direct assertion, it means that if A is B, it is not C, or if A 
 is C, it is not B, or if it is not B, it is C. This is the reason 
 that it is usually classed as a form of conditional judgment. 
 But if it be closely examined it will be found to contain both 
 assertory and conditional elements. It is categorical in Us form, 
 and conditional in its matter or meaning. 
 
 2. PROPOSITIONS ACCORDING TO QUALITY. Propositions may 
 be divided into Affirmative and Negative, according as they 
 affirm or deny the agreement between the subject and the 
 predicate. This relation is called or determines their quality. 
 An affirmative proposition asserts an agreement between sub- 
 ject and predicate ; as, " Gold is yellow," or, " Doves are birds." 
 A negative proposition is one which denies an agreement be- 
 tween subject and predicate ; as, " Men are not trees," or, 
 "Gas is not heavy." 
 
 3. PROPOSITIONS ACCORDING TO QUANTITY. Propositions ac- 
 cording to quantity are divided into Universal and Particular. 
 The distinction between them is determined by the question
 
 110 ELEMENTS OF LOGIC 
 
 whether the predicate is affirmed or denied of the whole of the 
 subject. Hence a universal proposition is one in which the 
 predicate is said to be affirmed or denied of the whole of the 
 subject ; as, " All men are mortal," or, " No men are trees." A 
 particular proposition is one in which the predicate is said to 
 be affirmed or denied of a part of the subject ; as, " Some 
 men are wise," or, " Some snow is not black." But the diffi- 
 culty with this definition is that there is a sense in which 
 the predicate is affirmed or denied of the whole of the sub- 
 ject in the particular proposition. For according to what 
 has been said of the nature of the subject it may include what 
 is known in grammar as the " logical subject," which consists of 
 all the terms constituting a complex conception and standing 
 in the relation of " subject " to the proposition. In this sense 
 the predicate of a particular proposition is affirmed or denied 
 of the whole of its logical subject, but of only a part of the 
 grammatical subject. If therefore we could say that a uni- 
 versal proposition affirms or denies the predicate of the whole 
 subject, grammatical and logical, and a particular proposition, 
 of a part of the grammatical subject only, the difficulty would 
 be removed. But it returns again in such propositions as 
 " All good men are respected," which would be particular ac- 
 cording to the definition. For the predicate is affirmed of 
 only a part of the grammatical subject. 
 
 It would, therefore, be better for the purposes of definition 
 either to divide propositions into Definite and Indefinite, or 
 define universal propositions as affirming or denying the pred- 
 icate of the whole of a definite subject, and particular proposi- 
 tions, of an indefinite subject. This is what is really meant by 
 universal and particular propositions, and hence, with the pro- 
 viso that they shall be identical in meaning with definite and 
 indefinite, we shall adopt them as expressing the division of 
 propositions according to quantity. 
 
 But this twofold division is the result of a reduction from 
 a division which is frequently^iue/bW. Propositions are fre- 
 quently divided, according to quantity, into Universal, Singu- 
 lar, General, Plurative, and Particular. The first and the last
 
 PROPOSITIONS OR JUDGMENTS 111 
 
 have been adequately defined and illustrated. The intermediate 
 three may be reduced to one or the other of the first and the 
 last, as their definition will prove. A singular proposition is 
 one in which the subject is a singular term, and hence definite 
 in its meaning ; as, "Louis XIV. was king of France." Here 
 the predicate is affirmed of the whole of a definite subject, and 
 hence for all logical purposes the proposition is universal. 
 That is, the same laws of reasoning, mediate or immediate, 
 will apply to singular or apply to universal propositions. A 
 general proposition is one in which the distribution of the sub- 
 ject is ambiguous; as, "Metals are useful," "Man is intelli- 
 gent." It is not stated whether " All metals are useful," or 
 " All men are intelligent," or whether some are so. The prop- 
 ositions are capable of either interpretation, and according as 
 we think of all or some, are universal or particular. Plurative 
 propositions are undoubtedly particular. They are intro- 
 duced by the word most, or its equivalent ; as, " Most rumi- 
 nants are horned," and require mention only because of- a pe- 
 culiar syllogism which is valid in spite of its composition from 
 particular premises ; of which again. But they affirm or 
 deny the predicate definitely of more than half the subject, but 
 indefinitely in regard to which of the two halves is exhausted 
 in the term most. They are, therefore, classed as particular 
 propositions. A summary of this reduction appears in the 
 following table : 
 
 Propositions 
 
 ( Universal ) 
 Definite < Singular >- Universal. 
 
 | General j 
 
 K 
 
 I 
 
 Indefinite J Plurative >- Particular. 
 ( Particular ) 
 
 The mark of a universal proposition usually consists of some 
 adjective denoting quantity, such as, all, every, each, any 
 (meaning all individually), and whole. But wherever we find 
 the predicate referring definitely to the whole of the subject 
 we may treat the proposition as universal. This merely im- 
 plies that some propositions may be universal in their matter, 
 but indefinite in their form. The signs of particular propo-
 
 112 ELEMENTS OF LOGIC 
 
 sitions are also certain adjectives of quantity, such as some, 
 certain, a few, many, most, any (meaning an indefinite indi- 
 vidual), or such others as denote at least a part of a class. 
 
 The signs of a negative proposition are no, not, and none, 
 the first and last being prefixed to the subject, and the second 
 joined to the copula. Examples of them are, " No metals are 
 animals," "None of the rebels were punished," and " Men are 
 not quadrupeds." The term " no " is one which denotes both 
 universal quantity and negative quality in propositions. 
 
 The quality and quantity of propositions may be combined 
 in classifying them, and we shall have universal affirmative 
 propositions, universal negatives, particular affirmatives, and 
 particular negatives. It has been usual to choose an abbre- 
 viated symbol to denote each of these classes. The first four 
 vowels of the alphabet A, E, I, O have been chosen for this 
 purpose. A is the symbol of a universal affirmative, I of 
 a particular affirmative, E of a universal negative, and O of 
 a particular negative. Henceforth we shall employ them with 
 this denotation whenever it is most convenient. It will be in- 
 teresting to remark that A and I occur in the Latin affirmo, 
 and E and O in the Latin nego. There is no significance in 
 this, save perhaps as a mnemonic aid. The following table 
 summarizes results : 
 
 4. ANALYTIC AND SYNTHETIC PROPOSITIONS. Another division 
 separates propositions into Analytic, Essential or Explicative, 
 and Synthetic or Ampliative. An analytic proposition affirms of 
 its subject a predicate, which is implied in the very conception 
 of the subject. Thus, " Matter is extended," " Water is moist," 
 "Living beings are organic," "Wood is a substance," are all 
 analytical judgments because the subject cannot be represented 
 to the mind without thinking implicitly or explicitly of the 
 notion expressed by the predicate. The use of the term " es- 
 sential " to describe such judgments means that the property 
 expressed by the predicate is an essential one, which is neces-
 
 PROPOSITIONS OR JUDGMENTS 113 
 
 sary to conceiving the subject. The term/' explicative," de- 
 scribing the same judgment, means merely that the predicate 
 develops or unfolds what is involved in the thought of the 
 subject. On the other hand, a synthetic proposition or judg- 
 ment is one in which the predicate conveys information not 
 necessarily implied in the conception of the subject. Ex- 
 amples of them are, " Water is a conductor of sound," " Plato 
 was aristocratic," " Some men are honest," " The Popes were 
 patrons of art." In these instances the predicate is not neces- 
 sarily associated with the subject. It is no part of our concep- 
 tion of water that it conducts sound, nor of the Popes that they 
 should be patrons of art. It would seem from this, therefore, 
 that analytic judgments assert essential qualities of the sub- 
 ject, and synthetic judgments accidental qualities of it. If so, 
 the distinction is a very clear, and perhaps a very useful one. 
 
 But the division of propositions into analytic and synthetic 
 has little or no importance for Formal Logic. Its chief im- 
 portance is in the domain of psychology and philosophy. The 
 laws of reasoning, mediate or immediate, are not affected by it. 
 Besides this there is often a great difficulty in distinguishiag 
 between the two classes of judgment so named, because of the 
 confusion to which we are liable in distinguishing between an 
 essentiiil property which is universal, and a universal property 
 which is accidental. Indeed it may be gravely doubted 
 whether any universal property can be accidental. At least 
 some would doubt it, and it may be a mere matter of our 
 knowledge as to whether a given property is essential or acci- 
 dental. If so, the distinction between analytic and synthetic 
 propositions will only express the difference between our mode 
 of representing a concept uniformly and the accidental asso- 
 ciation of some other property with it, less frequent in our ex- 
 perience. That is, the proposition " Body is extended " may 
 appear analytic to the mind who has always or most frequently 
 experienced it in connection with the idea of extension, while 
 the want of frequent experience in connection with its sono- 
 rousness might make the proposition " Body is sonorous " a 
 synthetic proposition. On the other hand, the limitation of
 
 114 ELEMENTS OF LOGIC 
 
 experience to hearing might make the latter proposition ana- 
 lytic, and the former synthethic. 
 
 It will be seen, therefore, that the distinction is not only a 
 relative one, but is mainly of psychological importance. We 
 may, consequently, dismiss it from farther consideration. 
 
 5. MISCELLANEOUS PBOPOSITIONS. There is a species of prop- 
 ositions called Tautologous or Truistic. They are those which 
 affirm the subject of itself, and so may be regarded, in form at 
 least, as a kind of analytic judgment. They are such as " A 
 is A," "Whatever is, is," "Man is man," "A beast is a beast," 
 etc. Some of them, after all, are synthetic, and although the 
 predicate is the same word as the subject, it conveys a slightly, 
 or even wholly, different meaning ; as, for instance, " A man's 
 a man," " The king is king," etc. They are tautologous in form, 
 but instructive in matter. In reasoning we require to be on 
 the alert for such ambiguity. Otherwise the consideration of 
 truistic propositions has no logical importance. 
 
 There is another division of propositions into Pure and 
 Modal. " The pure proposition simply asserts that the predi- 
 cate does or does not belong to the subject, while the modal 
 proposition states this cum modo, or with an intimation of the 
 mode or manner in which the predicate belongs to the subject. 
 The presence of any adverb of time, place, manner, degree, 
 etc., or any expression equivalent to an adverb, confers mo- 
 dality on a proposition. ' Error is always in haste,' ' Justice 
 is ever equal,' 'A perfect man ought always to be conquering 
 himself,' are examples of modal propositions in this accepta- 
 tion of the name. Other logicians, however, have adopted a 
 different view, and treat modality as consisting in the degree 
 of certainty or probability with which a judgment is made and 
 asserted. Thus, we may say, ' An equilateral triangle is neces- 
 sarily equiangular,' ' Men are generally trustworthy,' ' A falling 
 barometer probably indicates a coming storm,' ' Aristotle's lost 
 treatises may possibly be recovered ; ' and all these assertions 
 are made with a different degree of certainty or modality." 
 But this does not affect the nature and relations of the copula, 
 and if we remain by the definition of the predicate, we shall
 
 PROPOSITIONS OR JUDGMENTS 115 
 
 find that modal articles and terms simply modify attributives, 
 verbal or adjectival, and no special significance should be at- 
 tached to them when they do not affect the quantity of the 
 proposition. 
 
 Some logicians distinguish propositions into True and False. 
 But this has to do with their matter as valid, and not their form, 
 as a mode of thinking, and as Logic is of formal laws it is not 
 concerned with the material truth or falsehood of propositions. 
 A system of pure and formal Logic, correctly illustrating the 
 laws of thought, could be constructed upon materially false 
 propositions as well as upon true ones. It is not a science of 
 truth in general ; but only of the formal laws of thought. It 
 is, therefore, not concerned whether propositions be true or 
 false. 
 
 3d. Ambiguity of Propositions. Judgments are rendered 
 ambiguous in three ways : First, by the ambiguous use of certain 
 terms ; second, by the inverted position of certain terms and 
 clauses ; and third, by the double meaning of certain .proposi- 
 tions even when there is no ambiguity in any of the terms com- 
 posing it. The first and the third of these influences affect 
 propositions in the same way, giving them a double import, 
 in which one of the implied propositions is the complement of 
 the other. They may be called Duplex propositions because 
 they are susceptible of analysis into two distinct judgments. 
 Those due to the second cause may be called Inverted propo- 
 sitions. 
 
 1. INVERTED PROPOSITIONS. These are of two kinds, accord- 
 ing as the inversion is of the subject and predicate, or of 
 some relative clause. In regard to the first, an example, such 
 as may frequently be found in poetry, is, " Full short his 
 journey was," or, " Great is Diana of the Ephesians." In such 
 cases the order of subject and predicate must be rein verted 
 before the proposition can be dealt with logically according o 
 the formal rules of conversion and reasoning. In regard to 
 the second class, the subject may sometimes be mistaken for 
 the predicate when it is described by a relative clause stand- 
 ing at the end of the sentence ; as, " No man is honest who
 
 116 ELEMENTS OF LOGIG 
 
 cheats his neighbor," or, " No one is fit for a king who cannot 
 rule himself." The real subjects in these propositions are, 
 " No one who cheats his neighbor," and " No one who cannot 
 rule himself," and unless we keep this fact in mind such in- 
 stances would give trouble in determining the Figure of a syllo- 
 gism, as will appear when that subject is to be discussed. 
 
 2. DUPLEX PBOPOSITIONS. A duplex proposition is one which 
 is capable of a double meaning and can be analyzed into two 
 distinct judgments. There are three kinds : Partitive, Ex- 
 clusive, and Exceptive. The chief characteristic of these prop- 
 ositions is, that the complementary proposition implied by them 
 is of the opposite quality of that which is asserted in the given 
 instance. This will be very important to keep in mind, be- 
 cause the process of reasoning will be affected by the question 
 whether one or the other of them is the real one in the thought 
 of the reasoner, as will be illustrated. 
 
 (a) Partitive Propositions. These express a part of a whole, 
 of which the implied proposition is a complementary part, 
 and are determined by the ambiguous use of the terms "All 
 not," "Some," and "Few." "All not" is often conceived as 
 the same as " Not all," and hence when the proposition seems 
 to be universal it is really particular. As an illustration we 
 have, " All metals are not denser than water," or, " All men are 
 not red-haired," where we may mean that " Not all metals are 
 denser than water," and " Not all men are red-haired." Strictly 
 construed the original propositions are E in form, but in mat- 
 ter they are either I or O, with the other of the two implied 
 when one of them is distinctly intended. When I say that 
 " Not all men are red-haired," or "All men are not red-haired," 
 in the sense of the former, I mean that " Some men are red- 
 haired," and that " Some men are not red-haired." Whichever 
 of the two I have in thought, the other is implied as its com- 
 plement. 
 
 Again, the term " some " is subject to a similar ambiguity, 
 denoting some but not all, and some at least, and it may be all- 
 Thus the proposition "Some metals are precious," especially 
 if, in speaking, the emphasis be upon the word " some," may
 
 PROPOSITIONS OR JUDGMENTS 117 
 
 mean that "Some metals are precious," and "Some metals are 
 not precious." This is when the term is equivalent to not all, 
 or only a part. In such instances it implies its complementary 
 opposite, so that it means I and O at the same time. If it be 
 I, it implies O ; if it be O, it implies I. The strict and proper 
 import of the term, however, when describing particular prop- 
 ositions is that in which it denotes " some, and there may or 
 may not be all." The importance of this will appear in 
 considering the matter of Opposition. But in actual reason- 
 ing we must be on the alert for the ambiguity to which the 
 term is incident, and be ready to detect the fallacy which it 
 may occasion. 
 
 A third proposition of a partitive and duplex nature is that 
 introduced by the term "few ; " as, " Few cities are as large as 
 Vienna," or, " Few men can be President," etc., in which we 
 mean that " Most cities are not as large as Vienna," and " Most 
 men cannot be President." Such propositions imply a com- 
 plementary opposite, because the term " few " denotes some, 
 but not all, or a few, but not all. The expression " a few " 
 taken alone does not imply any complementary conception, but 
 is equivalent to the unambiguous use of the word " some." 
 " A few," therefore, introduces a proposition which will be 
 either I or O alone, unless " A few not" be regarded as am- 
 biguous like " All not." But " few " introduces a proposition 
 which has the meaning of I and O together, as the illustrations 
 given very clearly prove. The confusion to which such proposi- 
 tions may give rise will be seen in those forms of reasoning 
 where the validity of the conclusion turns upon the question 
 whether they are to be interpreted as I or O. Thus the danger 
 can be illustrated : 
 
 All philosophers are mortal. 
 
 Few representative men are philosophers. 
 
 . . Few representative men are mortal. 
 
 Taking " few " in its duplex import, the conclusion would 
 mean that " Most representative men are not mortal," when 
 we know that they are all so. Hence we cannot treat the
 
 118 ELEMENTS OF LOGIC 
 
 minor premise as a simple unambiguous proposition, but must 
 interpret it as meaning both I and O, in which the conclusion 
 would be valid with I, where "few" is equivalent to " some " 
 (unambiguous), but vitiated with O, for reasons that will ap- 
 pear when discussing the doctrine of fallacies. 
 
 (b) Exclusive Propositions. They are introduced or have 
 their meaning determined by such particles as only, alone, and 
 none but. They, therefore, limit the predicate to the subject, 
 and are illustrated by such propositions as " Only Caucasians 
 are white," " Giants alone can be seven feet tall," " Elements 
 alone are metals," " None but honest men can be trusted," etc. 
 When I say that " only elements are metals," I do not neces- 
 sarily mean that " all elements are metals," but that the class 
 "metal" belongs exclusively to the class "element," and that 
 it can be affirmed of no other class. Hence the meaning of 
 the proposition is either "All metals ai-e elements," or, "All 
 compounds (not-elements) are not metals." The first of these 
 is what is called the simple converse, and the second may be 
 called the complementary opposite of the exclusive proposition 
 in question. It is with one of these, or the conception ex- 
 pressed by one of them, with which we have to deal in reason- 
 ing, or in testing any case of reasoning involving an exclusive 
 proposition. Thus, " Only Caucasians are white " must be re- 
 duced either to " All white men are Caucasians," or to " Those 
 who are not Caucasians are not white," when testing the formal 
 process of the syllogism. The error to which we are liable in 
 using them without considering their duplex meaning is illus- 
 trated in the following argument : 
 
 Only elements are metals, 
 Oxygen is an element. 
 .'. Oxygen is a metal. 
 
 Now we know that oxygen along with a number of other sub- 
 stances is not a metal. Or again, in a better case : 
 
 Only men are allowed to vote. 
 
 Criminals are men. 
 
 .. Criminals are allowed to vote.
 
 PROPOSITIONS OR JUDGMENTS 119 
 
 But we know that criminals are not allowed to vote, and hence 
 to test the character of the reasoning we must use either the 
 converse or the complementary of the proposition in the major 
 premise. Thus if I say : 
 
 All who are allowed to vote are men. 
 Criminals are men, 
 
 it will be seen that I can draw no conclusions, for reasons to 
 be noted when we study the syllogism ; and so with the com- 
 plementary proposition ; as, 
 
 Those who are not men are not allowed to vote. 
 Criminals are men. 
 
 There are cases also where there might appear to be a formal 
 error in the reasoning, but which is perfectly correct when we 
 consider either the converse or the complementary of the ex- 
 clusive proposition expressed. Thus : 
 
 Only elements are metals. 
 
 Gold is a metal. 
 
 .. Gold is an element. 
 
 This is correct, but the reason for it will appear again. 
 
 It is important to say a few words about the quantity of ex- 
 clusive propositions, because the question may be asked 
 whether they are -universal or particular. The answer cannot 
 be made without an explanation of this peculiarity. Exclusive 
 propositions are of two forms. An ilhistration of the first form 
 is, " Only citizens can vote," and of the second form, " Only some 
 men are wise." The first may seem to be universal, and the sec- 
 ond particular. The second, which is a proposition in I, is a 
 particular proposition, and differs from the ordinary instance 
 only in implying the complementary opposite, O. The first 
 form is the most frequent, and has the peculiarity that the 
 terms "only," "alone," and "none but," have the effect of dis- 
 tributing the predicate while the subject is left undistributed. 
 The meaning of distribution will have to be ascertained in its 
 proper place, but we may say regarding it at present, that it
 
 120 ELEMENTS OF LOGIC 
 
 denotes that the whole extension of the concept distributed is 
 taken into account. When undistributed, a part of that ex- 
 tension is taken into account, and it may or may not be that 
 all of it is considered. This makes the exclusive proposition, 
 as it stands in the first form, a sort of inverted universal. 
 But it is neither a universal nor a particular in that form. 
 Hence we can determine its quantity only by taking either 
 its simple converse, or its complementary, and in either of 
 these cases the result is a universal. Consequently, the first 
 form of the exclusive proposition must be treated in terms of 
 what it implies, namely, a universal proposition. 
 
 The second form is the ordinary particular proposition and 
 must be treated accordingly, except that the signs " only," 
 " alone," etc., do not distribute the predicate. They have only 
 the effect of implying the complementary opposite proposi- 
 tion O. 
 
 One exception, however, must be made in regard to these 
 observations. It relates to the negative instances of ex- 
 clusive propositions, such as, "Only bad men are not wise." 
 They have a complementary opposite but no converse, while 
 the affirmative cases have both. Negative exclusive propo- 
 sitions are in reality particular propositions in O. The ex- 
 clusive particle ~does not distribute the subject and being 
 negative the predicate is distributed. Consequently they can- 
 not be converted without violating the rule that no term shall 
 be distributed in the converse which is not distributed in the 
 convertend. In the reduction of reasoning containing them, 
 therefore, we can use only the complementary opposite, and 
 not the converse as our substitute. 
 
 (c) Exceptive Propositions. These are such as are introduced 
 or modified by the terms All but, All except, All save, etc. 
 For example, "All except those under twenty-one years of 
 age are citizens," " All the planets, except Venus and Mercury, 
 are beyond the earth's orbit." Such propositions appear to be 
 universal, and simple at the same time. But they really con- 
 sist of two particular propositions ; namely, I and O ; as, 
 " Some men are citizens," and " Some men are not citizens," or,
 
 PROPOSITIONS OR JUDGMENTS 
 
 121 
 
 " Some planets are beyond the earth's orbit," and " Some planets 
 are not beyond the earth's orbit." If the class " men " or the 
 class " planets " were divided into two species with a name ac- 
 cording to t'ue two portions indicated by the nature of the 
 subject, exceptive propositions might be resolved into two 
 universals, A and E, instead of two particulars, I and O. Thus, 
 " All who are twenty-one years of age and over are citizens," 
 and " All who are below twenty-one are not citizens." But in 
 either case we have an illustration of the complementary nature 
 of the two propositions developed from the duplexity of an ex- 
 ceptive judgment. This class, however, is not so important in 
 Logic as the two previous classes, because fallacies are less 
 frequently incident to the use of them. We require only to 
 observe the peculiar nature of the conception involved in such 
 judgments, and to be on the alert for any disturbing influence 
 it is likely to exercise.* 
 
 The following outline is a resume of the chapter : 
 
 Definitions ( Indicative. 
 
 Grammatical < Interrogative. 
 ( Imperative. 
 
 ( Categorical. 
 
 Grammatico-Logical < Conditional. 
 ( Disjunctive. 
 
 ^ to ,^ lre | \jsj&.=i 
 
 T> v. i ~:~ i ! Analytic, 
 irsyctioloilical < xii j.- 
 I Synthetic. 
 
 ( Truistic. 
 
 Miscellaneous 4 Pure. 
 ( Modal. 
 
 ( Inverted. 
 
 Ambiguous propositions < ( Partitive. 
 
 ( Duplex < Exclusive. 
 ( Exceptive. 
 
 Propositions 
 
 Divisions 
 
 * General references on Propositions are the following: Mill: Logic, 
 Bk., I., Chaps., IV., V., and VI ; also Examination of the Philosophy of 
 Sir William Hamilton, Chap. XVIII ; Hamilton : Lectures on Logic, 
 Lect XIII. ; Venn : Empirical Logic, Chaps. IX. and X. ; De Morgan : 
 Formal Logic, Chap. IV. ; Wundt : Logik, Dritter Abschnitt, Cap. 
 II. ; Keynes : Formal Logic, Part II., Chap. I.
 
 CHAPTEK VHL 
 
 THE RELATION BETWEEN SUBJECT AND PREDICATE 
 
 1st. Nature of the Relation Between Subject and Pred- 
 icate. We have briefly alluded to the relation between sub- 
 iect and predicate without explaining it. This was in the case 
 of drawing a distinction between the judgments "Man is a 
 biped " and " Man is wise." But now this relation must be ex- 
 amined more carefully and illustrated by symbolic diagrams 
 which will be convenient for testing visibly certain forms of 
 reasoning and inference. 
 
 There has been much dispute about the nature of this rela- 
 tion. One set of logicians has claimed, and the other has 
 denied, that the relation can be expressed in terms of quantity ; 
 in other words, that the subject and predicate are considered 
 as expressing merely a relation of quantity. Thus the judg- 
 ment " Men are bipeds " is supposed to have its meaning indi- 
 cated in the notion that the subject expresses a number of 
 individuals not greater and possibly less than the number 
 indicated by the predicate. This conception refers us to the 
 matter already discussed ; namely, that of extension. In for- 
 mal Logic it has been customary to deal with all propositions 
 as if we had to take no other relation into account, and owing 
 to the peculiarly different character of such propositions as 
 "Man is wise" the correctness and accuracy of the general 
 practice has been impeached by some writers, and more par- 
 ticularly the mode of representing the relation by geometrical 
 figures. It is the merits of this question which we wish to ex- 
 amine with some care. 
 
 The student must remember what has been said about the ex- 
 tension of terms and the laws regarding the relation between
 
 RELATION BETWEEN SUBJECT AND PREDICATE 123 
 
 intension and extension. In the present problem it has been 
 customary to deal only with their relations in extension and 
 to choose geometrical figures to represent them in order to 
 show some of the characteristics affecting the process of rea- 
 soning. If propositions express a relation of quantity, of 
 equality, of more or less, between terms it would be natural to 
 represent them mathematically. But as the proposition "Man 
 is wise " does not seem to indicate a relation of quantity or ex- 
 tension between subject and predicate, but a relation of attri- 
 bute to its substance ; and as judgments like " Man is a biped '' 
 imply the connection of attribute and substance, whatever else 
 is thought of, it has been maintained that symbols of quantity 
 representing the relation are misleading. This is the problem 
 to be considered. 
 
 In examining the relation between subject and predicate, we 
 may adopt a division of propositions or judgments which has 
 not been mentioned, but which is based upon the distinction 
 between the extension, and the intension of concepts. Accord- 
 ingly all judgments may be divided into judgments of exten- 
 sion, and judgments of intension. An example of the former 
 is, " Man is a biped," or " Horses are animals ; " of the latter 
 examples are, " Man is wise," and " Trees are tall." A good 
 way to represent the difference is found in the following prop- 
 ositions, the first extensive and the second intensive : " Man 
 is a mortal," and "Man is mortal." The great difference be- 
 tween them is remarked in the nature of the predicate and its 
 relation to the subject. In all propositions the predicate is 
 either substantive or attributive. In judgments of extension 
 the predicate is substantive ; in judgments of intension it is 
 attributive. As we have already explained, such propositions 
 as "John struck James/' or " The king rules his subjects," the 
 verb expresses a function or attribute of the subject, or an 
 attribute that is attributive, and hence the presence of a sub- 
 stantive object does not affect the attributive nature of the 
 predicate, or the intensive nature of the judgment. But with 
 this uniform relation of a substantive predicate to extensive 
 and of an attributive predicate to intensive judgments, we may
 
 124 ELEMENTS OF LOGIC 
 
 remark another important distinction which is its corollary. 
 In the extensive proposition the subject is contained or compre- 
 hended in the predicate or excluded from it ; in the intensive 
 proposition the predicate is contained or comprehended in the 
 subject or excluded from it. The mode of comprehension, how- 
 ever, is distinct in each case. In the former it may be called 
 that of inclusion or exclusion ; in the latter that of inhesion or 
 non-inhesion. By inclusion of the subject in the predicate I 
 mean that it is contained as a species in the genus, or as one 
 class in another class term of equal or greater extension. The 
 exclusive proposition is no exception to this, because we found 
 that its logical meaning was the converse of its grammatical 
 form. Hence the predicate of the extensive judgment is a 
 class concept. By the inhesion of the predicate in the subject 
 I mean that it is contained, or inheres, in the subject as an 
 attribute in a substance, or rather expresses that relation. In 
 the extensive judgment, therefore, the number of individuals 
 denoted by the subject can never be greater than the number 
 denoted by the predicate. This establishes a relation of quan- 
 tity between them, and hence they may be called quantitative 
 judgments. The relation being quantitative can be represented 
 in some mathematical way, if not to indicate the nature of it, 
 certainly to indicate an accident quite uniform with the essen- 
 tial qualities and proportionally variable with them. In the 
 intensive judgment the predicate is an attribute or quality of 
 the subject, and is comprehended in it rather than the re- 
 verse, as in the extensive proposition. Intensive may, there- 
 fore; be called qualitative judgments, because the connec- 
 tion between subject and predicate is a relation of quality. It 
 is, therefore, a question whether they can be represented or 
 symbolized by any figures expressing relations of quantity. 
 
 The answer to this question will be found in the fact to be 
 shown that both kinds of judgment can be conceived in both 
 a quantitative and a qualitative form at the same time, and 
 qualitative in a double sense. If this can be proved, the sym- 
 bolization of one will be that of the other also. We proceed, 
 therefore, to the examination of this question.
 
 RELATION BETWEEN SUBJECT AND PREDICATE 125 
 
 First, in extensive judgments, although the predicate is a 
 substantive and class concept, it connotes certain attributes 
 which belong to the subject in the same way as in the inten- 
 sive judgment. Thus, when I say "Man is a biped," I not 
 only mean that the class of individuals or species " man " is 
 included in the equal or larger class " biped," but I also mean 
 that " man " is characterized by the quality of two-footedness, 
 common to the whole class of bipeds. I therefore affirm or 
 imply this attribute of him, and the extensive judgment be- 
 comes at the same time an intensive one. 
 
 Second, in intensive judgments, although the predicate is 
 an attribute term, it is generally assumed that it cannot stand 
 alone in thought, but must qualify some substantive. Thus, 
 when I say " Man is wise," I not only mean that wisdom is an 
 attribute of the subject, but I equally mean that he is a wise 
 something. I do not mean that " Man is a wise man," for this 
 is tautological. But I mean that he is a " wise creature," in 
 which case I have a substantive predicate, as implied in the 
 simple intensive form, and the intensive judgment becomes an 
 extensive one at the same time. It is true that we do not or- 
 dinarily, perhaps never, think of this class relation in such a 
 judgment as " Man is wise," but the fact that the extensive 
 conception of it coincides and is perfectly compatible with 
 the intensive conception, is sufficient to give it that double 
 logical construction, as in the case of the extensive judgments. 
 This is especially the fact when we reflect that in the extensive 
 judgments we may not ordinarily represent to our thought 
 the attributive relation between subject and predicate any 
 more than in the intensive judgment we represent the class 
 relation, and yet no one questions that the attributive relation 
 is implied in the extensive judgment. The extensive relation 
 is equally involved, or implied in the intensive judgment, al- 
 though it may not be thought of. At any rate, it is possible 
 to represent it so, in perfect compatibility with the attribu- 
 tive relation more particularly expressed by it. This will ena- 
 ble us to represent the relation quantitatively or mathemati- 
 cally as in the extensive judgment.
 
 126 ELEMENTS OF LOGIC 
 
 Another fact sustains the same conclusions. When I say 
 " Man is wise," there is nothing in the nature of this form of 
 statement to prevent my affirming wisdom of other beings as 
 well. Take the proposition " Trees are tall," and we may also 
 say, "Houses are tall" "Tallness" is not exclusively an attri- 
 bute of " trees," and hence " trees " belong to a larger class of 
 objects having the same quality, "tallness," as "man" belongs 
 to a larger class of beings having the quality " two-footed- 
 ness," expressed in the proposition " Man is a biped." Hence, 
 so far as form of statement is concerned, the judgment " Man 
 is wise," may admit that other beings are " wise " as well, and 
 as long as this is the case, formally, the judgment is extensive 
 as well as intensive. 
 
 It may be important to consider the relation between the 
 judgment " Man is a wise creature " and a definition. I have said 
 that the predicate " wise," in the simple intensive proposition, 
 may be affirmed of other individuals and species than the sub- 
 ject, and that this constitutes a significant resemblance to exten- 
 sive propositions, because their fundamental characteristic is 
 precisely this fact, that it is affirmable of other individuals be- 
 sides the given subject. But this is not the case with a defini- 
 tion. In a definition the subject and predicate are identical in 
 extension, and convertible with each other. Thus, if I define 
 man to be a rational animal, I can as well say that " Rational 
 animals are men." This is merely because I regard the property 
 " rational " as belonging exclusively to man, and so make it the 
 differentia, while the word " animal '' refers to the conferentia. It 
 is, therefore, the total predicate which is identical with the sub- 
 ject, while the generic term, with its conferentia, indicates an 
 extensive relation numerically greater than the subject, and it 
 is only the differentia that can make the total equal to the 
 subject. Now it is to be remarked that the conversion of a 
 simple intensive judgment, such as " Man is wise," into its 
 corresponding extensive judgment, such as " Man is a wise 
 creature," looks very much like a definition, and hence the 
 wisdom might not be predicable of anything else than the 
 subject. It might be argued that this is the possible case
 
 RELATION BETWEEN SUBJECT AND PREDICATE 127 
 
 with all intensive judgments, and if so their resemblance to ex- 
 tensive judgments is modified. But the reply to this is very 
 clear. 
 
 In the first place, extensive judgments do not require that 
 the predicate be greater in extension than the subject. It may 
 be equal to it and supply all the conditions necessary. A defi- 
 nition, therefore, may be an extensive judgment, and can always 
 be treated so. Farther, it is important to remark that, formally, 
 a definition has to be ti'eated as all other propositions, and it 
 is only materially, that is, when we consider the modifying 
 attributive as a differentia, that we can treat its predicate as 
 convertible with the subject. In the second place, the attri- 
 bute " wise " in this particular judgment, and the qualifying 
 term in any otheir proposition, may indicate either the confer- 
 entia or the accidentia ; and in either case involve a predicate of 
 broader extension than the subject, when a substantive is modi- 
 fied by them. It is only when the modifier expresses the differ- 
 entia that the predicate can ever be equal in extension to the 
 subject, assuming it to qualify a substantive. Hence all judg- 
 ments which are not definitions represent predicates of greater 
 extension than the subject, and as even definitions cannot be 
 formally distinguished from them as such, they must be treated 
 logically in the same way, although when materially known to 
 be definitions we may consider their extension as reduced in 
 reality to that of the subject. But even this does not prevent 
 them, as we have shown, from being extensive judgments, and 
 therefore, whenever an intensive proposition is assumed to 
 imply a substantive element in the predicate, it possesses 
 quantitative properties identical in character with those of 
 the so - called extensive judgment and may be represented 
 accordingly. The figured symbols representing the quanti- 
 ty, or relations of quantity in extension between subject and 
 predicate, shall be illustrated presently, and as soon as another 
 interesting feature of both kinds of judgment has I/sen 
 considered. 
 
 What we have considered up to this point in tiie two forms 
 of proposition is quantity of extension, expressed or implied.
 
 128 ELEMENTS OF LOGIC 
 
 "What we have still to consider is quantity of intension ; for this 
 is as marked a characteristic of judgments as any other prop- 
 erty, although it is not usual to regard it in symbolic Logic. 
 But take the simple intensive judgment, " Man is wise." We 
 say the predicate is here contained in the subject, or denotes 
 a property belonging to it. Not only is it to be remarked 
 that the same predicate may belong to other subjects also, but 
 this one is not the only property of the subject. As in the ex- 
 tensive proposition the subject can never represent a greater 
 number of individuals than the predicate, excepting in nega- 
 tive propositions as explained, so in the intensive proposition 
 the predicate can never represent a greater number of attri- 
 butes than the subject, except in negative propositions, as be- 
 fore. The quantity of intension, therefore, of the subject in 
 intensive judgments must be equal to, or greater than, that of 
 the predicate. Consequently the mode of symbolizing the 
 mathematical relation would be the reverse of that in the ex- 
 tensive proposition ; that is, with the same figures, but with a 
 reversed position for the signs of the subject and predicate. 
 Now, as we have shown that the extensive proposition has its 
 intensive interpretation ; thus, " Man is a biped " equals " Man 
 is two-footed," denoting qualitatively what is implied by the 
 extensive form, the quantity of intension between subject and 
 predicate is the reverse of the quantity of extension between 
 the same terms, and hence the mode of representing it sym- 
 bolically will be the reverse again. This we proceed now to 
 illustrate in full. But we must first explain how it is done, and 
 shall then represent the quantity of extension, as if it applied 
 only to extensive judgments. Afterward we can extend the 
 principle to intensive propositions. 
 
 We should, perhaps, remark a connection between subject 
 and predicate which is in some cases different from the two 
 we have discussed, and which wquld be considered preferable 
 to them by certain schools of thought known as Empiricists 
 and Positivists, or such as oppose all Metaphysics. Instead 
 of supposing that subject and predicate expressed real objects, 
 or things in which all predicates affirmed inhere as qualities or
 
 RELATION BETWEEN SUBJECT AND PREDICATE 129 
 
 attributes, they would say that judgments expressed the connec- 
 tion of coexistence or sequence between subject and predicate. 
 This view would get rid of the necessity of supposing the 
 subject to always express a substantive, and the predicate a 
 substantive or attributive conception, the latter distinctly indi- 
 cating the inhesion of a quality in a substance, and the former 
 implying it while affirming a class-whole to which the subject 
 belonged as a numerical part. In this way they would not 
 mean to imply any necessarily metaphysical connection, but 
 only one of coexistence or sequence between the two terms. 
 Thus in the proposition " Man wears clothing," or " Man is 
 a clothed being," we mean, they would say, that the essential 
 qualities or characteristics of man are accompanied by the ac- 
 cidental one of wearing clothing, not that a being has this as 
 an attribute. This is only to say that certain facts or phe- 
 nomena, say bipedality, bimanousness, rationality, etc., are ac- 
 companied by the other quality of being clothed. The same 
 thing, perhaps, could be said of any proposition, such as 
 " Gold is yellow," " Iron is hard," when we suppose that 
 " gold," and " iron " are names for certain qualities, among 
 which " yellow " may be found in one case and " hardness " in 
 the other. 
 
 It is true that many propositions seem to express, or to be 
 resolvable into, this kind of connection. But it is not opposed 
 to the kinds of connection we have previously investigated, 
 and may be said to be always coincident with them. Hence, 
 while we admit that such a connection is most apparent in 
 many cases, it does not exclude the idea that the relation be- 
 tween subject and predicate is that of a subject and attribute, 
 in which we include, but may not expressly think of, the rela- 
 tion of mere concomitance or non-concomitance. Besides, in 
 extensive judgments this relation of mere connection by co- 
 incidence or sequence is not easy to imagine, unless we resolve 
 the predicate into its attributive meaning. But it is just as 
 easy to conceive, and more suitable to the traditional forms of 
 logical discussion to admit, that we think of other relations 
 than of mere coincidence and sequence. Judgments of exten-
 
 130 ELEMENTS OF LOGIC 
 
 sion mean to indicate, when conceived attributively, as they 
 in reality always are, the conferential qualities of the subject, 
 and to ignore the differential. But judgments of intension 
 intend to express the inhesion of a quality in the subject with- 
 out distinction of essential or accidental. These connections 
 are simultaneous with that of mere coincidence or sequence, 
 and hence no important end is served by a controversy about 
 the question. 
 
 In order to represent the relation between subject and 
 predicate, mathematically, Euler chose circles whose area could 
 correspond to the suppositions already made about the equal 
 or greater extension of the predicate as compared with the 
 subject. As we have said, we shall first limit the symbolic 
 representation to judgments of extension, because there can 
 be no question about their quantitative nature and their math- 
 ematical representation accordingly. All the figures we shall 
 employ may apply to the extensive proposition, " Metals are 
 substances." We shall use the letter S to denote the subject, 
 and the letter P to denote the predicate.* Hence Swill stand 
 for " metals " and P for " substances." Now as the extension of 
 " metals " cannot be greater than that of " substances," the 
 area of the circle representing it must not be greater but may 
 be smaller. Hence proposition A may be represented in the 
 following manner : 
 
 PIG. 4. FIG. 5. 
 
 Proposition I, which would be " Some metals are sub- 
 
 * We mast not confuse this with the later use of the same symbols to 
 indicate the minw and major terms of the syllogism. They are respect- 
 ively the subject and predicate of the conclusion, but are not always 
 such in the premises.
 
 RELATION BETWEEN SUBJECT AND PREDICATE 131 
 
 stances," will be represented in the following manner, to be ex- 
 plained again : 
 
 FIG. 6. 
 
 FIG. 7. 
 
 FIG. 8. 
 
 FIG. 9. 
 
 Proposition E, " No metals are substances," would be repre- 
 sented in only one form, as follows : 
 
 FIG. 10. 
 
 Proposition O, which would be " Some metals are not sub- 
 stances," would require three distinct figures for its symboli- 
 zation, as follows : 
 
 FIG. 11. 
 
 FIG. 12. 
 
 FIG. 13. 
 
 In Fig. 4 the two circles are supposed to coincide and make 
 one, representing an equal extension between subject and
 
 132 ELEMENTS OF LOGIC 
 
 predicate. This, as remarked, is true of definitions, and 
 might be true of all other propositions in A, so far we knoiv 
 positively from the assertion. But the extension of the predi- 
 cate may be greater, as we happen to know it is in many cases, 
 and hence for that conception Fig. 5 has to be employed. It 
 means that the area or number of individuals denoted by 
 "metals" is less than that denoted by "substances," and it 
 may always be so in A propositions, so far as we can determine 
 from the form alone. 
 
 In proposition I, represented by Figs. 6, 7, 8, and 9, it is in- 
 teresting to remark that two of them are identical with. Figs. 
 
 4 and 5, and a third differs from Fig. 5 only in the position 
 of the letters S and P. But this resemblance is due to the 
 character of particular propositions. We have remarked that 
 the sign " some " properly denotes in Logic a part, and it may 
 or may not be all, and hence Figs. 6 and 7 very clearly indicate 
 this possibility. For if " All S is P," it is evident that " Some S 
 is P," although the former does not follow from the latter. But 
 the conditions of a particular proposition being what they are, 
 it is possible, so far as the statement is concerned, that " All S 
 is P," when " Some S is P." Figs. 6 and 7 provide for this 
 possibility. In Fig. 8 some portions of S and P are excluded 
 from each other, and hence when it is compared with Fig. 11, 
 it is found to represent both I and O, and hence it might be 
 taken to symbolize the duplex proposition, where " some " 
 implies its complementary opposite. But it does symbolize I, 
 whether we regard it as an ambiguous representation or not. 
 Fig. 9 has no ambiguity about it, if we regard carefully the 
 relation expressed by the position of the letters S and P. 
 But it has the fault of not admitting the possibility that " All 
 
 5 is P," at the same time, which is one contingency in proposi- 
 tion I. So far as we know from proposition I, proposition A 
 is also true, and Fig. 9 does not indicate this. 
 
 We must remark, before passing to the negative proposi- 
 tions, that affirmative judgments must express inclusion. A 
 must express total, and I partial inclusion at least. But when 
 we come to negative propositions the relation expressed must
 
 RELATION BETWEEN SUBJECT AND PREDICATE 133 
 
 be one of exclusion : E must express total, and O partial ex- 
 clusion at least. Hence in Fig. 10 we have the only pos- 
 sible symbol of E. The two excluding circles denote that 
 subject and predicate are not connected in a given respect, 
 and hence no part of the extension of one can be included in 
 that of the other. 
 
 In proposition O the exclusion must be at least partial. 
 Fig. 11 represents it, and is like Fig. 8 for I, although differ- 
 ent arcs or portions of the circles must be chosen to represent 
 the exclusion, as compared with the arcs in Fig. 8 to represent 
 inclusion. Fig. 11, however, does not indicate the possibility 
 that E may be true, which is the case, so far as we know from 
 the proposition. Hence for the same reason that Fig. 6 may 
 represent I, although also the symbol of A, Fig. 13 may rep- 
 resent O, although the symbol of E. Fig. 12 explains itself 
 as indicating that some of the circle S is not included in the 
 circle P ; but it is defective in not admitting the possibility of 
 E at the same time. What is desirable in all this symboliza- 
 tion is that the figures shall properly represent the differences 
 between A, E, I, and O, and at the same time represent the 
 possibility that A is true when I is, and that E is true when O 
 is. We desire also, at the same time, to represent the equal 
 possibility that A shall not be true when I is, and E when O 
 is. In other words, we require a symbol which will represent 
 our entire ignorance as to whether there is total or only par- 
 tial inclusion implied when I is affirmed, or whether there is 
 total or only partial exclusion when O is affirmed. If this be 
 possible the number of figures or symbols might be reduced. 
 As it is at present we escape confusion only by carefully 
 observing the relation between the various positions of S 
 and P. 
 
 Ueberweg has a representation which will simplify matters 
 very much. He reduces them all to four figures, representing 
 respectively the propositions A, E, I, and O. He employs a 
 system of dotted lines in order to express the various possi- 
 bilities involved in the expressed relation of subject and pred- 
 icate, but neither affirmed nor denied by the form of the judg-
 
 134 
 
 ELEMENTS OF LOG-IO 
 
 ment. Thus in proposition A, so far as we know, the exten- 
 sion of the predicate may be equal to or greater than that of 
 the subject. The form of the proposition does not say which 
 it is. We only know that it cannot be less. Fig. 4 implies 
 that it is equal when it may be greater, and Fig. 5 implies 
 that it is greater when it may be equal to that of the subject. 
 Hence not only the incompleteness of the symbol in each case, 
 but also the liability to confusion with those for other propo- 
 sitions. If, therefore, we can find symbols quite distinct from 
 each other, and yet expressing all the possibilities of the propo- 
 sitions, we can greatly simplify the problem. This Ueberweg 
 has done in the following manner, Fig. 14 standing for propo* 
 sition A, Fig. 15 for E, Fig. 16 for I, and Fig. 17 for O : 
 
 Fia. 14. 
 
 Fio. 15. 
 
 Fia. 16. 
 
 Fio. IT. 
 
 In Fig. 14, which we see is quite distinct in form from all 
 others, the dotted line means that so far as we can tell from 
 the proposition A, " All S is P," the extension of P may be 
 either equal to or greater than S. Fig. 15 leaves no room for 
 doubt. It always expresses total exclusion and nothing else. 
 Fig. 16 makes it unknown whether there be a difference of 
 extension between subject and predicate, and yet allows the 
 possibility of A being true when I is true. That is, it is stated 
 and indicated by the undotted lines that " Some S is P," while 
 it may also be true that " All S is P," as is apparent, if S be 
 exhausted within the area of the undotted lines. It is equally
 
 RELATION BETWEEN SUBJECT AND PREDICATE 135 
 
 possible that O be true if S includes the area of the dotted 
 lines. The same is true of the extension of P. It must be 
 equal to or greater than S, if proposition A be possible, and 
 must express some exclusion if O be possible. The doubt in 
 both cases is expressed by the dotted line. 
 
 Fig. 17 is much more complicated. We have to express 
 the partial exclusion of S and P, the possibility of their total 
 exclusion ; that is, proposition E ; the possibility of their par- 
 tial inclusion ; that is, proposition I ; and the possibility of 
 the total inclusion of P in S, as the larger dotted circle im- 
 plies, and all these at the same time. A little observation will 
 show that this has been done. Thus if S contain only what is 
 represented by the undotted curved line and the straight dot- 
 ted line, O is symbolized and E is also possible. But if S con- 
 tains the whole circle represented by the dotted and undotted 
 arcs, O is true and I is possible. On the other hand, suppos- 
 ing that S has or can have a larger extension than P, the 
 larger dotted circle represents again the possibility of both O 
 and I. Taken altogether these four symbols are the only com- 
 plete ones, and are the only representations which are not lia- 
 ble to confusion with each other. 
 
 An important observation in the quantitative relations of 
 subject and predicate is, that in negative propositions, E and 
 O, no comparison of equal, greater, or less can really be made. 
 When subject and predicate exclude each other, we do not in- 
 dicate anything about the relative extension or number of in- 
 dividuals denoted by them, and hence in negative proposi- 
 tions relations of extension are not commensurable. As a 
 consequence of this, negative propositions either compare spe- 
 cies which always exclude each other, or compare genus and 
 species in the inverted order of universal affirmative judgments. 
 E compares co-ordinate species, which are represented by 
 total exclusion because of their distinct differentia. O either 
 compares co-ordinate species and admits the possibility of E, 
 or total exclusion in regard to the same species, or it com- 
 pares a genus with a species, taking the genus for the subject 
 and the species for the predicate. Thus, under the genus
 
 136 ELEMENTS OF LOGIC 
 
 vertebrate, we can say, "No men are horses," = E, or "Some 
 men are not horses " = O, with the possibility of E ; or again, 
 "Some vertebrates are not men" = O. But in the last case 
 it is not implied that the extension of " vertebrates " can pos- 
 sibly be greater than that of the predicate, " men." Hence, so 
 far as the form of judgment is concerned, no definite com- 
 parison in the quantity of extension can be made between sub- 
 ject and predicate in negative judgments. 
 
 It is otherwise in affirmative propositions, because they 
 express a relation of inclusion, and hence always a relation 
 between genus and species, and never between species and 
 species ; as, " All men are vertebrates " and " Some men are 
 negroes." 
 
 Thus far we have symbolized only extensive judgments, and 
 it remains to see whether a similar representation can be em- 
 ployed for judgments of intension. It is a very simple matter 
 to solve this problem. "We have only to recall the previous 
 reduction of intensive propositions to the extensive form, in- 
 volving quantitative relations as well as qualitative between 
 subject and predicate, in order to see that the symbolization 
 we have already employed will apply equally to intensive judg- 
 ments. Hence the same figures will represent the quantitative 
 relations of extension in such propositions as " Men are wise " 
 and "Man is mortal," as in "Men are bipeds," or "Trees are 
 vegetables." At any rate, such a representation is very con- 
 venient for testing certain forms of reasoning, and is not in- 
 compatible with the relation of intension expressed by such 
 judgments. But assuming that all judgments are both quan- 
 titative and qualitative, we may represent the quantitative 
 without interfering with the qualitative relation, and this is all 
 that the symbols of Euler are intended to express. They com- 
 pare subject and predicate only in the quantity of their exten- 
 sion, marking either their inclusion or exclusion. 
 
 But when we come to compare subject and predicate in re- 
 spect of their quantity of intension, the matter is somewhat 
 different. Here we mark the relation of inhesion or non-in- 
 hesion, and it is a question whether we can symbolize it in any
 
 RELATION BETWEEN SUBJECT AND PREDICATE 137 
 
 mathematical manner or not. But since we have established 
 the fact that the quantity of extension in the relation between 
 subject and predicate is the direct reverse in the relation ex- 
 pressed by the quantity of intension, we may simply reverse 
 the positions of the letter S and P in order to symbolize the 
 mathematical relation in the intensive judgment as such. In 
 the proposition "Man is wise," we have seen that the predicate 
 can never be greater in intension, or quantity of intension, 
 than the subject, and that it is contained in the subject. This 
 quantitative relation is very clearly represented by Figs. 4 and 
 5, only in Fig. 5 the letter S will be placed in the larger, and 
 the letter P in the smaller circle, as in Fig. 9. As Fig. 5 stands 
 S is contained in P, as representing quantity of extension, or 
 the comprehension of the subject in the predicate. But as the 
 relation must be reversed for the quantity of intension, or the 
 comprehension of the predicate in the subject, P must be in- 
 cluded in S. This is the case for the proposition A, which 
 Figs. 4 and 5 represent. For the quantity of intension in 
 propositions E, I, and O, the same figures will serve as for the 
 quantity of extension, only we must, as in proposition A, re- 
 verse the positions of the letters S and P. Of course, in Figs. 
 4, 6, 8, 10, 11, and 13, this reversal of S and P is not neces- 
 sary. We require only to keep in mind the reversed order of 
 inclusion as compared with extensive judgments. It is to be 
 remarked, however, that in negative propositions the quantity 
 of intension is no more determinate than the quantity of ex- 
 tension, relatively considered, as .the same principles are ap- 
 plicable here as there. In the Ueberweg scheme we require 
 to reverse the position of the letters S and P only in Fig. 14, 
 and understand the reversed relation in the others, although 
 in all cases no harm will be done by actually changing them in 
 order to mark the contrast between quantity of intension and 
 quantity of extension. 
 
 It will be evident that the scheme for the quantity of inten- 
 sion will apply also to judgments of extension, so far as they 
 can be reduced, as we have shown, to judgments of intension. 
 " All men are bipeds," conceived as meaning " All men are
 
 138 ELEMENTS OF LOGIC 
 
 two-footed," can be represented in the same way as " All men 
 are wise." Consequently we not only have the same symboli- 
 zation for the two kinds of judgment, but a double system 
 according as each judgment is considered in the quantity of 
 its extension or the quantity of its intension. But it will not 
 be necessary for practical purposes to consider more than one 
 of them. The scheme for the quantity of extension is the one 
 usually employed, and as it avails to represent and test all 
 practical cases of reasoning, and the relations between terms, 
 we shall confine our method to it alone. It is the symboliza- 
 tion for the quantity of extension that is used to explain the 
 distribution of terms, which is the next topic for considera- 
 tion. 
 
 2d. The Distribution of Subject and Predicate. By 
 the distribution of a term we mean that something is said 
 about the whole of what it contains ; that is, about the whole 
 of its extension. Thus in the proposition " All men are mor- 
 tal " we state something about the whole class of men, and 
 hence the subject in this proposition is said to be distributed. 
 An undistributed term, therefore, is one in which we do not 
 say something about the whole class denoted by the term. 
 Thus in proposition I, " Some men are negroes," we do not as- 
 sert something of the whole class of men, and hence the sub- 
 ject is said to be undistributed. This appears very clearly in 
 the symbolization we have adopted. For instance, in Fig. 5 
 the circle S represents that something is said about the whole 
 of the class it denotes, and" se in any other figure symbolizing 
 a universal proposition, while the undistributed character of 
 the subject in particular propositions is equally evident when 
 allowance is made for their proper implications. But no am- 
 biguity in regard to this matter will be noticed in Figs. 14, 15, 
 16, and 17. 
 
 In regard to the predicate, it ma}' not be so easy to deter- 
 mine its degree of distribution from the figures without care- 
 ful explanation. But this, perhaps, may make it clear. In 
 proposition A, "All metals are elements," we perceive without 
 difficulty that something is said about the whole of the sub-
 
 RELATION BETWEEN SUBJECT AND PREDICATE 139 
 
 ject, and that it is therefore distributed. But nothing is said 
 or implied about the whole of the predicate, except as repre- 
 sented in Fig. 4, where we assume that the proposition is a 
 definition. But as this can never be assumed formally, and as 
 all propositions must be formally treated in Formal Logic, Fig. 
 5 and Fig. 14 are the only proper symbols of the quantitative 
 relation between subject and predicate. Nothing, then, is 
 definitely said about the whole of the predicate in proposition 
 A, because other substances besides "metals" may be included 
 in " elements." If we said anything about the whole of the 
 class " elements " in the proposition " All metals are elements," 
 we could reverse the order of subject and predicate and say, 
 "All elements are metals." But if other substances besides 
 metals are elements this latter proposition could not be true, 
 and hence as long as we can possibly say that other things be- 
 sides metals are elements, or as long as the proposition "All 
 metals are elements," is entirely silent about the extension of 
 the term " elements," we have asserted nothing about the whole 
 of it, as indicated by the larger circle P in Fig. 5, and hence 
 the predicate is not distributed. In proposition I, "Some 
 metals are elements," the same conclusion is apparent, and the 
 predicate is undistributed. 
 
 In proposition E, " No men are trees," for instance, not only 
 is something. said about the whole of the subject, but some- 
 thing is also said about the ivhole of the predicate. It is defi- 
 nitely excluded in its whole extension from the subject, as Figs. 
 10 and 15 indicate : That is, "men" are not any part of the 
 class " trees," so that the whole of the class " trees " is ex- 
 cluded from the subject, and we can as well say " No trees are 
 men," as "No men are trees." Hence in the negative proposi- 
 tion E the predicate is distributed, since something is said or 
 denied about the whole of it. In the negative proposition O 
 the same conclusion will be apparent, if we merely observe that 
 a part of the subject is definitely excluded from the whole of 
 the predicate, as is clear in Figs. 12 and 17. The predicate of 
 O, " Some elements are not metals " is therefore distributed. 
 We therefore summarize the rules for the distribution of sub-
 
 140 ELEMENTS OF LOGIC 
 
 ject and predicate as follows. We give two forms of statement, 
 and the student may adopt the most convenient : 
 
 Subject. Predicate. 
 
 f TT 1 J Affirmative A. Distributed. Undistributed. 
 
 . J 1 Negative E. Distributed. Distributed. 
 
 1 p Q7 4i 1a ( Affirmative L Undistributed. Undistributed. 
 
 [ ra lrv ( Negative O. Undistributed. Distributed. 
 
 AH Universal propositions, A and E, distribute the subject. 
 
 All Particular propositions, I and O, do not distribute the 
 subject. 
 
 All Affirmative propositions, A and I, do not distribute the 
 predicate. 
 
 All Negative propositions, E and O, distribute the predicate.* 
 
 The symbol which I shall adopt to indicate the distribu- 
 tion of a term will be a small circle placed around the sub- 
 ject or predicate, as the case may be. Thus the distribution 
 and non-distribution of terms in A, E, I, and 'O may be repre- 
 sented as follows, the cross indicating a negative proposition : 
 
 A, @ = P. E, (g) X (g). I, S = P. O, S X . 
 
 * General references on the relation between subject and predicate are 
 the following : Venn : Empirical Logic, Chaps. VIII. and IX. ; Symbolic 
 Logic, Chaps. I. and VII., inclusive; Bosanquet : Logic, Book I., Chaps. 
 I. and VII., inclusive; Keynes: Formal Logic, Part II., Chap. VI. ; 
 Wundt: Logik, Dritter Ahschnitt, Cap. I. and II.
 
 CHAPTER IX. 
 
 OPPOSITION 
 
 1st. Meaning of Opposition. Opposition treats of the 
 relations between the propositions A, E, I, and O, growing 
 out of their quantity and quality ; that is, out of the fact that 
 they are universal and particular on the one hand, and affirm- 
 ative and negative on the other. It has not to do with the 
 subject and predicate, or the elements of the proposition as 
 such, but with the propositions as a whole. When they con- 
 tain the same matter they have certain relations of agreement 
 or conflict which it is the business of the logician to exhibit. 
 Unless they do contain the same matter no such relation can 
 be determined. We can, in such cases, only decide upon their 
 quantity and quality, and so merely treat them as universal or 
 particular, affirmative or negative. But the various relations 
 of agreement and conflict between conceptions give rise to 
 corresponding relations between propositions containing them, 
 and hence we require to ascertain the rules which regulate the 
 extent to which any given proposition is true or false when 
 another is known to be true or false. Some propositions, if 
 true, interfere with the truth of others, and some do not. On 
 the other hand, some, if false, necessitate the truth of others, 
 and some do not. The sense and extent to*Vhich this is true 
 remains to be determined. 
 
 If " All horses are animals," it cannot be true at the same 
 time that " No horses are animals," or that " Some horses are 
 not animals." This we express by saying that if A is true, E 
 and O cannot be true at the same time ; it is inconsistent with 
 both of them. Also, if it be true that " No men are quadru- 
 peds," it cannot be true that " All men are quadrupeds," or 
 that "Some men are quadrupeds." This we again express by
 
 14:2 ELEMENTS OF LOGIC 
 
 saying that if E be true, A and I cannot be true at the same 
 time ; it is inconsistent with them. But it is important to 
 observe that if " All men are quadrupeds " is false, it follows 
 that " Some men are not quadrupeds." It may be true, also, 
 that " No men are quadrupeds," but the falsity of the universal 
 affirmative with the same terms does not prove that fact. It 
 can only prove the truth of the particular negative, and it re- 
 mains entirely unknown from that proposition whether its 
 universal negative is true or false. Hence if A be false, it fol- 
 lows that must be true, but it does not follow that E is true 
 or false. Now if it be false that " Some men are not mortal," 
 it must follow that "All men are mortal," and, as we have 
 shown in the first case, the negative of this, namely, " No men 
 are mortal," is false. This we express by saying that if O be 
 false, A is true and E is false. Similarly, if I be false, E must 
 be true and A must be false. In this way we find that if A 
 be true, O is false, and if A be false, O is true ; again, if E be 
 true, I is false, and if E be false, I is true. On the other hand, 
 if O be true, A is false, and if O be false, A is true ; and if I be 
 true, E is false, and if I be false, E is true. This kind of incon- 
 sistency between A and O, on the one hand, and E and I, on 
 the other, we call contradiction. In a loose sense the words 
 " contradiction " and " contradictory " are used to express any 
 kind of inconsistency which prevents two things from being 
 true at the same time. But the relations between propositions 
 A and E are so different from those between A and O, and E 
 and I, that the term " contradictory " has been chosen to indi- 
 cate that mutual inconsistency between A and O, and E and 
 I, by which only*bne of them can be true, and only one of 
 them false, at the same time. But A and E are called Con- 
 traries, because although the truth of A implies the falsity of 
 E, and, vice versa, the truth of E implies the falsity of A, yet 
 the falsity of A does not imply the truth of E, nor the falsity 
 of E the truth of A. The mutual inconsistency existing be- 
 tween the universals and their opposite particulars does not 
 exist between universals. Hence they are called contraries to 
 distinguish them from contradictories.
 
 OPPOSITION 143 
 
 It remains to determine the relations between A and I, E 
 and O, and I and O. If it be true that " All men are mortal," 
 it must be true that " Some men are mortal." So, if it be 
 true that " No men are trees," it must be true that " Some 
 men are not trees." This we express by saying that if A be 
 true, I is true, and if E be true, O is true, because the part 
 must be included in the whole. But if it be true that " Some 
 men are wise," it does not follow that "All men are wise ;" 
 and if it be true that " Some men are not wise," it does not 
 follow that " No men are wise." This we express by saying 
 that if I be true, A is indeterminate, and if O be true, E is in- 
 determinate. This is because we can affirm nothing of the 
 whole when we affirm something only of the part. If we were 
 to take cases supposing the falsity of A, we should find I in- 
 determinate, or the falsity of E, we should find O indetermin- 
 ate. But the falsity of I does not leave A indeterminate, nor 
 does the falsity of O leave E indeterminate. This variable 
 relation 'is expressed by calling A and I, or E and O, subalterns 
 of each other. But I and O are called subalternates, and A 
 and E are each called a subalternans. 
 
 When we compare I and O we find that they represent 
 propositions of opposite quality ; that is, one is affirmative 
 and the other negative, and in that respect they are the oppo- 
 site of each other. But the relation between them is the re- 
 verse of that between A and E. If it be true that " Some 
 metals are elements," the law of contradiction already estab- 
 lished between I and E will make the proposition " No metals 
 are elements " false, and by subalternation O, " Some metals 
 are not elements," will be indeterminate. That is, nothing 
 follows about O from the truth of I, and also nothing about I 
 from the truth of O. But if it be false that " Some men are 
 trees," it then follows by contradiction that the proposition 
 "No men are trees," is true, and by subalternation, "Some 
 men are not trees," is true also. This we express by saying 
 that if I be false, O is true, and if O be false, I is true. But 
 both cannot be false at the same time, and both may be true. 
 This relation is expressed by calling them subcontraries, as A
 
 144 ELEMENTS OF LOGIC 
 
 and E are called contraries because they cannot both be true 
 at the same time, but both may be false. 
 
 These various relations of the propositions A, E, I, and O 
 are represented by a diagram which has been but slightly 
 modified since Aristotle. It is called the Square of Opposi- 
 tion. 
 
 A Contraries. E 
 
 <% ? 
 
 I Subcontraries. O 
 
 The relations, as we have developed them, can easily be ap- 
 plied to this scheme. They are embodied in the following 
 rules, which it is important to keep in mind : 
 
 1. Of contradictory propositions, one must be true and the 
 
 other false. 
 
 2. Of contrary propositions, both cannot be true at the same 
 
 time, and both may be false. 
 
 3. Of subcontrary propositions, one only can be false, and 
 
 both may be true at the same time. 
 
 4. Of subalterns, both may be true or both may be false at 
 
 the same time. But if the subalternans be true, the 
 subalternate is true, and if the subaltemate be false 
 the subalternans is false ; if the subalternans be false, 
 the subalternate is indeterminate, and if the subalter- 
 nate be true, the subalternans is indeterminate. 
 2d. Application of Opposition and its Principles. 
 Nothing can be determined form ally about the relation of op- 
 position between the propositions A, E, I, and O, unless we 
 assume identity of matter. A difference of matter simply iso- 
 lates the two propositions and throws them out of all relation 
 to each other in the scheme of opposition. The question is 
 therefore suggested, What laws determine the nature of these 
 relations of contradiction, contrariety, and subalternation, and 
 what must be taken into account when applying them to actual
 
 OPPOSITION 145 
 
 discourse ? Keynes furnishes a complete answer to the first 
 of these questions. 
 
 " The inferences," he says, "based on the square of opposition, 
 may be considered to depend exclusively on the three funda- 
 mental Laws of Thought, namely, the Law of Identity A is 
 A ; the Law of Contradiction A is not A ; and the Law of Ex- 
 cluded Middle A is either B or not B." For example, from 
 the truth that "All men are mortal" I may infer by the Law of 
 Identity that " Some men are mortal," and by the Law of Con- 
 tradiction the falsity of the proposition that " Some men are 
 not mortal." By the Law of Excluded Middle we can infer 
 from the falsity of the proposition " All men are mortal," the 
 truth of the proposition that, " Some men are not mortal." 
 The Law of Identity means that a thing can be affirmed of it- 
 self, or conceptions which agree with each other can be af- 
 firmed in that sense. The Law of Contradiction means that 
 two conflicting or contradictory conceptions cannot be af- 
 firmed of a thing at the same time. Thus I cannot affirm that 
 a man is both " mortal " and " not mortal " at the same time. 
 The Law of Excluded Middle means that of two contradictories 
 one must be true. But a fuller discussion of these laws must 
 be postponed to a later chapter. This brief account will suf- 
 fice for the applications with which we have to deal at pres- 
 ent. 
 
 In discourse and controversy we have to be careful about 
 the real nature of our conceptions and propositions. We are 
 liable to mistake, at times, a contrary for a contradictory judg- 
 ment, or an indefinite for a definite judgment, or subcontra- 
 ries for contraries. This will particularly be the case when 
 propositions are one thing in form and another thing in mat- 
 ter. Thus singular propositions are treated as universal in 
 form and are therefore contraries. But in matter, subject and 
 predicate being the same and their quality the opposite of each 
 other, they are contradictories ; as, for example, " Socrates is a 
 man," the only possible negative of which is " Socrates is not a 
 man." Pure universals have two opposites, the contrary and the 
 contradictory, but singulars have only one, which is in reality 
 10
 
 146 ELEMENTS OF LOGIC 
 
 the contradictory, as will be seen in the case given. Thus if it 
 be true that " Socrates is a man," the negative, "Socrates is not 
 a man," is false, and vice versa. So far it seems like a case of 
 contraries. But if " Socrates is a man " be false, it is time 
 that " Socrates is not a man," and vice versa. This makes it a 
 case of contradiction, because if they were contraries they 
 might both be false at the same time. This would mean that 
 the assumed or proved falsity of the proposition " Socrates is 
 a man," would leave the proposition " Socrates is not a man " 
 indeterminate. But we observe that it cannot be so, and hence 
 singular judgments in respect to quality have to be treated as 
 contradictories, but in respect to quantity as universals. This 
 will determine the relation between the two propositions 
 " Socrates is a man " and " Socrates is a horse." They must 
 be regarded as contraries, not as contradictories. They con- 
 tain different matter in their predicates, but the same matter 
 in their subjects ; so that although the predicates are both 
 positive concepts they are mutually exclusive as species, and 
 so relatively negative in comparison with each other. The 
 two propositions are related as contraries because the truth 
 of either denies the other, while both may be false. If it 
 be true that " Socrates is a man," it cannot be true that 
 " Socrates is a horse," and so if he were a horse he could not 
 be a man. But if it be false that "Socrates is a man," it 
 does not follow that he is a horse, because he might be any- 
 thing else except a man. Hence terms representing co-ordi- 
 nate species will be contraries, not contradictories, in the 
 scheme of opposition. 
 
 But what will be made of the propositions " Socrates is a 
 man " and " Socrates is a Greek ? " Of course, formally neither 
 these nor the previous propositions can be treated of under 
 the principles of opposition, and I am not designing so to 
 treat them. I am endeavoring to give the purely formal rules 
 some modifications to suit their material application. The last 
 two propositions, although singular, as the two previous ones, 
 are somewhat different because of the relation between the 
 two predicates. These, instead of being co-ordinate species,
 
 OPPOSITION 147 
 
 are genus and species. Hence if the first be true it does not 
 follow that the second is either true or false ; but if it be false 
 the second is false. It is interesting to note from this that 
 they are subalterns, the proposition "Socrates is a man" being 
 the subalternate, and " Socrates is a Greek " being the subal- 
 ternans. We may generalize, therefore, in such cases, by say- 
 ing that in singular judgments with the same subject, when 
 the predicate is a genus in one and a species in the other, the 
 genus is the mark of the subalternate and the species of the 
 subalternans. 
 
 When the predicate is the same the subjects can never be a 
 genus and species, and the propositions remain singular at the 
 same time. Hence no relation of opposition in such cases is 
 determinable. Thus, in the propositions, "Socrates is a 
 Greek," and "Plato is a Greek," nothing can be said about 
 the truth or falsity of one when the other is either true or 
 false. 
 
 Hence, when the predicates are the same in both propo- 
 sitions, and the subject of one a genus, and of the other a spe- 
 cies, both propositions formally are universals, but materially 
 they are A and I, or universal and particular, and so are sub- 
 alterns again. But in this case the genus marks the subalter- 
 nans and the species the subalternate. Thus " All Greeks are 
 men," and " Socrates is a man." The truth of the first implies 
 that of the second, and the falsity of the first leaves the second 
 indeterminate (except on a condition to be discussed again), 
 while the truth of the second leaves the first indeterminate, 
 and the falsity of the second implies the falsity of the first. 
 We must keep in mind, however, that we assume all along 
 that Socrates is a Greek, and hence an individual or a species 
 of the genus. 
 
 The matter is still more complicated when we come to con- 
 sider universal propositions or judgments. Suppose we take 
 the first example with the same subject and with co-ordinate 
 species for predicates. Thus " All men are bipeds," and "All 
 men are rational." It is apparent in such cases that neither 
 agreement nor conflict between them can be inferred from
 
 148 ELEMENTS OF LOGIC 
 
 either the truth or the falsity of one of them. Hence a rela- 
 tion of opposition is not determinate here, any more than in 
 the singular propositions " Socrates is a man," and " Socrates 
 is blind." If the predicates be such inconsistent concepts as 
 " quadrupeds " and " newspapers," the propositions might be 
 treated as contraries, but there is no criterion for determining 
 this opposition. It requires to be a uniform concomitant of 
 some other characteristic. But if the predicates be genus and 
 species, and the subject identical, as "All men are vertebrates," 
 and " All men are bipeds," the problem is not different from 
 the previous one, because genus and species, or conferentia 
 and differentia, always agree and never conflict, but never 
 imply each other. No relation of opposition therefore can be 
 established between such propositions. 
 
 If the predicate remain the same, and the subjects are genus 
 and species, as " All Europeans are Caucasians," and " All 
 Frenchmen are Caucasians," the propositions are evidently 
 subalterns, the genus marking the subalternans, and the spe- 
 cies the subalternate, as already shown. But as between " All 
 Europeans are Caucasians," and " Some Frenchmen are not 
 Caucasians," we evidently have complex cases, and so also 
 with " All Frenchmen are not Caucasians," because the relation 
 between genus and species determines the particularity of the 
 last proposition in comparison with the first. Of course such 
 propositions as "All men are bipeds," and "All men are quad- 
 rupeds," or " All wise men are good," and " All wise men are 
 bad," will be contraries, owing to the nature of the concepts 
 biped and quadruped, and good and bad. But the form of the 
 terms cannot determine this fact, and hence no rule for esti- 
 mating them can be established. It will be important to ob- 
 serve, however, that good and bad, wise and ignorant, rational 
 and irrational, beautiful and ugly, etc., are often used as con- 
 tradictory conceptions, and sometimes only as contraries. The 
 latter is the true conception of them, as there is a third alter- 
 native between them, owing to the fact that they are both 
 positive terms. A true contradiction can exist only between 
 positive and negative terms, never between two positives, or
 
 OPPOSITION 149 
 
 between a positive and a nego-positive. The latter are only 
 conti-aries, and so establish contrariety between propositions 
 having them as predicates, but with the same subject. 
 
 It is very important to keep these facts and distinctions in 
 mind in order to understand the mental processes involved in 
 one man's asserting that " Washington was a good man," and 
 another controverting it by the counter-assertion that " Wash- 
 ington stole a horse," .or one man's affirming that " Americans 
 are a mercantile people," and another's asserting that " A. B. 
 C. of the Americans are great scientists." In the first two the 
 propositions are evidently contraries. In the second instances, 
 everything depends upon whether the indefinite proposition, 
 " Americans are a mercantile people," be regarded as a univer- 
 sal or a particular, A or I. If the former, they are contraries, 
 supposing that commerce and science are incompatible call- 
 ings, as in practice, generally at least, they seem to be, al- 
 though perhaps not ideally or theoretically so. But if the 
 proposition be I, they are subcontraries, under the same as- 
 sumption as before. But the cases only show how we are to 
 analyze the conceptions when a controversy is involved. 
 
 In the relation between subcontrary judgments we must be 
 careful not to admit the ambiguous use of the word some, as 
 this would at once imply the truth of the opposite proposition. 
 But in the true conception of opposition I never implies O, 
 and O never implies I. Hence the term " some " must always 
 mean a part, and it may or may not be all, as has already been 
 shown. 
 
 Errors in argument, so far as an assumption of the relation 
 between propositions may be concerned, are occasioned in two 
 ways ; first, in assuming an agreement between them when they 
 are inconsistent, and second, in assuming a conflict when they 
 are consistent with each other. The latter is always called 
 the fallacy of lynoratio Elenchi, which we shall have to ex- 
 plain again. But it will be illustrated in the course of the 
 present discussion. We are to examine proof and disproof of 
 judgments so far as that can be accomplished under the 
 scheme of opposition.
 
 150 ELEMENTS OF LOGIC 
 
 By the principles of opposition we can only prove I and O as 
 subalternates by assuming or proving A and E, and disprove 
 A and E by assuming or proving I and O as their contradic- 
 tories. Thus we prove I if we admit A, and we pi'ove O if we 
 admit E. But we disprove A if we admit O, and disprove E 
 if we admit I. In such formal propositions as " Some men are 
 mortal " and " All men are mortal," this is evident from the 
 rules. But in such propositions as " Europeans are white," 
 and " Germans are white," it is not evident until we notice 
 the logical relation existing between the conceptions " Euro- 
 peans " and " Germans." I prove the truth of the latter when 
 I admit the former, assuming that " Germans " are a species of 
 the genus " Europeans," because they are related as subalterns, 
 the former being the subalternans, and the latter the subalter- 
 nate, the proposition to be proved. I may prove that " Scien- 
 tists are learned," by assuming or proving that "Educated men 
 are wise," if " learned " and " wise " are identical, and " educated 
 men " be the genus of which " scientists " are the species. But 
 I cannot prove that " Germans are good " by assuming or 
 proving that " Europeans are not bad," without making the 
 second assumption that " good " and " not bad " are identical. 
 
 In disproof the matter is a more important and interesting 
 one. Here we have to do with A and E as conclusions, and I 
 and O as conditions of their validity or invalidity. "In order 
 to prove the falsity of A, it is sufficient to establish the truth 
 of O, and it is superfluous, even if possible, to prove E ; simi- 
 larly E is disproved by proving I, and it is superfluous to 
 prove A. Any person who asserts a universal proposition, 
 either A or E, lays himself under the necessity of explaining 
 away or disproving every single exception brought against it. 
 An opponent may always restrict himself to the much easier 
 task of finding instances which apparently or truly contradict 
 the universality of the statement, but if he takes upon himself 
 to affirm the direct contrary, he is himself open to easy at- 
 tack. "Were it to be asserted, for instance, that 'All Chris- 
 tians are more moral than Pagans,' it would be easy to adduce 
 some examples showing that ' Some Christians are not more
 
 OPPOSITION 151 
 
 moral than Pagans,' but it would be absurd to suppose that it 
 would be necessary to go to the contrary extreme, and show 
 that ' No Christians are more moral than Pagans.' In short, 
 A is sufficiently and best disproved by O, and E by I. It will 
 be easily apparent that, vice versa, O is disproved by A, and I 
 by E ; nor is there, indeed, any other mode of disproving 
 these particular propositions." The error in disproof, how- 
 ever, may lie in certain assumptions about the relations be- 
 tween the two propositions after the proper one has been 
 proved. Thus it may be no disproof of the assertion that 
 " John Smith is good," to prove that he is not civilized, be- 
 cause the conceptions " good " and " not civilized " are not 
 necessarily contradictory, nor even contrary. I simply evade 
 the issue. Again, it is no disproof of the assertion that 
 " Cromwell was a usurper," to say that " Foreign nations ac- 
 knowledged his authority," any more than it would be a proof 
 of his legitimacy to make the same statement. Likewise it is 
 no disproof of a man's badness to say that he is religious, any 
 more than we should prove he is white by showing that he is 
 not black. If I assert that " Government is a necessary insti- 
 tution," it is no disproof of it to show that some governments 
 are bad. Many arguments, however, are conducted upon just 
 such logic, where agreement or conflict are simply assumed 
 by both parties to exist between conceptions which may be 
 inconsistent in certain accidental relations, but not necessarily 
 so. Thus, if I assert that free-trade is the right policy for a 
 country, it is no disproof of it to show that protection helps 
 manufacturers ; nor is it a disproof of it to show that I have 
 myself instituted a policy of protection. The untrained mind, 
 however, is likely to suppose the argument valid in both 
 cases. In purely formal Logic this is all very easy, but in 
 applied Logic, or actual discourse, we require dexterity in dis- 
 covering the relations between conceptions in order to test 
 the connection of general principles with them. To illus- 
 trate, take the conceptions just mentioned in the last exam- 
 ple, protection and free-trade. Without some knowledge of 
 economics, perhaps, we should not suspect or detect any rela-
 
 152 ELEMENTS OF LOGIC 
 
 tion of either agreement or conflict. "Protection," as a 
 formal and logical conception, might suggest nothing more 
 than the idea of " self-defence" or "self-preservation," and so 
 carry everything before it by this association, while no con- 
 ceivable meaning might be attached to " free-trade," except 
 the vague sense of contradiction or opposition from the mere 
 fact that some one so considered it. But an examination of 
 their real meaning from a knowledge of economics shows them 
 to be contradictories in reality, and not merely contraries, al- 
 though they are both positive terms, in perhaps every other 
 connection than this one. This is made clear by the fact that 
 "protection" is a restriction upon free commerce, "free- 
 trade " is th'e absence of that restriction. We see in this dis- 
 tinction the presence and the absence of certain qualities 
 which defines respectively a positive and negative term. 
 
 In many cases the error can be tested by taking into ac- 
 count the assumptions made in any particular assertion. But 
 this introduces the syllogism into the problem, while it may 
 be possible to decide the matter upon the principles of oppo- 
 sition. This is all that we have here considered, and the stu- 
 dent should practise the application of these principles on 
 every possible occasion, not so much for the sake of familiar- 
 ity with the laws of formal Logic, as for a better understand- 
 ing of the means for dealing with the subject-matter of dis- 
 course and controversy. As an illustration of what a mental 
 process is, and what law of opposition is involved or implied, 
 I quote at random a statement from an article in a monthly 
 periodical : " Those who oppose nationalism on the ground 
 that the present social condition is, by reason of its privations, 
 a blessing, ought to understand that the exact opposite of a 
 false proposition is by no means certain to be the true one 
 though it is a favorite argumentative short-cut to assume this 
 to be the case." Now whether he knew it or not, the writer 
 was using the law regulating the relation between contraries 
 in asserting that "the exact opposite of a false proposition 
 was by no means certain to be the true one." But the asser- 
 tion charges a fallacy upon his opponent which would have
 
 OPPOSITION 153 
 
 been technically called an ignoratio elenchi. The case could, 
 perhaps, have been made stronger by connecting the error di- 
 rectly with the violation of a law of Logic. Illustrations of 
 the same kind are plentiful, and the student can profitably 
 spend his time in looking for them.* 
 
 * Special references on the subject of Opposition need not be indicated, 
 as the general agreement upon it leaves no points of importance in 
 doubt. The works previously mentioned deal with the subject very 
 briefly.
 
 CHAPTER X. 
 
 IMMEDIATE INFERENCE 
 
 1st. Definition and Divisions. An inference of any kind 
 is merely the explicit statement of what is implicit in a pre- 
 vious assertion or thought. If this account of it is not clear, 
 we may define it as drawing one conception or truth from an- 
 other, which implies the one drawn, or it is the carrying out 
 into a following proposition a thought which was virtually 
 contained in an antecedent judgment. This last definition is 
 Hamilton's, slightly modified. It can be illustrated in several 
 ways. I can infer that "Europeans are white," if I know that 
 Caucasians are white, and that Europeans are Caucasians. Or 
 I may infer that the weather will be clear, when I see the 
 clouds breaking. The first form of inference is said to be de- 
 ductive, and the second inductive. But for the present we 
 need only to know that a certain fact or facts may suggest to 
 consciousness other facts not explicitly stated or conceived in 
 the first case, and which are said to follow from it. An infer- 
 ence, then, is what follows from another thing, in so far as the 
 latter is a conceived truth. A case quite different from the pre- 
 vious illustrations is the succeeding one. From the proposition 
 " The sciences are useful studies," I can infer that " Some of 
 the useful studies v are sciences," or from "All negroes are 
 black," that "Those who are not black are not negroes." It is 
 evident that these forms of inference are quite different from 
 the previous instances. Hence we proceed to the two kinds 
 of inference. 
 
 There is, first, the inference known as Mediate Inference, or 
 Reasoning, which is illustrated by the first instances. It cle-
 
 IMMEDIATE INFERENCE 155 
 
 notes reasoning by the agency of a middle term. Thus in all 
 mediate inference we have first to compare two terms with a 
 third, called the middle term, and then infer that these two 
 terms can be compared with each other. But it is quite dif- 
 ferent with the second class of inferences. This is called Im- 
 mediate Inference, or Reasoning, and denotes reasoning with- 
 out the use of a middle term. Mediate inference requires two, 
 immediate inference but one proposition as a basis. This is 
 exemplified in the illustrations. Immediate inference is usu- 
 ally divided into Conversion, Obversion, Contraposition, Added 
 Determinants, and Complex Conceptions. But I shall slightly 
 modify this division so as to read Conversion, Obversion, Con- 
 traversion, Inversion, and Contribution, the last of which com- 
 prehends the last two in the previous classification. Opposi- 
 tion is not treated as a mode of inference. But it would not 
 be far amiss to so treat it, or at least to consider it as involv- 
 ing processes of immediate inference. For we pass directly 
 to certain conclusions from certain other conditions or pre- 
 mises. But it differs from other modes of inference in that it 
 requires the supposition of certain material truths in order to 
 render it possible, while in other forms of immediate inference 
 we require only a formal relation between subject and predi- 
 cate. Perhaps, however, if we regard the conditions of oppo- 
 sition as formal, it may be possible to give the transition from 
 A to O, or from E to I, by contradiction, or from A to E, by 
 contraries, as sufficiently formal and inferential to speak of 
 the process as involving a kind of immediate inference. Al- 
 though I believe this to be the case I have remained by tradi- 
 tional usage and discussed it by itself. Under immediate in- 
 ference, therefore, I begin with Conversion. 
 
 2d. Conversion. Conversion is the transposition of sub- 
 ject and predicate, or the process of immediate inference by 
 which we can infer from a given proposition another having 
 the predicate of the original for its subject, and the subject of 
 the original for its predicate. But there are certain limita- 
 tions under which the transposition can take place. For in- 
 stance, from the proposition that " All horses are animals," I
 
 156 ELEMENTS OF LOGIC 
 
 cannot infer that " All animals are horses ; " nor from " All 
 metals are elements " that " Some elements are not metals," 
 although this may actually be the case. The rules, therefore, 
 which limit the process of conversion are two. 
 
 (a) The quality of the converse must be the same as that 
 of the convertend. 
 
 (6) No term must be distributed in the converse which is 
 not distributed in the convertend. 
 
 These rules may be abbreviated so as to read : Do not 
 change the quality of the proposition, and Do not distribute an 
 undistributed term. We may indistribute a term which is dis- 
 tributed, but not vice versa. The Convertend is the proposi- 
 tion to be converted : the Converse is the converted propo- 
 sition. 
 
 The forms of Conversion are two, according as the quantity 
 of the proposition is changed or remains the same. If the 
 quantity of the converse remains the same as that of the con- 
 vertend, the conversion is called Conversio simplex, or Simple 
 Conversion ; if the quantity is changed, it is called Conversio 
 per accidens, or Limited Conversion, usually Conversion by 
 Limitation. We have only to illustrate them and to ascertain 
 the extent of their application to the several propositions A, 
 E, I, and O. 
 
 Take a proposition in A, " All apples are fruit." In this 
 proposition, as already shown, the predicate is not distributed. 
 This means, as illustrated in Fig. 5 (p. 130), that other things 
 also may be contained in the class " fruit," so far as can be de- 
 termined by the assertion given. Hence, if in transposing 
 subject and predicate we say, " All fruits are apples," we shall 
 be asserting more than the original proposition will permit. 
 In the original we have said nothing about the whole of the 
 term " fruit," and so cannot be permitted to do so in the con- 
 verse. Hence we can assert something only of a part of it, 
 if we can assert anything xat all. That we may assert some- 
 thing is evident from the fact that some degree of identity or 
 connection exists between the subject and predicate in the 
 convertend, and this same relation can be asserted in the
 
 IMMEDIATE INFERENCE 157 
 
 converse. By limiting our statement, therefore, to the part of 
 the predicate of which we actually affirm something, we are able 
 to infer from the original proposition that " Some fruits are 
 apples." This is evidently true, if the original be true. Here 
 the quantity of the proposition is changed, but its quality re- 
 mains the same ; that is, the quantity of the convertend is 
 universal and its quality affirmative, while the quantity of the 
 converse is particular and its quality affirmative. We have, 
 therefore, converted A into I. To convert " All apples are 
 fruit" into "All fruits are apples" would be to violate the sec- 
 ond rule. Hence A cannot be converted into A. To change 
 the quality of the proposition A in conversion, that is, into 
 either E or O, would be to violate the first rule, and it is ap- 
 parent that we cannot infer an exclusion between a subject 
 and predicate from an affirmed connection or identity between 
 them. Hence A cannot be converted into either E or O, and 
 we have found that A cannot be converted into A, but that it 
 can be converted into I, which is to say that propositions in 
 A cannot be converted simply, but only by limitation. 
 
 There is one exception in the conversion of A. This is the 
 case of definitions, and of singular propositions with a singu- 
 lar term for predicate. Definitions are universal affirmatives, 
 and are nevertheless convertible by simple conversion. This 
 is a case of A ^being converted into A. But this is only be- 
 cause the nature of the case enables us to know the extension 
 or distribution of the predicate. Formally the proposition is 
 like all others and not convertible simply. But materially we 
 know from the fact that the proposition is a definition, that 
 the extension of the subject and predicate is the same, and 
 hence convertible from A into A. Without assuming its ma- 
 terial nature, however, we could know nothing of this distri- 
 bution, and hence formally it would have to be treated as all 
 other propositions in A, which, being neither formally nor 
 materially distributed in the predicate, can only be converted 
 by limitation. Nevertheless it is important to observe that in 
 some of our material reasoning the mind may be correct in its 
 processes on the ground that its datum is a definition ; that
 
 158 ELEMENTS OF LOGIC 
 
 is, subject and predicate are conceived as identical, while for- 
 mally the reasoning may seem to be fallacious. 
 
 The exclusive proposition, although it appears to be a uni- 
 versal, may or may not be so. The " only " means some, and 
 it may and it may not be all, but certainly nothing else. When 
 converted, it becomes a universal, but not before this. 
 
 Proposition I, " Some men are vertebrates," can only be con- 
 verted simply, or by Simple Conversion. We cannot infer 
 from it that " All vertebrates are men," for the same reason 
 that we could not convert A into A. It is because the predi- 
 cate is undistributed in the convertend, and must not be dis- 
 tributed in the converse. It would violate rule second to con- 
 vert it into E and O, and for the same reason that A could not 
 be converted into E or O. Hence I must be converted into I, 
 " Some men are vertebrates " into " Some vertebrates are 
 men." This is Conversio simplex, or Simple Conversion, be- 
 cause the form of the converse is the same as that of the con- 
 vertend. 
 
 Proposition E, "No books are pens," can be converted either 
 simply or by limitation. In this the predicate is distributed, 
 and this fact will permit of its distribution in the converse. 
 Besides, since something is said excluding the whole of the 
 predicate from the subject, we can assert this in the converse. 
 Hence we can infer that " No pens are books. "- By subalter- 
 nation, from this we can infer " Some pens are not books." 
 The first is the simple, and the second the limited converse of 
 the original. Hence E is convertible into either E or O. O 
 might be called a weakened converse in this case, because E 
 might as well be inferred. 
 
 In regard to propositions in O, as, " Some men are not Cau- 
 casians," a peculiar difficulty exists. First, the converse must 
 be of the same quality as the convertend, according to the first 
 rule. It must, therefore, be negative. But second, negative 
 propositions distribute the predicate. Hence, whether we 
 convert "Some men are not Caucasians," by simple conversion 
 or by limitation, the predicate in the converse, which is un- 
 distributed rn the convertend, will be distributed, and hence
 
 IMMEDIATE INFERENCE 159 
 
 violates the second rule. O cannot, therefore, be converted 
 by the ordinary method. 
 
 It has been usual, however, to apply an indirect method, 
 called Conversion by Negation. Take, for example, " Some real- 
 ities are not material objects." If we infer that " Some or All 
 material objects are not realities," we commit a fallacy, because 
 the predicate of the converse is distributed, while it is not dis- 
 tributed as subject in the convertend, because the proposition 
 is particular. But if we attach the negation to the predicate 
 in the original we have "Some realities are not-material 
 objects," or "Some realities are immaterial objects." The 
 proposition thus becomes I, which we can convert simply into 
 "Some immaterial objects are realities." This proposition, 
 then, can be inferred from the original, and the process of 
 reaching it is called Conversion by Negation. The same pro- 
 cess is applicable to similar propositions. Thus " Some pleas- 
 ant acts are not just," would become " Some pleasant acts are 
 not-just, or unjust," and then by conversion, " Some unjust 
 acts are pleasant." " Some men are not agreeable " would be 
 converted into " Some disagreeable persons are men." 
 
 But it must be observed that the quality of this so-called 
 converse is affirmative while the convertend is negative, and 
 hence the process of conversion by negation is a violation of 
 the first rule. Besides, we have been led by it to affirm some- 
 thing positive about non-material or immaterial objects in the 
 assumed converse, when the convertend merely denies some- 
 thing about material objects. While this may be allowable by 
 some other process, it is not permissible by conversion. The 
 violation of the first rule decides that matter. Hence we con- 
 clude that proposition O is really not convertible at all. This 
 is the general opinion of logicians. In his smaller work Jevons 
 assumes the validity of the process, but in a later treatise he 
 concedes his error, and comes over to the general view. The 
 proposition which is supposed to be the converse in this case 
 is really the result of a double process, obversion and conver- 
 sion, and hence is the converse of the obverse, or the contra- 
 positive, called also the contraverse.
 
 160 
 
 ELEMENTS OF LOGIC 
 
 The forms of Conversion may be summarized in the follow- 
 ing table, showing also the impossible forms : 
 
 Convertend. 
 
 
 Converse. 
 
 
 Impossible Forms. 
 
 All S is P. 
 
 A 
 
 Some P is S. 
 
 I 
 
 A, E, and O. 
 
 Some S is P. 
 
 I 
 
 Some P is S. 
 
 I 
 
 A, E, and O. 
 
 No S is P. 
 
 E 
 
 No P is S. 
 
 E 
 
 A and I. 
 
 
 
 Some P is not S. 
 
 
 
 
 Some S"is not P. 
 
 
 
 (None.) 
 
 
 A, E, I, and O. 
 
 3d, Obversion. This is sometimes called "Immediate 
 Inference by Privative Conception." This will serve as a good 
 name when the propositions are affirmative, and when a priva- 
 tive term can be found for the purpose. But when the propo- 
 sition is negative, and when a privative term is not accessible, 
 the shorter name Obversion is much to be preferred. Tfie pro- 
 cess consists in negating the copula and the predicate without 
 conversion. That is, the quality of the obvertend must be 
 changed, and new matter introduced into the obverse, which 
 shah 1 be the negative concept of the predicate in the obvertend. 
 Thus the proposition " All men are mortal " must be ob- 
 verted by changing the quality of the proposition and Baking 
 the predicate of the obverse the negative of "mortal." Hence 
 the obverse of it will be, " All men are not not-mortal," or "All 
 men are not immortal," or "No men are immortal." Of the 
 proposition "All oaks are trees/' it will be "No oaks are not 
 trees." It will be apparent that this is, after all, identical in 
 meaning with the original proposition, although the meaning 
 is stated in a negative way for the sake of emphasis. The two 
 negatives together make an affirmative, and we can have a neg- 
 ative proposition of the obverse only on the supposition that 
 one of the negative pai-ticles qualifies the copula, and the other 
 the predicate, so as to make it a negative conception. In some 
 cases this negative particle can be joined to the predicate, in 
 others not. 
 
 In regard to the negative proposition the obversion is ac- 
 complished simply by connecting the negative particle with
 
 IMMEDIATE INFERENCE 161 
 
 the predicate, which both changes the quality of the proposi- 
 tion and the character of the predicate. Thus if I say " All 
 men are not angels," the obverse will be "All men are not- 
 angels." A better illustration, because of the negative term 
 "unpleasant," is, "All pain is not pleasant," the obverse of 
 which is " All pain is unpleasant," assuming for the present 
 that "not pleasant" and "unpleasant" are the same. This 
 result can be brought about by the more complicated process 
 described in the definition, which is really implied in the case 
 just mentioned. Thus, if I say "No pain is pleasant," I can 
 obvert, as required by the definition, in negating the copula 
 and predicate, as follows, " No pain is not not-pleasant," or 
 " No pain is not unpleasant." But this is very awkward, and 
 as two negatives make an affirmative by cancelling each other, 
 we assume this effect and have either the original proposition 
 or its obverse, as, " All pain is unpleasant." A negative propo- 
 sition is therefore most conveniently obverted by transferring 
 the negative particle to the predicate, which changes its qual- 
 ity. In regard to obversion, therefore, it is noticeable that it 
 can be applied to all four forms of propositions A, E, I, 
 andO. 
 
 4th. Contraversion or Contraposition. This process is 
 usually called by the latter name, but a few logicians have used 
 the former term, which seems decidedly preferable to the lat- 
 ter, because it indicates to some extent by the very name the 
 nature of the process involved. Jevons, using the term Con- 
 traposition, calls it a form of conversion, but this is only 
 partly true, because the conversion takes place after the pro- 
 cess of obversion has been performed, as will be shown after 
 defining what the process is. 
 
 Contraversion, or Contraposition, consists in the negation of 
 the copula and of the predicate Krith conversion. That is, we 
 negate the copula or proposition and the predicate, and then 
 convert. It amounts to the same thing to take the negative 
 of the predicate for the subject of the contraverse, and deny 
 the connection between it and the subject of the contravertend 
 if the contravertend be affirmative, and affirm the connection 
 11
 
 162 ELEMENTS OF LOGIC 
 
 if the contravertend be negative. This can be better explained 
 by an example. Take the proposition "All men are mortal." 
 By the very terms of this judgment the class "men" is wholly 
 included in the class " mortal," as in Fig. 5 or Fig. 14. Hence 
 it is necessarily excluded from everything outside the circle 
 "mortal." I can therefore affirm that "All men are not in the 
 class of those who are not mortal ; " or more briefly, that " No 
 men are immortal." By simple conversion of E we get " No 
 immortals are men." But noticing that the inclusion of 
 "men " among the " mortal" excludes those who are not mor- 
 tal from the class "men," I may as well affirm that fact directly, 
 and hence from the original I can infer at once, from " All 
 men are mortal," that "All not-inortals are not men." The 
 result is the same by either process, and hence we have only 
 to apply the principle to any similar proposition. Thus "All 
 oaks are trees," when contraverted will be " All not-trees are 
 not oaks," etc. We seem in all such cases to reach the result 
 without any roundabout process. But nevertheless that of 
 obversion is actually involved, and it was the failure to remark 
 this fact which has led Jevons to deny contraversion or contra- 
 position of I and O, and admit it only in an indirect way for E. 
 But if we observe that obversion is implied in the process, 
 to be followed by the conversion of the obverse, we shall find 
 that contraversion is applicable to E and O as well as to A, and 
 in the same manner, but is not applicable to I because its ob- 
 verse, O, cannot be converted. Thus take a case in E, " No 
 Caucasians are negroes," or " All Caucasians are not negroes." 
 Obverting this according to rule we have, " All Caucasians are 
 not-negroes," and then converting we have, " Some not-negroes 
 are Caucasians." Or, "No men are perfect." Obverted it is 
 " All men are not-perfect," and converted, " Some not-perfect 
 (imperfect) beings are men." Then again take O, " Some men 
 are not Americans." Obverting we have, " Some men are not- 
 Americans," and converting we have, "Some not-Americans are 
 men." It will be noticed in this that we have a result which 
 has been called Conversion by Negation, as already explained. 
 But having shown that O cannot in reality be converted, it re-
 
 IMMEDIATE INFERENCE 163 
 
 mains to observe that those who have assumed that it could 
 be converted in this indirect manner have confused the pro- 
 cess with contraversion or contraposition. O is, therefore, 
 contravertible. I remains to be considered. 
 
 Take the proposition " Some men are wise." Obvert it and 
 we have "Some men are not not-wise (foolish)." Thus we 
 have proposition O, " Some men are not foolish," and as we 
 have seen that O cannot be converted, we can proceed no 
 further with the case. I, therefore, cannot be contraverted. 
 A, however, is contraverted into E ; E into I ; O into I, and I 
 not at ah 1 . It might be added, also, that A can be contra- 
 verted into O as well as E, because O is the subalternate of E, 
 and so is equally true with it.* 
 
 * The first thing to be remarked about Contraversion is, that, unlike 
 Conversion, the process involves a change in the quality of the judgment, 
 a change from the affirmative in the contravertend to the negative in the 
 contraverse, and from the negative in the contravertend to the affirmative 
 in the contraverse. The accompaniment of this fact and the cause of it, 
 perhaps, is the second incident, which is that there is a change of matter 
 as well as form in the proposition. The subject in the contraverse is new 
 matter, and we may well question the validity of the whole process, since 
 immediate inference aims to deduce from a content already given noth- 
 ing more than it contains ; that is, immediate inference purports to give 
 the same kind of matter, not always the same quantity, in the conclusion 
 that is given in the premise. That new matter appears in contraversion 
 or contraposition is evident from the fact that in the original proposition 
 nothing is said about the negative of the predicate, and it may seem gra- 
 tuitous to introduce it into the contraverse. Thus, ' ' All men are mortal " 
 asserts nothing about the "not-mortal," or the "immortal." Why, then, 
 assert anything about them in the contraverse ? Again, a more important 
 objection is that the inference can be at best only formal, because we do 
 not know from " All men are mortal " that there are any " not-mortals." 
 Undoubtedly we can assert that <k All not-mortals are not men, "but if 
 we suppose that the contravertend expresses a real and material fact, 
 does it follow that the contraverse also expresses a real fact ? If so, it 
 would imply that there are real " not-mortals," about which, as a fact, 
 we know nothing. Of course, if there are any beings other than "mor- 
 tal," the contraverse will be materially true, but this existence of them 
 cannot be inferred from the original proposition. In immediate inference 
 the conchision should be the same really and materially as the premise ; 
 that is, it should be materially as well as formally true if the premise is
 
 164 
 
 ELEMENTS OF LOGIC 
 
 Conversion and Contraversion may be compared in the fol- 
 lowing table, and it will be seen at once how they contrast 
 with each other : 
 
 Original 
 Proposition. 
 
 Converse. 
 
 Impossi- 
 ble. 
 
 Contraverse. 
 
 Impossi- 
 ble. 
 
 All S is P . A. 
 
 Some S is P . I. 
 No S is P . E. 
 
 Some S is not P . O. 
 
 Some P is S . I. 
 
 Some P is S . L 
 No P. is S . E. 
 Some P is not S . O. 
 (None.) 
 
 A. E. O. 
 A. E. O. 
 
 A. X* 
 
 A, E, I, O. 
 
 All not-P is not S . E. 
 Some not-P is not S . O. 
 (None.) 
 
 Some not-P is 8 . 1. 
 Some not-P is S .-I. 
 
 A, I. 
 A, E, I, 0. 
 
 A, E, O. 
 A, E, O. 
 
 First, it is noticeable that there is a change of quality in 
 Contraversion. Then where we convert A into I only, we 
 
 both. Of coarse, if the premise is only formal, or consists of pure con- 
 ceptions, the conclusion cannot assert more. But this only shows that 
 we cannot go beyond the matter of the original proposition. What, then, 
 becomes of the validity of Contraversion ? 
 
 The only possible reply to this question is, that we are always assured 
 by the law of contradiction between the predicate and its negative that 
 they are mutually exclusive, and that what is affirmed of one can be de- 
 nied of the other, and vice versa ; and second, that practicutty there are 
 real existences, in the case of the largest number of propositions, which 
 belong to a class that is the negative of the predicate. But this reply is 
 not satisfactory, because the law of contradiction, as mentioned, can only 
 give a formal inference and not a material one, and because the principle 
 does not justify a material inference where other objects than those in 
 the predicate are not known to exist ; and their existence is not implied 
 by the material truth of the original. However, it is to be remarked 
 that, if any objects exist which are excluded from the predicate of the 
 original proposition, the inference will then be material!}' true, and that 
 this inference is contingent upon that condition, including, on the one 
 hand, the existence of certain objects other than the predicate, and, on 
 the other, the exclusion or contradiction between them and the given 
 predicate. We, therefore, conclude from the existence of this condition 
 that Contraversion is, at least nuiteriatty considered, a conditional infer- 
 ence ; not conditioned upon the material truth of the contravertend, but 
 upon the material existence of data that are excluded from the predicate 
 of the original, or contravertend. In conversion and obversion the con- 
 verse and obverse are always materially the same in kind as the proposi- 
 tions from which they are drawn, so that, if the originals are materially 
 true, the derivatives are also. But in Contraversion, as we have seen, 
 the material truth of the contraverse is not deducible from anything act-
 
 IMMEDIATE INFERENCE 165 
 
 can contravert it into both E and O, as we can convert E 
 into both E and O, and contravert it into I only. Again, as I 
 can be converted into I, and O cannot be converted at all, so 
 I cannot be contraverted, and O can be into I. The impossible 
 cases denote that the forms included under that head cannot 
 be derived from the original proposition by the process indi- 
 cated. 
 
 In the practical application of Contraversion we must be 
 careful not to confuse nego-positive with infinitaled negative 
 conceptions. Thus if we say that " All just acts are expedient," 
 the contraverse or contrapositive is, " All not-expedient acts 
 are not just." But we are not entitled to say "All not- expedient 
 acts are unjust," because all that we know from the original 
 assertion is that they are excluded from the " just," and the 
 negative of this, which is " not-just," includes the "unjust," 
 and those which are neither just nor unjust, namely, indif- 
 ferent acts, such as physical actions, and we do not know from 
 the original whether the " not-expedient " acts are excluded 
 from those which are neither just nor unjust, or included in 
 the " unjust." We only know that they are excluded from the 
 "just." Besides, to say that "All not-expedient acts are un- 
 just " is to produce a proposition in A, when the contraverse 
 should be E. The conception "unjust " is not necessarily co- 
 extensive with the negative " not-just," which is infinitated. 
 If it were, there would be no objection to their identification, 
 and then the original contraverse, "All not-expedient acts are 
 
 ually affirmed in the contravertend, but is conditioned solely upon the 
 existence of objects other than the predicate of the latter. It may then 
 be regarded as formally true, but materially indeterminate. The same 
 will be true of Inversion, of which we have yet to speak, and would be 
 true of Obversion except for the fact that it expresses merely in a negative 
 way the same as the original proposition. In Conversion, however, the 
 converse is both formally and materially true ; that is, materially true 
 when the convertend is so. The following is a classification of immediate 
 inferences based upon the above considerations : 
 
 Immediate | Formally and materially true { ^^^ 
 I Formally true, but materially indeterminate
 
 166 ELEMENTS OF LOGIC 
 
 not just," could be obverted into " All not-expedient acts are 
 unjust " (not-just). But only on that condition can it be 
 done. The same remarks apply to the substitution of inex- 
 pedient for "not-expedient." A better and moi-e self-evident 
 illustration of what is here maintained can be seen in the 
 proposition " All human actions are free," the contraverse of 
 which must be, " All not-free actions are not human," not " All 
 not-free actions are inhuman." Or again, " Human kindness 
 is a virtue," the contraverse of which is " All that is not virtue 
 is not human kindness," not " All that is not- virtue is in- 
 human kindness." The absurdity of the last is palpable, but 
 it is precisely what we have to guard against. 
 
 5th. Inversion. Inversion is not often treated by logicians, 
 and it is not a process of any practical importance. Besides, 
 it has first to be obtained through such combinations of the 
 three previous processes as not to be apparent at first sight. 
 But the fact that it is possible requires mention in a com- 
 plete treatment of immediate inferences. Inversion is the 
 process of inferring from one proposition another which shall 
 contain for its subject the negative of the subject in the orig- 
 inal, and for its predicate the predicate of the original. The 
 result can be obtained in more than one way. We may alter- 
 nate conversion and obversion, or use contraversion with 
 either or both the other processes. As contraversion, how- 
 ever, is merely a combination of conversion and obversion, we 
 may use only the former method. We may begin with either 
 conversion or obversion. 
 
 Take the proposition "All horses are animals." Convert 
 and we have " Some animals are horses." Then obvert this 
 result, which gives " Some animals are not not-horses." Here 
 comes the turn for conversion, but as the proposition is O we 
 can proceed no further. But if we begin with obversion the 
 first transformation gives us "No horses are not-animals," 
 then convert and we have "All not-animals are not horses." 
 Obvert again and we have "All not-animals are not-horses" 
 (remembering that obversion is only a change of quality in the 
 proposition), and then converting this obverse we have " Some
 
 IMMEDIATE INFERENCE 
 
 167 
 
 not-horses are not-animals." Obverting this again we have 
 "Some not-horses are not animals," which is the required 
 proposition, having the negative of the original subject for its 
 subject, and the original predicate for its predicate. 
 
 If we take proposition E, " No men are quadrupeds," and 
 begin with conversion, we have " No quadrupeds are men." 
 Obvert this and we get " All quadrupeds are not-men," and 
 applying conversion we have " Some not-inen are quadru- 
 peds," the required inverse. If we begin with ob version we 
 shall find that the result cannot be obtained. 
 
 Without going through the process with I and O, which the 
 student can do for himself, we shall content ourselves with 
 the assertion that they cannot be inverted. A and E are the 
 only two invertible, A into O, and E into I. It must be re- 
 marked, however, that the material truth of the conclusion in 
 either case is dependent upon the same conditions as in con- 
 traversion. Take the case of the inversion of A into the form 
 " Some not-S is not P," from " All S is P," and it is clear 
 from "All S is P" that, so far as we know, all things outside of 
 S are included in P, and if so, we could hardly conclude that 
 some of them were not so included, and hence " not P." But 
 if there are any objects outside of P, then it is clear from the 
 subalternate of the contraverse of " All S is P," that " Some 
 not-S is not P." But it can be materially true only on that 
 condition. Hence I have placed inversion in the class of in- 
 ferences which are materially indeterminate. It has no prac- 
 tical importance, however, and is applicable only to A and E, 
 and not to I and O. It should be observed that it involves a 
 change of quality in the deduced proposition. All four pro- 
 cesses can be compared in the following table : 
 
 Original. 
 
 Converse. 
 
 Contraverse. 
 
 Inverse. 
 
 All 8 is P . A. 
 No S is P . E. 
 
 Some S is P . I. 
 Some S is not P . O. 
 
 Some P is S . I. 
 
 No P is S . E. 
 Some P is not S . O. 
 Some P is S . I. 
 (None.) 
 
 All not P is not S . E. 
 Some not P is not S . O. 
 
 Some not P is S . I. 
 (None.) 
 Some not P is S . I. 
 
 Some not-S is not P . O. 
 
 Some not-S is P . I. 
 (None.) 
 (None.)
 
 168 ELEMENTS OF LOGIC 
 
 6th. Contribution. I have called two processes of imme- 
 diate inference by this name because the same general princi- 
 ple is involved in both. They are called Immediate Inference 
 by added Determinants, and Immediate Inference by Complex 
 Conceptions. The essential characteristic in common between 
 them is that what is affixed to the subject as a modifier may also 
 be affixed to the predicate in the same sense. What is contri- 
 buted to one may be contributed to the other. Its simplest 
 illustration is in mathematics. Thus if# a, x + I = a + I. 
 
 (a) Inference by Added Determinants. This consists in 
 merely adding some adjective or similar term to both subject 
 and predicate. Thus if " A brick house is a dwelling," " A 
 good brick house is a good dwelling." There is one important 
 limitation to this process, however, and it is that the addition 
 must represent the same quality and quantity to both subject 
 and predicate. Thus, " Dogs are quadrupeds " may be modi- 
 fied by adding " useful " to both terms, but not by saying 
 " The largest dogs are large quadrupeds." Nor can we infer 
 from " Dwarfs are men," or " A cottage is a building," that 
 " Large dwarfs are large men," or " A large cottage is a large 
 building." The reason for the error in these cases is that 
 the addition of quantity in each case is not the same. When 
 the superlative degree is added it implies a comparison be- 
 tween the special individuals mentioned or described and the 
 whole class which is thus limited, so that the same addition is 
 not made to subject and* predicate. It is the same with such 
 terms as " large," " long," '" small," " short," etc., in certain 
 cases, namely, when an unequal comparison is suggested. 
 Terms, therefore, expressing quantity must be used carefully 
 in this form of inference. Terms expressing quality, however, 
 can be used with perfect freedom, provided they are not used 
 equivocally. 
 
 (b) Inference by Complex Conception. This consists in the 
 addition of words, phrases, or clauses that make the subject 
 and predicate complex in their nature. Thus from " Apples 
 are fruit" I may infer that "A box of apples is a box of fruit," 
 or from " Pigeons are birds," " A flock of white pigeons is a
 
 IMMEDIATE INFERENCE 169 
 
 flock of white birds," etc. But here again we have to be 
 warned against the addition of expressions which indicate the 
 addition of a greater quantity to one side than to the other. 
 Thus we cannot infer from " Voters are men," that " The ma- 
 jority of voters is the majority of men." The same principle 
 holds here as in inference by added determinants. 
 
 Inferences by contribution have little importance, as they 
 are seldom employed, and if fallacies occur in this use they 
 are so easily detected and corrected that little provision re- 
 quires to be made against their occurrence. 
 
 7th. Antithesis. There is a very common error in dealing 
 with propositions, which it is necessary to remark. We have 
 seen that an exclusive proposition implies a complementary 
 opposite, and it frequently occurs that we infer from an or- 
 dinary universal, A or E, proposition one which is its opposite 
 both in regard to the subject and predicate. Thus, if we assert 
 that " All white objects are visible," many persons think them- 
 selves entitled to infer that "All not-white objects are not 
 visible;" or from "All good men are wise," that "All bad 
 men are not wise." But such an interpretation and inference 
 is a case of illicit distribution of the predicate. If the propo- 
 sition were either exclusive or a definition, the inference might 
 be drawn. Formally, however, no one ever has the right so 
 to treat it. In ordinary antithesis this relation is assumed, as 
 is so frequent in the Book of Proverbs ; for example, " In the 
 multitude of words there wanteth not transgression ; but he 
 that refraineth his lips doeth wisely." And in all ordinary 
 cases where subject and predicate are coterminous, the same 
 can be done. In such instances we are entitled to infer the 
 complementary opposite. But we should not generalize a 
 privilege of this kind and apply it to all universal propositions. 
 In the universal propositions as defined, we have no data for 
 asserting anything outside the subject, unless we assume 
 something to exist outside the predicate, which we have no 
 right to assume from the given proposition. But if it be 
 known or assumed that something exists outside the predicate, 
 we may then infer a contrapositive or the contraverse, and the
 
 170 ELEMENTS OF LOGIC 
 
 limited converse of this which has been shown to be the inverse 
 of the original proposition. Any other inference than those 
 based upon this supposition, is an illicit assumption of distri- 
 bution of the predicate when the original proposition is affirm- 
 ative, and an illicit distribution of the negative of the subject 
 when the proposition is negative. Thus, in the first case, if 
 we infer from "All men are mortal " that "All not men are not 
 mortal," we assume a distribution of the predicate which is not 
 stated in the original. In the second case, if we infer from 
 "No men are quadrupeds" that "All not-men are quadru- 
 peds," we assume a distribution of the negative of the subject 
 " men." This cannot be true, because although we know by 
 the infinitation of the concept "not-men" that some things 
 excluded from " men " are included in " quadrupeds," there is 
 nothing in this idea to justify the assertion that all "not- 
 men " are so included. The subalternans cannot be inferred 
 from the truth of the subalternate. This error may be called 
 illicit contrast or illicit antithesis.* 
 
 * References on Immediate Inference are the following : Keynes : For- 
 mal Logic, Part II., Chaps. IIL, IV. and V.; Jevons: Studies in Deductive 
 Logic, Chap. V.
 
 CHAPTER XL 
 
 PRINCIPLES OF MEDIATE REASONING 
 
 1st. Definition. Mediate inference, as already defined, is 
 reasoning by means of a middle term. A middle term is one 
 which is compared with two others, and on the_ground of 
 which a connection or relation can be established between the 
 two terms thus compared with it. Suppose I wish to prove 
 that " Machines are useful." This is the conclusion, and its 
 proof will depend upon two other propositions containing the 
 two terms in the conclusion and comparing them with a mid- 
 dle term. If, therefore, I can obtain assent, first, to the prop- 
 osition that " Mechanical arrangements for the application of 
 power are useful," and second, to the proposition that "Ma- 
 chines are mechanical arrangements for the application of 
 power," the conclusion will follow as a matter of course. The 
 terms " machines " and " useful " are compared with the single 
 term " mechanical arrangements," etc., and found to agree 
 with it, and so it must be inferred that they agree with each 
 other. The conclusion is thus mediated by a middle term. 
 The reasoning involved in this process is called the Syllogism, 
 or Syllogistic Reasoning. It differs from immediate inference 
 in requiring more than one proposition, and more than two 
 terms to start from. The two rules which determine or con- 
 dition the character of the syllogism are : 
 
 (a) Every syllogism should have three, and only three terms. 
 
 (b) Every syllogism should have three, and only three proposi- 
 
 tions. 
 
 The three terms are called the Major, the Middle, and the 
 Minor terms. Of the three propositions two are called the 
 Premises, and one the Conclusion. The Major term is the 
 predicate of the conclusion ; the Minor term is the subject of 
 the conclusion. The Middle term is found only in the prem-
 
 172 ELEMENTS OF LOGIC 
 
 ises, and may be either the subject or the predicate as the 
 case requires. The Major Premise contains the Major and 
 Middle terms, and the Minor Premise the Minor and Middle 
 terms. Without the conclusion being expressed there is no 
 absolute rule for determining which of the premises is the 
 major and which the minor. We are at liberty to choose either 
 of them and to try the consequences by the laws of the syllo- 
 gism. The usual method of the logician is to state the major 
 premise first, the minor premise second, and the conclusion last. 
 But in common discourse and reasoning the minor premise 
 may come first, and the major second ; or the conclusion may 
 even come first, and the two premises after it, as reasons for 
 its truth. As an illustration of a syllogism having the minor 
 premise first, the following is an instance : 
 
 The earth revolves around the sun. 
 
 Bodies revolving around the sun are planets. 
 
 Therefore the earth is a planet. 
 
 It is evident from the conclusion, and from what has been 
 said about the terms which it contains, that the minor premise 
 is the first and the major premise the second. As a case of 
 the conclusion being stated first, the following is an instance : 
 
 " Comets consist of matter, for they obey the law of gravita- 
 tion, and whatever obeys the law of gravitation is matter." 
 
 The symbols which we shall employ for the several terms of 
 the syllogism are the letters S, M, and P, with a slight modifi- 
 cation of their previous usage, but yet essentially the same. 
 S shall stand for the minor term, which is the subject of the 
 conclusion, and P for the major term, which is the predicate of 
 the conclusion. They thus stand respectively for subject and 
 predicate, as heretofore, but only in the conclusion, since S or the 
 minor term is not always the subject in the minor premise, 
 and P, the major term, is not always the predicate of the major 
 premise. M will stand for the middle term. The combination 
 of these terms will represent the premises and conclusion. 
 
 Term*. Propositions. 
 
 M = Middle term. M is P = Major Premise. 
 
 8 = Minor terra = Subject of Conclusion. S is M = Minor Premise. 
 
 P = Major term = Predicate of Conclusion. S is P = Conclusion.
 
 PRINCIPLES OF MEDIATE REASONING 173 
 
 2d. Rules of the Syllogism. The rules of the syllo- 
 gism are classified in slightly different forms, but substan- 
 tially in the same manner. It is usual to include among them 
 the two which I have already mentioned, and to include in one 
 of them a statement about the ambiguous middle. But I pre- 
 fer to specify two classes of rules affecting different aspects of 
 the syllogism. The first two already mentioned, and with 
 them a precaution against an ambiguous middle, relate to the 
 logical matter of the syllogism. The remaining six relate to 
 tke quantity and quality of propositions, and the distribution 
 of terms, as effecting the conclusion. We shall classify them 
 according to these divisions. 
 
 (A) Rules affecting the logical matter of the syllogism. 
 
 1. Every syllogism must have three terms, and only 
 
 three. 
 
 2. Every syllogism must have three propositions, and 
 
 only three. 
 
 3. No term shall be used ambiguously. 
 
 The violation of the first of these rules gives rise to the fal- 
 lacy known as Quaternio Terminorum, or the Fallacy of Four 
 Terms. The third rule is a corollary of the first, because an 
 ambiguous term is a word with two meanings, and hence is 
 equivalent to two terms, so that it is only a form of introduc- 
 ing four terms into the syllogism, producing in effect a fallacy 
 of Quaternio Terminorum. 
 
 (B) Eules affecting the quantity and quality of propositions 
 
 and the distribution of terms. 
 
 4. The middle term must be distributed at least once 
 
 in the premises. 
 
 5. No term must be distributed in the conclusion 
 . which was not also distributed in one of the 
 
 premises. 
 
 6. No conclusion can be inferred when both premises 
 
 are negative. 
 
 7. If one premise be negative the conclusion must be 
 
 negative and, vice versa, in order to prove a negative 
 conclusion one of the premises must be negative.
 
 174 ELEMENTS OF LOGIC 
 
 8. No conclusion can be drawn when both premises 
 
 are particular. 
 
 9. If one of the premises be particular the conclu- 
 
 sion must be particular. 
 
 The last two are frequently considered as corollaries of the 
 preceding. All of them are so important that they should be 
 thoroughly committed to memory by the student for constant 
 reference. 
 
 3d. Symbolization of the Syllogism and Representa- 
 tion of Formal Fallacies. The syllogism can be symbol- 
 ized in the same manner as propositions, except that it re- 
 quires a larger number of circles for each syllogism, because 
 there are three terms and three propositions. The represen- 
 tation depends upon the quantitative relation between subject 
 and predicate, and can be used to indicate whether any two 
 terms agree or exclude each other. When the relation be- 
 tween subject and predicate is properly observed, and when 
 the proper distribution of terms is made, the conclusion will 
 be valid. But if the middle term be not distributed in one of 
 the premises, or if the major or minor terms be distributed in 
 the conclusion when they are not distributed in the prem- 
 ises, a fallacy occurs, which is called a Formal Fallacy, 
 because it is a violation of the formal principles of the syllo- 
 gism. Hence such reasoning will be formally invalid. The 
 conclusion may happen to be materially true, but under the cir- 
 cumstances it has not been correctly obtained or deduced 
 from the premises, as a conclusion. These formal fallacies are 
 called Illicit Processes, respectively, of the major, middle, and 
 minor terms. If the major term be distributed in the conclu- 
 sion and is not distributed in the premise, the inference is 
 an Illicit Process of the Major Term. If the middle term be 
 not distributed at least once in the premises, the inference is 
 an Illicit Process of the Middle Term. If the minor term be 
 distributed in the conclusion and is not distributed in the 
 premise, the inference is an Illicit Process of the Minor Term. 
 
 There are two ways in which we can represent symbolically 
 the valid and the illicit processes of formal inference. We
 
 PRINCIPLES OF MEDIATE REASONING 175 
 
 shall employ both of them. We must remember the distinc- 
 tion, however, between the formal truth or falsehood of the in- 
 ference and the material truth or falsehood of the propositions. 
 The two may have no relation to each other. Thus we may 
 reason correctly from false premises, or falsely from correct 
 premises. Hence, as we are dealing here only with the manner 
 of getting the conclusion, we may set aside all questions about 
 the nature of the premises materially considered. The stu- 
 dent must always be prepared to detect the difference between 
 the truth of the conclusion and the validity or invalidity of the 
 process by which we obtain it. It is a common fault of our 
 minds to accept the reasoning if we accept the conclusion, 
 and to impeach the reasoning if we do not accept the conclu- 
 sion. But this is a fallacy of itself, and we must dispel such 
 assumptions from our minds, or we shall not be able in all 
 cases to detect the true' or false character of the reasoning. 
 On the other hand, we must not be induced by the purely 
 formal correctness of the reasoning to accept the material 
 truth of the conclusion on that account. The most perfect 
 reasoning is that in which the premises are materially correct 
 and the reasoning formally correct. This will give a conclu- 
 sion which is both formally and materially valid. Now to 
 illustrate only the formal process, and to symbolize it accord- 
 ingly, we shall take first a valid case, the ordinary syllogism : 
 thus "Metals are elements," "Iron is a metal," "Therefore 
 iron is an element," will be represented by the circles in Fig. 
 18, and the distribution of terms in the accompanying diagram. 
 
 
 FIG. 18. Representation of Distribution. 
 
 In Fig. 18, " metals " are represented as included in the class 
 "elements," and "iron" in the class "metals." It must, 
 therefore, follow that " iron " is included in the class " ele- 
 ments," as the lesser is included in the greater, or the part in
 
 176 
 
 the whole. The diagram shows that the middle term has been 
 distributed at least once, and that no term has been distrib- 
 uted in the conclusion which was not distributed in the prem- 
 ises. The propositions are, of course, affirmative. But we 
 may take a case where one of them is negative, and so have 
 an instance of exclusion. Thus, "Iron is a metal," and 
 " Wood is not a metal" Therefore " Wood is not iron." This 
 syllogism will be represented also by three figures, but one of 
 them is excluded from the other two, as required by the nega- 
 tive proposition. 
 
 (?) = M 
 
 x 
 
 (D x 
 
 Representation of Distribution. 
 
 FIG. 19. 
 
 Fig. 19 represents " iron " as included in class " metal," and 
 " wood " as excluded from the same class, so that it must be 
 excluded from all contained within the class "metal." The 
 diagram shows the observation of the rules in another form. 
 
 Instead of a dotted line we might use a smaller circle in 
 either case, as is usual, and perhaps better, in most instances. 
 When the predicate is not exhausted in the subject the com- 
 plete smaller circle is much more convenient, and hence in the 
 future it will be employed, especially in testing the forms of 
 illicit process. 
 
 Had it been used in the previous symbols Fig. 18 would 
 stand as Fig. 20, and Fig. 19 as Fig. 21. 
 
 Fio. 20. 
 
 FIG. 21. 
 
 The student or the teacher may vary the forms of figures to 
 suit other valid modes of reasoning. We turn next to test
 
 PRINCIPLES OF MEDIATE REASONING 177 
 
 and represent the forms of illicit process. Take first the case 
 of illicit middle. It occurs in the following syllogism, " Men 
 are mortal." "Horses are mortal" "Therefore horses are 
 men." It is represented in Fig 22. 
 
 P = 
 
 FIG. 22. Representation of Distribution. 
 
 As both propositions of the premises are affirmative, the 
 predicate, which in this case is the middle term, is not dis- 
 tributed, and hence an attempt to draw the conchision, 
 " Horses are men," or S is P, is illicit or invalid, because a vio- 
 lation of Rule 4. It is represented by Fig. 22, in that both 
 " horses" and "men" are included in the class mortal, but we 
 have no right to infer therefrom that " horses " are included 
 in the class " men." The fact that we know it is not the case 
 helps us to perceive thfe impossibility of it. But even in cases 
 where the conclusion would be true as a fact, we should have 
 no right to draw it in this manner. Thus we may say, " All 
 Americans are men," and " All Virginians are men," but we 
 should have no right to infer that " All Virginians are Ameri- 
 cans," although it is true as a matter of fact. The fallacy here 
 is that of an Illicit middle term. 
 
 Illicit process of the major may occur as follows, and is 
 likely to be a very frequent fallacy. If from " Men are mor- 
 tal," and "Horses are not men," we infer that "Horses are not 
 mortal," we have more in our conclusion than is 
 
 ' /TT/rN _ p 
 
 given in the premises. Fig. 22 will also represent v*y ~ 
 
 this form. But the diagram for distribution of (g> x (M) 
 terms will be different. It will be as follows, and 
 
 represents the major term as distributed in the (&) * (P) 
 
 conclusion when it is not distributed in the prem- Representation 
 
 of Distribution. 
 
 ises. The major premise is affirmative, and hence 
 the predicate, which in this case is the major term, is not dis- 
 12
 
 178 ELEMENTS OF LOGIC 
 
 tributed. But the minor premise being negative, the conclu- 
 sion, according to Eule 7, must be negative. We have seen that 
 all negative propositions distribute the predicate, and as this 
 is the major term in the conclusion, it must be distributed, if 
 drawn at all, although it is not distributed in the major prem- 
 ise. The error is, therefore, an illicit process of the major term. 
 Fig. 22 represents it by showing that the class " horses " is 
 excluded from the class "men," as asserted by the minor 
 premise, and yet is not excluded from the class " mortal," as 
 the conclusion would indicate. The reasoning is therefore a 
 violation of Rule 5. 
 
 Illicit process of the minor term is quite as easily repre- 
 sented. It occurs in the following syllogism: "All horses 
 are quadrupeds," and " All quadrupeds are animals." " There- 
 fore all animals are horses." It is represented in Fig. 23. 
 
 (P) = M 
 
 @ = S 
 
 Fia. 23. Representation of Distribution. 
 
 In the minor premise the predicate S, or "animals," is not 
 distributed, because the proposition is affirmative, but in the 
 conclusion it is distributed, because the proposition is univer- 
 sal, and thus to distribute it is violation of Rule 5. The im- 
 possibility of the conclusion is evident in the representation, 
 on the ground that the larger class cannot be included in the 
 smaller, or the whole in the part, as it is represented in the 
 conclusion actually drawn. Had we said " Some animals are 
 horses," the conclusion would have been valid as conform- 
 ing to all the rules. But the minor term is distributed in 
 the conclusion first drawn, and is not distributed in the 
 premise. 
 
 The violation of Rule 6 occurs in the following syllogism, 
 and is represented by either of Figs. 24 and 25, " No solids
 
 PRINCIPLES OF MEDIATE REASONING 179 
 
 are liquids," and " No metals are liquids." " Therefore no 
 metals are solids," or " Metals are solids." 
 
 Fid. 24. 
 
 Fia. 25. 
 
 X (M) 
 
 s)x@ 
 
 g) X @ or (|) = M 
 
 Diagram of Distribution. 
 
 Fig. 24 represents the premises as 
 negative propositions, in which the sub- 
 jects " solids " and " metals " are both 
 excluded from " liquids," and excluded 
 from each other. But their common 
 exclusion from the middle term does 
 not necessarily imply their exclusion from each other, as is ap- 
 parent in Fig. 25, where they are both excluded from "liquids," 
 and yet one of them is included in the other. All that is 
 necessary to make the premises negative is to have the major 
 and minor terms excluded from the same middle term, and 
 this leaves the connection between the major and minor terms 
 entirely indeterminate. We can, therefore, draw no conclusion 
 at all, either affirmative or negative, because there is nothing 
 said or denied in the premises to imply one or the other. 
 
 There is an apparent exception to Rule 6 in such cases as 
 the following, which require analysis. 
 
 That which is not wise cannot be useful. 
 
 Intemperance is not wise. 
 
 . . Intemperance cannot be useful. 
 
 Here it seems very apparent that we have negative premises, 
 and yet the mind does not wince at the conclusion, nor feel 
 that anything wrong has occurred in drawing it. The reason- 
 ing resists all attempts to invalidate it. But the reason for 
 this is, that although the propositions seem to be EEE, they 
 are in reality EAE of the first figure, and therefore valid. 
 The first proposition is made negative by the "cannot," but in
 
 180 ELEMENTS OF LOGIC 
 
 the second the nature of the middle term is such that in order 
 to have it the same in both premises the predicate of the mi- 
 nor premise must be negative, and the " not " in its real im- 
 port must go with the " wise." The logical subject in the 
 major premise is the whole thought expressed by the middle 
 term, which is " the not- wise acts," and in the minor premise 
 this conception can be brought out only by construing the 
 predicate in the same way, and so regarding it as equivalent 
 to " that which is not wise," or " one of those things which is 
 not wise," and therefore " a not-wise act," and we find that the 
 proposition becomes A with a negative conception for the 
 predicate. That is, we simply obvert the minor premise, so 
 that it becomes in reality affirmative. "Whatever we may think 
 of the form of the proposition, therefore, it becomes mate- 
 rially an affirmative one, or is thought as such, and the rea- 
 soning is determined accordingly. 
 
 The fact calls attention to the double use of the negative 
 particle in propositions which must be reckoned with in deal- 
 ing with them. It may be attached to the copula, denying 
 the connection between subject and predicate, and making the 
 quality of the judgment negative, or it may be attached to the 
 predicate, making the quality of the judgment affirmative and 
 the predicate negative. This is but a case of obversion, but it 
 shows that we may often mistake the mere form of a proposi- 
 tion for its matter. We must, therefore, be on our guard 
 against mistaking what is really a negative attribute for a neg- 
 ative assertion. 
 
 Rule 7 is proved and illustrated by Figs. 19 and 21. A viola- 
 tion of it would appear in the attempt to draw the affirmative 
 conclusion, " Wood is iron," because in the premises " wood " 
 is excluded from "metals," and so must be excluded from 
 whatever is contained in that class. 
 
 Rules 8 and 9 can be best tried and illustrated when we 
 come to consider the Moods and Figures of the syllogism in 
 the next chapter. The circles would take up more space than 
 is necessary, *while practically few fallacies are incident to the 
 8th Rule.
 
 CHAPTER XH. 
 
 / 
 
 MOODS AND FIGURES OF THE SYLLOGISM 
 
 1st. Moods. The Mood of a syllogism is that characteris- 
 tic of it which is determined solely by the quantity and quality 
 of its propositions. The Mood can never be separated from 
 the Figure, but it is not determined by the same properties. 
 It is necessary to consider both of them in order to ascertain 
 the valid and invalid forms of reasoning, but this result can 
 be most easily and most briefly attained by first considering 
 the Moods. Every syllogism, as we have seen, must contain 
 three propositions, and only three. But there are four forms 
 of propositions, namely, A, E, I, and O, from which to choose, 
 or which may be combined in various ways to produce the re- 
 quisite number for a syllogism. Thus it is possible to take all 
 three propositions in A. In such a case the major and minor 
 premises, and the conclusion, would each and all be proposi- 
 tions in A. Or, we might have one in A, one in E, and one in 
 O. It is conceivable that all three propositions should be 
 E, or I, or O. We happen to know, however, from Rules 6 
 and 8, that such forms are not valid ; but apart from those 
 rules we could not so decide the matter. Hence we must rep- 
 resent at present all the conceivable moods, and test them af- 
 terward. Apart from these rules, therefore, any combination 
 of three propositions is possible, as representing the conceiva- 
 ble ways in which a syllogism might be formed. If the three 
 propositions were in A, the syllogism for AAA would repre- 
 sent a mood in which the major premise was A, the minor 
 premise A, and the conclusion A. This order represents the 
 order of the premises and the conclusion. The mood AEO 
 would therefore mean that the major premise was A, the mi- 
 nor E, and the conclusion O. Now as the conceivable moods 
 may represent all possible combinations, either of the same 
 kind or of different kinds of propositions, AEO being one
 
 182 ELEMENTS OF LOGIC 
 
 mood an EAO being another, there will be a large number 
 of possible moods. When completed we find the possible 
 number to be 64. They appear as follows : 
 
 AAA AEA AIA AOA EAA EEA EIA EGA 
 
 AAE AEE AIE AOE EAE EEE EIE EOE 
 
 AAI AEI AQ AOI EAI EEI EH EOI 
 
 AAO AEO AIO AGO EAO EEO EIO EGO 
 
 IAA 
 
 IEA 
 
 HA 
 
 IOA 
 
 OAA 
 
 OEA 
 
 OIA 
 
 OOA 
 
 IAE 
 
 IEE 
 
 HE 
 
 IOE 
 
 OAE 
 
 GEE 
 
 OIE 
 
 OOE 
 
 IAI 
 
 IEI 
 
 III 
 
 IOI 
 
 OAI 
 
 OEI 
 
 on 
 
 OOI 
 
 IAO 
 
 EEO 
 
 no 
 
 IOO 
 
 GAG 
 
 OEO 
 
 OIO 
 
 GOO 
 
 But these are not all valid. Some of them violate one rule, 
 and some another. For example, EEA violates Eule 6 ; EAI 
 violates Rule 7 ; IOA violates Rules 7 and 8. By thus ap- 
 plying the rules of the syllogism to each one we find that 
 a large number of them are to be rejected as pseudo-moods, or 
 false moods, because no conclusion can be valid in them. In 
 this way 52 are rejected because they violate some one or two 
 of the Rules 6, 7, 8, and 9. Some of them also violate Rule 5, 
 but this law is not applied until we come to test the moods in 
 the Figures of the syllogism. There remain, therefore, 12 
 possibly valid moods ; we say possibly, because it is not proved 
 that they are valid when they are not found to conflict with 
 any of the above four rules. They must first be tested in 
 the Figures, and it will then be found that some of them are 
 valid and some are invalid in one or more of the Figures, and 
 one mood, IEO, proves to be invalid in all of them, on the 
 ground of an illicit process of the major term. The follow- 
 ing are the 12 moods, with IEO bracketed for the reason just 
 specified : 
 
 AAA EAE IAI OAO 
 
 AAI EAO [IEO] 
 
 AH EIO 
 
 AEE 
 
 AEO 
 
 AOO
 
 MOODS AND FIGURES OF THE SYLLOGISM 183 
 
 These remain to be tested by the Figures, and as there are 
 four Figures there will be 48 forms which have yet to be con- 
 sidered, four of which have to be rejected as involving LEO, 
 and leaving 44 cases in which the reasoning might be valid. 
 But as a large number are found to violate Rule 5, in one or 
 more of the Figures, they have to be rejected. What remains 
 will appear in a moment. 
 
 2d. Figures. The Figure of a syllogism is that character- 
 istic of it which is determined by the position of the middle 
 term. As there are two propositions, each with a subject and 
 predicate, in the premises of every syllogism, there are four 
 possible positions for the middle term. It may be the subject 
 of both, the predicate of both, the subject of the major, and 
 the predicate of the minor, or the predicate of the major and 
 subject of the minor premise. These positions are repre- 
 sented in the following diagrams : 
 
 1st Fig. 2d Fig. 3d Fig. 4th Fig. 
 
 M=P P=M M = P P = M 
 
 S = M S = M M = S M = S 
 
 S = P S = P S = P S=P 
 
 Each of the 12 moods can be tested in the four Figures, 
 making, as already said, 48 possible forms in all, which might 
 be valid, were not some of them violations of Rule 5 in one 
 or more of the Figures. Thus in mood AAA we may have 
 four different positions of the middle term, which is repre- 
 sented in the following diagrams, showing those in which it 
 is valid and those in which it is invalid : 
 
 let Fig. 2d Fig. 3d Fig. 4th Fig. 
 
 A@ = P A(p) = M A@ = P A(|) = M 
 A(g)=P A(S) = P Ar) = P A(g) = P 
 
 Valid. Invalid. Invalid. Invalid. 
 
 In the second Figure, as shown, AAA gives rise to the fal- 
 lacy of undistributed or Illicit Middle ; in the third Figure, 
 to the fallacy of Illicit Minor ; and also Illicit Minor in the
 
 184 ELEMENTS OF LOGIC 
 
 fourth Figure. It is therefore valid formally only in the first 
 Figure. Take another example, the mood AEE : 
 
 1st Fig. 2d Pig. 3d Fig. 4th Fig. 
 
 A @ = P A (g) = M A @ = P A (f>) = M 
 
 E X EX(M) E(M)X(S) E(M)x(S) 
 E(S)X(P) EX(P) EfSJX Ex(p) 
 
 Invalid. Valid. Invalid. Valid. 
 
 We find AEE valid only in the second and fourth Figures. 
 In the first and third Figures it gives rise to the fallacy of Il- 
 licit Major. By testing each mood in this way we reject a 
 large number of them as invalid in some Figures, although 
 valid in others. Only one is not valid in any of them, and 
 this, as remarked, is IEO. When all the valid moods, there- 
 fore, are selected out of the 44, we find there are 24 of them, 
 which are given below. Of this number 5 are called Weak- 
 ened Conclusions, because, although valid, they are of no prac- 
 tical use, because a particular conclusion is drawn when a uni- 
 versal one might as well have been drawn. They are placed 
 in brackets. 
 
 1st Fig. 
 
 2dFig. 
 
 3d Fig. 
 
 4th Fig. 
 
 AAA 
 
 AEE 
 
 AAI 
 
 AAI 
 
 All 
 
 AOO 
 
 All 
 
 AEE 
 
 EAE 
 
 EAE 
 
 IAI 
 
 IAI 
 
 [EAO] 
 EIO 
 
 [EAO] 
 EIO 
 
 EAO 
 EIO 
 
 EAO 
 EIO 
 
 [AAI] 
 
 [AEO] 
 
 OAO 
 
 [AEO] 
 
 Several other lists with a slightly different order might be 
 given, but the present one is probably convenient enough 
 for the memory. The arrangement has been made as nearly 
 as possible to indicate in the same line, and with as little space, 
 those moods which are valid in more than one Figure, and at 
 the same time to collect allied moods together, so far as per- 
 missible. But since it may appear a difficult task to memor- 
 ize the whole 24 forms, and to locate them in their right Fig- 
 ures. I have adopted a mnemonic system which may be con-
 
 MOODS AND FIGURES OF THE SYLLOGISM 185 
 
 venient for many students, although I do not find it necessary 
 to use it for myself. The meaning of the system can be easily 
 explained. The vowels indicate the Moods; the consonants 
 B, C, D, and M represent, respectively, the first, second, third, 
 and fourth Figures, M being chosen instead of F, solely for 
 the sake of euphony. L and N are simply connecting conso- 
 nants where we cannot use B, C, D, or M. There are only 
 eleven words in all in the system, corresponding to the 
 eleven valid moods. Each word indicates the mood and Fig- 
 ures in which a given form is valid. They are as follows : * 
 
 Balana Caleme Dilami 
 
 Badami Calemo Dolano 
 
 Badini Calono 
 
 Becane 
 
 Becadom 
 
 Becidom 
 
 The shortest possible list of the valid moods is the following, 
 and it will not appear so formidable to the memory as the 
 previous instances. It ignores the weakened conclusions be- 
 cause it is apparent that I can be drawn wherever A is drawn, 
 an O wherever E is drawn. The numbers following each mood 
 signify the Figures in which it is valid. These moods can 
 probably be best remembered without resort to a mnemonic 
 system. AAI, 3. 4 and EAO, 3. 4 are omitted. 
 
 AAA, 1. AEE, 2. 4. EAE, 1. 2. IAI, 3. 4. 
 
 AH, 1. 3. AGO, 2. EIO, 1. 2. 3. 4. OAO, 3. 
 
 In examining the list of valid moods and Figures, we ob- 
 serve that the first Figure is the only one which can have A for 
 a conclusion, and is the only Figure in which all of the four 
 propositions A, E, I, O can be proved. In the second Figure 
 we observe only negative conclusions. No affirmative conclu- 
 
 * This is not intended to take the entire place of the usual mnemonic 
 verses to be considered under the reduction of the Moods and Figures, 
 but only to aid in the detection of those which are valid in more than one 
 Figure. Those who adopt the common system need not adopt the one I 
 have given.
 
 186 ELEMENTS OF LOGIC 
 
 sion is possible in this Figure because the premises would 
 have to be affirmative in order to give it. But since the middle 
 term is the predicate in both premises of this Figure, it is un- 
 distributed whenever the proposition is affirmative, and hence 
 an inference would be a case of illicit middle. The third 
 Figure gives only particular conclusions. If the major prem- 
 ise be negative, a universal conclusion would be an illicit 
 minor ; if the minor premise be negative, any conclusion would 
 be an illicit major. Hence only particular conclusions can be 
 drawn, and that only when both premises are affirmative, or 
 the major premise negative, except in the Mood OAO. The 
 fourth Figure is peculiar in being easily reducible into the 
 first Figure, and hence it is of little practical use. We have 
 only to change the position of the premises, or mutate them, 
 as it is called, in order to convert the fourth Figure into the 
 first. It requires mention, however, because certain moods 
 are valid in it which are not valid in the first. 
 
 In regard to the practical importance and usefulness of the 
 Figures there is some difference between them. We have al- 
 ready remarked that the first Figure is the only one to give con- 
 clusions in all four propositions, A, E, I, and O ; the second will 
 give them in E and O ; the third in I and O ; and the fourth 
 in E, I, and O. The first Figure, therefore, is the only one to 
 give universal affirmatives, or A. This fact makes it the most 
 important and useful of the forms of reasoning, because the 
 demands of human belief and conduct require universal truths 
 of some kind which do not give exceptions such a place and 
 influence as to impair the value of general principles. Thus if 
 the needs of society require that " All men should do right," 
 and this proposition was not accepted on its bare statement, 
 we should require to prove it, and upon premises of which it 
 was a part or an equal. But if we could prove from those 
 premises only that " Some men should do right," the truth 
 would be so indefinite that there would be no telling whether 
 the obligation was incumbent upon a sufficient number of 
 men to make the principle worth asserting. If the " some " 
 means two or three, or a hundred, the immunities which the
 
 MOODS AND FIGURES OF THE SYLLOGISM 187 
 
 remaining majority may have, so far as premises and con- 
 clusion are concerned, make the principle perfectly ineffect- 
 ive, especially as the particularity and indefiniteness of the 
 conclusion is such as to make it, practically, a universal nega- 
 tive ; that is, if we only prove that " Some men should do right," 
 and are not able to specify what " some " or portion of them 
 are under that obligation, the indefiniteness of the assertion 
 prevents us from asserting the obligation of anybody in par- 
 ticular. But if we can prove the universal, the conclusion ap- 
 plies definitely to everybody. Hence the first Figure has an 
 importance corresponding to its efficiency for establishing 
 such conclusions. 
 
 There is another point of interest in this Figure to which 
 Keynes calls attention. It is that only in it have we both the 
 subject of the conclusion as the subject in the premises, and 
 the predicate of the conclusion as the predicate in the prem- 
 ises. This makes it unnecessary for the mind to change its 
 conception from one relation to the other in coming to its 
 conclusion ; it can still think of its terms in the conclusion as 
 having the same functions and logical relations as in the prem- 
 ises. It, therefore, accounts for the fact that reasoning so 
 generally seems more natural in the first Figure than in any 
 other. We resort to it for greater clearness and effective- 
 ness. 
 
 The second Figure is especially adapted to disproof, or the 
 proof of universal negatives, or propositions in E. It is so 
 adapted, partly because one of the premises must be negative, 
 and partly because of the kind of comparison which can be es- 
 tablished by it between subjects and predicate. It is clear that 
 if two things cannot agree in their predicate they cannot agree 
 with each other. Hence to disprove an assertion we have only 
 to prove that one instance, or more, included in the general 
 statement does not agree with the predicate and we have the 
 contradictory established. Thus, if it be asserted that "All 
 forms of government are beneficial," we may disprove this by 
 showing that "An arbitrary despotism is not beneficial," 
 which would involve in a comparison of the two statements a
 
 188 ELEMENTS OF LOGIC 
 
 syllogism of the second Figure, whose conclusions would be, 
 " Arbitrary despotisms are not governments," which is the op- 
 posite of what is implied in the nature of the case, and hence 
 the opponent will have the option of giving up the universal- 
 ity of his assertion that " All governments are beneficial," or 
 of maintaining that despotisms are not forms of government. 
 
 This seems, however, like a mere proof of an exception, and 
 not of the total falsity of the universal. A better instance of 
 the case is the following. Suppose it be asserted that " Strikes 
 are justifiable." The most complete disproof of this would 
 be the truth of the proposition that "No strikes are justifi- 
 able." Now, if we can assert that " All interferences with the 
 rights of property and capital are not justifiable," and can have 
 it taken for granted that strikes are such interferences, we have 
 proved the direct opposite of the original The syllogism 
 would stand thus with the original proposition as the minor 
 premise, instead of the major, as in the first case. 
 
 Interferences with the rights of property and capital are not 
 
 justifiable. 
 
 Strikes are justifiable. 
 Strikes are not justifiable. 
 
 Now as the minor premise is asserted by one person and 
 the major by another, with the supposition that it will be ad- 
 mitted by the first person, the conclusion intended to be en- 
 forced is, " Strikes are not interferences with the rights of 
 property and capital," while it is assumed that it is clear they 
 are such interferences. Hence if they are such, and yet the 
 major premise is admitted, it will follow that "All strikes are 
 not justifiable," the direct opposite of the original proposition 
 asserted. The second Figure is, therefore, well adapted to the 
 method of disproof. 
 
 The third Figure is adapted to the proof of exceptions to 
 universals, or disproves universals by their contradictories, as 
 the first Figure disproves them by their opposites. Or, when 
 it does not prove exceptions, it proves certain particular truths 
 which will be of considerable value against the assertion of
 
 MOODS AND FIGURES OF THE SYLLOGISM 189 
 
 this universal contradictory. As an illustration of this latter 
 fact, as well as the former, take the assertion that "No philos- 
 ophers are wise." Now, if we wish to disprove this, and at the 
 same time prove that " Some philosophers are wise," we have 
 only to prove the following : 
 
 Plato, etc., were wise. 
 
 Plato, etc., were philosophers. 
 
 .*. Some philosophers were wise. 
 
 Here we both prove that " Some philosophers are wise," and 
 disprove the universal negative which had been asserted 
 against this possible fact. A particular exception is proved. 
 Therein lies the value of the third Figure, when it is found 
 difficult or impossible to prove the universal opposite of a 
 given assertion. 
 
 The fourth Figure is not regarded by logicians as having any 
 great practical value. It is not so frequently used as the 
 others, and can so often be changed into the first Figure by 
 mutating the premises, that its practical importance does not 
 require special notice.
 
 CHAPTER 
 
 REDUCTION OF MOODS AND FIGURES 
 
 1st. Direct Reduction. The number of valid Moods and 
 Figures is so great, and the apparently irregular character of 
 them so perplexing that the mind often finds it difficult to re- 
 member them correctly. The old logicians, therefore, in- 
 vented a mnemonic device in aid of the memory, and which 
 at the same time would contain some indications of the vari- 
 ous processes required for what is called their reduction. 
 The mnemonic verses consist of barbarous Latin terms added 
 to a few genuine ones, indicating all the Moods and Figures 
 that are valid, except the instances of weakened conclusion. 
 The verses are given below, with the indications of the accent 
 to be employed in reading them according to the laws for 
 scanning Latin poetry : 
 
 Barbara, Celiirent, Darn, Ferioquc prioris : 
 Cesarg, Camestres, Festino, Baroko, secundse : 
 Tertia, Darapti, Disumls, Datlsi, Felapton. 
 Bokardo, Ferlson, Mbet : Quarta insuper addit 
 Bramantlp, Camcnes, Dlmarls, Fesapo, FresTson. 
 
 The first line indicates the moods of the first Figure, the 
 second the moods of the second Figure ; the third, and the 
 first two words of the fourth line, the moods of the third Fig- 
 ure, and the last line those of the fourth Figure. The first 
 Figure is called by logicians the perfect, and the remaining 
 three the imperfect Figures, because they are to be reduced 
 by certain processes to the first. The letters indicating how 
 this is to be done require to be explained. 
 
 The capital letters B, C, D, F, in the last four lines, indicate
 
 REDUCTION OF MOODS AND FIGURES 191 
 
 the mood in the first Figure to which the mood in the other 
 Figures beginning with that letter is to be reduced. Thus 
 Camestres is to be reduced to Celarent, etc. 
 
 The vowels a, e, i, o indicate the mood of the syllogism. 
 Thus in Ferison the mood is EIO etc. 
 
 p indicates that the preceding proposition is to be converted 
 per accidens or by limitation. 
 
 s at the end of a word indicates that the conclusion of the 
 new syllogism has to be converted by simple conversion in 
 order to obtain the given conclusion : in the middle of a word 
 it denotes that the preceding proposition is to be converted 
 simply in the process of reduction. This difference between 
 the use of s at the end, and its use in the middle of a word is 
 remarked by Keynes. But I am inclined to see little impor- 
 tance in the distinction, because s in both cases denotes that 
 the preceding proposition is to be converted simply, so that 
 the converted conclusion of the reduced mood becomes the 
 proper conclusion of the corresponding " perfect " Figure and 
 Mood. Thus in Camestres we are to reduce the mood, as al- 
 ready said, to Celarent, or AEE of the second Figure to E 
 AE of the first. The m denoting that the premises are to be 
 transposed, we have to convert the minor premise and make 
 it the major premise of the new syllogism, and make the ma- 
 jor of Camestres the minor of the new. Then, if we convert 
 the conclusion of Camestres 'by simple conversion, we shall 
 have the proper conclusion of the new. 
 
 Camestres. Celarent. 
 
 AisC } ( CisnotB 
 
 B is not C > Reduced to -I A is C 
 . . B is not A ) ( . . A is not B. 
 
 Keynes also makes the same distinction between the use of 
 p at the end, and its use in the middle of a word. But as in 
 the case of s, I do not find it important. In all cases it de- 
 notes that the preceding proposition is to be converted by lim- 
 itation, and so to appear in the new syllogism. Thus in Felap- 
 ton A is to be converted per accidens or by limitation. In
 
 192 ELEMENTS OF LOGIC 
 
 Bramantip, however, the subalternans of the converse is taken 
 after the conversion. 
 
 77i denotes that the premises of the " imperfect " moods are 
 to be transposed or mutated in order to form the premises of 
 the new and "perfect" mood. Thus in Camenes A and E must 
 be transposed so as to become E and A of Celarent. 
 
 k signifies that the mood is to be reduced indirectly ; and 
 the position of the letter is, as affirmed by Keynes, to indicate 
 that in the process of indirect reduction the first step is to 
 omit the premise preceding it ; that is, we take instead of it 
 the contradictory of the conclusion as our premise. Baroko 
 and Bokardo are the moods to which indirect reduction is 
 usually applied, although the process can be applied to others. 
 The present mnemonic lines, however, do not designate how 
 this can be done in the case of the other moods. They can be 
 more easily reduced directly, and it is superfluous to apply the 
 indirect process. 
 
 One or two more examples of direct reduction will suffice 
 for illustration, and we can pass to consider the indirect pro- 
 cess. Take the cases of Festino and Bramantip, which are to 
 be reduced respectively to Ferio and Barbara. If we perform 
 the processes as designated by the several letters they will 
 stand as follows : 
 
 Festino. Ferio. 
 
 A is not C } | C is not A 
 
 Some B is C > Eeduced to < Some B is C 
 
 . . Some B is not A ) ' . . Some B is not A. 
 
 Bramantip. Barbara. 
 
 A is C ] ( C is B 
 
 C is B > Reduced to < A is C 
 
 Some B is A ) ( . . Some A is B, 
 
 or All A is B. Subalternans. 
 
 In the case of Bramantip there is apparent a defect in the 
 mnemonic verse, as the term ought to indicate the process of 
 changing the converted I into its subalternans A. One or 
 two concrete examples will be helpful to a better understand-
 
 REDUCTION OF MOODS AND FIGURES 193 
 
 ing of the process. We may take Cesare and Disamis, to be 
 reduced to Celarent and Darii. Following the directions, we 
 have in Cesare and Celarent the two succeeding syllogisms : 
 
 Cesare. Celarent. 
 
 Men are not quadrupeds. \ ( Quadrupeds are not men. 
 
 Horses are quadrupeds. > Reduced to < Horses are quadrupeds. 
 . . Horses are not men. ) ( . . Horses are not men. 
 
 Disamis. Darii. 
 
 Some men are negroes. } ( All men are vertebrates. 
 
 All men are vertebrates. j- Reduced to < Some negroes are men. 
 
 . . Some vertebrates are negroes. ) ( . . Some negroes are vertebrates. 
 
 The process can easily be applied to the other moods with- 
 out further illustration. It will be good practice for the stu- 
 dent to choose his own examples. 
 
 2d. Indirect Reduction. Some difficulty attends the re- 
 duction of Baroko and Bokardo, because there are no moods 
 in the first Figure representing AOO and OAO, the former 
 giving an illicit major and the latter an illicit middle. Besides, 
 no conversion of O is possible, and hence if the reduction of 
 these two moods be possible it must be in some indirect way. 
 
 There are two ways in which it is done, and there is a dif- 
 ference of opinion as to whether the first form is indirect or 
 not. Jevons speaks of it as indirect ; Keynes as direct. But 
 inasmuch as the major premise in both moods is conti-a verted, 
 and the minor premise of Baroko obverted, it will hardly be 
 amiss to follow Jevons's opinion, because neither of these pro- 
 cesses are provided for in the mnemonic system we have 
 adopted. If the matter of reduction were of great practical 
 importance, it might be incumbent upon us to discuss this 
 question. But the chief matter of interest to the scientific 
 student of Logic is the fact and the mode of reduction, and 
 we content ourselves with stating them. 
 
 We take up the first method of treating Baroko and Bokar- 
 do, illustrating them as before : 
 
 Baroko, Ferio. 
 
 A is C } C All not-C is not A 
 
 Some B is not C / Reduced to < Some B is not-C 
 .. Some B is not A ) (. .-. Some B is not A. 
 
 13
 
 194: ELEMENTS OF LOGIC 
 
 It should first be remarked in this case that the mnemonic 
 Baroko is named so and introduced with the capital B to suit 
 its indirect reduction to Barbara. But in reducing it to Ferio, 
 it should be called Faroko. The obversion of the minor prem- 
 ise is accomplished as usual by connecting the negative par- 
 ticle with the predicate, making the proposition affirmative. 
 We give a concrete illustration of the same : 
 
 Baroko. Ferio. 
 
 Metals are elements. \ ( All not-elements are not metals. 
 
 Some solids are not elements. >- < Some solids are not-elements. 
 
 . . Some solids are not metals. ; ( . . Some solids are not metals. 
 
 The reduction of Bokardo by the same method is as follows, 
 except that we have to transpose the premises : 
 
 Bokardo. Darii. 
 
 Some A is not C ) ( All A is B 
 
 All A is B r Reduced to -s Some not-C is A 
 
 /. Some B is not C ) ' .'. Some not-C is B. 
 
 A similar remark should be made about Bokardo as was 
 made about Baroko. To suit the reduction we have given, it 
 should be called Dokardo. We also give a concrete illustra- 
 tion of the process : 
 
 Bokardo. Darii. 
 
 Some trees are not oaks. ) R d d ( ^ trees are or S anisms - 
 
 All trees are organisms. -s Some not-oaks are trees. 
 
 . ' . Some organisms are not oaks. ) ( . . Some not-oaks are organisms. 
 
 The conclusion in the form to which Bokardo is reduced is 
 also contraverted, as can be observed by the student. 
 
 The second method of reducing the same moods is un- 
 doubtedly indirect, as, instead of depending upon the forms 
 of conversion, obversion, or contraversion, it resorts to the 
 square of opposition for a contradictory of a premise for the 
 conclusion, and of the conclusion for a premise in Barbara, as 
 the illustrations below will distinctly show : 
 
 Baroko. 
 
 AisC } 
 
 Some B is not C > Reduced to 
 Some B is not A )
 
 REDUCTION OF MOODS AND FIGURES 195 
 
 In this process the major premise remains the same. But 
 instead of taking any form of the minor premise, converted or 
 otherwise, the contradictory of the conclusion is assumed. In 
 Baroko the conclusion is O ; its contradictory is A. This is 
 done on the supposition that if O, " Some B is not A," is not 
 true, then " B is A " is true, and if this be assumed hypo- 
 thetically in the premises we shall have a syllogism in Bar- 
 bara with a conclusion which is the contradictory of the 
 omitted minor premise, " Some B is not C." This contradic- 
 tory is " B is C," and we must either admit one of our prem- 
 ises to be false or allow that our original conclusion is true. 
 But as we assume that our original premises and conclusion 
 are true, the impossibility of their contradictories being true 
 is an indirect proof of them, as it is always regarded an in- 
 direct proof of a proposition to prove the impossibility of its 
 opposite or contradictory, and reduction is proving one mood 
 and Figure by another. Bokardo may be treated in a manner 
 similar to Baroko. 
 
 Bokardo. Barbara, 
 
 Some A is not C ) ( All B is C 
 
 All A is B > Reduced to -j All A is B 
 
 . . Some B is not C ) (. . All A is C. 
 
 In this instance the major premise is omitted, and in its 
 place stands hypothetically the contradiction of the conclu- 
 sion in Bokardo for the major premise of Barbara. We, there- 
 fore, have a conclusion which is the contradictory of " Some A 
 is not C," the major premise of Bokardo. If this premise be 
 assumed to be true, the impossibility of the truth of its con- 
 tradictory in the reduced form is an indirect proof of it, and 
 we have the indirect reduction as before. The method is 
 sometimes called the reductio per impossibile, or the proof of 
 a thing by showing its contradictory to be impossible. 
 
 Little or no practical importance can be attached to any 
 forms of reduction, except when we find it necessary to test a 
 conclusion which does not seem so clear to us in one Figure 
 or form of reasoning as in another. They may sometimes be
 
 196 ELEMENTS OF LOGIC 
 
 helpful in completing certain enthymemes which can be 
 formed in either the first or the second Figures, and be valid 
 in the first, but invalid in the second. But as reduction does 
 not apply to invalid moods it is only the principle of it that 
 can be applicable in such cases. Hence we do not generally 
 find the process to have more than a purely scientific interest 
 and importance.
 
 CHAPTER XIV. 
 
 FORMS OF SYLLOGISTIC REASONING 
 
 THE syllogism, as it has already been explained, appears to 
 have a very simple form, and to consist of three propositions 
 arranged in a particular way. While this is the regular form 
 of reasoning, and expresses the real nature of the mental pro- 
 cess involved in all instances of ratiocination, arguments are 
 not always formulated in the regular way. Some propositions 
 may be omitted in expression, although included in the 
 thought of the reasoner, and in other cases, two or more syl- 
 logisms may be involved in the argument. These two condi- 
 tions give rise to two divisions of the syllogism, besides the 
 simple form already considered. They are the incomplete and 
 the complex forms of it. We shall classify them before dis- 
 cussing them in detail. The following is an outline of them : 
 
 Co lete I Dimple = The ordinary form already discus 
 " ( Complex = Prosyllogism and Episyllogism. 
 
 ( 1 st order. 
 
 ( Simple = Enthymeme < 2d order. 
 Incomplete J ( 3d order. 
 
 * Complex = 
 
 cussed. 
 
 Syllogisms < 
 
 The complete forms of the syllogism are, of course, an ex- 
 plicit statement of all that is involved in the mental process. 
 They very seldom appear in ordinary discourse, and then only 
 to give it more cogency. The usual mode of stating an argu- 
 ment is either to state the facts and to allow the inference to 
 be drawn by others, or to abbreviate the process by assuming 
 the most apparent of the premises. But when we wish to in- 
 dicate with perfect clearness all that the mind takes into ac- 
 count we formulate our thoughts into complete syllogisms. 
 The first of these is the simple syllogism of three propositions,
 
 198 ELEMENTS OF LOGIC 
 
 and which has already been discussed. We then take up the 
 second to consider it briefly. 
 
 1. PROSTLLOGISM AND EPISYLLOGISM. This consists of two 
 syllogisms, the conclusion of one, the prosyllogism, being a 
 premise in the other, the episyllogism. The following are 
 illustrations : 
 
 A is B Men are vertebrates. 
 
 C is A Europeans are men. 
 
 . . C is B . . Europeans are vertebrates. 
 
 D is C Italians are Europeans. 
 
 . . D is B . . Italians are vertebrates. 
 
 In the formal illustration here given the prosyllogism ends 
 with the conclusion " C is B," and the episyllogism begins 
 with the same proposition as a premise. In the material in- 
 stance the proposition, "Europeans are vertebrates," is the 
 conclusion of the prosyllogism and a premise of the episyllo- 
 gism. 
 
 2. EKTHTMEME. An enthymeme is an incomplete syllogism 
 in which either one of the premises, or the conclusion, may be 
 omitted. If the major premise be the one which is omitted 
 the enthymeme is said to be of the first order ; if the minor 
 premise be omitted, it is of the second order, and if the con- 
 clusion be omitted, it is of the third order. The signs of the 
 enthymeme are such words as indicate a reason for the truth 
 of a proposition. They are for, because, since, inasmuch as, 
 and in the conclusion, therefore, consequently, etc. The last 
 two words denote that an inference is drawn from some pre- 
 ceding fact or statement, and are signs of an enthymeme only 
 when a single premise is given. Those which give a reason 
 for a truth are usually necessary when the syllogism is not 
 complete, or when the conclusion is stated first and requires 
 to have its dependence upon the premises indicated. As an 
 illustration of an enthymeme we have the proposition "Atmos- 
 pheric air must have weight because it is a material substance." 
 The conclusion in this example is, "Atmospheric air must have 
 weight." If we were stating a mere fact, it would not require
 
 FORMS OF SYLLOGISTIC REASONING 199 
 
 proof, but we often desire to support such assertions by rea- 
 sons that will show their truth apart from the acceptance of 
 them on authority, and so we give a reason for them. In this 
 case the reason for the truth or assertion that "Atmospheric 
 air must have weight " is, that " it is a material substance." 
 In this reason it is tacitly assumed that " all material sub- 
 stances have weight." It is the omission of this premise that 
 makes the reasoning an enthymeme. Whenever such proposi- 
 tions are taken for granted it is because they either require no 
 proof or they are sufficiently evident to justify their omission, 
 and the process of thought or reasoning can be abbreviated or 
 economized. It is the usual form in ordinary speech. When 
 resolved into the form represented by their proper logical 
 order and dependence, the propositions take the order laid 
 down for the common syllogism. The above instance can 
 easily be resolved by the student into a simple syllogism. 
 
 There are forms of the enthymeme in which the signs are 
 not expressed, but which have to be determined by the evi- 
 dent relation of the thoughts stated. Every sentence, how- 
 ever, which is definitely introduced by a particle denoting a 
 reason for something else is an express statement of reason- 
 ing. But such forms of proof become stilted and inelegant, 
 and hence discourse may sometimes be best conducted by af- 
 firming facts or truths which of themselves imply the de- 
 pendence of something else upon them. We require to be on 
 the alert in such cases in order either to detect the existence 
 of reasoning at all, or to discover the form in which it is im- 
 plied. This may also be true of statements involving com- 
 plete syllogisms. An interesting instance of reasoning in the 
 form of an enthymeme without the usual signs is the follow- 
 ing: 
 
 " The high prices caused by the new tariff law have severely 
 taxed the mind of the Chronicle-Telegraph, but it has at last 
 evolved this curious and interesting explanation : ' For a long 
 time past everything, or nearly everything, entering into the 
 daily consumption of the people has been unusually cheap. But 
 close observers of the trend of events believe we are approach-
 
 200 ELEMENTS OF LOGIC 
 
 ing an era of higher prices, and that it is near at hand, and 
 that it is to be looked upon as a natural reaction from the pe- 
 riod of constant decline in the value of merchandise which 
 has lasted for several years.' " The reasoning involved in this 
 case is apparent when we enunciate that the present high 
 prices are a reaction from previous low prices, and that such 
 reactions are natural, and not caused by tariff laws. The pas- 
 sage beginning with the words, "For a long time past," is in- 
 tended to express a fact, from which it follows that high prices 
 must occur without reference to tariff influences, and hence 
 we have in thought a process of reasoning without the use of 
 any of the signs. The premise which is omitted is, that " All 
 periods of extremely low prices are followed by a natural re- 
 action." If the other premise be formulated we should all 
 see how the conclusion would follow, which is intended to be 
 the contrary of the first proposition in the passage. Most ar- 
 guments in ordinary discourse are of this general kind, and 
 the student would do well to formulate them for the sake of 
 discovering where the reasoning is. 
 
 3. EPICHEIREMA. An epicheirema is a syllogism in which 
 one or both of the premises is supported by a reason which 
 implies an imperfectly expressed syllogism : in other words, it 
 is a syllogism in which one or both of the premises is an en 
 thymeme either of the first or of the second order. The epi- 
 cheirema may be single or double. It is single when only one 
 of the premises is an enthymeme, and double when both are 
 enthymemes. The following is an illustration of the single 
 
 epicheirema : 
 
 A is B, for it is P 
 C is A 
 . . C is B. 
 
 A material form of the same is the following, but the enthy- 
 meme in this case is in the minor premise : 
 
 Vice is odious. 
 
 Avarice is a vice, because it depraves. 
 
 Therefore avarice is odious.
 
 201 
 
 The double epicheirema takes the following form : 
 
 A is B, because it is P 
 C is A, because it is Q 
 . . C is B. 
 
 Or, Man has a mind, because he is rational. 
 
 Europeans are men, because they are civilized. 
 Therefore Europeans have minds. 
 
 The single epicheirema when resolved or completed be- 
 comes a prosy llogism and an episyllogism, or two syllogisms. 
 The double epicheirema when completed becomes three syllo- 
 gisms, representing two prosyllogisms and two episyllogism s, 
 one of the three being both a prosyllogism and an episyllo- 
 gism, the former in relation to the following, and the latter 
 in relation to the preceding syllogism. The student may 
 practise their resolution until familiar with the processes in- 
 volved in them. The resolution of the double epicheirema 
 above is as follows : 
 
 Man has a mind : I Whatever is rational has a mind, 
 for he is rational = < Man is rational. 
 ( Man has a mind. 
 
 Europeans are men : I Whatever is civilized is man. 
 for they are civilized = -j Europeans are civilized. 
 
 ( Europeans are men. 
 Therefore Europeans have minds. 
 
 The third syllogism appears in combining the conclusions 
 of the first two to form its premises, from which we obtain 
 the conclusion that " Europeans have minds." 
 
 4. SOKITES. A sorites is so called because the propositions 
 form what is regarded as a " chain," or a continuous series, 
 of premises from which a conclusion is drawn at the end of 
 the series. It consists of enthymemes of the third order, as 
 the epicheirema consists of enthymemes of the first and sec- 
 ond orders. When completed, therefore, it forms a prosyllo- 
 gism and episyllogism. To complete it we have only to sup- 
 ply the intermediate conclusions implied by the proper prem-
 
 202 ELEMENTS OF LOGIC 
 
 ises. The form of the sorites is twofold. It may be progres- 
 sive or regressive. The following are illustrations : 
 
 Progressive series. 
 
 A is B Bucephalus is a horse. 
 
 B is C A horse is a quadruped. 
 
 C is D or, A quadruped is an animal 
 D is E An animal is an organism. 
 
 . . A is E . . Bucephalus is an organism. 
 
 Regressive series. 
 
 A is B An animal is an organism. 
 
 C is A A quadruped is an animal. 
 
 D is C or, A horse is a quadruped. 
 E is D Bucephalus is a horse. 
 
 . . E is B . . Bucephalus is an organism. 
 
 The difference between the progressive and regressive so- 
 rites is merely in the appearance of the Figure of the syllo- 
 gism. In the progressive series it seems to be of the fourth 
 Figure, and the propositions are expressed in that order, but 
 the actual reasoning is performed by a virtual transmutation 
 of the premises, when we choose the minor term from the first 
 proposition and the major term from the last, and hence is of 
 the first Figure. In the regressive series both the form of ex- 
 pression and the reasoning are in the first Figure, and repre- 
 sents the major premise as the first in order. 
 
 The completion of a sorites may be illustrated as follows : 
 The incomplete form is sometimes called the occult, and the 
 complete the manifest form. 
 
 A is B Bucephalus is a horse. 
 
 B is C A horse is a quadruped. 
 
 . . A is C or, . ' . Bucephalus is a quadruped. 
 
 C is D A quadruped is an animal. 
 
 . . A is D . . Bucephalus is an animal. 
 
 The sorites may be constructive or destructive. It is con- 
 structive when the premises and conclusion are affirmative,
 
 FORMS OF SYLLOGISTIC REASONING 203 
 
 and hence must be represented by the moods AAA, AAI, 
 and All, of the first Figure. But it is destructive when the 
 conclusion is negative, and therefore when the major prem- 
 ise is negative. The destructive sorites must then be rep- 
 resented by the moods EAE, EAO, and EIO, of the first 
 Figure. 
 
 There is a dispute among logicians as to whether a sorites 
 can exist in the second and third Figures. I agree with Mill 
 and Keynes that it is possible, but its importance is so slight, 
 and its occurrence in practice so infrequent, that I do not 
 think it deserves any special attention. 
 
 The rules for the valid forms of the sorites should be stated, 
 because the number of cases in which this mode of reasoning 
 is valid is exceedingly limited. Any number of premises may 
 be used, but the quantity and quality of the propositions con- 
 stituting them are under strict limitations. The rules for the 
 process, therefore, are two, and are as follows : 
 
 1. Only one premise can be negative, and this must be the 
 
 prime major. 
 
 2. Only one premise can be particular, and this must be the 
 
 final minor. 
 
 Since the sorites is an incomplete prosyllogism and episyllo- 
 gism, there are intermediate major and minor premises. The 
 prime major, therefore, will denote the last premise in the pro- 
 gressive, and the first premise in the regressive, series. The 
 final minor will denote the first premise in the progressive, and 
 the last in the regressive, series. If any other premise than the 
 prime major be negative, there will be a fallacy of illicit major, 
 and if any other than the final minor be particular, there will 
 be an illicit middle.
 
 CHAPTEE XV. 
 HYPOTHETICAL REASONING 
 
 THE forms of reasoning may be divided in the same manner 
 as propositions. In fact they are determined by the nature of 
 the propositions which constitute them. We have already ob- 
 served that propositions may be divided into Categorical, 
 Hypothetical, and Disjunctive. Arguments or farms of rea- 
 soning are divided according as the premise or premises are 
 categorical, hypothetical, or disjunctive. A categorical syllo- 
 gism is one in which all the propositions are categorical. Of 
 this kind we have already treated, and it remains to consider 
 the hypothetical form. 
 
 1. HYPOTHETICAL SYLLOGISMS. A hypothetical syllogism, or 
 form of reasoning, is one in which one or more of the propo- 
 sitions is hypothetical or conditional. Most frequently it is 
 the major premise alone that is conditional, while the minor 
 premise and conclusion are categorical The major premise 
 consists of two propositions, one of which is dependent upon 
 the other. The proposition expressing the condition is called 
 the antecedent, and is introduced by some such words as if, 
 supjwse, alloiv, granted that, provided that, etc., all of which 
 indicate that a statement is made under conditions restricting 
 its application. The proposition depending upon this condi- 
 tion is called the consequent, and is either a categorical or a 
 disjunctive proposition. An illustration of an antecedent and 
 consequent so defined is : "If the sea is rough, it is danger- 
 ous." "If the sea is rough "is the antecedent, and "it is 
 dangerous " is the consequent. The rules regulating the va- 
 lidity of hypothetical inferences depend upon understanding 
 the use of these terms. 
 
 The general form of the hypothetical syllogism is that of an
 
 HYPOTHETICAL REASONING 205 
 
 antecedent and consequent, or a conditional and a categori- 
 cal proposition for the major premise, a categorical proposi- 
 tion for the minor premise, and a categorical proposition for 
 the conclusion. But as there are two propositions in the 
 major premise, the minor premise may affirm or deny either 
 the antecedent or the consequent. This gives four forms to 
 be considered, two of which are valid and two of which are 
 invalid modes of reasoning. The two valid moods are called, 
 respectively, the modus ponens and the modus tollens, or the 
 constructive and the destructive hypothetical syllogism. The 
 modus ponens means that if the antecedent be affirmed the 
 consequent will follow. The major premise only asserts this 
 relation conditionally. But if the antecedent be affirmed 
 categorically in the minor premise the conclusion will follow 
 categorically. The only effect of a hypothetical minor premise 
 is to make the conclusion hypothetical. The modus tollens 
 means that if the consequent be denied, the antecedent must 
 be denied. The two forms can be illustrated as follows : 
 
 Modus ponens. Constructive hypothetical syllogism. 
 
 If A is B, C is D If iron is impure it is brittle. 
 
 A is B or, Iron is impure. 
 
 . . C is D . . Iron is brittle. 
 
 Nodus tollens. Destructive hypothetical syllogism. 
 
 If A is B, C is D If the sun shines it is light. 
 
 C is not D or, It is not light. 
 
 . . A is not B . . The sun does not shine. 
 
 The rule, therefore, for testing the validity of hypothetical 
 reasoning is, that either the antecedent must be affirmed or the 
 consequent denied. If this rule be violated in either of its 
 conditions a fallacy occurs.. This is illustrated in the two re- 
 maining, but invalid, moods of the hypothetical syllogism : 
 
 If A is B, C is D If it rains it is cloudy. 
 
 C is D or, It is cloudy. 
 
 . * . A is B . . It is raining.
 
 206 ELEMENTS OF LOGIO 
 
 This is a case of the fallacy of affirming the consequent. The 
 fact that " A is B," or that " it is raining," is not the only con- 
 dition that C should be D, or that it should be cloudy. Some 
 other condition may be true, so that if " A is B " follows from 
 the affirmation of the consequent, the other condition would 
 follow also. But there is nothing to determine that, and 
 hence the known condition must remain indefinite so far as 
 drawing it from the truth of the consequent is concerned. 
 If any other condition, however, may exist as a determinant 
 of the consequent, that condition may be the very opposite of 
 the one specified. Thus, C may be D, if A is not B, or it may 
 be cloudy if it is not raining, and in this case it will be ap- 
 parent that we cannot equally draw opposite conclusions from 
 the truth of the consequent. Hence, we have no right to 
 draw any inference whatever. A concrete illustration will 
 make this still clearer. " Thus, if a man's character be avari- 
 cious, he will refuse to give money for useful purposes ; but 
 it does not follow that every person who refuses to give 
 money for such purposes is avaricious. There may be many 
 proper reasons or motives leading him to refuse ; he may 
 have no money, or he may consider the purpose not a useful 
 one, or he may have more useful purposes in view." No in- 
 ference, therefore, can be drawn from an affirmation of the 
 consequent. 
 
 A second fallacy comes from denying the antecedent, as rep- 
 resented in the following form : 
 
 If A is B, C is D If gold were cheap it would be useful. 
 
 A is not B or, Gold is not cheap. 
 . . C is not D . . Gold is not useful. 
 
 The fallacy is due to the same causes as before. The ante- 
 cedent is not the only possible condition of the consequent, 
 and hence no conclusion denying the existence or truth of the 
 consequent can be drawn from the denial of the antecedent. 
 It is apparent in the concrete case that other qualities besides 
 cheapness might make gold useful, and therefore the absence 
 of this quality would not remove the usefulness of the metal
 
 HYPOTHETICAL REASONING 207 
 
 Difficulties appear in some cases in which it would seem 
 that a negative conclusion does follow the denial of the ante- 
 cedent. To take a case we have : 
 
 If men are white they are Caucasians. 
 Men are not white. 
 . . They are not Caucasians. 
 
 In supposing whiteness as the only condition of being a 
 Caucasian, as we generally do, the absence of it would imply 
 that the person was not a Caucasian. And so perhaps with 
 the following instance : 
 
 If fire is hot it will burn. 
 Fire is not hot. 
 . . It will not burn. 
 
 The usual assumption is that fire and the power to burn are 
 the same, so that to deny the heat of the former is to deny the 
 capacity of it to burn. In such cases a denial of the antece- 
 dent seems to involve an inference to the negation of the con- 
 sequent. 
 
 But we must not be deluded by such instances. They are 
 in effect cases in which the major premise is an exclusive hy- 
 pothetical proposition. When converted it becomes equiva- 
 lent to a modus tollens, in which the consequent instead of the 
 antecedent is denied. This explains how the reasoning is 
 valid. Thus if whiteness is the only consideration of being 
 Caucasian, the hypothetical proposition would be " If only 
 men are white they are Caucasian," which is equivalent to say- 
 ing, "If men are Caucasians they are white." Hence when we 
 affirm that "they are not white," in the minor premise, we 
 should be denying the consequent and not the antecedent. 
 Materially, therefore, the reasoning is correct, and it would 
 always be so when the antecedent expresses the only condi- 
 tion of the consequent. But formally we cannot consider it 
 so, as there is nothing in the form of the proposition to indi- 
 cate whether the antecedent expresses the only condition or 
 not. Since we are treating only of formal reasoning, we have
 
 208 ELEMENTS OF LOGIC 
 
 to regard all propositions alike which are only formal in their 
 character. If the exclusive nature of the hypothetical propo- 
 sition were expressed, we might formulate a rule for it, but 
 when it is not so expressed, we must consider it formally as 
 under the general law. Nevertheless in practical reasoning 
 we should always be prepared to distinguish when the mind 
 tacitly supposes the material conditions which might make the 
 material reasoning correct while it is formally wrong. In this 
 way we could admit the truth of the conclusion and yet show 
 that it has not been obtained in accordance with the formal 
 law of hypothetical reasoning, or that we are liable to frequent 
 fallacies, if we allow the correctness of this material reasoning 
 to lead us into the indiscriminate use of its privileges in hy- 
 pothetical syllogisms at large. 
 
 There are three forms of hypothetical propositions and syllo- 
 gisms which require notice because of the misunderstanding 
 to which they may give rise. They consist of negative propo- 
 sitions, while those we have illustrated consist of affirmative 
 propositions. We must show that the case is not altered by 
 the use of negative propositions, but that the whole matter 
 turns upon the connection between the antecedent and the 
 consequent. There are, therefore, three more forms in which 
 the major premise of the hypothetical syllogism may be ex- 
 pressed. They are : 
 
 (a) If A is B, C is not D 
 (6) If A is not B, C is D 
 (c) If A is not B, C is not D. 
 
 The peculiar characteristic to be remarked about these prop- 
 ositions is their quality and the mode of affirming and denying 
 the antecedent or the consequent. In proposition (a) the an- 
 tecedent will be treated in the same manner as in previous 
 instances, but the consequent will be affirmed in the minor 
 premise by saying " C is not D," in which case the fallacy of 
 affirming the consequent is committed. But the consequent 
 would be denied by saying " C is D," and then we should be 
 obliged to draw the conclusion that " A is not B." In propo-
 
 HYPOTHETICAL REASONING 209 
 
 sition (b) the consequent being an affirmative proposition 
 would be treated as before, and the antecedent would be 
 affirmed by making the minor premise to be " A is not B," 
 when we should have a modus ponens. But it would be denied 
 by the form "A is B," and the usual fallacy would be com- 
 mitted. In proposition (r) both the antecedent and the con- 
 sequent must be treated as we have treated the consequent 
 in proposition (a) and the antecedent in proposition (b). 
 
 2. REDUCTION OF HYPOTHETICAL SYLLOGISMS. It does not ap- 
 pear from the manner in which hypothetical syllogisms have 
 been discussed that the process of reasoning involved can be 
 reduced to the forms of categorical syllogisms. This, however, 
 is the fact, and in order to understand how the ordinary laws 
 of reasoning are applicable it is necessary to reduce them to 
 the categorical form. They are convenient, often, for the pur- 
 pose of emphasizing the conditional character of the major 
 premise, and insuring the acceptance of the conclusion on 
 those conditions when the minor premise is accepted. The 
 usual object, however, is to have the major premise accepted 
 formally, or the connection between antecedent and conse- 
 quent, and then to show that the antecedent is true or the 
 consequent false, in order to obtain a conclusion which is not 
 clear or admitted at the outset of the argument. 
 
 Nevertheless, in spite of the superior convenience, at times, 
 of the hypothetical forms of reasoning, they can all be reduced 
 to the categorical form. In all cases we may regard the ante- 
 cedent of the hypothetical major premise as the subject of the 
 categorical proposition, and the consequent of the hypothetical 
 proposition as the predicate of the categorical. In some in- 
 stances this change is a very simple one ; in others it can be 
 effected only by a circumlocution. It can be done simply 
 when the terms of the antecedent and consequent can be made 
 to form a phrase representing a noun and its modifiers. 
 Thus we have the examples : 
 
 If iron is impure it is brittle, \ i Impure iron is brittle. 
 
 It is impure. Iron is impure. 
 
 . . It is brittle. ) ( . . Iron is brittle. 
 
 14
 
 210 ELEMENTS OF LOGIC 
 
 If the weather is stormy sea travel will be dangerous. 
 The weather -is stormy. 
 . . Sea travel will be dangerous. 
 
 This may be reduced to the following : 
 
 Stormy weather is a cause of dangerous sea travel. 
 The present weather is stormy. 
 . . The present weather is a cause of dangerous sea travel. 
 
 In all such instances we practically supply a new minor term 
 in order to complete the categorical form, but it is only a par- 
 ticular case under the general in the major premise. 
 
 But all instances of the hypothetical syllogism are not so 
 easily reduced. In many of them we have to resort to a cir- 
 cumlocution in such phrases as "the case of" "the circum- 
 stances that" etc. Thus in the hypothetical syllogism below 
 we must use this means of its conversion : 
 
 If Aristotle is right, slavery is a proper form of society. 
 But slavery is not a proper form, of society. 
 .'. Aristotle is not right. 
 
 By using the phrase " the case of" this becomes in the cate- 
 gorical form : 
 
 The case of Aristotle being right is the case of slavery be- 
 ing a proper form of society. 
 But slavery is not a proper form of society. 
 .. Aristotle is not right. 
 
 This is clearly a syllogism in the second Figure of the 
 mood AEE. It is evident, therefore, that we may easily de- 
 termine the valid and invalid forms of hypothetical reason- 
 ing in terms of the categorical syllogism. We shall illustrate 
 all four forms in order to make this clear. First, the modus 
 ponens : 
 
 If water is pure it is good, "j ( Pure water is good. 
 It is pure. V -\ This water is pure. 
 
 . . It is good. ' . *. This water is good.
 
 HYPOTHETICAL REASONING 211 
 
 In this we have a syllogism in Barbara, or AAA of the first 
 Figure, and therefore valid. 
 
 The modus tollens will appear as follows : 
 
 If water is pure it is good. } ( Pure water is good. 
 It is not good. Y -\ This water is not good. 
 
 .'. It is not pure. J ' .*. This water is not pure. 
 
 Here we have again a case of Camestres, or AEE in the 
 second Figure, and valid. 
 
 The case of denying the antecedent is as follows : 
 
 If water is pure it is good. } ( Pure water is good. 
 It is not pure. > < This water is not pure. 
 
 .'. It is not good. J ' .'. This water is not good. 
 
 In this instance we have a case of AEE in the first Fig- 
 ure, which is invalid. The fallacy is that of illicit process of 
 the major term. The major term is not distributed in the ma- 
 jor premise, but is distributed in the conclusion. Hence, 
 when we attempt to draw a conclusion after denying the ante- 
 cedent the fallacy is that of an illicit major term. 
 
 The next is an illustration of affirming the consequent : 
 
 If water is pure it is good. } ( Pure water is good. 
 It is good. > -< This water is good. 
 
 .'. It is pure. ' .. This water is pure. 
 
 "We have in this illustration a case of AAA in the second 
 Figure, and invalid because the middle term is xindistributed. 
 Hence all cases of affirming the consequent are instances in 
 which we commit the fallacy of illicit process of the middle 
 term, or undistributed middle. The valid forms, therefore, 
 are AAA of the first, and AEE of the second Figure. The 
 invalid forms are AEE of the first, and AAA of the second 
 Figure. 
 
 When the hypothetical propositions are negative, we may 
 either obvert them into their corresponding affirmatives, or 
 consider the invalid forms as due to attempts to reason with 
 negative premises.
 
 CHAPTEE XVI. 
 
 DISJUNCTIVE SYLLOGISMS 
 
 A DISJUNCTIVE syllogism is one which is determined by the 
 presence of a disjunctive proposition in one of its premises, 
 and sometimes in the conclusion also. A disjunctive proposi- 
 tion we have already learned to be one which contains alterna- 
 tive or mutually exclusive conceptions between which the 
 choice of the mind is to be made, and which are accompanied 
 by the disjunctives either and or. Wherever these terms are 
 found in such propositions they are meant to imply that only 
 one of the two things can be affirmed, and the other denied. 
 Thus when I say that " The weather is either clear or cloudy," 
 I mean that it cannot be both at once, but that if it is one it 
 cannot be the other. It is this fact which determines the 
 right to draw the inference in the disjunctive syllogism. 
 
 "It is a disputed question whether in a disjunctive proposi- 
 tion the alternatives should be regarded as in all cases mu- 
 tually exclusive ; whether, for example, in the proposition ' A 
 is either B or C,' it is necessarily implied that A cannot be 
 both B and C. There are really involved here two questions 
 which should be distinguished. 
 
 "(1) In ordinary speech do we intend that the alterna- 
 tives in a disjunctive proposition should be necessarily under- 
 stood as excluding one another? A very few instances, I 
 think, will enable us to decide in the negative. Take, for ex- 
 ample, the proposition, ' He has either used bad text-books, or 
 he has been badly taught : ' would any one understand this to 
 exclude the possibility of his having been badly taught and 
 having used bad text-books as well ? Or, suppose it laid down 
 as a condition of eligibility for some appointment that every 
 candidate must be a member either of the University of Ox-
 
 DISJUNCTIVE SYLLOGISMS ,213 
 
 ford, or of the University of Cambridge, or of the University of 
 London. Would any one regard this as implying the ineligi- 
 bility of persons who happened to be members of more than 
 one of these Universities ? Jevons instances the following 
 proposition, ' A peer is either a duke, or a marquis, or an earl, 
 or a viscount, or a baron.' We do not consider this statement 
 incorrect because many peers, as a matter of fact, possess two 
 or more titles. 
 
 " (2) Still this does not definitely settle the question. Grant- 
 ed that in common speech the alternatives of a disjunction 
 may or may not be mutually exclusive, it may neverthe- 
 less be maintained that this is only because common speech is 
 elliptical, that in Logic we should be more precise, and that 
 the statement ' A is either B or C ' (where it may be both) 
 should therefore be written, ' A is either B and not C, or C and 
 not B, or both B and C.' 
 
 " This is a question of interpretation or method, and I do not 
 apprehend that any burning principle is involved in the an- 
 swer that we may give. For my own part I do not find any 
 siifficient reason for diverging from the usage of every-day lan- 
 guage. On the other hand, I think that if Logic is to be of 
 practical utility, the less logical forms diverge from those of 
 ordinary speech the better. And further, condensed forms 
 of expression do not conduce to clearness, or even ultimately to 
 conciseness. For where our information is meagre, a con- 
 densed form is likely to express more than we intend, and in 
 order to keep within the mark we mur,t indicate additional al- 
 ternatives." * 
 
 The purport of these remarks is that the disjunction "either 
 or" is capable of a double import. The first is, that the 
 terms may denote alternatives, either one of which may be suf- 
 ficient to satisfy the terms of the proposition, although both 
 may exist in the same connection. Thus in the case of a can- 
 didate's eligibility depending upon membership in either the 
 University of Oxford, or the University of Cambridge, etc., we 
 mean that membership in any one of them is sufficient, and 
 * Keynes's Formal Logic, Part II., Chap. IX., p. 167.
 
 214 ELEMENTS OF LOGIC 
 
 that non-membership in the others will not be an obstacle iv 
 that case. The second meaning is that the two alternatives 
 shall be mutually exclusive. This is the form which is neces- 
 sary for correct formal disjunctive reasoning, and as it fre- 
 quently occurs, we have to take it into account in a complete 
 exposition of the syllogism. The former case, when " either 
 or" does not express mutual exclusion between the alter- 
 natives, gives rise to what is called an incomplete disjunction. 
 The fallacy incident to this fact will be noticed again. At 
 present we have only to consider those cases where " either 
 or " expresses mutually exclusive alternatives. Upon this as- 
 sumption definite rules can be established for disjunctive rea- 
 soning. In the meantime we shall use the terms either 
 or as the only accessible symbols for a formal disjunction. 
 
 Before enunciating these laws and illustrating them we shall 
 classify the forms of the disjunctive syllogism. There are two 
 general divisions, the categorical and the dilemmatic, or the 
 definite disjunctive syllogism and the dilemma. The former 
 consists of a disjunctive proposition in the major premise, and 
 a categorical in the minor premise, giving a categorical conclu- 
 sion. The dilemma consists of a hypothetical proposition in 
 the major premise, and a disjunctive in the minor premise. The 
 subdivisions of these two general forms is illustrated in the fol- 
 lowing table : 
 
 Disjunctive syllogisms 
 
 n * i ( Modus ponendo tollens. 
 Categorical - Modug ollendo ponena 
 
 The first form of the categorical disjunctive syllogism is 
 called the modus ponendo tollens, because it means that by 
 affirming one of the alternatives we must deny the other. This 
 is the meaning of the Latin phrase denominating it. It is il- 
 lustrated as follows : 
 
 A is either B or C ] I Oak trees are either tall or short 
 
 But A is B > They are tall. 
 
 . . A is not C ) ( . . They are not short.
 
 DISJUNCTIVE SYLLOGISMS 215 
 
 The modus tollendo ponens, which is the second form, is so 
 named because by denying one of the alternatives we must affirm 
 the other. It is illustrated thus : 
 
 A is either B or C ) I The air is either cool or warm. 
 A is not B It is not cool. 
 
 . . A is C ( . . It is warm. 
 
 In these cases we assume that the alternatives are mutually 
 exclusive, and that the subject cannot be both at once, or that 
 there can be no other alternative. If this assumption were 
 not made the conclusion would be invalid, as a case of incom- 
 plete disjunction. This is illustrated in the following in- 
 stance : 
 
 Macaulay either had great talents or he was very studious. 
 He had great talents. 
 . . He was not very studious. 
 
 This conclusion does not necessarily follow because the alter- 
 natives are not necessarily exclusive of each other. A man 
 may be both talented and studious. Hence when the disjunc- 
 tion is incomplete in the major premise, it gives rise to a fal- 
 lacy which is a petitio principii, and which will be explained 
 again. This fallacy, however, will not occur, if we assume the 
 disjunction to be complete. If we really assume that Macau- 
 lay was either one or the other of the two alternatives, and not 
 possibly both of them, or anything else, the conclusion is valid. 
 Very frequently in such cases we mean that the disjunction 
 shall be perfect, and hence the reasoning cannot be criticised. 
 Thus, if I say " All birds are either white or black," and then, 
 after affirming that " they are not white," infer that " they are 
 black," I would be wrong only because I was wrong in the ma- 
 jor premise. If I really meant that these were the only two 
 alternatives, the conclusion would be true. We see, therefore, 
 that there is no formal fallacy in disjunctive reasoning, but 
 that it occurs in the matter of the assumption in the major 
 premise.
 
 216 ELEMENTS OF LOGIC 
 
 The laws of disjunctive syllogisms seem to be quite different 
 from the categorical and the hypothetical. We seem to infer 
 a negative conclusion from affirmative premises, and an affirm- 
 ative conclusion when one of the premises is negative ; a neg- 
 ative conclusion in the modus ponendo tollens, and affirmative 
 in the modus tollendo ponens. But it can easily be shoAvn that 
 this is not exceptional. This can be done in two ways. First, 
 the major premise, which is a disjunctive proposition, con- 
 tains both an affirmative and a negative assertion, with the im- 
 plication that one is true and the other false. The reasoning is, 
 therefore, based upon the law of contradiction in the square of 
 opposition, so that a negative conclusion is involved in the nega- 
 tion expressed, or implied in the major premise, and an affirma- 
 tive conclusion when the minor premise is negative. But the 
 clearer exposition of the case is the second. As we have already 
 remarked a disjunctive proposition is one which is categorical 
 in its form and conditional or hypothetical in its matter. Its 
 meaning, therefore, must be determined by reducing it to its 
 equivalent, and we shall see that the disjunctive syllogism can 
 be resolved into the hypothetical, and this hypothetical into the 
 categorical, so that, after all, the regular laws of reasoning apply 
 to the disjunctive syllogism, although in a modified and less ap- 
 parent form. 
 
 When we say that " A is either B or C," and imply that there 
 are no other alternatives, we mean that if A is it is not C. 
 This, we see, is a hypothetical proposition with a negative con- 
 sequent. Or we may mean that if A is not B it is C, in which 
 case we have a negative antecedent. We have then only to 
 state the minor premise, as in the disjunctive syllogism, and 
 the reasoning becomes hypothetical. This can be illustrated 
 in the following manner : 
 
 A is either B or C ) ( - If A is B, it is not 
 A is B A is B 
 
 . . A is not C ; ( . . A is not C. 
 
 Li the hypothetical form we have, therefore, a case of modus
 
 DISJUNCTIVE SYLLOGISMS 217 
 
 ponens which is valid. After the reduction of the disjunctive 
 form the modus tollens appears thus : 
 
 A is either B or C ) ( If A is B, it is not C 
 A is C A is C 
 
 . . A is not B ) ( . . A is not B. 
 
 But as the nature of the disjunctive syllogism is such that 
 we can always make it a modus ponens in the hypothetical, and 
 as it is always formally valid, we do not require to test its 
 laws by either the modus tollens or the invalid forms of 
 hypothetical reasoning. It therefore suffices to convert it 
 always into the one form for the purpose of discovering the 
 law underlying its logical process, and this is the law of the 
 hypothetical syllogism, which we have already ascertained to 
 be the same as that of categorical syllogism. 
 
 The dilemma, or dilemmatic disjunctive syllogism, is subject 
 to the laws of hypothetical reasoning, because its major prem- 
 ise is hypothetical. The first form is that of the simple con- 
 structive dilemma. 
 
 If A is B, C is D ; and if E is F, C is D 
 
 But either A is B, or E is F 
 . . C is D. 
 
 We observe in this and all cases of the simple constructive 
 dilemma that the consequent is the same for both antecedents. 
 This gives as its distinctive mark a categorical conclusion. A 
 concrete illustration is the following : 
 
 " If a science furnishes useful facts, it is worthy of being 
 cultivated ; and if the study of it exercises the reasoning 
 powers, it is worthy of being cultivated ; but a science either 
 furnishes useful facts, or its study exercises the reasoning 
 powers ; therefore it is worthy of being cultivated." 
 
 The second form of the dilemma is the complex constructive 
 
 dilemma. 
 
 If A is B, C is D ; and if E is F, G is H 
 But either A is B, or E is F 
 . . Either C is D, or G is H.
 
 218 ELEMENTS OF LOGIC 
 
 This is different from the simple constructive dilemma in that 
 the consequents are different, and this fact gives as its dis- 
 tinctive mark a disjunctive conclusion. As an instance of it 
 we have the following argument : 
 
 " If a statesman who sees his former opinions to be wrong 
 does not alter his course he is guilty of deceit ; and if he does 
 alter his course he is open to the charge of inconsistency ; 
 but either he does not alter his coiirse or he does ; therefore 
 he is either guilty of deceit or he is open to the charge of in- 
 consistency." 
 
 The destructive dilemma is supposed always to be complex, 
 because it can otherwise be resolved into two distinct hypo- 
 thetical syllogisms, and because no disjunctive 'proposition 
 occurs in it. If a simple destructive dilemma occurred, it 
 would be in the following form : 
 
 If A is B, C is D ; and if E is F, C is D 
 But C is not D 
 . . Neither A is B, nor E is F. 
 
 This form of reasoning is possible, and it is only a question of 
 definition as to whether we shall call it disjunctive and 
 dilemmatic. But we should have to change the conception of 
 " disjunction " in order to include it in that form. The com- 
 plex dilemma, therefore, is the only one that complies with 
 the conditions. It is as follows : 
 
 If A is B, C is D ; and if E is F, G is H. 
 But either C is not D, or G is not H. 
 . . Either A is not B, or E is not F. 
 
 A concrete illustration is found in the following argument: "If 
 this man were wise, he would not speak irreverently of Script- 
 ure in jest ; and if he were good, he would not do so in ear- 
 nest ; but he does it either in jest or in earnest, therefore he is 
 either not wise or not good." 
 
 The fallacy incident to the dilemma is the same as in hy- 
 pothetical reasoning, and does not require special discussion.
 
 CHAPTER XVIL 
 
 CLASSIFICATION OF FALLACIES 
 
 1st. Definition. The term "fallacy" is from the Latin 
 fallo, denoting deception, illusion, error. In Logic it must be 
 distinguished from such words as illusion. An illusion is a 
 misinterpretation of the data of sense perception : a fallacy is 
 an error in reasoning. The term, however, is often applied 
 to those errors which are liable to occur in the interpretation 
 of ambiguous propositions, made so by the displacement of a 
 word or a phrase. But in the true logical sense these errors 
 are not fallacies. They may give rise to fallacies in reasoning 
 by rendering the data uncertain and ambiguous, but they are 
 not errors in reasoning itself, they are only errors in inter- 
 pretation. As logicians, however, have uniformly included 
 them in their treatment and classification of fallacies, we shall 
 continue this practice for the sake of the practical convenience 
 they possess in ordinary reasoning, although the proper place 
 to deal with them is in Rhetoric. 
 
 In discussing the laws of the syllogism we have been trying 
 to ascertain the rules or laws which regulate right reasoning. 
 We have now to examine the illegitimate modes of inference, 
 or the mental processes which result in fallacies, or erroneous 
 reasoning. We require some means of knowing when the ra- 
 tional faculty is liable to go astray, as well as when it has con- 
 formed to the true principles of reasoning. In order to do 
 this we must classify and explain the various forms of fallacy. 
 
 2d. Divisions. As already indicated, the term fallacy is 
 used in a broad sense to cover both errors of interpretation, 
 or of grammatical and rhetorical form, and errors of infer- 
 ence, or logical reasoning. This gives rise to a twofold di- 
 vision of fallacies, into Hermeneutic and Logical fallacies. I
 
 220 ELEMENTS OF LOGIC 
 
 employ the terra " hermeneutic " to denote that they are 
 errors of interpretation, and hence of the perception of the 
 meaning of a proposition. The error is intellectual, but not 
 ratiocinative. This class of error or fallacy will require very 
 brief consideration. I recognize but two forms of it. First, 
 the so-called Fallacy of Amphibology, and second, the Fallacy 
 of Accent. 
 
 We quote the language of Jevons upon each of these forms 
 of error : " The Fallacy of Amphibology consists in an am- 
 biguous grammatical structure of a sentence, which produces 
 misconception. A celebrated instance occurs in the prophecy 
 of the Spirit in Shakespeare's Henry VI. : ' The Duke yet 
 lives that Henry shall depose,' which leaves it wholly doubtful 
 whether the Duke shall depose Henry, or Henry the Duke. 
 This prophecy is doubtless an imitation of those which the 
 ancient oracle of Delphi is reported to have uttered ; and it 
 seems that this fallacy was a great resource to the oracles who 
 were not confident in their own powers of foresight. The 
 Latin language gives great scope to misconstructions, because 
 it does not require any fixed order for the words of a sen- 
 tence, and when there are two accusative cases with an infini- 
 tive verb, it may be difficult to tell, except from the context, 
 which comes in regard to sense before the verb. The double 
 meaning which may be given to ' twice two and three ' arises 
 from amphibology ; it may be 7 or 10, according as we add 
 the 3 after or before multiplying. In the careless construc- 
 tion of sentences it is often impossible to tell to what part any 
 adverb or qualifying clause refers. Thus, if a person says, ' I 
 accomplished my business and returned the day after,' it may 
 be that the business was accomplished on the day after as 
 well as the return ; but it may equally have been finished on 
 the previous day. Any ambiguity of this kind may generally 
 be avoided by a simple change in the order of the words ; as, 
 for instance, 'I accomplished my business, and on the day 
 after returned.' Amphibology may sometimes arise from 
 confusing the subjects and predicates in a compound sen- 
 tence, as if in the sentence, ' Platinum and iron are very rare
 
 CLASSIFICATION OF FALLACIES 221 
 
 and useful metals,' I were to apply the predicate useful to 
 platinum and rare to iron, which is not intended. The word 
 ' respectively ' is often used to show that the reader is not at 
 liberty to apply each predicate to each subject." 
 
 " The Fallacy of Accent consists in any ambiguity arising 
 from a misplaced accent or emphasis thrown upon some word 
 of a sentence. A ludicrous instance is liable to occur in read- 
 ing Chapter XHL of the First Book of Kings, verse 27, where 
 it is said of the prophet, 'And he spoke to his sons, saying, 
 Saddle me the ass, and they saddled him.' The italics indi- 
 cate that the word him was supplied by the translators of the 
 authorized version, but it may suggest a very different mean- 
 ing. The Commandment, ' Thou shalt not bear false witness 
 against thy neighbor,' may be made by a slight emphasis of 
 the voice on the last word to imply that we are at liberty to 
 bear false witness against other persons. Mr. De Morgan, 
 who remarks this, also points out that the erroneous quoting 
 of an author, by unfairly separating a word from its context, 
 or italicising words which were not intended to be italicised, 
 gives rise to cases of this fallacy. 
 
 " It is curious to observe how many and various may be the 
 meanings attributable to the same sentence according as em- 
 phasis is thrown upon one word or another. Thus the sen- 
 tence, ' The study of Logic is not supposed to communicate a 
 knowledge of many useful facts,' may be made to imply that 
 the study of Logic does communicate such a knowledge al- 
 though it is not supposed to do so ; or that it communicates 
 a knowledge of a few useful facts ; or that it communicates a 
 knowledge of many useless facts. This ambiguity may be 
 explained by considering that if you deny a thing to have the 
 group of qualities A, B, C, D, the truth of your statement will 
 be satisfied by any one quality being absent, and an accented 
 pronunciation will often be used to indicate that which the 
 speaker believes to be absent. If you deny that a particular 
 fruit is ripe and sweet and well-flavored, it may be unripe and 
 sweet and well-flavored ; or ripe and sour and well-flavored ; 
 or ripe and sweet and ill-flavored ; or any two or even all
 
 222 ELEMENTS OF LOGIC 
 
 three qualities may be absent. But if you deny it to be ripe 
 and sweet and well-flavored, the denial would be understood 
 to refer to the last quality. Jeremy Bentham was so much 
 afraid of being misled by this fallacy of accent that he em- 
 ployed a person to read to him, as I have heard, who had a 
 peculiarly monotonous manner of reading." 
 
 As already remarked, although these errors in the interpre- 
 tation of propositions are not strictly fallacies, according to 
 the present usual acceptation of that term, it may be well to 
 have given them this consideration, because they are often the 
 source of logical fallacies in giving wrong assumptions to start 
 from. But they are not the result of violating any logical laws 
 such as have been laid down. It is these violations with which 
 the proper discussion of fallacies is concerned. Hence we turn 
 to the second class, which we have denominated Logical Falla- 
 cies. 
 
 Logical Fallacies are errors in reasoning or inference, and 
 not of interpretation. An error of interpretation is an error 
 of intellectual perception ; an error of reasoning is an error of 
 judgment in the passage from one proposition or conception 
 to another assumed to be contained in the former. The data 
 or premises may be correctly interpreted and yet the infer- 
 ence be a wrong one. As an illustration take the simple con- 
 version of propositions in A. We may be correct in our con- 
 ception of the proposition "All nations are aggregates of 
 men," but wrong in the inference from it, by simple conver- 
 sion, that " All aggregates of men are nations." And so on 
 with all other forms of inference where the conclusion is not 
 legitimately deduced from the premises. 
 
 Logical fallacies are divided into formal and material, ac- 
 cording as the error is in the form of the reasoning or in the 
 subject-matter of reasoning. A formal fallacy is an error 
 which arises from a violation of the formal laws of inference. 
 It is incident to the mere form of statement, or, as it is often 
 said, is a fallacy in dictione or in voce. It requires only a 
 knowledge of what the formal laws of reasoning are to detect 
 such fallacies. On the other hand, a material fallacy is one
 
 CLASSIFICATION OF FALLACIES 223 
 
 which is due to some peculiarity in the matter of the reasoning, 
 and hence arises independently of the form of statement, and 
 so is said to be extra dictionem. The formal laws may be con- 
 formed to, but owing to some ambiguity of meaning or assump- 
 tion of facts which" are not true the conclusion may be materi- 
 ally vitiated in spite of the correctness of the formal reasoning. 
 The material fallacy can be detected only by those who are 
 familiar with the subject-matter of the discourse or argument. 
 In Political Economy, for instance, any one familiar with the 
 laws of reasoning might be able to detect formal errors in 
 reasoning, but in order to discover the fallacies due to ma- 
 terial considerations, that is, to the matter of the subject, the 
 student must understand Political Economy. It is the same 
 with all other subjects when the question regards material 
 fallacies. 
 
 The further classification or subdivisions of formal and ma- 
 terial fallacies must be considered in separate paragraphs. 
 We take u"p briefly the formal fallacies. 
 
 1. FORMAL FALLACIES. These have been sufficiently defined 
 as mere violations of the principles of the syllogism which we 
 have previously enunciated. They are determined by the 
 number of terms, the distribution of terms, and the nature 
 of the premises in the syllogism. Each species may be con- 
 sidered briefly. 
 
 (a) Fallacy of Four Terms, or Quaternio Terminorum. 
 One rule of the syllogism is that it shall not contain more 
 than three terms : the presence of a fourth term vitiates the 
 conclusion, because it prevents that comparison with a middle 
 term which is necessary to reasoning. A simple illustration 
 of the Quaternio Terminorum is the following : 
 
 Men are mortal. 
 Socrates is a Greek. 
 . * . Socrates is mortal. 
 
 The impossibility of drawing an inference in such cases is 
 so apparent, and the temptation to do it is so unlikely that 
 errors of this kind scarcely deserve notice. They are not
 
 224 ELEMENTS OF LOGIC 
 
 common enough to require any special warning against them. 
 It is only in the modified form of Equivocation that they are 
 frequent. This occurs when the form and matter of a term 
 are different, that is, when the same term has different mean- 
 ings. There are, of course, cases where the terms are not 
 grammatically the same, but which are logically identical in 
 meaning. These would only apparently be cases of four terms. 
 The one circumstance which determines a case of four terms 
 is a distinction of material import that is not likely to be con- 
 fused with any form of a concept. This will distinguish such 
 instances from Equivocation, which is a modified form of Qua- 
 ternio Terrninorum. 
 
 There is, perhaps, a sense in which the fallacy of four terms 
 is a material fallacy, in that new matter is introduced into the 
 syllogism besides what is necessary to give it legitimacy. But 
 this aspect of it is hardly worth serious consideration, al- 
 though it may deserve mention for the purpose of recogniz- 
 ing the possibility. 
 
 (6) Illicit Process of the Middle Term. This fallacy has al- 
 ready been explained and illustrated. It is due to a failure 
 to distribute the middle term at least once in the premises. 
 It may occur in several ways. One illustration of it will suf- 
 fice : 
 
 M = P Some Pennsylvanians are Americans. I ^j 
 
 fg) = M All Philadelphians are Pennsylvanians. A I pj g j 
 
 . ' . fS> = P All Philadelphians are Americans. A j 
 
 (c) Illicit Process of the Major Term. This is due to the 
 distribution of the major term in the conclusion when it is 
 not distributed in the premises. This also may occur in sev- 
 eral ways : 
 
 (M) = P All men are mortal A 1 
 
 S X (M) Some animals are not men. O I Fig. I. 
 
 * S X (p) ' Some animals are not mortal. O I 
 
 (d) Illicit Process of the Minor Term. This fallacy is due to 
 the distribution of the minor term in the conclusion when it
 
 CLASSIFICATION OF FALLACIES 225 
 
 is not distributed in the premises. One of the many ways in 
 which this occurs is the following : 
 
 (M) = P All Germans are Caucasians. A 1 
 (M) = S All Germans are men. A \~Ficr. III. 
 
 . . (s) P . . All men are Caucasians. A 
 
 J 
 
 (e) Fallacy of Negative Premises. This is due to an attempt 
 to draw a conclusion when both premises are negative and re- 
 quires no illustration. 
 
 (/) Fallacy of Particular Premises. This is due to an at- 
 tempt to reason with particular premises. This case when 
 tested turns out to be a fallacy due to illicit distribution of 
 terms, either illicit middle or illicit major. 
 
 We might also include in the formal fallacies breaches of 
 Kules 7 and 9 (p. 173), or attempts to draw an affirmative con- 
 clusion when one of the premises is negative, and to draw a 
 universal conclusion when one of the premises is particular. 
 These errors, however, have not received any special name. 
 
 2. MATERIAL FALLACIES. Material fallacies, as defined, are 
 due to something in the matter of reasoning. They are, ex- 
 cepting one instance, the Petitio Principii, cases of introduc- 
 ing new matter into the syllogism, while the form remains the 
 same, and so are modifications of the Quaternio Terminorum. 
 This introduction of new matter may be either in the prem- 
 ises or in the conclusion. If it be in the premises it must be 
 in connection with the middle term, which will give some form 
 of Equivocation as the first material fallacy. If it be in the 
 conclusion, it must be in connection with the major or minor 
 terms, which will give some form of Inconsequence. Those of 
 Equivocation correspond to the formal fallacies of Undistrib- 
 uted Middle and Quaternio Terminorum. Those of Inconse- 
 quence correspond to the formal fallacies of Illicit Major and 
 Illicit Minor. The exceptional case of Petitio Principii, which 
 we have mentioned, is an instance of assuming matter which is 
 not admitted or not proved. There seems, therefore, to be 
 three forms of material fallacy. But when we consider that 
 15
 
 226 ELEMENTSJ)F LOGIC 
 
 fallacies of Inconsequence are assumptions of matter not in 
 the premises, as the Petitio Principii is an assumption of un- 
 proved premises, we may reduce the last two to what may be 
 called Fallacies of Presumption, and take as the first class 
 those of Equivocation. Material^fallacies we therefore divide 
 into two classes, those of Equivocation and those of Pre- 
 sumption. These two classes require some further explana- 
 tion. 
 
 The fallacies of Equivocation are all due to the equivocal or 
 ambiguous use of terms. They most frequently occur in con- 
 nection with the middle term of the syllogism, although some 
 logicians consider them possible in connection with the major 
 and minor terms. I prefer to limit them to the middle term 
 for the sake of convenience, although we might admit Equivo- 
 cation in the major and minor terms as at the same time a 
 fallacy of Inconsequence. The whole matter, and our reasons 
 for the classification, will not appear clear until the material 
 fallacies have been explained in detail. At present it must 
 suffice to obtain their classification. 
 
 The fallacy of Equivocation with logicians generally is lim- 
 ited to what is called Ambiguous Middle, and is not identified 
 with the two fallacies of Accident, and the two of Composition 
 and Division. But since they all turn upon an equivocal use of 
 the middle term, and since, in my own experience with students 
 of Logic, these various fallacies are constantly confounded 
 with each other, I am convinced that they should be classed 
 together under a common principle. I, therefore, use the term 
 Equivocation in a much more comprehensive sense than is 
 usual with writers on Logic, and so to include the fallacies of 
 Accident, Simple and Converse, of Composition and Division, 
 and the ordinary case of Ambiguous Middle, which I shall call 
 Specific Accident. In this way the student has only to deter- 
 mine, first, whether the fallacy turns upon the use of an equiv- 
 ocal term, or upon the presumption of matter in the premises 
 or the conclusion, and then he can proceed to determine the 
 special form of Equivocation. The peculiarities of these forms 
 will be examined in the next chapter.
 
 CLASSIFICATION OF FALLACIES 
 
 227 
 
 In the so-called fallacies of Presumption, as already indi- 
 cated, we may assume the truth of the premises when they 
 should be proved, or we may assume new matter in the con- 
 clusion not contained in the premises, which may be admitted. 
 This distinction gives rise to two general divisions of the fal- 
 lacies of Presumption, namely, the Petilio Principii, or Beg- 
 ging the Question, and the Non Sequitur, or False Consequent. 
 They will be discussed in the next chapter. The following 
 table summarizes our classification : 
 
 ( Tim-mo (Fallacy of Amphibology Ambiguous grammatical structure of the sen- 
 tence. 
 
 (Fallacy of Accent = Ambiguity due to misplaced accent, 
 f Quaternio Terminorum. 
 
 Illicit Middle. 
 Formal J L 1 !! ^ Major. 
 
 Logical 
 
 Material 
 
 Illicit Minor. 
 Negative Premises. 
 Particular Premises. 
 
 i TTI QnanHtv (Composition. 
 Equivo- j ID Q nantlt y {Division, 
 cation ) ( Simple Accident. 
 
 ( In Quality or Accident ! Converse Accident. 
 (Specific Accident. 
 
 f PetiHo ( P et iti f Assumptionon probata. 
 Presump- j Principii 
 
 Non Sequitur
 
 CHAPTER 
 
 MATERIAL FALLACIES 
 
 1st. Fallacies of Equivocation. These have been de- 
 fined as caused by the equivocal use of terms. I have divided 
 them into two classes, those of quality, or Accident, and those 
 of quantity. Those of quality or accident are so called because 
 the fallacy arises from some confusion due to differences of 
 meaning in regard to the attributes denoted by a term in a 
 proposition. Thus, if I say " Iron is a metal," I affirm " metal " 
 of it in its proper form, as an aggregate of certain qualities or 
 attributes. Now, if I also say " Rust is iron," I use the term 
 "iron "in a slightly different sense, affirming that the sub- 
 stance, or generic, not the specific, qualities of it are identical 
 with " rust ; " that is to say " rust " is " iron " only in its sub- 
 stance not in its form. This fact prevents me from drawing 
 the conclusion that "Rust is a metal." The fallacies of quan- 
 tity are so called because they are due to the different senses 
 in which a merely numerical aggregate of individuals can be 
 taken. Thus, " All the trees " m&y be taken collectively or dis- 
 tributively, and so .give rise, as we shall see, to an equivoca- 
 tion. We consider this form of fallacy first in order, and it is 
 perhaps the easier to detect. It is that of Composition and 
 Division. 
 
 1. FALLACY OF COMPOSITION AN$ DIVISION. Both fallacies 
 arise from the confusion of a collective and a distributive term, 
 but one of them is the converse of the other. The mode of de- 
 termining them can be expressed in the following formula : 
 
 ( In the major premise the middle term is used distributively. 
 Composition J 
 
 ( In the minor premise the middle term is used collectively. 
 
 f\- j In the major premise the middle term is used collectively. 
 
 ( In the minor premise the middle term is used distributively.
 
 MATERIAL FALLACIES 229 
 
 In the fallacy of Composition it will thus be seen that we 
 argue from a distributive to a collective use of the term ; 
 and, vice versa, in Division we argue from the collective to the 
 distributive use of a term. Probably a simpler means of de- 
 termining the matter in each case would be to observe wheth- 
 er, as a whole, the proposition was used distributively or col- 
 lectively, and not to make the decision more difficult by look- 
 ing for this distinction in the middle term of the premises. 
 
 One of the best illustrations of a fallacy of Composition is 
 the following : 
 
 All the angles of a triangle are less than two right angles. 
 
 A, B, C are the angles of a triangle. 
 
 Therefore A, B, C are less than two right angles. 
 
 In the major premise the proposition is true, if we suppose 
 that the expression " all the angles of a triangle " is taken dis- 
 tributively ; that each angle taken alone is less than two right 
 angles ; for taken together they are equal to two right angles. 
 The conclusion, therefore, cannot be true, unless A, B, C are 
 taken distributively. For if we mean in the major premise 
 that each angle is less than two right angles, and in the con- 
 clusion that all together are less than two right angles, we infer 
 what we have no right to infer ; that is, we argue from what 
 is distributively true of A, B, C, to what is supposed wrongly to 
 be collectively true of them. A similar case often occurs in 
 arguments like the following : 
 
 Thirteen and seventeen are prime numbers. 
 Thirty is thirteen and seventeen. 
 Therefore thirty is a prime number. 
 
 In the major premise " thirteen and seventeen " are used dis- 
 tributively, and in the minor premise collectively. Thirty not 
 being identical with " thirteen and seventeen," considered dis- 
 tributively, cannot be identical with that which is identical 
 with them in this sense, and hence the fallacy of composition. 
 In the first illustration the fallacy grows out of the ambigu-
 
 230 ELEMENTS OF LOGIC 
 
 cms use of the word all, which in such cases may have either a 
 collective or a distributive signification. Thus if we were to 
 argue that because " All the peers derive their titles from the 
 crown," and " The House of Parliament consisted of all the 
 peers," therefore " The House of Parliament derived its title 
 from the crown," we should be committing again the fallacy of 
 Composition. " We must not argue that because every mem- 
 ber of a jury is very likely to judge erroneously, the jury as a 
 whole are very likely to judge erroneously ; nor that because 
 each of the witnesses in a law case is liable to give false or 
 mistaken evidence, no confidence can be reposed in the con- 
 current testimony of a number of witnesses." And we may 
 add that we cannot argue from the truth of all the incidents 
 in a story to the truth of the story as a whole. A novel may 
 interweave a large number of true facts and incidents and yet 
 not be true or historical in its totality. " It is by a fallacy of 
 Composition that protective duties are still sometimes up- 
 held. Because any one or any few trades which enjoy protec- 
 tive duties are benefited thereby, it is supposed that all trades 
 at once might be benefited similarly ; but this is impossible, 
 because the protection of one trade by raising prices injures 
 all others." 
 
 The best illustration of the fallacy of Division is the con- 
 verse of the one given for Composition. It is as follows : 
 
 All the angles of a triangle are equal to two right angles. 
 
 A is an angle of a triangle. 
 
 Therefore A is equal to two right angles. 
 
 In the major premise the middle term is used collectively, as 
 in no other way could we say that "all the angles of a tri- 
 angle are equal to two right angles." Hence we mean, not 
 that each individual angle is so, but only that all together are. 
 In the minor premise the middle term is distributive, and 
 hence in the conclusion we show that we have argued from 
 what is true collectively, or of an aggregate, to what is true 
 only in a distributive sense. The fallacy is, therefore, one of
 
 MATERIAL FALLACIES 231 
 
 Division. If I were to argue from the fact that Congress or 
 Parliament had voted a subsidy that Mr. A. or Lord B. had 
 voted for the same, I should be committing the same fallacy. 
 So also would be the argument that because houses make a 
 city a given mansion would make a city because it is a house. 
 We commit this fallacy when we imagine that because the ag- 
 gregrate of expense is large, the number of individual items of 
 expense will be large. 
 
 2. FALLACIES OF ACCIDENT It is important to keep these 
 distinct from the fallacies of Composition and Division. The 
 latter have to do with numerical or mathematical aggregates 
 and individuals, the former with logical or metaphysical wholes 
 which represent totals of attributes. Unless we keep this in 
 view we are liable to confuse them. But if we remember that 
 Composition and Division turn upon the collective and distrib- 
 utive use of terms, and the fallacies of Accident upon the con- 
 fusion of essentia and accidentia, or genus and species (confer- 
 entia and differentia), or of the abstract and concrete, we shall 
 have no difficulty in the judgment of particular cases, We 
 divide the fallacies of Accident or Quality into three kinds, 
 namely, 
 
 (a) Simple Accident, or argument from the essence or con- 
 ferentia to the accident or differentia. Its dictum in old Latin 
 is, a dicto simpliciter ad dictum secundum quid, meaning 
 " from an absolute or unconditioned statement to one which 
 is conditioned or accidental." 
 
 (6) Converse Accident, or argument from an accident or dif- 
 ferentia to the essence or conferentia. Its dictum in old Latin 
 is, a dicto secundum quid ctd dictum simpliciter, meaning the 
 reverse of that for Simple Accident. 
 
 (c) Differential or Specific Accident, an argument from ac- 
 cident to accident, or from differentia to differentia. Its dic- 
 tum would be, a dicto secundum quid ad dictum secundum quid, 
 meaning "from a conditioned to a conditioned assertion."* 
 
 * This classification of Ambiguous Middle with the fallacies of Accident 
 is entirely new, so far as I know, but I think the exposition of it will 
 quite justify the fnnovation.
 
 232 ELEMENTS OF LOGIC 
 
 Jevons defines the fallacy of Simple Accident to be an argu- 
 ment "from a general rule to a special case" and the fallacy of 
 Converse Accident, "from a special case to a general one." 
 There is considerable ambiguity in this account of the case, 
 because the expression " general rule " is equivocal. It may 
 denote what is numerically or mathematically " general," or 
 what is logically "general." In the former instance it simply 
 denotes what is true of a large number or the majority of a 
 class, but in the latter it denotes what is true of the genus, 
 essentia, or conferentia. If we accept the former meaning we 
 are liable to confuse the formal fallacy of an undistributed 
 middle with the material fallacy of accident. Thus if we were 
 to argue that because "men have the right to vote," and 
 " criminals are men ; " " therefore criminals have the right to 
 vote," our reasoning would be perfectly correct as long as the 
 major premise was regarded as a universal proposition. But 
 as it stands it is what is called a "general " or indefinite prop- 
 osition, and may simply denote that men as a "general rule" 
 have a right to vote. In such a case the proposition is a par- 
 ticular one and the middle term is undistributed. But we 
 should be arguing from a " general " to a special case, and yet 
 the fallacy is not one of Accident. We prefer, therefore, to de- 
 fine the fallacies of Accident more accurately by indicating 
 that the " general rule " or case must mean the genus, essence, 
 or conferentia, and the " special case " must mean the species, 
 accident, or differentia. This fallacy, therefore, is an argu- 
 ment from one of these properties or group of properties to 
 the other ; that of Simple Accident from genus to species, from 
 conferentia to differentia, from essence to accident, from ab- 
 stract to concrete, etc. ; that of Converse Accident from spe- 
 cies to genus, etc., and that of Differential Accident from spe- 
 cies to species, or differentia to differentia, etc. There can be 
 no fallacy in arguing from genus to genus, or essence to es- 
 sence, because these always represent the same or identical 
 properties. 
 
 One of the oldest examples of Simple Accident is the fol- 
 lowing :
 
 MATERIAL FALLACIES 233 
 
 What you bought yesterday you eat to-day. 
 You bought raw meat yesterday. 
 Therefore you eat raw meat to-day. 
 
 De Morgan humorously remarks of this ancient illustration : 
 "This piece of meat has remained uncooked, as fresh as ever, 
 a prodigious time. It was raw when Eeisch mentioned it in 
 the ' Margarita Philosophica ' in 1496 ; and Dr. Whately found 
 it in just the same state in 1826." It is not so accurate an il- 
 lustration as is desirable according to the definition, because 
 the subject of the major premise is so indefinite, and is 
 hardly a genus. But in the conclusion the predicate is as- 
 serted of the subject, with the accidental quality of rawness 
 added, while in the major premise that predicate is asserted 
 only of the substance or essence of what was bought, and 
 hence we mistakenly argue from meat in general, and without 
 qualification to meat in a particular form. Another and per- 
 haps better illustration is the following : 
 
 Pine wood is good for lumber. 
 
 Matches are pine wood. 
 
 Therefore matches are good for lumber. 
 
 Here the predicate of the major premise is asserted of the 
 substance or essence of " pine wood," not of all forms of it, 
 while matches are pine wood not only in essence, but in a par- 
 ticular form or accident. We cannot affirm of this differen- 
 tial accident what is true only of the essence or conferentia. 
 So also we cannot argue from the fact that oxygen and hy- 
 drogen will burn, that water will burn because it is oxygen 
 and hydrogen. " It would be a case of the simple fallacy of 
 Accident to argue that a magistrate is justified in using his 
 power to forward his own religious views, because every man 
 has a right to inculcate his own opinions. Evidently a magis- 
 trate as a man has the rights of other men, but in his capacity 
 of a magistrate he is distinguished from other men, and he 
 must not infer of his special powers in this respect what is 
 true only of his rights as a man." All fallacies which attempt
 
 234: ELEMENTS OF LOGIC 
 
 the" substitution of a particular thing for the generic form be- 
 long to this head. 
 
 An illustration of the fallacy of Converse Accident is the 
 following : 
 
 Intoxicating liquors act as a poison. 
 
 Wine is an intoxicating liquor. 
 
 Therefore wine acts as a poison. 
 
 In this case we are arguing from the excessive use to all uses 
 of wine, an inference that is fallacious. The major premise is 
 true only of a particular mode of using liquors, or of the ex- 
 cessive use of them, while the conclusion, unless interpreted 
 with a similar qualification, asserts the same thing of all forms 
 of using them. " It is undoubtedly true that to give to beg- 
 gars promotes mendicancy and causes evil ; but if we inter- 
 pret this to mean that assistance is never to be given to those 
 who solicit it, we fall into the converse fallacy of Accident, 
 inferring of all who solicit alms what is true only of those who 
 solicit alms as a profession." Another formulated instance 
 appears in the following illustration : 
 
 Loyalty to the government is the duty of all citizens. 
 Loyalty to Charles I. was loyalty to the government. 
 Therefore loyalty to Charles I. was the duty of all citizens. 
 
 We may look at this instance in more than one way. In the 
 first place, the major premise means that loyalty is a duty to 
 legitimate governments or to such as execute the law, while 
 the minor premise asserts the fact that loyalty to Charles L 
 was loyalty to the government whatever its nature was, and 
 hence the conclusion asserts loyalty to Charles I. to be a duty 
 without qualification, and without distinguishing between him 
 as a magistrate and as a man. In the second place, loyalty to 
 Charles L may have been loyalty to him as a private person, 
 say by his servants, while all citizens could not be loyal to 
 him in this capacity, and so it is an error to argue from this 
 particular kind of loyalty to every form of it including civil 
 allegiance.
 
 MATERIAL FALLACIES 235 
 
 When these fallacies of Accident occur we may formulate 
 the following means of determining one form from the other, 
 the simple from the converse, and vice versa. The principle 
 is the same as in the fallacies of Quantity, Composition, and 
 Division. 
 
 fin the major premise the middle term must be a genus, es- 
 sentia, or conferentia, the predicate affirmed of these 
 
 making an abstract proposition. 
 Simple Accidents T ,, 
 
 In the minor premise the middle term must be a species, 
 
 accidentia, or differentia, the predicate so affirmed of 
 I the subject making a concrete proposition. 
 
 fin the major premise the middle term must be a species, 
 accidentia, or differentia, the predicate affirmed of 
 
 these making a concrete proposition. 
 Converse Accident < , ., . ,, .,,, , 
 
 In the minor premise the middle term must be a genus, 
 
 essentia, or conferentia, the predicate so affirmed of 
 the subject making an abstract proposition. 
 
 These rules are worded for the first Figure of the syllogism, 
 and are designed to indicate a clear means of deciding the 
 case when it cannot be done without such help. But the 
 syllogistic form of inference is not the only, and probably not 
 the most frequent, form of committing this fallacy, although 
 the process can no doubt be thrown into the form of the syl- 
 logism. The fallacy may often occur in the immediate infer- 
 ence by subalternation and by added determinants and com- 
 plex conceptions, or if not by these, by a process which very 
 much resembles them. If we infer from the contemptible char- 
 acter of one "reformer" in a particular cause, that all "re- 
 formers " are bad, we are committing the fallacy of converse 
 Accident. On the other hand, if we infer from the exchangea- 
 ble value of money, that the old Confederate currency has ex- 
 changeable value, because it is money, we commit the fallacy 
 of Simple Accident. 
 
 Without reference to the distinction between the two forms 
 of these fallacies, it may be said that wherever we attempt to 
 make an interchange of essence and accidence, or abstract and 
 concrete, under the same term, a fallacy of Accident is com- 
 mitted. An assertion sometimes seems to be made of the
 
 236 ELEMENTS OF LOGIC 
 
 whole of a concrete subject, when in fact it is made only of 
 its essential or of its accidental forms. Thus, when we say that 
 sulphuric acid is poisonous, we can assert this predicate, not of 
 its essence, but of a particular accidental quantity of it, be- 
 cause it can be taken with impunity in certain forms or 
 amounts. But we can neither argue from its poisonous char- 
 acter in large amounts to its injury in small amounts, nor to 
 its harmfulness in small quantities from its dangers in large 
 quantities. So of the assertions that " Governments are use- 
 ful," "Truth is sublime," " Charity is a virtue," etc. This 
 will be the case in many, if not nearly all abstract ideas and 
 propositions. Indeed we might say that we are extremely 
 liable to commit fallacies of Accident in arguing from the 
 concrete to the abstract and from the abstract to the con- 
 crete. We are certain to do so when the concrete and ab- 
 stract are viewed logically and not mathematically, as pre- 
 viously explained. When we say "Governments are useful," 
 we really affirm the predicate of governments in the abstract, or 
 perhaps generally of actual governments. But we mean usually 
 to speak of certain ideal forms of social organization and not 
 necessarily of any or all particular concrete forms of them. 
 Hence we speak of them in their essence or conferentia, and so 
 are not allowed to infer by subalternation that what is true of 
 them in this sense is true of them in the concrete. On the 
 other hand, if we asserted the predicate of them in the con- 
 crete as bad, we could not immediately infer that the same 
 was true of them in the ideal or abstract sense. Although I 
 have spoken of these inferences as apparently immediate, they 
 may be converted into mediate arguments by supplying a sup- 
 pressed and perhaps implied premise. Their immediacy ap- 
 pears in the assumption that what can be affirmed of govern- 
 ment in general can be affirmed of governments in particular 
 included in the class. 
 
 But the error lies precisely in this assumption, which does 
 not allow for the two senses in which the conceptions " gov- 
 ernment," truth," " charity," etc., can be used. It is here that 
 we can put to practical use the distinction between logical and
 
 MATERIAL FALLACIES 23T 
 
 mathematical generals, or between the logical and the mathe- 
 matical genus ; that is, between the genus as the sum of the 
 species, and the genus as a name for the conferentia. The 
 latter is the logical, as the genus, taken as the sum of the spe- 
 cies, is the mathematical conception of a class. If we say that 
 " governments are useful," in the mathematical sense, we mean 
 all individual and particular governments in the class, and so no 
 fallacy will be committed in an inference to the species. But 
 if we say it in the logical and abstract sense, we are using the 
 term in a sense which may not be true of any individual case 
 whatever, and hence the fallacy. 
 
 Another illustration will perhaps make the matter still 
 clearer. When we say "Men are mortal," we have made an as- 
 sertion which applies to all individual men. The predicate is 
 asserted mathematically of men. It is not asserted of man in 
 the abstract, but of all men in the concrete, all forms and 
 conditions of men. Hence when we form a minor premise in- 
 volving any number or species of men, the conclusion follows 
 necessarily because they are included in the same sense in the 
 major premise. The middle term has only a mathematical 
 signification and so admits of no fallacy. But suppose we af- 
 firm " Meat is healthy food," here is a statement which may 
 be taken either mathematically to denote all specific kinds of 
 meat, or in an abstract logical sense to denote that the sub- 
 stance so called is healthy food, and so it would be spoken of 
 in its essence, essential qualities, or conferentia, while we would 
 not intend to include the same matter in its raw state, its 
 stale or decayed condition, or in unlimited quantities. Hence 
 we could not argue from the universal truth in the first case 
 to the particular case in the second. This has already been 
 illustrated in the first syllogism representing the fallacy of 
 Simple Accident. In fact such statements are meant to affirm 
 the predicate of certain well-known, perhaps usual and nor- 
 mal forms of the subject, and so exclude the cases involved in 
 the conclusion when the fallacy of accident is committed. 
 But it is not apparent from the form of statement, because 
 the mathematical conception of universal propositions is the
 
 238 ELEMENTS OF LOGIC 
 
 usual and the most natural one. But it is just such substitu- 
 tions that the student must be on guard against, as liable, in 
 more serious situations than we have illustrated, to lead him 
 astray. 
 
 The technicalities of law offer a very rich field for the falla- 
 cies of Accident. It is the difference between one case and 
 another that occupies the barrister in his attempts to show 
 that they are not included in the general rule. In philosophy 
 the Cartesians committed the fallacy by denying that hard- 
 ness, weight, etc., were essential qualities of matter, and then 
 inferring that a cubic foot of iron had no more matter in it 
 than a cubic foot of air, because space or extension was re- 
 garded as the essence of matter. De Morgan quotes an amus- 
 ing story from Boccaccio which illustrates the fallacy, but in 
 too obtrusive a form to deceive any one, and yet it illustrates 
 the whole case : 
 
 "A servant who was roasting a stork for his master was pre- 
 vailed upon by his sweetheart to cut off a leg for her to eat. 
 When the bird came upon the table the master desired to 
 know what had become of the other leg. The man answered 
 that storks never had more than one leg. The master, very 
 angry, but determined to strike his servant dumb before he 
 punished him, took him next day into the fields, where they 
 saw storks, standing each on one leg, as storks do. The ser- 
 vant turned triumphantly to his master ; on which the latter 
 shouted, and the birds put down their other legs and flew 
 away. ' Ah, sir,' said the servant, ' you did not shout to the 
 stork at dinner yesterday ; if you had done so, he would have 
 shown his other leg too.' " 
 
 Not all fallacies of Accident are so easily detected as this, 
 but they illustrate the same principle and the same logical 
 characteristics. They are perhaps as frequent as any other 
 form of logical error, and in fact the inclination to make those 
 substitutions of two different things under the same name, 
 and separated only as essence and accident, is so common 
 that it has been well worth the pains to dwell upon the sub- 
 ject at great length.
 
 MATERIAL FALLACIES 239 
 
 It remains to show that the fallacy of Ambiguous Middle, 
 or what I have called Differential Accident, is rightly included 
 under the general head of Accident. We have intimated that 
 all are forms of equivocation, that is, substitutions of one mean- 
 ing of a term for another, and now we have to show that a 
 closer relation than is usually recognized by logical writers ex- 
 ists between the fallacies of Accident and the ordinary Am- 
 biguous Middle. An illustration will be the best means of 
 proving the case. 
 
 The end of life is its perfection. 
 Death is the end of life. 
 Therefore death is the perfection of life. 
 
 The ambiguity of the word " end " is perfectly apparent, and 
 we might be content with calling the fallacy merely one of 
 equivocation. But if we observe closely, although the word 
 end denotes two different things, there is a common idea at 
 the basis of them which makes the equivocation possible. 
 This common characteristic constitutes the generic or confer- 
 ential idea of the word. But it is the differential quality 
 which in each case determines the nature of the assertion. In 
 the major premise "end" means the object or purpose of life 
 of which perfection is asserted. In the minor premise it 
 means the termination of life, which is made identical with 
 death. Now, the common idea or conception which enables 
 us to apply the word " end " in both cases is the notion of 
 limit, or the point of interruption in a line, beyond which we 
 need not go for a given purpose. Hence the notion of object 
 is the differentia of one use, and termination that of the other, 
 so that the attempt to argue from one to the other, on the 
 ground of a common medium, is an attempt to pass from one 
 accident to another. It will be the same in all equivocal 
 terms where the confusion is not due to a mistaking of the ge- 
 nus or conferentia for the species or differentia, and vice versa. 
 Thus, again to use Jevons's example, 
 
 All criminal actions ought to be punished by law. 
 
 Prosecutions for theft are criminal actions. 
 
 Therefore prosecutions for theft ought to be punished by law.
 
 240 ELEMENTS OF LOGIC 
 
 Both the terms " criminal" and "action " are used in a double 
 sense. In the major premise " criminal " denotes what is im- 
 moral, and " action " a form of conduct, as an act of the will. 
 In the minor premise "criminal" denotes merely pertaining to 
 a crime, without implying any judgment upon its character, 
 and "action" denotes, after its old Latin use, a suit at law. 
 These are simply differential or specific meanings of the term, 
 which has no generic application apart from such as are given, 
 and so the argument is from one of these to the other through 
 the common conception implied in the terms. The fallacy is 
 a modified form of Quaternio Terminorum. But we should 
 call it the fallacy of Differential or Specific Accident in order 
 to classify it correctly and in order to understand its charac- 
 teristics. The expression Ambiguous Middle should be re- 
 served for a more comprehensive use, as equivalent to equivo- 
 cation, the two terms to be used interchangeably. 
 
 2d. Fallacies of Presumption. According to our pre- 
 vious explanation of these fallacies, something is presumed or 
 assumed which we have no right to take for granted in the 
 terms of the syllogism. They are presumptions in regard to 
 the matter or contents of the reasoning. The presumption 
 may be regarding the material truth of the premises, or it 
 may be regarding the introduction of new matter into the 
 conclusion when the premises are admitted. However cor- 
 rect the formal reasoning may be, the conclusion may be viti- 
 ated materially, either by assuming the premises when they 
 should be proved, or by introducing a fourth term into the 
 conclusion. We have then, as indicated in the classification of 
 fallacies, two kinds of materially false inferences of Presump- 
 tion, the Petitio Principii and the Fallacia Consequentis, or 
 Non Sequitur. 
 
 1. FALLACY OF PETITIO PRIXCIPII. This is ordinarily called 
 Begging the Question, and means the assumption of a fact or 
 a premise without proof, or, as in the argument called reason- 
 ing in a circle, is an attempt to prove a proposition by itself. 
 This is a form of assuming it when it should be proved by 
 some more general and accepted truth. The Petitio Principii
 
 MATERIAL FALLACIES 241 
 
 we divide into two distinct forms, the Petitio Argumenti, which 
 is committed in the presentation of an argument or when at- 
 tempting the proof of a proposition, and the Ignoratio Elenchi, 
 which is committed in the refutation or the attempt to dis- 
 prove a proposition ; it is simply a little more complicated 
 petilio principii. The Petitio Argumenti we again subdivide 
 into two forms, the assumptio non probata, or assumption of 
 unproved premises, which may be different from the conclu- 
 sion, and the circulus in probando, or reasoning in a circle, the 
 assumption of premises which are the same as the conclusion. 
 
 The assumptio non probata can be illustrated by any syllo- 
 gism whatever. Thus if we were trying to prove that " All 
 men were mortal," and assumed that " All organic beings are 
 mortal," with the minor premise that " All men are organic 
 beings," we could be charged with begging the question by one 
 who did not admit the proposition "All organic beings are 
 mortal." He might admit that the formal reasoning was per- 
 fectly correct, and that the conclusion would be true if the 
 premises were ; but he would insist that the material inference 
 was false because the premise was not admitted or not proved. 
 It does not matter which premise is disputed, the effect is the 
 same. We can charge a petitio principii upon a man when we 
 dispute the major and admit the minor premise, and vice versa, 
 or when we dispute both premises. It is sufficient to question 
 one of the conditions of the conclusion. 
 
 It is not merely the failure to prove one's premises that con- 
 stitutes the fallacy of begging the question. This failure must 
 be one which occurs when proof is needed or demanded. It 
 is, perhaps, most frequent when trying to convince some one 
 else of a given truth, although it may occur whenever we 
 are trying to prove to our own minds a conclusion without 
 assuring ourselves sufficiently of the stability of the premises 
 upon which the conclusion rests. But it is most frequent in 
 arguments with others, because the one condition of proof in 
 such cases is that an opponent or reader admits the principles 
 upon which the conclusion is to be established. We cannot 
 prove to him a truth with premises he does not admit. If we 
 16
 
 242 ELEMENTS OF LOGIC 
 
 assume these without his acceptance, our reasoning has no 
 cogency, and he is at liberty to say that we are begging the 
 question, and this without disputing either the formal accu- 
 racy of our process or the truth of our proposition. He 
 merely claims that the case is not proved. A proposition in 
 a conclusion may be true, although it has not been proved 
 in the premises. The advantage of proving it lies in mak- 
 ing it a special case, included under a general law or class, 
 so that when a person has admitted the larger he must per- 
 force admit the smaller. But there are instances in which we 
 may dispute the universality of a principle or premises either 
 to show that the conclusion may, so far as we know, be an ex- 
 ception, or to assert that it is not proved by such a case, how- 
 ever true it may be in reality. Suppose we wish to prove that 
 " All cattle have cloven feet." If, in order to do so, we assert 
 that "All ruminants are cloven-footed," and "All cattle are ru- 
 minants," the conclusion will follow, provided the premises 
 are accepted. But we can be charged with begging the ques- 
 tion if the major premise, "All ruminants are cloven-footed," 
 is not true, although it may be true that " All cattle are rumi- 
 nants," and also that they are all cloven-footed. But the prop- 
 osition is not proved except by the universality of the major 
 premise in this case. It is one thing to perceive the truth of 
 a proposition as a matter of fact, and it is another to prove it 
 by means of a higher condition. The charge of petitio princi- 
 pii, then, must not be construed as properly meaning that the 
 conclusion is denied, but only that it is not proved. We 
 should be committing a counter-fallacy if we supposed that 
 this error was a disproof of the proposition in question. 
 
 We may too hastily impute the fallacy of begging the ques- 
 tion. This is virtually done when we demand proof for a 
 premise merely because we see that the conclusion must be 
 accepted if the premise is admitted. It is often employed in 
 order to evade the issue and escape conviction. It may be 
 permissible sometimes to carry the demand for proof back 
 through several steps, but the danger is that it will most fre- 
 quently be dishonestly done, or be the mark of a weak cause.
 
 MATERIAL FALLACIES 243 
 
 De Morgan describes the case in the following language : 
 " There is an opponent fallacy to the pelitio principii which, I 
 suspect, is of more frequent occurrence ; it is the habit of 
 many to treat an advanced proposition as a begging of the 
 question the moment they see that, if established, it would 
 establish the question. Before the advancer has more than 
 stated his thesis, and before he has had time to add that he 
 proposes to prove it, he is treated as a sophist, on his oppo- 
 nent's perception of the relevancy of his first step." In such 
 emergencies the person presenting the argument must ascer- 
 tain whether his opponent admits in each case that the con- 
 clusion will follow if the premises are true, and by continuing 
 this process he will either expose the motive of his opponent 
 or morally weaken his demand for proof. 
 
 This fallacy is very likely to occur in the disjunctive syllo- 
 gism, and especially in the dilemma ; for we may assume the 
 disjunction to be complete when it is not. There may be 
 more than the two alternatives usually assumed in the case. 
 An instance of this occurs in the sophism which was used by 
 early Greek philosophers to prove the impossibility of motion. 
 It was said that a thing must either move where it was, or 
 where it was not. It was absurd to suppose that it could 
 move where it was not, and if it moved it could not be in the 
 place where it was, and therefore it was inferred that its mo- 
 tion was impossible. But this conclusion, or the premises 
 rather, lost sight of the third alternative, namely, that a body 
 might move from the place where it was to a place where it was 
 not, or had not been the moment before. The omission of 
 this alternative in the premise made the argument a petitio 
 principii. There is a traditional answer to this argument 
 which we shall notice under the Ignoratio Elenchi. 
 
 "Jeremy Bentham pointed out that the use even of a single 
 name may imply a petitio principii. Thus in a church assem- 
 bly or synod, where a discussion is taking place as to whether 
 a certain doctrine should be condemned, it would be a petitio 
 principii to argue that the doctine is heresy, and therefore it 
 ought to be condemned. To assert that it is heresy is to beg
 
 244 ELEMENTS OF LOGIC 
 
 the question, because every one understands by heresy a doc- 
 trine which is to be condemned. Similarly, in Parliament, a 
 bill is often opposed on the ground that it is unconstitutional 
 and therefore ought to be rejected ; but as no precise definition 
 can be given of what is or is not constitutional, it means little 
 more than that the measure is distasteful to the opponent. 
 Names which were used in this fallacious manner were aptly 
 called by Benthain Question-begging Epithets." 
 
 The circulus in probando is a species of petitio principii, 
 which consists in " arguing in a circle," or in assuming as 
 proof of a proposition some assertion which is identical with 
 it in its import, or in. trying to prove a proposition by itself. 
 Thus to say that " Man is wise because he is rational," is to 
 argue in a circle, because " rational " is substantially identical 
 in meaning with " wise." So also would it be to argue that 
 " The weather is warm because it is summer, and it is sum- 
 mer because it is warm," and " Men never practise excess be- 
 cause they are not immoderate in their habits." Je von s's illus- 
 tration is the following : " Consciousness must be immediate 
 cognition of an object ; for I cannot be said really to know a 
 thing unless my mind has been affected by the thing itself." 
 Here " to know " and " immediate cognition," are identical in 
 import and cannot be used to prove each other. 
 
 It is mostly in long arguments that this fallacy can be com- 
 mitted without ready detection. When we argue that a per- 
 son should submit himself to the guidance of his party, or his 
 government, because they maintain what is right, and then 
 proceed to prove this by asserting they are right because 
 they ought to be submitted to ; or if we argued that lead had 
 more matter in it than a given amount of wood, because it 
 was heavier, and that it was heavier because it had more 
 matter in it, the circle would be so narrow that it would be 
 easy of detection. But when the circular petitio principii oc- 
 curs at the end of a long discourse, as it often may do, it may 
 be committed without easy discovery.- Only the closest obser- 
 vation can secure us against it. It is likely to be committed 
 by the use of synonyms which are taken to express more than
 
 MATERIAL FALLACIES 245 
 
 the conception involved. Jevons and Whately have remarked 
 that the English language, being composed of two or more 
 languages, is liable to this fallacy, because it frequently has 
 several synonymous terms for the same conception. 
 
 The Ignoratio Elenchi is the second general class of fallacies 
 which we have included under the head of Petitio Principii. 
 It may not seem clear why we have chosen to consider it a 
 species of begging the question. The reason for so doing can- 
 not be fully appreciated until the fallacy has been denned. It 
 has been called by Whately and others the fallacy of Irrele- 
 vant Conclusion. This is a true enough description of it, ex- 
 cept that the definition does not exclude thefallacia consequen- 
 tis or non sequitur. We prefer, therefore, to define it as Igno- 
 rance of the Issue or Argument, and hence it consists in argu- 
 ing to the wrong point, or in proving one thing in a way that 
 seems to prove something else ; or proving something which is 
 not the contradictory of the thing asserted. It will be appar- 
 ent from the account of it that it occurs in the process of refu- 
 tation, or in proving something which is supposed to be the 
 opposite of what is believed or affirmed by an opponent. We 
 commit the fallacy by assuming the conclusion we reach to 
 be in contradiction with that against which we are arguing, 
 when it is not a contradictory. In refutation it is our busi- 
 ness to prove a contradictory of a given assertion, but if we 
 prove something which is not denied by our opponent, we are 
 evading the issue, and proving something that is irrelevant. 
 Thus, in assuming the contradiction which is not a contradic- 
 tion, or something to be denied by our opponent which is not 
 denied by him, we indirectly beg the question. This fact is, 
 our reason for classing the Ignoratio Elenchi with the Petitio 
 Principii. It is much more complicated than the simple case, 
 but when we consider that it is merely the counter-petition of 
 one who is adducing an argument in refutation to that of the 
 person producing proof of a proposition, we shall perceive 
 the right to regard it as we have done. It is the assumption 
 of what is not a fact, or of what is not admitted to be a fact 
 by an opponent, and such an assumption is of the nature of
 
 ELEMENTS OF LOGIC 
 
 a petitio principii. But the fact is concealed and complicated 
 by the circumstance that two syllogisms and the law of con- 
 tradiction are involved in the explanation of the case. That 
 the issue is evaded can generally be determined without re- 
 solving the fallacy into a petitio principii. But it is this kind 
 of fallacy nevertheless. 
 
 A good illustration of the Ignoratio Elenchi is the follow- 
 ing : Suppose a man is accused of being a thief, and I prove 
 that he is not a thief. Now the proper disproof of this asser- 
 tion is to prove the contradictory, namely, that he is a thief. 
 But if my opponent instead of proving this, proves that the 
 man is a rogue, he commits the ignoratio elenchi, because I 
 have not denied the latter proposition, or asserted the contra- 
 dictory of it. He virtually begs the question by assuming 
 that to prove him a rogue is to prove him a thief, and that he 
 has proved the contradictory of my assertion when his propo- 
 sition is not denied. Omitting the premises which might be 
 involved in establishing either side, and those involved in 
 proving the proportion assumed to be disproof, the whole re- 
 lation may be represented as follows : 
 
 Proof. Disproof. Ignoratio Elenchi. 
 
 .'. A is not a thief. .'. A is a thief. .-. A is a rogue. 
 
 In asserting that the man is a rogue the opponent intends 
 to avail himself of certain presumptions which might follow 
 from the fact that the man was a rogue. The proof that A 
 was not a thief might imply, to untrained minds, that he was a 
 good man, or the disproof might be such as it was not easy to 
 counteract the effect of. Hence if the man can be proved to 
 be a rogue, it is assumed that a presumption against the valid- 
 ity of the disproof is established, or that to prove him a rogue 
 is to prove him a thief. The fallacy, nevertheless, is apparent, 
 in that we may say that all thieves are rogues, but not that all 
 rogues are thieves. The fallacy, then, is in assuming the con- 
 vertibility of the two conceptions, " rogue " and " thief," and
 
 MATERIAL FALLACIES 247 
 
 that his assertion contradicts the proposition involved in the 
 original statement. 
 
 " The fallacy is the great resource of those who have to 
 support a weak case. It is not unknown in the legal profes- 
 sion, and an attorney for the defendant in a lawsuit is said to 
 have handed to the barrister his brief marked, 'No case; 
 abuse the plaintiffs attorney.' " In all the attacks on a per- 
 son, or his character, when the question regards a doctrine, 
 the fallacy is the same as in the case of the above attorney. 
 Thus if I praise a man's poetry or his philosophy, it is no refu- 
 tation of him to show that his life has been bad, or that he 
 has lost his mind. 
 
 De Morgan mentions a good instance : " If a man were to 
 sue another for debt, for goods sold and delivered, and if de- 
 fendant were to reply that he had paid for the goods furnished, 
 and plaintiff were to rejoin that he could find no record of 
 that payment in his books, the fallacy would be probably 
 committed. The rejoinder, supposed true, shows that either 
 defendant has not paid, or plaintiff keeps negligent accounts ; 
 and is a dilemma, one horn of which only * contradicts the 
 defence. It is the plaintiff's business to prove the sale from 
 what is in his books, not the absence of payment from what 
 is not ; and it is then the defendant's business to prove the 
 payment from his vouchers." 
 
 The observations of Whately and Mill are well worth quot- 
 ing in this connection at some length, since they furnish so 
 clear an exposition of this fallacy. Says the former: "Va- 
 rious kinds of propositions are, according to the occasion, sub- 
 stituted for the one of which proof is required. Sometimes 
 the particular for the universal ; sometimes a proposition with 
 different terms ; and various are the contrivances employed to 
 effect and to conceal this substitution, and to make the conclu- 
 sion which the sophist has drawn answer practically the same 
 purpose as the one he ought to have established. I say ' prac- 
 tically the same purpose,' because it will often happen that 
 
 * De Morgan in placing " only " in the position which it occupies in 
 the sentence makes his statement liable to a fallacy of accent.
 
 248 ELEMENTS OF LOGIC 
 
 some emotion will be excited some sentiment impressed on 
 the mind such as shall bring men into the disposition requi- 
 site for your purpose, though they may not have assented to, 
 or even stated distinctly in their own minds, the proposition 
 which it was your business to establish. Thus if a sophist has 
 to defend one who has been guilty of some serious offence, 
 which he wishes to extenuate, though he is unable distinctly 
 to prove that it is not such, yet if he can succeed in making 
 the audience laugh at some casual matter, he has gained prac- 
 tically the same point. 
 
 " So also if any one has pointed out the extenuating circum- 
 stances in some particular case of offence so as to show that it 
 differs widely from the generality of the same class, the soph- 
 ist, if he find himself unable to disprove these circumstances, 
 may do away with the force of them by simply referring the 
 action to that very class, which no one can deny that it belongs 
 to, and the very name of which will excite a feeling of disgust 
 sufficient to counteract the extenuation ; e.g., let it be a case 
 of peculation, and that many mitigating circumstances have 
 been brought forward which cannot be denied ; the sophisti- 
 cal opponent will reply, ' Well, but after all, the man is a 
 rogue, and there is an end of it ;' now in reality this was by 
 hypothesis never the question ; and the mere assertion of 
 what was never denied ought not, in fairness, to be regarded 
 as decisive ; but practically, the odiousness of the word, aris- 
 ing in great measure from the association of those very circum- 
 stances which belong to most of the class, but which we have 
 supposed to be absent in this particular instance, excites pre- 
 cisely the feeling of disgust which in effect destroys the force 
 of the defence. In like manner we may refer to this head all 
 cases of improper appeals to the passions, and everything else 
 which is mentioned by Aristotle as extraneous to the matter 
 in hand (l^w TOV Trpay/iaros). 
 
 "In all these cases, as has been before observed, if the fal- 
 lacy we are now treating of be employed for the apparent 
 establishment, not of the ultimate conclusion, but, as it very 
 commonly happens, of a premise, then there will be a com-
 
 MATERIAL FALLACIES 249 
 
 bination of this fallacy with the last mentioned (undue as- 
 sumption). 
 
 " For instance, instead of proving that ' this prisoner has com- 
 mitted an atrocious fraud,' you prove that ' the fraud he is ac- 
 cused of is atrocious ; ' instead of proving, as in the well-known 
 tale of Cyrus and the two coats, that ' the taller boy had a right 
 to force the other boy to exchange coats with him/ you prove 
 that ' the exchange would have been advantageous to both ; ' 
 instead of proving that ' a man has not a right to educate his 
 children or dispose of his property in the way he thinks best,' 
 you show that ' the way in which he educates his children or 
 disposes of his property is not really the best ; ' instead of prov- 
 ing that 'the poor ought to be relieved in this way,' you prove 
 that ' they ought to be relieved ; ' instead of proving that ' an 
 irrational agent whether a brute or a madman can never be 
 deterred from any act by the apprehension of punishment,' as, 
 for instance, a dog from sheep-biting, by fear of being beaten, 
 you prove that ' the beating of one dog does not operate as an 
 example to other dogs,' etc., and then you proceed to assume 
 as premises, conclusions different from what have really been 
 established," you commit the fallacy of ignoratio elenchi. But 
 it is in a modified form, because it appears less as a refutation 
 than as an attempted confirmation of some position. They 
 can, however, be conceived in the usual form by supposing 
 that the thesis to which it is assumed the conclusions are op- 
 posed is suppressed. Besides, one of them, the instance about 
 relieving the poor, might be considered a case of converse acci- 
 dent. But as something is proved with the assumption that 
 it is identical with another position, while it is in reality op- 
 posed to it, we have the ignoratio elenchi in the converse form. 
 
 " A good instance of the employment and exposure of this 
 fallacy occurs in Thucyclides, in the speeches of Cleon and Dio- 
 dotus concerning the Mitylenaeans ; the former, over and above 
 his appeal to the angry passions of his audience, urges fhe jus- 
 tice of putting the revolters to death, which, as the latter re- 
 marked, was nothing to the purpose, since the Athenians were 
 not sitting in judgment, but in deliberation ; of which the proper
 
 250 ELEMENTS OF LOGIC 
 
 end is expediency. And to prove that they had a right to put 
 them to death, did not prove this to be an advisable step." 
 
 Mill observes that " the works of controversial writers are 
 seldom free from this fallacy. The attempts, for instance, to 
 disprove the population doctrines of Malthus have been mostly 
 cases of ignoratio elenchi. Malthus has been supposed to be 
 refuted if it could be shown that in some countries or ages 
 population has been nearly stationary ; as if he had asserted 
 that population always increases in a given ratio, or had not 
 expressly declared that it increases only in so far as it is not 
 restrained by prudence or kept down by poverty and dis- 
 ease." 
 
 Dr. Johnson's refutation of Berkeley's idealism by kicking 
 against a stone is a similar fallacy. And so are all cases of 
 appeal to consequences supposed to contradict a given asser- 
 tion before proving that such a contradiction exists. In such 
 instances the opponent may accept the consequences, unless 
 the contradiction between them and his assertion is first 
 proved. 
 
 In addition to the general form of the Ignoratio Elenchi, 
 there are several special forms which it is important in a 
 treatise of Logic to consider. The valid process, of which the 
 ignoratio elenchi is the invalid, is called the argumentum ad 
 rem. The special invalid forms or cases of evasion are the ar- 
 gumentum adjudicium, argumentum ad populum, argumentum 
 ad hominem, argumentum ad ingorantiam, and the argumentum 
 ad verecundiam. 
 
 The argumentum adjudicium is an appeal to general or uni- 
 versal belief, and so is based upon the common judgments of 
 mankind. The dictum of such an appeal is the admitted or 
 assumed truth of what all men everywhere believe. The con- 
 troversialist appeals to this maxim because he supposes it is 
 admitted and that it contradicts some conclusion which an op- 
 ponent is trying to maintain. Thus if I deny the existence of 
 an external world, of spirit, or of an unseen world, it would 
 be an argumentum ad judicium to show that all men have 
 everywhere believed in their existence. This universal belief
 
 MATERIAL FALLACIES 251 
 
 may create a presumption or make it necessary to consider 
 the matter seriously, but it does not prove it. 
 
 The argumentum ad populum is an appeal to public opinion, 
 or to the passions and prejudices rather than to the intelli- 
 gence of people. 
 
 The argumentum ad hominem is an appeal to the practice, 
 profession, or principles of the person to whom or against 
 whom an argument is directed. It is an effective method of 
 silencing an opponent, but it does not prove the case. 
 
 The argumentum ad ignorantiam is an appeal to a man's ig- 
 norance in order to produce conviction upon his inability to 
 dispute the case. 
 
 The argumentum ad verecundiam is an appeal to authority, 
 or an accepted body of doctrines. 
 
 These several forms of argumenia are essentially the same 
 in their principles and their import. Four of them appeal to 
 certain admitted or assumed principles which are supposed to 
 prove the case because they are assumed to contradict the 
 opposite, which is the position to be disproved by the proof of 
 its alternative. But in no case, unless we except the ad homi- 
 nem instance, are we assured either that the dicta upon which 
 we depend are admitted by an opponent, or that they are 
 necessarily contradictory to the point in question. They are 
 thus evasions of the issue. 
 
 But it is important to remark that they are not always ir- 
 relevant or illegitimate merely because they are evasions. 
 There are circumstances in which it is perfectly legitimate to 
 use them ; only we must not suppose that this legitimacy im- 
 plies that they are methods of real proof. Although they 
 have a proper application they are not argumenta ad res. It 
 is important to take this fact into account in order not to 
 infer, from their fallacious nature as arguments to the point, 
 that their illegitimacy either impeaches the proposition in 
 question or excludes them from a certain relevancy for another 
 purpose. They are invalid only as proofs or disproofs of a 
 matter in discussion, but they are not invalid as means of es- 
 tablishing a contradiction between two propositions. Com-
 
 252 ELEMENTS OF LOGIC 
 
 mon discourse assumes that a man is refuted if we show 
 that he has contradicted himself and that he must accept a 
 given conclusion if the opposite contradicts his profession or 
 his practice. But this is not the fact. It does place him in a 
 position that compels him to choose between the two contra- 
 dictories, but it does not decide which of the alternatives he 
 must select. Hence the charge or proof of a contradiction in 
 a man's discourse is no disproof of his assertion, unless he 
 still holds to its contradictory. If he denies the contradiction 
 he may hold both alternatives. Hence the several argumenta 
 non ad res, in merely proving a contradiction somewhere, are 
 fallacies of ignoratio elenchi, in the relation of assuming that 
 they prove anything. But we must distinguish between this 
 and their valid use for establishing a contradiction. An illus- 
 tration in the case of the argumentum ad hominem will make 
 this position clear. We quoted the instance of incomplete 
 disjunction by the ancient Greek philosophers, who sought to 
 prove the impossibility of motion by trying to limit our con- 
 ception of it either to a change where a thing is, or a change 
 where it is not. Tradition has it, says De Morgan, that the 
 originator of this disjunction called in a physician to set a 
 dislocated shoulder, and the physician turned his argument 
 upon the philosopher to prove that his shoulder was not hurt. 
 He argued that the shoulder must be put out of place either 
 where it was, or where it was not. But as it could neither be 
 put out of place where it was, nor where it was not, it could 
 not be dislocated at all This is an excellent case of the argu- 
 mentum ad hominem, both in its legitimate and its illegitimate 
 relation. It is an admirable exposure of the absurdity of the 
 Greek philosopher's argument, but it neither disproves the 
 impossibility of motion nor proves its existence. Nor is it a 
 refutation of the assertion which is imputed by inference to 
 the philosopher, namely, that his shoulder was out of place. 
 It only establishes a contradiction between his philosophic 
 doctrine about motion and his present belief about the dislo- 
 cation of his shoulder. The philosopher would have only to 
 say either that it was not a case of motion, or that his shoulder
 
 MATERIAL FALLACIES . 253 
 
 was not displaced, in order to indicate that his argument or 
 position was not overthrown, while admitting that the reason- 
 ing of the physician was correct. Nevertheless he would not 
 escape the charge of a contradiction somewhere, and although 
 his assertion is not disproved by the ad hominem argument, 
 he is under obligation to explain the contradiction or to give 
 up one of the alternatives. This is the value of the argumen- 
 tum ad hominem, and of the other similar forms of appeal to 
 admitted principles. 
 
 2. FALLACY OF NON SEQUITUB. As already indicated this is 
 generally called thefallacia consequents, or False Consequent. 
 It arises in connection with the conclusion, and not in connec- 
 tion with the premises. It is, therefore, the introduction of 
 new matter into the conclusion, which is not contained in the 
 premises. There is no special necessity for subdividing it 
 into distinct forms, except that one class has received a separ- 
 ate name for the sake of particular convenience, and perhaps 
 because of its peculiar frequency. If we must distinguish 
 them at all, it must be into the common non sequitur, and the 
 non causa pro causa, or false cause, often called the post hoc, 
 ergo propter hoc, fallacy. The form in which the fallacy usu- 
 ally occurs can be represented in the following manner : 
 
 All men are rational. 
 Socrates is a man. 
 Therefore Socrates is noble. 
 
 It is evident that this conclusion cannot follow from the prem- 
 ises unless we regard " noble " as identical with " rational," 
 which it is not intended to be. The fourth term is here in the 
 conclusion. De Morgan's illustration of the fallacy is less 
 simple. It is : 
 
 Episcopacy is of Scripture origin. 
 
 The Church of England is the only Episcopal church in 
 
 England. 
 Therefore the church established is the church that should 
 
 be supported.
 
 254 . ELEMENTS OF LOGIC 
 
 It is evident that nothing has been said about supporting 
 the church in the premises, and hence it does not follow from 
 them. The fallacy is determined wholly by the presence of 
 new matter in the conclusion. It closely resembles the formal 
 fallacies of illicit major and minor. The difference is that in 
 the latter the addition is quantitative, while in the non sequitur 
 it is qualitative. 
 
 The literal meaning of non sequitur would apply to any fal- 
 lacy whatever, because the fallacy means that the conclusion 
 does not follow from the premises. But technically logicians 
 mean or should mean by this particular term that the conclu- 
 sion does not follow from the premises, although they are 
 true. The petitio principii vitiates the conclusion because of 
 false premises ; in the non sequitur the premises are not dis- 
 puted, but are admitted, at least for the sake of the argument. 
 It is important to observe in this connection that both falla- 
 cies are possible at the same time and in the same syllogism. 
 We may question the premises and so charge a petitio principii 
 upon the conclusion ; or we may say that even if the premises 
 are true the conclusion does not follow, in which case we im- 
 pute a non sequitur to the reasoning. Therefore whenever we 
 can make the error turn upon false assumptions in the prem- 
 ises, we charge the former fallacy against the reasoning, and 
 whenever it turns upon false assumption in the conclusion, in- 
 dependently of the premises, we charge the latter fallacy. 
 
 Very frequently the fallacy of non sequitur is due to appar- 
 ent cases of immediate reasoning, which are in reality enthy- 
 memes. Thus if we were to say " History is authentic because 
 mankind has accepted its statements," or " Philosophy is use- 
 less because it bakes no bread," we might be charged with a 
 non sequitur on the ground that the conclusion was not in- 
 cluded in the premise. But since the argument is an enthy- 
 meme we can complete it in the usual way, so that the conclu- 
 sion after all might be included in the terms of the suppressed 
 premise. Thus the major premise of the first enthymeme is 
 " Whatever mankind has accepted is .authentic," and of the 
 second, " Whatever bakes no bread is useless." When these
 
 MATERIAL FALLACIES 255 
 
 are supplied we find that the conclusion is valid unless we 
 can impeach the premises, but to question them turns the fal- 
 lacy into a petitio principii. We thus discover that what may 
 be regarded as a non sequitur in one relation may be a petitio 
 principii in another ; what is not involved in one premise may 
 be begged in the other. 
 
 We have therefore to be careful in deciding when a fallacy 
 is a non sequitur alone. The pure and simple form of it oc- 
 curs when both premises are admitted, either in reality or for 
 the sake of argument. In such cases as we have just indi- 
 cated it coincides with the petitio principii, and may be re- 
 duced to it. But when it occurs in its pure form this cannot 
 be done. 
 
 The fallacy of False Cause, or non causa pro causa, is the mis- 
 take of imagining a necessary connection where there is none, 
 or of confusing a causal connection with a mere coexistence or 
 sequence. It occurs when we argue that a certain thing is 
 the cause of another when we find them occurring together. 
 Thus if we were to argue that a change of the weather was 
 due to the occurrence of a new or full moon, because they 
 coincided, or because the former immediately followed the 
 latter ; or if we attributed a pestilence to the occurrence of 
 a comet ; or a death in the family to an eclipse of the sun, we 
 should be committing this fallacy. The Latin phrase, post hoc, 
 ergo propter hoc, indicates the manner in which the conclu- 
 sion is drawn, and upon what it depends. " When things 
 are seen together," says De Morgan, " there is frequently an 
 assumption of necessary connection. There is, of course, a 
 presumption of connection : if A and B have never been seen 
 apart, there is probability (the amount of which depends upon 
 the number of instances observed) lhat the removal of one 
 would be the removal of the other. It is when there is only 
 one instance to proceed upon that the assumption falls under 
 this fallacy ; were there but two, inductive probability might 
 be said to begin. The fallacy could then consist only in es- 
 timating the probability too high." But a probability is no 
 proof. The inference may be a deductive fallacy, however
 
 256 ELEMENTS OF LOGIC 
 
 great the probability, and in spite of the inference being in> 
 ductively legitimate. No number of mere coexistences or se- 
 quences contains the statement of the cause of phenomena, 
 and we are not entitled to infer it from them. Necessary con- 
 nection is not involved in the mere fact of connection. If it 
 were, I might argue that night was the cause of day, or vice 
 versa, because we find that one invariably precedes the other ; 
 or I might argue that the flight of birds was the cause of 
 springtime, because it accompanies the latter. 
 
 If we analyze the cases of non causa pro causa, however, we 
 shall find that they too may coincide with a petitio principii, 
 and perhaps they should be classified with that form of fallacy. 
 Thus we might say that the inference that night was the cause 
 of day was a non sequitur when drawn from their invariable 
 connection, but when we complete the syllogism by supplying 
 the suppressed premise the major premise would be, " All that 
 precedes day is the cause of it." The minor premise would be, 
 " Night precedes day," and the conclusion would follow as in- 
 volved in the major, although not in the minor, premise. But 
 we ma}' charge the major premise with begging the question, 
 and hence, as before, this case, which appears a non sequitur in 
 relation to the minor premise, is a petitio principii in relation 
 to the major premise. All post hoc, ergo propter hoc, fallacies 
 can be reduced in this way, and hence it might seem best to 
 include them as a species of begging the question. But as they 
 usually occur with an enthymeme where the conclusion in 
 such cases is not included in the premise, the conveniences of 
 controversy make it best to regard the fallacy as a non sequitur, 
 although it is one which coincides with a pelitio principii, or 
 may so coincide with it. But the most perfect form of non 
 sequitur will occur when both premises are unquestionable. 
 
 3d. General Observations. The first observation to be 
 made regarding the fallacies which we have just considered is 
 that they are not always distinct from each other. This is 
 apparent in the fact that the last two often coincide, and that 
 the non causa pro causa may be resolved into a petitio principii 
 when the suppressed premise is supplied. A similar reduction
 
 MATERIAL FALLACIES 257 
 
 might be possible with some of the others. For example, take 
 the fallacies of Accident, and in particular the illustration of 
 the use of pine wood : 
 
 Pine wood is good for lumber. 
 
 Matches are pine wood. 
 
 Therefore matches are good for lumber. 
 
 We gave this as a fallacy of Accident. But in fact it may be 
 resolved in two other ways at the same time, which may show 
 why the fallacy of Accident occurs. In the first place, the 
 major premise is an indefinite or general proposition, and we 
 have already learned that such propositions are very frequently 
 particular in their real import. This is, in fact, the real mean- 
 ing of the statement. It is not true, nor would it be intended 
 to assert, that all pine wood, that is, all forms of it, are good 
 for lumber, but only that some pine wood is good for lumber. 
 But thus to convert the real import of the major premise into 
 a particular proposition, making the syllogism IAA of the 
 first Figure, prevents the distribution of the middle term, so 
 that the fallacy would virtually be a formal one. Many of the 
 fallacies of Accident can be so reduced. But it is only because 
 we are viewing the premises in their quantitative signification 
 instead of their qualitative. For it is true, qualitatively, that 
 "All pine wood is good for lumber," that is, in substance, 
 but not in every form, and hence the case of Accident can be 
 brought against this conception of it. But interpreting the 
 case mathematically, what would be regarded logically and 
 qualitatively a fallacy of Accident becomes formally and quan- 
 titatively an illicit middle. 
 
 In the second place, since we suppose the material meaning 
 of the major premise to be that " Some pine wood is good for 
 lumber," we impeach the truth of the proposition universally, 
 and it is upon its universal truth that the conclusion depends. 
 Hence, in considering the premise or premises doubtful we 
 can regard the fallacy as a petitio principii. There are thus 
 two fallacies, one formal and the other material, which can be 
 17
 
 258 ELEMENTS OF LOGIC 
 
 imputed to this syllogism, besides that of Accident. In fact 
 we can make it a fallacy of Accident only upon the supposition 
 that the major premise is universally true of the essentia of pine 
 wood, but not of its accidentia, while the predicate of the con- 
 clusion would connect with one of its accidents what had been 
 connected in the major premise only with the essence. But 
 aside from this interpretation either an illicit middle or a 
 petitio principii can be imputed to the syllogism. 
 
 Perhaps a similar resolution of Composition and Division 
 could be made, because the premises in syllogisms committing 
 those fallacies are capable of a double interpretation. It is 
 sufficient to suggest the possibility, and the actual achieve- 
 ment of it can be left to the student. And it will not be nec- 
 essary to say more on the close relation between the non seaui- 
 tur and the petitio principii in many cases, after having shown 
 that the two may be applied to the same conclusion, but in 
 different relations, one of them indicating assent to a premise, 
 but not to the inference, and the other indicating that one of 
 the premises vitiates it. The two will not coincide when they 
 are imputable, one of them only to false premises and the 
 other only to a false inference. 
 
 One more remark, which has been alluded to, it is impor- 
 tant to make. The imputation of a fallacy in the reasoning 
 does not necessarily imply that the proposition in the conclu- 
 sion is a false one. In many cases the falsehood of the propo- 
 sition and the existence of the fallacy go together ; but it is 
 not always the fact, and we must learn to recognize this fact 
 because although a proposition may be true, it may lead to 
 error to have it connected falsely with another proposition as 
 proof when that proposition may not be true, or when the 
 conclusion is not an inference from it. We commit a fallacy 
 when we suppose that an error in reasoning is a sufficient dis- 
 proof of a proposition. The fallacy thus committed is an igno- 
 ratio elenchi. All that the existence of a fallacy can establish 
 is a mistake in the mode of proving a proposition, unless it 
 serve as the means of disco vering the actual error in our 
 propositions. We usually discover the error, in fact, before
 
 MATERIAL FALLACIES 259 
 
 we find why or how it has been committed. But the fallacy 
 in reasoning is an error growing mainly out of an attempt to 
 deduce one connection of terms from another, and so will not 
 always be an intlex of material errors of fact. They are, of 
 course, accompanied by error somewhere of a material kind 
 frequently enough, but not necessarily implying it where the 
 ordinary mind assumes it ; we require to be on our guard 
 against committing a fallacy when imputing one to others. 
 We must always distinguish between the error in fact which 
 we may first perceive, and the error in reasoning to which 
 such a discovery may have led us. 
 
 Another important remark is that it is not necessary to put 
 the argument into the form of a syllogism in order to discover 
 what the fallacy is. We have only to observe the manner of 
 substituting one term for another. Most frequently in actual 
 discourse arguments are either stated in the form of enthy- 
 memes, or the premises are so expressed as to effectually con- 
 ceal the Mood and Figure of the syllogism, and we are left en- 
 tirely to depend upon the manner in which we use certain 
 terms. Then, since in enthymemes we can construct them 
 into syllogisms, at least, of the first or of the second Figure, 
 as we please, in one of which the same matter may be valid 
 which is invalid in the other, and since the three Figures can 
 be reduced to the first at pleasure, it will not be necessary to 
 consider the form in detail, but only how we substitute one 
 term for another. If the fallacy be a formal one, it will be 
 most easily detected in some cases by observing the form of 
 the argument, but in some cases this is not necessary. Be- 
 sides formal fallacies are not so often committed as the mate- 
 rial. When any doubt, however, exists about the nature of 
 an illegitimate inference, it is best to throw the argument into 
 the form of a syllogism, and then ascertain its relation to the 
 general rules. 
 
 But in many, if not in most instances of material fallacy, we 
 can determine the error by observing the two or more senses 
 in which a term is used without stopping to consider whether 
 the form of the syllogism is expressed correctly or not, be-
 
 260 ELEMENTS OF LOGIC 
 
 cause it may either be thought in a different manner from the 
 expressed relation, or it may be materially what it is not for- 
 mally. Thus in the following syllogism : 
 
 
 
 White men are Caucasians. 
 The Germans are Caucasians. 
 Therefore the Germans are white. 
 
 we should be guilty of a formal fallacy of illicit middle. But 
 since we may have stated the major premise in a form in which 
 it was not thought, namely, in the inverse form, the proposi- 
 tion being a definition, the reasoning may be in the form of 
 the first instead of the second Figure, as it is stated, and 
 hence perfectly valid. We cannot, of course, always rely 
 upon this method of dealing with an argument, but in cases 
 of material reasoning we either use the first Figure most com- 
 monly, unless we are proving a negative, or our data can be 
 so easily reduced to it, that we can generally depend upon the 
 mere form of substitution of one term for another in order to 
 determine the nature of the fallacy. Thus if we try to prove 
 that a man should give alms to a particular person on the 
 ground of his duty to be charitable, we commit a fallacy of Ac- 
 cident, because we argue from the genus " charity " to a par- 
 ticular case of it where an accidental or differential circum- 
 stance may modify the obligation. Similarly with the fallacies 
 of Quantity, if I argue from the effect of a forest in producing 
 a thick shade to a similar effect from a single tree I commit the 
 fallacy of Composition. We have seen how the non sequitur 
 may be imputed without considering, at least in some cases, 
 more than the statement which is assigned as its ground, al- 
 though a completion of the argument by supplying a sup- 
 dressed premise may convert it into a petitio principii. This 
 is only an illustration of observing whether the conclusion is 
 deduced from the given data or not. In the ignoratio elenchi we 
 never require to construct the syllogism, but only to observe 
 whether the conclusion evades the question or not, or whether 
 the assumed contradiction is a true one or not. Since all
 
 MATERIAL FALLACIES 261 
 
 material fallacies, with the possible exception of the petitio 
 principii in some cases, are a modified form of Quaternio 
 Terminorum, we have only to see whether terms are used 
 throughout an . argument in an identical sense or not, in 
 order to determine the nature of the fallacy committed in any 
 particular case.* 
 
 * On Fallacies consult De Morgan : Formal Logic, Chapter XIII. ; Mill : 
 Logic, Book V. , especially Chapters V., VI., and VII.; Whately: Ele- 
 ments of Logic, Book III. ; Hamilton : Lectures on Logic, Lecture XXIII.
 
 CHAPTER XIX. 
 
 QUANTIFICATION OF THE PREDICATE 
 
 A TREATISE on Logic is hardly complete that omits an ac- 
 count of recent doctrines regarding what is called the " quan- 
 tification of the predicate." The usual expositions of the 
 subject are confined to the forms left by Aristotle, and which, 
 sufficing for practical purposes, are best adapted to the actual 
 usages of language. But language does not always express 
 explicitly what thought involves implicitly, and hence many 
 logicians have felt it necessary to correct this defect by an 
 ideal scheme of logical doctrine which might enable us better 
 to understand logical processes in their pure forms, and then 
 to modify this scheme to suit the exigencies of defective usage. 
 Sir William Hamilton, Professor De Morgan, and George 
 Bentham, all about the same time conceived the propriety, or 
 at least the possibility, of modifying logical doctrine by the 
 " quantification of the predicate." This view we shall pro- 
 ceed to explain, with its importance in practical reasoning. 
 
 We have already shown what the quantification of the subject 
 is, although it has not been stated under that name. But to 
 quantify it is only to say whether the whole or the part of it 
 is taken into account. Its quantification refers to its distri- 
 bution or non-distribution. Hence to quantify the predicate 
 is to state whether the whole or only the part of it agrees 
 with the subject, or differs from it. We have seen that the 
 proposition " Men are wise," is an indefinite one so far as its 
 explicit statement is concerned, and that we cannot deal with 
 it logically or with any degree of assurance as to the results 
 unless we first know whether it means " all men " or " some 
 men are wise." This is quantifying the subject in order to
 
 QUANTIFICATION OF THE PREDICATE 263 
 
 bring the proposition into a definite form for logical use. 
 Thus far the ordinary Logic proceeds, but no further. But 
 why not also quantify the predicate in a similar manner, in 
 order to evade the equivocations incident to its indefinite 
 forms? In the above proposition it is said to be undistributed, 
 because nothing is stated or implied to indicate whether the 
 whole or only a part of its extension is taken into account. 
 It is, therefore, perfectly indefinite. But in some cases, in 
 spite of this mode of statement, we happen to know that the 
 meaning of the predicate is definite ; that it is identical with 
 the subject in its quantity. As already explained this is the 
 case with all definitions in which subject and predicate are con- 
 vertible terms. Supposing that we define man as a rational 
 animal, we can say with equal truth " All men are rational 
 animals," and " All rational animals are men," and so with any 
 other definition. The same happens to be true of the prop- 
 osition " White men are Caucasians." We can convert it sim- 
 ply into "All Caucasians are white," but only because we 
 happen to know that whiteness and Caucasian are convertible 
 terms. According to the formal laws enunciated regarding 
 the form of such propositions, it would have to be converted 
 per accidens : so also even with definitions. But as the form 
 of a definition and that of an ordinary assertion cannot be 
 distinguished in general usage, why would it not be an im- 
 provement in the science of Logic to treat the predicate as we 
 have treated the subject, and to state explicitly what is implic- 
 itly involved in it? The answer to this question will appear 
 in the sequel, after we have shown how the quantification can 
 be effected and what are its consequences to the ordinary 
 forms of reasoning. Jevons's exposition suffices for the pur- 
 pose. 
 
 " In the proposition ' All metals are elements/ the subject 
 is quantified, but the predicate is not ; we know that all metals 
 are elements, but the proposition does not distinctly assert 
 whether metals make the whole of the elements or not. In 
 the quantified proposition 'All metals are some elements,' the 
 little word some expresses clearly that in reality the metals
 
 264 ELEMENTS OF LOGIC 
 
 form only a part of the elements.* Aristotle avoided the use 
 of any mark of quantity by assuming, as we have seen, that all 
 affirmative propositions have a particular predicate, like the 
 example just given ; and that only negative propositions have a 
 distributed or universal predicate. The fact is, however, that 
 he was entirely in error, and thus excluded from his system 
 an infinite number of affirmative propositions which are uni- 
 versal in both terms. It is true that ' All equilateral triangles 
 are all equiangular triangles,' but this proposition could not 
 have appeared in his system except in the mutilated form, 'All 
 equilateral triangles are equiangular.' Such a proposition as 
 ' London is the capital of England,' or ' Iron is the cheapest 
 metal' had no proper place whatever in his syllogism, since 
 both terms are singular and identical with each other, and 
 both are accordingly universal 
 
 " As soon as we allow the quantity of the predicate to be 
 stated the forms of reasoning become much simplified. We 
 may first consider the process of conversion. In our dis- 
 cussion of the subject it was necessary to distinguish between 
 simple conversion and conversion by limitation. But now one 
 simple process of simple conversion is sufficient for all kinds 
 of propositions. Thus the quantified proposition of the form 
 A, ' All metals are some elements,' would be simply converted 
 into ' some elements are all metals.' " The quantified form of 
 A, " All metals are all elements," would be simply converted 
 into " All elements are all metals ;" and so on with all propo- 
 sitions. We could simply proceed upon the rule that what- 
 ever we do with one term we could do with the other. Their 
 meaning is made definite by their explicit quantification. 
 
 " The doubly universal proposition is of most frequent oc- 
 
 * If the ordinary canon about the signification of some, as previously 
 defined (p. 116), is to be enforced here, Jevons is wrong in saying that the 
 word denotes only a part. It merely asserts distinctly a part, and does 
 not assert or even imply anything about the whole. But it does indicate 
 that we are not to assume anything about the whole of the predicate. If 
 we adopt its use with the implication that it denotes only a part, we 
 should have to define it so.
 
 QUANTIFICATION OF THE PREDICATE 265 
 
 currence, as in the case of all definitions and singular prop- 
 ositions. I may give as instances, ' Honesty is the best 
 policy,' 'The greatest truths are the simplest truths,' 'Vir- 
 tue alone is happiness below,' ' Self -exaltation is the fool's 
 paradise.' 
 
 "When affirmative propositions are expressed in the quanti- 
 fied form all immediate inferences can be readily drawn from 
 them by this one rule, that whatever we do with one term we 
 should do with the other term. Thus, from the doubly universal 
 proposition, 'Honesty is the best policy,' we infer that ' what 
 is not the best policy is not honesty,' and also, 'what is not 
 honesty is not the best policy.' From this proposition, in fact, 
 we can draw two contrapositives ; * but the reader will care- 
 fully remember that from the ordinary unquantified proposi- 
 tion A we can only draw one contrapositive. Thus if ' metals 
 are elements,' we must not say that ' what are not metals are 
 not elements.' But if we quantify the predicate thus, ' All the 
 metals are some elements,' we may infer that 'what are not 
 metals are not some elements.' Immediate inference by added 
 determinant and complex conception can also be applied in 
 either direction to quantified propositions without fear of the 
 errors noticed under those heads." 
 
 The quantification of the predicate adds four more propo- 
 sitions to those of the quantified subject, A, E, I, O, and 
 Thompson employed new symbols for them, U and Y for the 
 affirmative, and the Greek letters 77 and w for the negative. U 
 and Y represent the predicate as distributed in the affirma- 
 tive propositions, and 77 and o> as undistributed in the nega- 
 
 * Jevons here uses the terra contrapositive in a sense which is different 
 from the usual definition of it, and to denote two distinct processes. We 
 wish logicians could adopt it to denote the inference to complementary 
 propositions, and limit the use of the term contraversion to what is gener- 
 ally called contraposition. Instead of speaking of "two contrapositives," 
 therefore, Jevons might say a contrapositive and a contraverse. The 
 conditions are different for inferring that " what is not honesty is not 
 the best policy," from those in which we can infer that " what is not the 
 best policy is not honesty," except with a quantified predicate. See also 
 Antithesis (p. 169).
 
 266 
 
 ELEMENTS OF LOGIC 
 
 tive. The following table represents the forms of the eight 
 propositions : 
 
 U All Sis all P 1 
 
 A All S is some P 
 
 T c, a . > Affirmative propositions. 
 
 I borne o is some P 
 
 Y Some S is all P. 
 
 ' 
 
 E No S is any P 
 
 n No S is some P 
 
 c, . _. > Negative propositions. 
 
 u> Some S is not some P 
 
 O Some S is no P. 
 
 The existence of these eight propositions makes it neces- 
 sary to extend the number of valid moods from 24, or rather 
 19, omitting the weakened conclusion, to 108, without the 
 fourth Figure, and 144 with that Figure. Sir William Ham- 
 ilton did not include this Figure in his table and notation, nor 
 did Thompson, to whose table we add the moods of the fourth 
 Figure : 
 
 TABLE OF MOODS OF THE SYLLOGISM. 
 
 
 FIGURE I. 
 
 FlGUBB El. 
 
 FIGURE m. 
 
 FIGURE IV. 
 
 Affirm- 
 ative. 
 
 Neg- 
 ative. 
 
 Affirm- 
 ative. 
 
 Neg- 
 ative. 
 
 Affirm- 
 ative. 
 
 Neg- 
 ative. 
 
 Affirm- 
 ative. 
 
 Neg- 
 ative. 
 
 I 
 
 uuu 
 
 AY I 
 A A A 
 YYY 
 All 
 I Y I 
 UY Y 
 AUA 
 UA A 
 YUY 
 U I I 
 I U I 
 
 EUE 
 UEE 
 r) Y to 
 AOto 
 
 n A n 
 A n r\ 
 YO 
 YOO 
 
 n I co 
 
 A u u 
 
 (0 Y CO 
 
 I O to 
 EYO 
 UOO 
 1 U r, 
 AE T, 
 EAE 
 U T, n 
 UO 
 Y EE 
 E I O 
 
 U <O CO 
 
 to U to 
 I E 77 
 
 UUU 
 YYI 
 YA A 
 AY Y 
 Y I I 
 I Y I 
 UY Y 
 YU A 
 UA A 
 AUY 
 U I I 
 I U I 
 
 EUE 
 UEE 
 O Y to 
 YO to 
 O Ai? 
 Y n ij 
 r, YO 
 AOO 
 
 O I 0) 
 
 Y 0) tc) 
 
 to Y to 
 I to 
 EYO 
 UOO 
 OU ij 
 Y E r, 
 EA E 
 U >, 4 
 
 7 UO 
 
 AEE 
 E I O 
 
 U to to 
 
 CO U to 
 
 I E r, 
 
 UUU 
 A A I 
 AY A 
 YA Y 
 All 
 I A I 
 UA Y 
 AUA 
 UYA 
 YUY 
 U I I 
 I U I 
 
 EUE 
 UEE 
 
 17 A to 
 
 A r\ to 
 
 n Y n 
 AO 17 
 O A O 
 Y 17 O 
 17 I to 
 A to to 
 O A to 
 I 7 co 
 E AO 
 U 17 O 
 v U 77 
 AE 17 
 EYE 
 UO 17 
 OUO 
 YEE 
 E I 
 TJ to to 
 to U to 
 
 I E n 
 
 UUU 
 Y A I 
 A AY 
 YYA 
 Y I I 
 I A I 
 UA Y 
 YU A 
 UYA 
 AUY 
 U I I 
 I U I 
 
 EU E 
 UEE 
 
 to A to 
 
 Y r, to 
 Y 17 
 
 YO 
 n A 
 A r;O 
 
 I CO 
 
 Y to to 
 O A to 
 I 17 to 
 E AO 
 U 
 
 O U T, 
 
 YE n 
 EYE 
 
 U O .7 
 
 T, UO 
 
 AEE 
 E I 
 
 U to co 
 
 CO U CO 
 
 I E 17 
 
 II 
 
 m 
 
 IV 
 V 
 
 VI 
 VII .... 
 VIII . . . 
 IX 
 X 
 
 XI 
 XII . . . .
 
 In this table the columns marked affirmative and negative 
 represent respectively the affirmative and negative conclusions 
 which the moods of the quantified predicate will give. It 
 omits the cases of weakened conclusion. It is interesting to 
 remark that the negative conclusions are twice as many as the 
 affirmative. The large number of both of them adds very much 
 to the difficulties of remembering those that are valid and 
 those that are invalid. It would seem that instead of simpli- 
 fying the process of reasoning the quantification of the predi- 
 cate very much complicates it. In some cases it certainly 
 does simplify it, but in so many other cases it complicates it 
 that little is to be gained by the system. But its theoretical 
 principle should be considered. 
 
 Hamilton's statement of its value is brief and to the point. 
 He says the fact " that we can only rationally deal with what 
 we already understand, determines the simple logical postu- 
 late To state explicitly what is thought implicitly. From the 
 consistent application of this postulate, on which Logic ever 
 insists, but which logicians have never fairly obeyed, it fol- 
 lows, that, logically, we ought to take into account the quan- 
 tity, always understood in thought, but usually, and for mani- 
 fest reasons, elided in its expression, not only of the subject, 
 but also of the predicate of a judgment. This being done, 
 and the necessity of doing it will be proved against Aristotle 
 and his repeaters, we obtain, inter alia, the ensuing results : 
 
 1. " That the preindesfynate * terms of a proposition, whether 
 subject or predicate, are never, on that account, thought as 
 indefinite (or indeterminate) in quantity. The only indefinite 
 is particular, as opposed to definite quantity ; and this last, as 
 it is either of an extensive maximum undivided, or of an ex- 
 
 * Hamilton employs the terms predesir/nate and preindesignate to de- 
 note the two subdivisions of Particular propositions or terms. " Predesig- 
 nate " denotes what we have called General terms and propositions, as 
 " Man is wise," where we may mean all or some, the quantity being in- 
 differently expressed. " Preindesignate " denotes the ordinary particu- 
 lar proposition, as, "Some men are wise," where the subject is definitely 
 indefinite.
 
 268 ELEMENTS OF LOGIC 
 
 tensive minimum indivisible, constitutes quantity universal 
 (general) and quantity singular (individual). In fact, definite 
 and indefinite are the only quantities of which we ought to 
 hear in Logic ; for it is only as indefinite that particular, it is 
 only as definite that individual and general, quantities have 
 any (and the same) logical avail. 
 
 2. " The revocation of the two terms of a proposition to their 
 true relation ; a proposition being always an equation of its 
 subject and predicate. 
 
 3. " The consequent reduction of the Conversion of Proposi- 
 tions from three species to one ; that of Simple Conversion. 
 
 4. " The reduction of all the General Laws of Categorical 
 Syllogisms to a Single Canon. 
 
 5. " The evolution from one canon of all the species and va- 
 rieties of Syllogism. 
 
 6. " The abrogation of all the Special Laws of Syllogism. 
 
 7. " A demonstration of the exclusive possibility of Three 
 syllogistic Figures, and (on new grounds) the scientific and 
 final abolition of the Fourth. 
 
 8. " A manifestation that Figure is an unessential variation 
 in syllogistic form ; and the consequent absurdity of reducing 
 the syllogisms of the other figures to the first. 
 
 9. "An enouncement of one Organic Principle for each Fig- 
 ure. 
 
 10. " A determination of the true number of legitimate 
 Moods; with 
 
 11. " Their amplification in number (thirty-six). 
 
 12. " Their numerical equality under all the figures ; and 
 
 13. " Their relative equivalence, or virtual identity, through- 
 out every schematic difference." 
 
 The remaining points of modification and advantage are 
 mainly of interest to advanced scientific Logic, and need not 
 be repeated here. Even some of those we have quoted are 
 not of any apparent importance to practical Logic. But all 
 of them present a formidable number of consequences imputed 
 to the quantification of the predicate. The first four and the 
 eighth are important simplifications of reasoning, and the re-
 
 QUANTIFICATION OF THE PREDICATE 2G9 
 
 maining are at least interesting modifications. But there are 
 two or three difficulties in the theory which Hamilton seems 
 not to have noticed, and which go far to offset all the advan- 
 tages of quantifying the predicate. 
 
 The first of these is an incident in the use of the word some. 
 This term in the quantification of the subject, according to the 
 Square of Opposition, must mean some, and it may or it may 
 not be all, and must not mean some and only some. If it meant 
 the latter, the assertion of I would always prevent the asser- 
 tion of A. Hence, for the purposes of Logic it must take 
 the former meaning, so that A may remain indeterminate and 
 possible upon the assertion of I. But in the quantification 
 of the predicate, if some mean an indefinite part, and it may or 
 not be all of the predicate, the first thing to be remarked 
 is that the predicate in reality remains as undistributed as 
 before, and the second is, that the antitheses or complemen- 
 taries of such propositions are no more possible than be- 
 fore, a ' possibility which was said to be consequent upon 
 the quantification of the predicate. On the other hand, if 
 we use some to mean a part, and only a part, we obtain the 
 quantification of the predicate in affirmative propositions at 
 the expense of the regular rule about its use in the quan- 
 tification of the subject ; and in negative propositions a se- 
 rious ambiguity arises which it is hard to overcome. Thus 
 propositions 17 and w may each have two meanings. "No 
 S is some P," may mean either that S is not any por- 
 tion of P, or that it is not a certain indefinite portion of P. 
 So in the proposition " Some S is not some P." This am- 
 biguity appears most distinctly when we come to draw the 
 antitheses or complementaries of such propositions. Thus 
 in proposition U, " All men are ah 1 rational," we can imme- 
 diately infer, as in exclusive propositions, " All not-men are 
 not rational." When the use of some in the predicate means 
 a part, and only a part, the complementary can be inferred 
 that all which is not the subject is not that portion of the 
 predicate. But the terms indefinitely considered would imply 
 that it was not the other portion also. Thus, " All metals
 
 270 ELEMENTS OF LOGIC 
 
 are some elements," ought to give " All not-metals are not 
 some elements," and so it will, if we mean the particular some 
 identified with the subject. But the complementary may 
 mean not any " some " of the elements, which cannot possibly 
 be true. These ambiguities are a serious difficulty in the way 
 of the Hamiltonian theory. 
 
 There is a second important difficulty which shows how lit- 
 tle the accepted rules of syllogistic reasoning can be relied 
 upon to determine the validity of the new cases introduced 
 by the quantification of the predicate. It is remarked by 
 Keynes. In ordinary reasoning the distribution of the middle 
 term in at least one of the premises, and the retention in the 
 conclusion of the same quantity of the major and minor terms 
 as in the premises, are a guarantee of valid inference. But in 
 the mood AUA of a quantified predicate and of the second 
 Figure, the observance of these conditions is no security 
 against a fallacy, as will be apparent in the following syllogism 
 and its symbolic representation in Fig. 26. 
 
 All P is some M 
 All Sis all M 
 .-. All S is all P or some P. 
 
 The circles will bring out the fallacy more clearly. 
 
 FIG. 26. 
 
 It is evident from this representation that we cannot infer 
 from the premises that all S is either all or some P. All that 
 we could infer is that " Some S is all P," which is Y or " Some 
 S is some P, which is w ;" and yet neither illicit middle, nor 
 illicit major, nor illicit minor, is committed in the propositions 
 which, it seems, ought to follow, namely, " All S is all P," etc. ; 
 as a consequence of this, Keynes finds it necessary to add the
 
 271 
 
 following new rule to the list already given for determining 
 valid syllogisms : If one premise is U, while in the other premise 
 the middle term is undistributed, then the term combined with 
 the middle term in the U premise must be undistributed in the 
 conclusion. 
 
 But this addition of a rule only complicates, instead of sim- 
 plifies, the process of reasoning as we desire to use it in prac- 
 tical life ; and the immense number of valid moods has the 
 same effect, while the ordinary rules of reduction to the first 
 Figure, and the common practice of reasoning in this Figure 
 makes the matter as simple as it can be made, taking into ac- 
 count the nature of language and the defects of expression in 
 our thought. These, then, with other incidents, are defects 
 in the practical importance of the quantification of the predi- 
 cate. 
 
 Nevertheless there is one consideration of great interest 
 and importance in it, which, if we cannot make as much prac- 
 tical use of it as might be desirable, does explain the simpli- 
 city of mathematical reasoning the use of exclusive proposi- 
 tions and definitions, and the habitual tendency of ordinary 
 minds to use predicates as if they were definitely quantified. 
 Hamilton's dictum is a correct one, namely, that ready rea- 
 soning is greatly facilitated, and rendered less liable to fallacy 
 by stating explicitly what is implicitly thought. This means 
 that we should state definitely what is definitely thought, and 
 not state it indefinitely. If I mean that " AH men are wise," 
 " All governments are good," I should say so, and not conceal 
 a subterfuge under the indefinite " Man " or " Government," 
 etc. But in such propositions the predicate remains as in- 
 definite in extent or quantity as the subjects just mentioned. 
 If only I knew in such statements whether the subject was the 
 whole or the part of the predicate, I could know better how 
 to use them. Thus if I mean by the first proposition that 
 "All men are all the wise," I know first that "All who are 
 not-men are not wise," and second, that the two terms are 
 perfectly convertible. This is precisely what occurs in ex- 
 clusive propositions and definitions. In the former class the
 
 272 ELEMENTS OF LOGIC 
 
 predicate is definitely, and, we might say, formally, quantified 
 by the use of only qualifying the subject, and which indicates 
 that the quantity of the predicate is not greater than that of 
 the subject. In definitions the predicate, although not for- 
 mally, is materially quantified, in being made identical with 
 the subject, and so convertible with it. The effect of such 
 propositions upon the reasoning is to make it valid in some 
 instances where there appears to be a distinct formal fallacy. 
 Take an illustration with an exclusive proposition : 
 
 Only virtue is praiseworthy. 
 Courage is praiseworthy. 
 Therefore courage is a virtue. 
 
 This appears from the form of the propositions to be a syllo- 
 gism in AAA of the second Figure, and therefore invalid be- 
 cause of the undistributed middle, and yet in spite of this we 
 cannot resist the impression that the inference is correct 
 The reason for this is that the virtual quantification of the 
 predicate in the major premise totally modifies the reasoning 
 as compared with the ordinary rules. We can therefore ex- 
 plain the case in two waya First, since the subject and 
 predicate are made coextensive by the use of only, the major 
 premise is equivalent in meaning to its converse, namely, 
 " What is praiseworthy is virtue," and this change turns the 
 syllogism, in spite of its present form, into AAA of the first 
 Figure, which is valid. We may conceive that this is the or- 
 der of thought, in spite of the order of expression, and the 
 right to so conceive it depends wholly upon the definitely 
 quantified predicate. Second, since the quantification of the 
 predicate in the major premise, making it coextensive with 
 the quantity of the subject, has the effect of distributing it, 
 we have a syllogism of the second Figure which distributes 
 the middle term at least once in the premises, in spite of the 
 affirmative character of the propositions. This distribution 
 enables us to draw the inference which appears in the con- 
 clusion. In the case of definitions the illustration would be 
 the same.
 
 QUANTIFICATION OF THE PREDICATE 273 
 
 Common reasoning often treats propositions as if they 
 were definitions or mathematical propositions, and it is be- 
 cause the thought of the person reasoning is conceived as 
 representing a certain identity between subject and predicate 
 which may not be expressed, but is in mind. Hence the dis- 
 position to substitute them for each other. Now this substi- 
 tution can be done with impunity whenever the predicate is 
 made quantitatively equal to the subject, and so the need of 
 distinguishing between propositions which intend to dis- 
 tribute and those which do not intend to distribute the predi- 
 cate. If common usage could adopt a symbol which could 
 serve this purpose many logical difficulties would be over- 
 come. The theory of pure Logic would thus find its condi- 
 tions supplied in practice. But as it is, all propositions, ex- 
 cept definitive, exclusive, and mathematical propositions, are 
 alike in their form, while some of them in their matter, or the 
 way in which the mind thinks them, may be quantified, so that 
 the syllogisms containing them cannot be judged by the for- 
 mal laws of reasoning imtil we know what content they are sup- 
 posed to have. Thus we may say, " Houses are residences," and 
 formally the predicate is not quantified, or is not distributed, 
 and cannot be used with an affirmative minor premise in the 
 second Figure. But if we have in thought the notion that the 
 predicate is identical with the subject it is quantified, and the 
 reasoning is altered. This, then, in brief, is the advantage 
 of quantifying the predicate. It indicates explicitly whether 
 all or only a part of it is taken into account. 
 
 It must be observed, however, that in propositions I and Y 
 no special advantage is gained, because they leave the predi- 
 cate undistributed, except that we know definitely from their 
 form how to understand them. The chief importance attaches 
 to the distinction between A and U, which, if the quantifica- 
 tion were practicable in most cases, would enable us to know 
 the extent of the predicate as well as that of the subject. But 
 even if not of much service practically, the theory shows very 
 clearly what can be done with definite propositions, and how 
 the common mind often acts in its reasoning, when the trained 
 18
 
 274 ELEMENTS OF LOGIC 
 
 t 
 logician would impute a violation of the formal laws of Logic 
 
 to it. Besides, it explains a common tendency of the mind 
 to carry on the substitution of one term for another without 
 stopping to consider their distribution. The substitution in 
 an unqualified manner is, no doubt, an error, but it is rather 
 an error of assumption in many cases than of inference.* 
 
 * References on the doctrine of the Quantification of the Predicate are 
 as follows: Hamilton : Lectures on Logic, Appendix V.; De Morgan: 
 Formal Logic; Thompson: Laws of Thought, Part II., Sections 77-79 ; 
 Mill : Examination of Sir William Hamilton's Philosophy, Chap. XXII. 
 
 NOTE. The remarks in this chapter which seem to imply the recipro- 
 cal and convertible coincidence in quantity of subject and predicate in 
 exclusive propositions must not be misunderstood. In as much as nega- 
 tive exclusive propositions are very seldom used at all, I have limited my 
 statements to the affirmative form. Affirmative exclusive propositions re- 
 semble mathematical propositions in the fact that the predicate, although 
 of an affirmative instance, is definitely quantified or distributed, and can 
 be simply converted. But this convertibility into a universal is not recip- 
 rocal with the subject and predicate ; that is, we cannot convert and re- 
 convert them as often as we please unless we retain the exclusive form 
 with every reconversion of the universal. The illusion which they give 
 rise to comes from the actual distribution of the predicate and the fact 
 that the extension of the subject is at least equal to that of the predicate. 
 That is to say in the exclusive propositions mentioned the extension of the 
 subject must be equal to, and may be equal to, that of the subject, but it 
 is not distributed, while the extension of the predicate cannot be greater 
 and may be equal, but is distributed. Consequently in the first degree 
 of conversion they are, like mathematical propositions and definitions, 
 capable of simple conversion. As already indicated they are really in- 
 verted universals and their complementary opposites are the coutraverse 
 of the real universals. In negative exclusive propositions the predicate 
 is distributed without reference to the exclusive particle, and hence it has 
 no function practically exopt to imply the complementary opposite. The 
 quantification is the same as in the affirmative form, but conversion is not 
 applicable because the distribution is the same as in particular negatives, 
 and as we have learned propositions in O are not convertible. But in the 
 affirmative forms not only are the propositions capable of simple conver- 
 sion, but also are equivalent to the complementary opposite in which both 
 subject and predicate are distributed and which represent a more dis- 
 tinctly mathematical proposition.
 
 CHAPTER XX. 
 
 MATHEMATICAL AND OTHER REASONING 
 
 THE principles involved in the quantification of the predi- 
 cate, and for that matter, also of the subject, will help us to 
 understand the certitude and exemption from error displayed 
 in mathematical reasoning, and also the difference between 
 this and other reasoning. We shall define it more carefully 
 after illustrating it and pointing out its relation to the princi- 
 ples discussed in the preceding chapter. 
 
 The first important observation to be made is that in mathe- 
 matical reasoning all propositions are U or E propositions, or 
 their equivalents ; that is, are universally and definitely quanti- 
 fied in their subject and predicate. The effect of this fact is that 
 which was remarked by Hamilton in the eighth observation 
 we quoted from him (p. 268). It makes subject and predi- 
 cate quantitatively equal, so that we can dispense with the 
 variations of Figure in Syllogisms, in so far as the validity of 
 the reasoning is concerned. All the Figures are valid alike, 
 and from what has been said about the nature of the propo- 
 sitions in mathematical reasoning the Moods will be only 
 three, namely, UUU, UEE, and EUE, all of which are valid in 
 all the Figures. In the following illustrations the sign of 
 equality serves both as the copula and the sign of equivalence, 
 and the negative symbol previously explained (p. 140) denotes 
 inequality as well as negation. The letters denote the various 
 terms as before, only they stand for numbers or quantities. 
 Mathematical reasoning, then, will appear in the following 
 forms indifferently : 
 
 1st Fig. 2d Fig. 3d Fig. 4th Fig. 
 
 UUU UUU UUU UUU 
 
 S =M S=M M = S M=S 
 
 .-.S=P .-.S=P .-.S=P .-.S=P.
 
 276 ELEMENTS OF LOGIC 
 
 1st Fig. 2d Fig. 3d Fig. 4th Fig. 
 
 UEE UEE UEE UEE 
 
 M = P P = M M = P P = M 
 
 SxM SxM MxS M x S 
 
 .-.s x P .-.s x P .-. s x P .-. s x P. 
 
 If we turn to the table of Moods with the quantified predicate 
 we shall find that these two are valid in all four Figures, a 
 fact which is due to understanding definitely that the predi- 
 cate is limited to the subject, or made equal to it in exten- 
 sion. The process for all four Figures can be represented 
 symbolically by the following circles : 
 
 FIG. 27. FIG. 28. 
 
 Fig. 27 represents the Mood UUU, and Fig. 28 the Mood 
 UEE. Mood EUE would be the same as Fig. 28, only the 
 letters S and P would change places. It is very evident from 
 this how simple mathematical reasoning is, and how little re- 
 gard we need to pay to Figure in the form of presenting it. 
 If all reasoning were the same, it would be less exposed to 
 formal fallacies, and undoubtedly it is the tendency of com- 
 mon minds to construe their reasoning mathematically, that 
 explains both their liability to formal fallacies and the neces- 
 sity of being on our guard for the existence of material fal- 
 lacies in it when it is formally correct. The untrained mind, 
 accustomed to deal with its terms and their import in a 
 mathematical sense, very easily confuses with them other 
 terms and propositions having the same form as the mathe- 
 matical, and is at a loss to discover its liability to error. 
 
 The illustrations and their symbolic representation enable 
 us to define mathematical reasoning and to distinguish it from 
 the species which might be called logical, but for the fact that 
 all forms of reasoning are logical. It could, however, be called
 
 MATHEMATICAL AND OTHER REASONING 277 
 
 " logical " in the sense in which we have distinguished be- 
 tween the "mathematical" and the "logical" genus. But 
 without pressing this meaning, mathematical reasoning is 
 that which deals purely with quantity. It maybe called pure 
 quantitative reasoning, in distinction from the second form, 
 which may be called mixed qualitative and quantitative. Math- 
 ematical reasoning is based upon relations of quantity which 
 expresses no variations or differences except in amount ; 
 quanto-qualitative reasoning is not only based upon rela- 
 tions of quality but of quantity also, and in addition to 
 differences of degree or amount, represents differences of 
 kind. This distinction between the two processes merely 
 means that the conceptions employed in mathematical rea- 
 soning either deal exclusively with the abstract ideas of 
 quantity, or deal with the concrete objects, only in numerical 
 terms of time, space, force, etc., or commensurable quantities. 
 They are not viewed as a group of qualities, but as individual 
 wholes. When we say " all men," "all animals/' " all citizens," 
 we speak of a number of individual persons as individuals, and 
 do not take into account their various differences. They may 
 be of different sizes, different color, or of different powers, 
 but the " all " does nothing but describe them numerically. 
 When the predicate is thus quantified, the identity between 
 subject and predicate becomes purely a quantitative identity, 
 and so the terms are equated. But when we say "man," " an- 
 imal," " citizen," etc., we think of attribute wholes, so that any 
 identity between them and a predicate is qualitative, and in my 
 judgment is not of the nature of an equation at all. Its real 
 import may be expressed in one of two ways, according as the 
 predicate is an attribute or a class term. We have already 
 spoken of the two kinds of judgment, the intensive and the 
 extensive (p. 123). The former, as for example, " Man is 
 wise," expresses by its predicate that a certain group of quali- 
 ties called man is accompamed'by the quality wise, or that amid 
 that group of qualities will be found one denoted by the term 
 "wise." The so-called "identity," or "agreement" between 
 the subject and predicate in such cases, is not one of quan-
 
 278 ELEMENTS OF LOGIC 
 
 trty. The relation should rather be called that of connection, a 
 relation that cannot be dealt with quantitatively, except as it 
 may be common or invariable. The extensive judgment, as 
 " Man is an animal," expresses by its predicate a distinct rela- 
 tion of identity between the subject and predicate, namely, that 
 the two have certain common qualities by which they may be 
 classified together. But no relation of quantity appears in 
 this case any more than in the intensive judgment. In both, 
 however, there is nothing said to prevent any number of other 
 and different beings from being connected with, or included 
 in, the predicate. The relation, therefore, is one which is 
 based upon the real or the possible coexistence of resem- 
 blances and differences between certain objects connected as 
 subject and predicate. This is only to say that in what we 
 have called qualitative judgments and reasoning the concep- 
 tions with which we have to deal are genera and sjyecies, es- 
 sentia and accidentia, or conferentia and differentia. These con- 
 ceptions do not appear as such in mathematical reasoning. 
 They prevent that definite quantification, especially of the 
 predicate, which is necessary to make the reasoning mathe- 
 matical 
 
 But there is a class of judgments, intensive or extensive, 
 just as we choose to regard them, in which the identity be- 
 tween subject and predicate is such as to preclude the admis- 
 sion of any other than the existing subject in the same con- 
 nection. They are such judgments as " Virtue is goodness," 
 "Quadrupeds are four-footed animals," "Honesty is the best 
 policy," "Government is social organization, "etc. Here there 
 is a qualitative connection, or identity of qualities, expressed 
 by the propositions. But it is an absolute identity. The 
 terms are either synonymous or represent definitions. The 
 relation expressed is not that between genus and species, 
 or the connection between essentia and accidentia, or con- 
 ferentia and differentia, but between genus and genus, es- 
 sentia and essentia, conferentia and conferentia, accidentia 
 and accidentia, or differentia and differentia. But such a 
 relation coincides exactly with the quantitative relation, and,
 
 MATHEMATICAL AND OTHER REASONING 279 
 
 so far as reasoning is concerned, can be made convertible 
 with it. Hence, in all such cases the reasoning with such 
 propositions is as assured as that in mathematics, because 
 their quantitative meaning coincides with their qualitative, 
 or their qualitative with their quantitative, and so can be 
 substituted in its stead. It will be self-evident from this fact 
 that the whole process might be greatly simplified if all terms 
 could be made to represent a similar connection between the 
 subject and predicate. 
 
 The peculiarity of the relation between the two terms in 
 these two classes of propositions and in all mathematical judg- 
 ments might justify the use of the name Traduction for this 
 species of reasoning, in contrast with ordinary Deduction on 
 the one hand, and Induction on the other. It does not differ 
 essentially, however, from Deduction, as the principle is the 
 same. Derived from trans and duco, to lead over, it might 
 denote the substitution which is characteristic of all reason- 
 ing that involves a predicate in its propositions quantitatively 
 identical with the subject. Jevons alludes to the use of this 
 term in connection with a species of syllogistic form and 
 reasoning which we have yet to consider. But he does not 
 employ it to denominate the essential characteristic of mathe- 
 matical reasoning, which can carry on its substitution without 
 regard to the question of Figure, as we have already seen. 
 All its propositions are definite, and so are its terms. Even 
 those conceptions which denote a part of a whole are definite. 
 Mathematical conceptions are not qualified by the indefinite 
 some, which may denote any portion whatever less than the 
 whole, but they are expressed by some definite fraction of 
 the whole, which is equivalent to a universal notion in its im- 
 port, whenever a portion of some larger total is to be reckoned 
 with. Even the unknown quantities of Algebra are no ex- 
 ception to this principle. They always represent definite 
 quantities, which are called " unknown " because they may be 
 used for any fixed number we please. In all instances of such 
 terras, however, having a mathematical import, we have for 
 our propositions an equation or equations in which the pro-
 
 280 ELEMENTS OF LOGIC 
 
 cess of transition from one term to another is substitution or 
 traduction. Mathematical reasoning thus embodies in a per- 
 fect form the principles most clearly enunciated in the doc- 
 trine of the quantification of the predicate, although implied 
 in the quantification of the subject. It is simply the substi- 
 tution of one term for another which involves it, because they 
 are quantitatively identical at the same time that they may 
 be necessarily connected in another way. 
 
 The principle thus established can be used to explain a 
 form of reasoning which Jevons regards as irregular, and yet 
 valid in spite of its real or apparent violation of the formal 
 laws of the syllogism. The first illustration chosen by him is 
 not formally incorrect, but is regarded by him as irregular. 
 It is as follows : 
 
 The sun is a thing insensible. 
 
 The Persians worship the sun. 
 
 Therefore the Persians worship a thing insensible. 
 
 The only apparent irregularity in this example is the use of a 
 part of the predicate of the minor premise in the conclusion, 
 and the consideration of the other part of it as the middle 
 term. But logically I can see no objection to this process, nor 
 a reason in it for regarding the syllogism as in any way ir- 
 regular. It is true that, grammatically considered, the subject 
 of the major premise is " the sun," and the predicate of the 
 minor premise is " worship the sun," so that being AAA of 
 the first Figure, they occupy the place of the middle term. 
 The difference between them might seem to make a fourth 
 term, and therefore a fallacy of Quaternio Terminorum. This 
 would undoubtedly be the case if we attempted to draw the 
 conclusion, " The Persians are a thing insensible." But the 
 whole matter is altered when we transfer the term " worship " 
 to the conclusion, because it signifies that the real middle 
 term of thought is " the sun," and the reasoning becomes per- 
 fectly valid, and is no exception to the ordinary form of syllo- 
 gistic inference. But it is not a case of mathematical substi-
 
 tution, except as a part of the apparent middle term is trans- 
 ferred to the conclusion. 
 
 The next instance is somewhat similar, and yet is as dif- 
 ferent in other respects. Jevons is correct when he asserts 
 that a great deal of our reasoning is done after this manner. 
 The following is his illustration : 
 
 The Divine Law commands us to honor kings. 
 
 Louis XIV. is a king. 
 
 Therefore the Divine Law commands us to honor Louis XIV. 
 
 The peculiarity of this instance is that it seems to be a case of 
 AAA in the second Figure, namely, an example of illicit mid- 
 dle, and yet valid in spite of the fact. As before, we substi- 
 tute in the conclusion a part of the predicate in the minor 
 premise, and this leaves the term " king " as the middle term. 
 Why, then, does the reasoning impress us irresistibly with its 
 validity, although the form is against this feeling ? If we 
 stated the case as follows there would be no difficulty in re- 
 jecting the conclusion : 
 
 Louis XIV. commands us to honor kings. 
 The Divine Law commands us to honor kings. 
 Therefore the Divine Law is Louis XTV. 
 
 The fallacy is palpable in this instance. Why is it not so in 
 the former? 
 
 Two explanations can be given to the case which will an- 
 swer the question. The first is a reduction of the thought 
 expressed to the proper Mood and Figure. Thus the minor 
 premise of the original example can be interpreted as mean- 
 ing " Kings are those whom the Divine Law commands us to 
 honor." This is precisely the same thought as before, and it 
 gives AAA of the first Figure, with the conclusion, " Louis 
 XIV. is he whom the Divine Law commands us to honor." 
 In the premise and conclusion, therefore, we have simply the 
 passive for the active form of expression, while the logical 
 import is precisely the same as before. In this way we an-
 
 282 ELEMENTS OF LOGIC 
 
 swer our question by showing that logically the apparently 
 invalid syllogism may really be valid by being of an actually 
 different form in thought from what it is in expression. 
 
 But there is a second way of dealing with the case. In the 
 proposition " The Divine Law commands us to honor kings,'' 
 it is evident that the predicate is not distributed, namely, that 
 other things might command the same. But in such state- 
 ments we usually quantify or universalize certain terms help- 
 ing to constitute the predicate. Thus in making this asser- 
 tion we are most likely to mean all kings. Now, since a part 
 of it is substituted in the conclusion, this quantification of 
 the term "king" has the effect of distributing it, that is, of 
 indicating that Louis XTV. is included in the general class as 
 soon as he is affirmed to be a king. The reasoning, there- 
 fore, becomes a simple case of substitution of Louis XIV. for 
 "king," which is already included by implication or asser- 
 tion in the whole class. If we had distinctly said or thought 
 that " the Divine Law commands us to honor some kings," 
 and then asserted that " Louis XIV. is a king," we should 
 have had no inclination to draw the conclusion that Louis 
 XTV. is to be honored, and would have very quickly perceived 
 the fallacy of attempting to do so, simply for the reason that 
 it would appear to be a case of undistributed middle. 
 
 But it remains to show whether the case can be resolved by 
 the regular laws of the syllogism. If we regard the univer- 
 salizing or distribution of the term " king " as bringing the 
 case under the quantification of the predicate, we should have, 
 or should seem to have, an instance of UAA in the second 
 Figure, which we have found to be formally valid, because 
 of the distribution of the middle term. But it can be shown, 
 in spite of its appearance, that the reasoning is, after all, not of 
 the second Figure. Thus if the minor premise were negative, 
 "Louis XIV. is not a king," we should have, according to our 
 supposition, UEE, which is valid according to the table. But 
 it requires only a glance at the conclusion, that " The Divine 
 Law does not command us to honor Louis XIV.," to see that 
 it is fallacious, because, so far as the assertion is concerned, the
 
 MATHEMATICAL AND OTHER REASONING 283 
 
 Divine Law may command us to honor all other persons as 
 well as kings. If we examine the fallacy carefully we shall 
 find that it is the same as AEE of the first Figure. Let it 
 be represented by the following diagram : We observe in 
 this representation that " kings " are included 
 in those whom the Divine Law commands us 
 to honor, and also that Louis XIV. is so con- 
 tained, but excluded from the class "kings." 
 The representation is thus precisely like that 
 
 which symbolizes the fallacy of AEE in the 
 
 FIG 29 
 first Figure. The placing of " Divine Law " in 
 
 a circle is only to indicate that the entire predicate is not dis- 
 tributed. For considering the fallacy we require only to take 
 the other circles into account. But the fallacy, being shown 
 to be equivalent to that of AEE in the first Figure, creates 
 the suspicion that the original syllogism, with the affirmative 
 premise, is really AAA of the first Figure, although it appears 
 to be more nearly relatsd to the second Figure. The next 
 diagram will illustrate this supposition. In this representa- 
 tion the relation between the terms is that of 
 the first Figure, and it also indicates what is 
 implied in the proposition that " the Divine 
 Law commands us to honor kings," namely, 
 that it may command us to honor others also. 
 But it brings out most distinctly what is in- 
 
 FlO 30 
 
 volved in the quantification of a term and the 
 relation implied by it between this term and some other con- 
 ception in the argument, namely, that a substitution can take 
 place in any particular instance included under a class. It is 
 done in this case without regard to the apparent Figure of the 
 Syllogism, and in so far resembles the process of mathemati- 
 cal or quantitative Logic. But the reasoning is neither a case 
 of traduction nor one of the ordinary syllogistic form as it is 
 expressed. It is what older logicians called the complex syl- 
 logism, and it has generally been maintained that it could not 
 be reduced to the regular form. But this we shall show can 
 be done, and the process may bring out more clearly the dif-
 
 284: ELEMENTS OF LOGIC 
 
 ference between mathematical and ordinary qualitative rea- 
 soning. 
 
 We have shown that the reasoning cannot be of the second 
 Figure, although it appears to be. It remains to show that it is 
 of the first, and so to confirm the suspicion created by the nat- 
 ure of the fallacy involved in supposing it to be of the second 
 figure. The two diagrams show the complexity of the rela- 
 tions to be dealt with. In the first place, under the ordinary 
 interpretation of the proposition " Divine Law " is the subject, 
 and is distributed, and " commands us to honor kings " is the 
 predicate, and undistributed. This relation is represented by 
 the largest circle and the small one containing " Divine Law," 
 and merely denotes that other sources of the same command 
 might exist, in so far as the proposition says nothing to the 
 contrary. But the predicate is a complex one, and it is not 
 upon the whole of it that the reasoning depends. This is evi- 
 dent from the substitution of apart of it in the conclusion. The 
 part which is not found in the conclusion, then, is the real 
 middle term. This is the word " kings." " Commands us to 
 honor," is not a part of the middle term, but of the minor or 
 major term, as the case may be. " Kings " is the term upon 
 which the reasoning turns, and as the circles represent it, is to 
 be included in the larger, because other persons may be in- 
 cluded under the same command. This relation is not ex- 
 cluded by the universal quantification of the word, so that it 
 cannot be conceived as a predicate, because its universal 
 quantification in that case would imply its coextension with 
 the subject ; which would imply that the divine law could 
 not command us to honor any one else. But since the 
 proposition does not exclude honor to others besides " kings," 
 and " kings " is the middle term and distributed, the only pos- 
 sible way of representing the fact is to conceive it as subject. 
 But to conceive it as subject makes the proposition the major 
 premise, and the other proposition, " Louis XTV. is a king," 
 the minor premise, and the form becomes AAA of the first 
 Figure, as we have already asserted it must be, in order to 
 make the reasoning valid.
 
 MATHEMATICAL AND OTHER REASONING 285 
 
 But in order to effect this transformation a change in the 
 order of the statement is required. Since the object of thought 
 is mainly the middle term, as in such cases it always is, there 
 is no objection to the process which explains how the mind 
 really treats it, although the form of statement appears to 
 make it different. The reason, however, for the change is in 
 the peculiar nature of the proposition, together with the quan- 
 tification of the term " kings." If we observe the two propo- 
 sitions we shall discover that only one of them connects the 
 subject and the predicate by means of the ordinary copula, 
 But this is of minor importance compared with the fact that 
 the two terms so connected are, one of them an individual or 
 singular and the other a class term. This makes the proposi- 
 tion quantitative, or an extensive judgment, its intensive mean- 
 ing being implied. We have already explained how all prop- 
 ositions may be either intensive or extensive, according as we 
 choose to view them. This double conception of them gener- 
 ally enables us to proceed in an argument without regard to the 
 other fact, which we remarked of them, namely, that the 
 quantity of intension is represented in the reverse order of that 
 in extension. In other words, according to Hamilton and oth- 
 ers, judgments of extension represent the subject as con- 
 tained in the predicate, and judgments of intension represent 
 the predicate as contained in the subject. Practically, how- 
 ever, this inversion of the order of comprehension has no im- 
 portance so long as the judgments are convertible from the 
 intensive to the extensive, and vice versa, by the mere thinking 
 of those qualities and without a change in the order of the 
 terms. This is because the propositions usually chosen for 
 illustration are simple ones. But the proposition, " The Di- 
 vine Law commands us to honor kings," is not a simple one 
 in the ordinary logical sense. In the first place, it is an in- 
 tensive judgment. The predicate, " commands us to honor 
 kings," is not a class term, but expresses a quality inherent in 
 the "Divine Law." But nevertheless, as explained before, it 
 can be immediately transformed into an extensive judgment 
 by saying, " The Divine Law is that which commands," etc.
 
 286 ELEMENTS OF LOGIC 
 
 This is the form in which the diagrams represent it. But at 
 the same time it is apparent, both from the reasoning and from 
 the diagrams, that the inference does not depend upon this 
 feature of its intension and extension. 27ie quantitative element 
 which determines the reasoning is in connection with the term 
 "kings." It is this that renders the proposition logically 
 complex. There is a double quantification of conceptions. 
 But being an intensive proposition, with the assertion or im- 
 plication that " kings " is to be taken in its extension, and 
 because the whole reasoning depends upon the latter, we may 
 ig.aore the relation of intension and extension expressed be- 
 tween " Divine Law " and " commands," etc., because this is 
 reproduced in the conclusion, and reconstruct the proposition 
 formally so as to express the way in which the mind deals 
 with it. In other words, we transform it from its intensive 
 form into an extensive judgment like the other premise in the 
 syllogism, and having the quantitative conceptions implicitly 
 used in the process explicitly stated in the order to indicate 
 the conformity of the reasoning to the regular laws of the 
 syllogism. The proposition " The Divine Law commands us 
 to honor kings," becomes, considering that " kings " is univer- 
 salized in its meaning, " All kings are those whom the Divine 
 Law commands us to honor." The Mood and Figure is appar- 
 ent from this construction, and it is brought out by observing 
 that the judgment is an extensive one, with its quantity, so far 
 as it affects the reasoning, in one of the terms, and not in the 
 whole predicate. 
 
 Now, as we have remarked, a large amount of our ordinary 
 reasoning is of this kind, namely, is reasoning with proposi- 
 tions involving complexities of intension and extension that 
 require analysis to bring them under the regular rules. As 
 indicated, we do this by making both premises extensive prop- 
 ositions. Even if one of them is intensive it must be capable of 
 an extensive conception or expression which indicates the proper 
 relation of inclusion or exclusion betiveen it and the other prem- 
 ise, and this is making it in effect extensive. 
 
 That this is a common state of matters in reasoning can be
 
 MATHEMATICAL AND OTHER REASONING 287 
 
 seen by more examples than the one we have been discussing. 
 We shall give some, both in the valid and the invalid form. 
 Take the first : 
 
 Men show a disposition to respect the brave. 
 
 Leonidas was brave. 
 
 Therefore men show a disposition to respect Leonidas. 
 
 This instance is precisely like the previous one, and. appears 
 to be of the second Figure, but is resolved into the first by 
 the same process, and for the same reasons as before. The 
 major premise becomes " the brave are those whom men show 
 a disposition to respect." That the reduction is dependent 
 upon the peculiarly intensive nature of the proposition is ap- 
 parent from the following instance which has the same form, 
 but does not resolve so easily : 
 
 Men show a disposition to be noble. 
 
 Lincoln was noble. 
 
 Therefore men show a disposition to be Lincoln. 
 
 This is manifestly absurd. But when reconstructed it gives 
 the conclusion, " Lincoln is one of those whom men show a 
 disposition to be," which is rational enough. Why does it 
 not appear so in the first case, where we seem to proceed as 
 before ? 
 
 The answer to this is found in the difference of import be- 
 tween the use of be and a transitive verb. " To honor kings," 
 " to respect the brave," are intensive conceptions whose ex- 
 tension can be brought out only as in the diagrams, by revers- 
 ing the order of expression, as, " Kings are to be honored," 
 "The brave are to be respected." But " to be brave," " to be 
 a king," may be intensive or extensive, as we choose to regard 
 them, and without any alteration of order. In the former the 
 two qualities can be made to coincide only by the inversion 
 we have indicated, and hence such intensive conceptions or 
 propositions must be transformed in order to show the formal 
 process of reasoning actually involved in their use.
 
 288 ELEMENTS OF LOGIC 
 
 An instance of an invalid form may be produced in order to 
 confirm in a negative way the reduction we have made : 
 All oxen are animals. 
 
 The Levitic law permits us to use oxen for food. 
 Therefore the Levitic law permits us to use animals for food. 
 
 If in the conclusion we mean "some animals," the reason- 
 ing is valid, but if we mean " all animals," as we have ex- 
 plained to be usual in such propositions, the inference is not 
 valid. We have distributed the term " animals " in the con- 
 clusion when it is not distributed in the premises. But it is 
 noticeable that the form of the syllogism is that of AAA of the 
 first Figure, and yet is not valid except on the condition that 
 we say or mean " some animals" in the conclusion. When re- 
 duced it becomes in reality AAA of the third Figure, which is 
 a case of illicit minor term. Thus, 
 
 All oxen are beings which the Levitic law permits us to use 
 for food. 
 
 All oxen are animals. 
 
 Therefore all animals are beings which the Levitic law per- 
 mits us to use for food. 
 
 Here the illicit minor is quite apparent. But if we had said 
 " Some animals are beings," etc., the conclusion would be 
 valid, and it would explain why to have said or meant " some 
 animals " in the first instance would have been valid reason- 
 ing. As is usual in the third Figure, it does not matter 
 which of the propositions is taken for the major premise. 
 The conclusion will be substantially the same in all cases. 
 We placed them in the above order solely for the convenience 
 of getting the conclusion in an order that would dispense 
 with conversion for comparing it with the original. 
 
 The first of the cases can be solved also according to the 
 same principles. The proposition, " The Persians worship the 
 sun," can be reduced to " The sun is the object which the Per- 
 sians worship." This will make the syllogism AAA of the 
 third Figure instead of the first, and the conclusion will be, 
 " The object which the Persians worship is a thing insensible."
 
 MATHEMATICAL AND OTHER REASONING 289 
 
 But there is no necessity for this reduction in this instance, 
 because the middle term is singular and the predicate of the 
 major premise is made indefinite or left unquantified by the 
 particle "a," which prevents it from being universalized in 
 the conclusion, and which has the effect of equating it with 
 the subject in the minor premise. Were it universalized in the 
 conclusion we should have an instance of the same fallacy as 
 that which we have just exposed. But the case is an inter- 
 esting one as showing the use of substitution whenever singu- 
 lar terms are employed, and the peculiar influence which may 
 be exercised by the particles "a" or "an," and " the," the 
 former denoting an individual which is at the same time a 
 part of a class and the substitutive equivalent of the term 
 which it defines. The latter often denotes that the term 
 which it agrees with is quantified universally, or that a certain 
 definite number of objects is considered, which enables us to re- 
 sort to substitution. They do not, however, alter the form of 
 a proposition, and do not always produce the effect described. 
 The meaning, as affected by them, is a material factor of the 
 proposition. But it is important to consider it for the reason 
 that it may make the reasoning actually valid in cases where, 
 tested by purely formal laws it would appear invalid. 
 
 In all these instances of complex syllogisms we have clear 
 illustrations of an apparent resemblance to mathematical rea- 
 soning in that there is at least an apparent disregard of the 
 Figure of the syllogism. But with the reality of this disre- 
 gard denied the illusion is dispelled, and we have illustrations 
 of reasoning in which traduction is not possible unless the 
 terms used are singular. The contrast, therefore, between 
 purely quantitative reasoning, in which the subject and pred- 
 icate are identical in extension, and qualitative reasoning 
 combined with the quantitative, where there is a disparity of 
 extension between subject and predicate could not be better 
 brought out than in these cases. And as much of our rea- 
 soning is, perhaps, of the type we have just been considering, we 
 see the liability to fallacy incident to it because of its variation 
 
 from the form where certitude and assurance are guaranteed. 
 19
 
 CHAPTER XXL 
 
 THE LAWS OF THOUGHT 
 
 THE Laws of Thought do not require any elaborate treat- 
 ment in an elementary treatise upon Logic, but the manner in 
 which they have been assumed, or in which they underlie all 
 our reasonings makes it necessary to state them and their 
 meaning very briefly. We have already explained what a 
 " law of thought " means, in our statement that it denotes the 
 uniform way in which we think and must think. In all our 
 reasoning we take these laws for granted. They are condr 
 tions of our reasoning and of the relation expressed between 
 subject and predicate, antecedent and consequent, in proposi- 
 tions. We do not require to announce them as premises in 
 our processes of transition from proposition to proposition, 
 because they are either universally assumed without question, 
 or they are the conditions of the formal and material truth of 
 the data themselves, which it is not the business of formal 
 Logic to investigate. Besides, they are of that axiomatic 
 nature which renders it necessary to admit them before we 
 could construct an objection or an argument against them. 
 We do not require, therefore, to investigate them to determine 
 their validity, but only to state what they are, their meaning 
 and their functions. 
 
 The Laws of Thought may be divided into two classes, the 
 Primary or Fundamental, and the Secondary or ^Derived. The 
 primary laws are those which regulate all thought, whatever, 
 whether of Conception, Judgment, or Reasoning. The second- 
 ary are simply those modified forms of the primary laws which 
 are formulated in a particular way to suit the contingencies of 
 syllogistic reasoning. We shall consider them in their order. 
 
 1st. The Primary Laws. As defined, they are the funda-
 
 THE LAWS OF THOUGHT 291 
 
 mental laws of all thinking ; that is, of conceiving percepts, 
 concepts, and judgments in relation to each other. There 
 are four of these general laws, The Law of Identity, The Law 
 of Contradiction, The Law of Excluded Middle, and TJie Law of 
 Sufficient Reason. 
 
 1. THE LAW or IDENTITY. This law expresses the right of 
 the mind to affirm that a thing is identical with itself. Thus, 
 A is A, or " Whatever is, is," that is, any existence is equal to 
 itself. This is the usual form of statement for the law. But 
 the law is the general principle at the foundation of all affirm- 
 ative judgments, whether the subject and predicate are quanti- 
 tatively or qualitatively equal or not. Hence there are two 
 kinds of identity, absolute or total, and relative or partial iden- 
 tity. Absolute identity is represented by the truistic or tau- 
 tological proposition, where both the form and the matter of 
 the terms in subject and predicate are the same. Kelative or 
 partial identity is that of the ordinary proposition where there 
 is a difference of extension between subject and predicate, or 
 where the form of the terms makes it possible to identify them 
 with others also. Thus to illustrate both instances : " Man is 
 man," "Animals are animals," are cases of absolute identity 
 in which a conception can be affirmed to be equal to itself. 
 But of partial identity we have " Man is an animal," " Horses 
 are quadrupeds." In these instances there are certain ele- 
 ments of identity, but the terms are not convertible with each 
 other ; that is, the propositions cannot be converted simply. 
 
 From what has just been said it might be inferred that ex- 
 clusive propositions and definitions, as well as synonymous 
 terms, are illustrations of absolute identity. This is true ; but 
 it is identity of matter and not of form. The terms are such as 
 can be used in other than truistic propositions. 
 
 2. THE LAW OF CONTRADICTION. This law Hamilton observes 
 should be called the law of non-contradiction, because it de- 
 notes that an object cannot be affirmed to be what it is not. 
 It is sometimes denned as denoting that a thing cannot exist 
 and not exist at the same time ; or that it cannot be affirmed 
 to be one thing and its opposite or contradictory at the same
 
 292 ELEMENTS OF LOGIC 
 
 time. This law is the complement of the law of identity, and 
 may be said to be the same law for thought as the law of im- 
 penetrability is for matter. It is the principle which deter- 
 mines the relations and inferences in the Square of Opposition 
 and the drawing of negative conclusions in the syllogism. 
 
 3. THE LAW OF EXCLUDED MIDDLE. This law denotes that 
 of two contradictions only one can be true. Thus of the 
 propositions, " Man is an animal," and " Man is not an ani- 
 mal," only one can be true. They are supposed to represent 
 a completely dichotomous division of all objects into two 
 contradictory classes, and to achieve this the subject and 
 predicates must be so related that if the law of identity ap- 
 plies to the relation in one case, that of contradiction ap- 
 plies in the other. Hence only one can be true, and the other 
 must be false. The law is a combination of the first two 
 laws, those of identity and contradiction, and is at the basis of 
 disjunctive propositions and syllogisms. 
 
 4. THE LAW OF SUFFICIENT REASON. This law is briefly de- 
 fined as denoting that every phenomenon, event, or relation 
 must have a sufficient reason or cause for being what it is. 
 There is some dispute about the right to regard this law as a 
 law of reasoning, and it is certain that it seems more appro- 
 priate an assumption or postulate for the physical sciences 
 than for the logical. But it nevertheless dominates certain 
 modes of thought which are occupied, not with the identity or 
 non-identity of objects or concepts, but with their connection. 
 We shall not discuss the merits of this question, since it does 
 not belong to the elementary plan of the present work to do 
 so. Hence, we shall only explain the meaning attached to the 
 law by those who regard it as a law of thought. 
 
 The law has been formulated to mean that we should affirm 
 or infer nothing without a reason or ground. This reason or 
 ground is the condition upon which the truth of a proposition 
 or the reality of an event depends. The condition may be 
 called the antecedent and the resultant the consequent. 
 Hence the law will appear to determine the relation expressed 
 in hypothetical propositions and syllogisms. But as these are
 
 THE LAWS OF THOUGHT 293 
 
 reducible to the categorical form the law must either be re- 
 duced to that of identity, or be found to apply to categorical 
 judgments. Perhaps it is the dependence of all individual 
 truths upon general truths or principles that represents the 
 application of this law as well as that of identity in all ordi- 
 nary propositions and reasoning. If so, it can have an inde- 
 pendent place in Logic. But not intending to decide this 
 question, pro or con, we must be content with recognizing 
 what is frequently regarded as a law of thought, and what 
 certainly expresses a relation of dependence quite fundamen- 
 tal to our mental processes. 
 
 2d. The Secondary Laws. These are simply derivatives 
 of the,primary laws of Identity and Contradiction. They are 
 often called the axioms of Logic, and are simply formulas for 
 justifying the inferences in formal reasoning. They are four, 
 two of them for immediate inference and two of them for me- 
 diate reasoning. After what has been said in previous chap- 
 ters they may be stated without further explanation. 
 
 1. If two concepts agree in one relation, they may be 
 stated to agree in the converse relation. 
 
 2. If two concepts differ or contradict in one relation, they 
 will disagree or contradict in the converse. 
 
 These rules regulate the process of immediate inference or 
 Conversion, and their principle is simply assumed in every 
 case of convertible and non-convertible terms. 
 
 3. If two terms agree with one and the same third term, 
 they agree with each other. 
 
 4. If of two terms one agrees and the other disagrees with 
 one and the same third term, they do not agree with each 
 other. 
 
 It must be observed, however, that the latter two axioms 
 are applicable in their purity and in an unqualified sense, 
 only to mathematical syllogisms, or reasoning with universally 
 or definitely quantified subjects and predicates. Hence for 
 ordinary syllogisms they have to be modified by the following 
 rule or law : 
 
 5. The terms which agree or disagree in the conclusion
 
 294: ELEMENTS OF LOGIC 
 
 must have no greater or the same distribution as in the prem- 
 ises, and they can agree or disagree only when the middle 
 term is properly distributed in the premises. 
 
 This last law is to provide against the commission of the 
 fallacies of Illicit Middle, Illicit Minor, and Illicit Major terms, 
 when the third and fourth laws have been conformed to in 
 their qualitative relations. The fifth law or rule specifies the 
 quantitative conditions affecting the conclusion.
 
 CHAPTER XXTL 
 
 INDUCTIVE REASONING 
 
 1st. The Nature of Induction. It has been usual to de- 
 fine Induction in a manner contrasted with Deduction. But 
 there are some peculiarities in connection with the meaning 
 of the term which must be considered before we define the 
 process with which we are at present concerned. Usage has 
 not given the term a uniform conception, such as belongs 
 quite generally to the word "Deduction." All classes of 
 thinkers are tolerably agreed in regard to the process of de- 
 duction, whether they are logicians or scientific investigators. 
 It is assumed to be the process of finding the proof of a par- 
 ticular truth in a general principle or proposition already con- 
 taining it, explicitly or implicitly. Induction is often de- 
 scribed as the reverse of this process, namely, as inferring 
 general truths from the particular. But this is not the only 
 conception of the term, and it is because this conception is 
 not the only one in use that there is. so little agreement about 
 the nature, functions, and importance of induction. It will, 
 therefore, be necessary to examine the several imports of the 
 term in order to make possible a true conception or theory of 
 the process denoted by it. 
 
 There are three different applications of the term " Induc- 
 tion," which are generally assumed to mean the same thing. 
 To explain what they are we have to produce the usual di- 
 visions of the subject, which are the so-called kinds of induc- 
 tion. They are " Perfect Induction " and " Imperfect Induc- 
 tion." The first meaning of the term applies to the first kind, 
 and the other two are modifications of what is implied in im- 
 perfect induction. We would not suspect a difference of 
 meaning from this general fact alone, but if we examine care-
 
 296 ELEMENTS OF LOGIC 
 
 fully the illustrations chosen to describe the nature of the 
 process as thus distinguished into different kinds, we shall 
 discern very clearly the great differences of real meaning at- 
 taching to the term. Thus " Perfect Induction " is simply an 
 enumeration of the particulars which form a class. It is the 
 process which characterized the method of Socrates in reach- 
 ing his definitions, and which Aristotle remarked was a new 
 method compared with the argumentation of his predecessors. 
 An example of perfect induction is the following : " Mercury 
 revolves on its axis ; so do Venus, the Earth, Mars, Jupiter, Sat- 
 urn, and Neptune. But these are all the planets, and therefore 
 all the planets revolve on their axes." Although this is stated 
 in the form of reasoning, it is not reasoning at alL This fact 
 is apparent in the nature of the conclusion, which is that " All 
 the planets revolve on their axes," not on the axis of Mercury, 
 although the same thing would be true if we had said " on the 
 axis of Mercury." But the special proof of its not being a 
 case of reasoning is in the fact that the so-called conclusion is 
 merely a universal statement of what had been enumerated 
 in detail in the premises. We are supposed to have observed 
 the individual fact that " Mercury revolves on its axis," and 
 then again that "Venus revolves on its axis," and so on 
 throughout the entire number of planets. Hence when we 
 say, " All the planets revolve on their axes," we but univer- 
 salize our particular observations we use the terms " all 
 planets" as an economical device to avoid repeating the 
 proper name of each planet. But we do not infer anything, 
 or reason from one proposition to another. We do not es- 
 tablish any new connections of thought by the process, as we 
 do in syllogistic reasoning, as already explained, but we only 
 generalize what we had observed in detail. It is precisely the 
 same with all enumerations of individuals or particulars into 
 a whole or class with a general name denoting those enumera- 
 tions only. They may be called " Inductions " if we choose 
 so to name them ; but they are not reasoning. They are only 
 generalizations as opposed to or distinct from reasoning, while 
 the term " Induction," as now used by logicians, denotes a
 
 INDUCTIVE REASONING 297 
 
 process of inference or reasoning of some kind. It is quite 
 generally agreed since the time of Bacon that the so-called 
 "Perfect Induction" is not properly called "Induction" 
 because it is not a mode of reasoning. It has been the 
 name for the Socratic process of obtaining universal concep- 
 tions and definitions. But the contingencies of the growth 
 of knowledge and the demand for a method \vhich would take 
 the place of the Aristotelian Logic suggested the term " In- 
 ductive" as opposed to "Deductive," and the rejection of 
 " Perfect Induction " on the ground that it was not ratiocina- 
 tive in its nature, implied that Induction must be a process of 
 reasoning in order to compare it with Deduction. 
 
 This second general meaning is the more important of the 
 two, and was called "Imperfect Induction" because the con- 
 clusion contained more than the premises. Thus if I had in- 
 ferred that " All the planets revolve about their axes," from 
 the mere fact that one of them did so, I should have drawn 
 an inductive inference. I should not in this case have merely 
 generalized the particulars of my observation or experience, 
 but have conjectured or inferred that what was true of one 
 case would turn out to be true of all the objects known upon 
 other grounds to belong to the same class. But this conclu- 
 sion has no definite certainty such as the mind desires, and 
 hence to give this conjecture greater probability I must vary 
 my observation of facts in connection with the several planets, 
 and find whether they agree or disagree with my supposition. 
 If, for instance, I observed that certain of them presented an 
 absolutely invariable appearance, such as a particular spot al- 
 ways in sight and in the same place, the fact would be at least 
 a presumption against the supposition of the planet's axial 
 revolution. On the other hand, if the spot presented certain 
 regular changes of position and periodical disappearance and 
 reappearance, the fact would be in favor of the hypothesis. 
 This mode of repeating and varying observations or experi- 
 ments in the case of the experimental sciences, according to 
 certain methods, which are called the " Method of Agree- 
 ment," the " Method of Difference," the " Method of Concom-
 
 298 ELEMENTS OF LOGIC 
 
 itant Variations," etc., has been called the Inductive Method in 
 general, as a mode of ascertaining certain truths in a manner 
 quite distinct from the ordinary syllogistic and deductive 
 reasoning. This is the third meaning of the term, with which 
 a theory of Induction has to reckon. It remains for us to se- 
 lect which of them we are to deal with under the heading of 
 this chapter. It is the confusion of all three under the same 
 term that leads to the uncertainties about the process itself. 
 
 It will conduce to clearness in the discussion of Induction 
 if we sketch briefly the history of the general meaning of the 
 term, and indicate specially the implication carried along with 
 it which is no necessary part of its import as describing a logi- 
 cal process. This latter fact is not sufficiently taken into 
 account by logicians when treating of the subject, and yet it 
 involves grave consequences to their theory of Induction. 
 
 The most general meaning of the term " Induction," as 
 used, especially by popular writers, since the time of Bacon, is 
 that of a process which adds to our knowledge. The accusation 
 which Bacon and his admirers have brought against the " de- 
 ductive method," or Aristotelian Logic, was that it could not 
 give us our premises, and therefore neither assured us of our 
 data for reasoning, nor, when we were assured of them, could 
 it add anything to our knowledge. In other words, as we 
 have seen, our conclusion depends wholly upon knowing our 
 premises, and if we already know the premises, the conclusion 
 adds nothing to what we know. The deductive method is, 
 therefore, useless for giving us knowledge, say its opponents. 
 It only manipulates in various ways that which we are sup- 
 posed to have in an implicit, if not an explicit, form. How, 
 then, do we get the knowledge we have ? How do we ever 
 make any additions to our general knowledge, or to the prem- 
 ises with which deduction deals and which it assumes? 
 
 This was the question which Bacon attempted to answer, 
 and he employed the term " Induction " to define the method 
 of acquring new data and principles of truth. The Aristote- 
 lian method was rejected as useless, because it could never ad- 
 vance beyond what was already given, and hence, in adopting
 
 INDUCTIVE REASONING 299 
 
 the new method, which was to contrast with the old, the term 
 "Induction" took on the meaning which that contrast im- 
 plied, namely, that process which produced an increment to the 
 knowledge which we already have at any given time. If we take 
 the first meaning of the term, that describing the Socratic in- 
 duction, and called " Perfect Induction," we shall see that it 
 involves to some extent this implication ; not that the gener- 
 alization at the end of our observations expresses any addi- 
 tion to our previous knowledge, but that the enumeration 
 of individual experiences or observations, or perhaps better, 
 the accumulation of them, is such a process. It is quite 
 as apparent that the inference from one or more known 
 facts to a greater number is an addition, or the necessary 
 step to such an addition. But if it be called "induction" 
 it must be regarded as quite of a different nature from the 
 process which we have just mentioned. It nevertheless re- 
 sembles it in the one quality of providing an increment to ex- 
 isting knowledge. And again, the " methods of Induction " 
 which seek by repeated observations and experiments to verify 
 a supposition or conjecture already made, represent means of 
 adding to previous knowledge. But they represent also some j 
 thing more than an inductive inference. They are corapli j 
 cated with direct experience and observation, and with deduc- 1 
 tive principles, assumptions, and inferences, so that they are 
 not simply cases of inductive reasoning, -although they are ihff 
 proper means of adding to our knowledge and widening ouf 
 generalizations. 
 
 But Logic does not immediately treat of increasing out 
 knowledge, or of the material means for applying scientific 
 methods. It has to do with thought or reasoning and with 
 the manner in which one idea is inferred from another, not 
 with the complicated methods of verifying this inference after 
 it is made. Hence in so far as " Induction " is a logical pro- 
 cess in contrast with Deduction, we must confine the term to a 
 certain kind of reasoning, and not extend it to all modes of 
 adding to our previous knowledge. We, therefore, choose the 
 second of the three meanings explained, as the proper one to
 
 300 ELEMENTS OF LOGIC 
 
 represent " Induction " in so far as formal Logic has to deal 
 with it, and which is the only meaning that will enable us to 
 contrast the process with Deduction or consider it exclusively 
 as a process of reasoning. This is very clear in the case of 
 the so-called "Perfect Induction," which we have found to be 
 simply observation and generalization, and not properly rea- 
 soning at all, and which is as much a condition of Deduction 
 as of Induction. 
 
 The third form is a process of verification of conjectures, or 
 probable inferences already made, and although such addi- 
 tional inferences may be connected with these various methods 
 of verification, the methods themselves are not pure inductive 
 inferences, as is admitted by more than one writer on Logic. 
 But to these points we shall return again, when they may be 
 treated more fully. It has been important here only to fix 
 upon the meaning which the term " Induction " is to have 
 when comparing it with deductive reasoning, and to establish 
 the fact that the notion or implication so common with popu- 
 lar writers, and tacitly admitted by logicians themselves, 
 namely, that " Induction " is any process of adding to what 
 we already know at any given time, is not the main conception 
 with which we have to do when considering it as a form of 
 reasoning. It does imply addition ; but it is by way of infer- 
 ence, not by observation or verification. 
 
 Having fixed upon the second of the three meanings as the 
 proper one for the term, so far as formal Logic is concerned 
 with it, we may indicate, before defining the process more 
 carefully, the three methods and their characteristics by a ter- 
 minology to some extent new. The first we shall call Observa- 
 tion and Generalization, the second, Induction, and the third, 
 Verification or Scientific Method. We use the term "Induc- 
 tion," therefore, to describe a process of inferring one truth 
 from another in a manner somewhat different from Deduction, 
 and are now prepared to define it more accurately. 
 
 We have already defined Induction as reasoning from the 
 particular to the universal in contrast with Deduction as rea- 
 soning from the universal, to the particular. It is also fre-
 
 INDUCTIVE REASONING 301 
 
 quently defined as reasoning from effects to causes, from the 
 known to the unknown, from the actual to the possible and 
 probable. From two of these conceptions it is often supposed 
 that the process is the inverse of Deduction, and this is true 
 in some respects, but not in all. For instance, to argue from 
 the known to the unknown, or from the actual to the possi- 
 ble and probable, is not the inverse of Deduction, for if 
 it were, we should be obliged to regard the latter as reasoning 
 from the unknown to the known, or from the possible to the 
 actual, a process which is the very opposite of the real case. 
 Induction is only the inverse of Deduction in certain respects, 
 and these are with reference to the extension of the concep- 
 tions involved. In one we narrow our conceptions as we pro- 
 ceed to the conclusion, and in the other we widen them. In 
 other respects we cannot say that the two processes are the 
 inverse of each other, but only that they are different froia 
 each other. Thus in Deduction we argue from the known to 
 either the known or the unknown ; to the known, if the con- 
 clusion is an actually known fact, but not seen in all its rela- 
 tions to general principles until these relations are enun- 
 ciated ; and to the unknown, if the conclusion happens to be 
 a truth implicitly contained in the premises, but not explicitly 
 realized in the consciousness of the reasoner or the hearer 
 until stated. But it is never a process of reasoning from the 
 unknown to the known. Both Induction and Deduction, 
 therefore, appear to be reasoning from the known to some- 
 thing else. But there are two differences between them. In 
 the first place, the known data are different as syllogistic mat- 
 ter. In Deduction the known is either a universal principle 
 in the abstract, or certain universally known concrete facts 
 which will enable us to affirm the same thing of all individ- 
 ual instances included under the universal. In the second 
 place, it is not strictly true in either of them that we argue to 
 the unknown. In Deduction we argue to what is necessarily 
 included in the nature of the premise, whether known or un- 
 known, explicit or implicit. In Induction we argue to the 
 possible and probable from known facts, not principles, and
 
 302 ELEMENTS OF LOGIC 
 
 so to more universal truths, which may in their turn become 
 deductive data when verified. These facts explain why it is 
 best to define Induction as proceeding from the particular to 
 the general or universal, or from effect to cause. The process 
 gets the last conception of its nature from the frequency with 
 which it is occupied in determining the causes of known 
 phenomena, which can never be ascertained deductively be- 
 cause more than one cause may produce the same known ef- 
 fect, and when the cause is known we can directly know what 
 the effect will be. But as our knowledge begins with matters 
 of fact we usually have to argue to their probable causes by 
 inductive inference or conjecture, no doubt after some ac- 
 cumulated experience and observation, and then verify our 
 inference by additional methods. 
 
 It is sometimes said that Induction is reasoning from par- 
 ticular to particular, and some writers on Logic admit or as- 
 sert that nearly all the reasoning of common life is of this 
 order. It is certainly true that we do reason from one partic- 
 ular instance, or set of instances, but when we examine into the 
 case more closely, the result or conclusion in its real meaning 
 is a universal broader than our original premise and containing 
 both what we had reasoned from as a particular and what we 
 had reasoned to as a particular. This, I think, will be clear, 
 when illustrated, to all who study the process. But in order 
 to include such cases in the definition, Inductive reasoning may 
 be defined as reasoning from what is known in a certain fact, 
 or facts, to the possible or probable truth of the same thing in 
 other facts where it has not been observed or proved, or is rea- 
 soning from actual to necessary connection. 
 
 As an illustration of this inductive reasoning we may take 
 the case of discovering gravitation by Sir Isaac Newton. 
 The story about his having been moved to the discovery by a 
 falling apple is probably legendary, but it may be used as if it 
 were a fact because it is to the purpose and might have been 
 employed by him as an illustration. But supposing his atten- 
 tion to have been arrested by the fall of an apple while reflect- 
 ing on the position of the heavenly bodies, he had before him
 
 INDUCTIVE REASONING 303 
 
 several facts. There is, first, the fixed and suspended position 
 of the apple before it breaks loose from the tree. Then there 
 is its falling to the earth under the influence of its weight. 
 The supposition that it is attracted by the earth is another fact 
 accounting for the weight of the apple. A fourth fact is that 
 the moon, the sun, the planets, and other heavenly bodies are 
 suspended in space, somewhat in the same relation to the earth 
 as the apple which we are considering. But it is not known 
 that any force from the earth is exerted to hold them in their 
 position, or create in them a tendency to fall toward it. But 
 assuming their relation to the earth and the attraction which 
 caused the apple to fall, we may infer the possibility that the 
 same force of attraction extends to the moon, sun, etc., inas- 
 much as we can see no reason in the nature of space to pre- 
 vent this action. As in the case of the moon, for example, the 
 resemblance between it and the apple, in regard to relative po- 
 sitions, was such that it was natural to infer from the attrac- 
 tion exerted upon the apple that the moon was similarly 
 affected, and so the other planets, by reason of their likeness 
 to the moon in qualities concerning the matter at issue. But 
 there was one circumstance in the case that prevented the in- 
 ference from being verified by observation. It was the fact 
 that the moon and the planets, with their satellites, did not fall 
 toward the earth, or that each did not fall to the body toward 
 which it gravitated. In the case of the apple there was no diffi- 
 culty, because as soon as it was released from its support on 
 the tree, the phenomenon of its fall was an observable fact, and 
 attraction was presumed to be the cause. But the moon did 
 not perceptibly move toward the earth, although it was sus- 
 pended in space, and without visible support. Observa- 
 tion, therefore, could do nothing directly in pi-educing or ver- 
 ifying the belief that attraction was exerted upon the moon, 
 but the belief was an inference from resemblances of relation 
 and material qualities to a resemblance in one other quality 
 already known of the apple and the earth. Thus far the in- 
 ference, however, expresses no other degree of certitude than 
 a possibility, and it remains to ascertain how a greater degree
 
 304 ELEMENTS OF LOGIC 
 
 might be given it in connection with the same or the same 
 kind of reasoning. 
 
 There was another known circumstance in connection with 
 the case which may have helped to suggest the inference, and 
 which certainly gives it greater probability. The motion of 
 the planets about the sun, and of the satellites about their re- 
 spective planets in elliptical orbits, was an admitted fact at 
 the time of Newton. Now he knew that such motion involved 
 the existence of a tendency on the part of all moving bodies 
 to keep in a straight line unless drawn from it by some other 
 force than that of their impulse, and also that bodies moving 
 in circular and elliptical orbits tend to fly off from the centre 
 unless held in their place by some external force. As the 
 planets were moving in an elliptical orbit, and the satellites 
 about their central bodies, there must be this tendency to 
 fly off from their centres, and unless they deviated from this 
 line there must be some force to sustain them in their place. 
 Gravitation, therefore, came in as the complementary centrip- 
 etal force to counterbalance the action of centrifugal tenden- 
 cies. The existence of some such influence was presumed in 
 the nature of the case, but that it should be the same gravity 
 that pulled the apple to the earth was not proved by that fact, 
 and hence it remained a probable inference of greater or less 
 degree, according as the nature of the case would make it. 
 The inductive nature of the inference lies precisely in this 
 fact, that it is not necessarily conclusive, but only that the 
 cause inferred is adequate to the effect, and that the circum- 
 stances render it probable that the supposition is true, or that 
 it has more in its favor than any other hypothesis. The mere 
 possibility of some other conditions to the same effect prevents 
 the case from being proved by the circumstances which occa- 
 sioned the inference. But it will increase in probability with 
 the number of incidents consistent with it, or which it aids in 
 making intelligible. In the case before us the probability of 
 gravitation was greatly increased by the mere circumstance 
 that a centripetal force was needed, and the conjecture ex- 
 tending to the moon the agency which caused the apple to fall
 
 INDUCTIVE REASONING 305 
 
 precisely supplied this want. In getting a clear notion of 
 what the inductive inference was, however, the student has 
 only to remember that it consisted in the extension to the va- 
 rious planets of the same force known or supposed to control 
 the movements of the apple, and under circumstances which 
 prevented the action of such a force from being an observed 
 fact. 
 
 Any number of similar illustrations might be chosen. For 
 instance, if we find that two or three gases are compressed 
 into liquids under certain degrees of temperature and press- 
 ure, we might infer the same of other gases not yet so com- 
 pressed. The conjunction of a solar eclipse with the dark of 
 the moon would suggest the inference that the moon was the 
 cause of the phenomenon. The known fact that intervening 
 bodies cast a shadow is the basis of supposing that the same 
 effect will take place when the moon is between the earth and 
 the sun. Our ignorance of the exact position of the moon 
 would make the inference merely conjectural, but a knowl- 
 edge of the fact that its latitude and longitude corresponded 
 exactly with the position of the sun at the place would be 
 equivalent to a demonstration of the inference, or would make 
 it so highly probable that a very peculiar combination of cir- 
 cumstances would be required to weaken it. But it must be 
 noticed that the circumstance rather verifies than instigates 
 the inference, and becomes the basis of predictions of the 
 eclipse. Again, the peculiar change in the shape of sun spots 
 leads to the inference both of the rotundity and the axial revo- 
 lution of the sun. They move across the visible plane of the 
 sun's surface, and appear elliptical or elongated as they ap- 
 proach the edge of the sun. The known fact of such elon- 
 gation in connection with spherical bodies would suggest sphe- 
 ricity in the sun in connection with the elongated character of 
 the spots, and the conjunction of this form in the sun spots 
 with their motion across the sun's surface suggests axial revo- 
 lution according to well-known facts. And again, the frequent 
 conjunction of a certain kind of cloud and rainfalls will lead 
 
 me to suppose that this kind of cloud is a cause, and that 
 20
 
 306 ELEMENTS OF LOGIC 
 
 it will be invariably accompanied with rain. To Benjamin 
 Franklin certain resemblances between lightning and electric- 
 ity led to the inference that they would resemble each other in 
 conduction over a wire and in charging a Leyden jar. The 
 identification of the two, and hence the verification of his sup- 
 position, was the result of an experiment suggested by this 
 inference, and is too well known to be repeated. 
 
 These illustrations suffice to indicate the nature of induc- 
 tive inferences, and it remains to show the syllogistic form 
 which they assume. The premises consist of certain facts 
 and assumptions, and the conclusion, of something wider than 
 the known facts in the premises. In representing them, how- 
 ever, we cannot follow the usual order of the Figures in the 
 syllogism without too much preliminary explanation. We take 
 first, therefore, the form most frequently adopted. 
 
 2d. The Form of the Inductive Syllogism. The usual 
 form of inductive reasoning is that of the third Figure. My 
 own opinion, however, is that it can be stated in all the Fig- 
 ures. But we take first the Third Figure, because it is un- 
 doubtedly the one in which the inductive inference occurs 
 most frequently. Thus : 
 
 Mars, Venus, etc., revolve around the sun. 
 
 Mars, Venus, etc., are planets. 
 
 Therefore the planets revolve around the sun. 
 
 In deductive reasoning we have learned that this would be a 
 case of illicit process of the minor term, because it is undis- 
 tributed in the minor premise, but distributed in the conclu- 
 sion. But we must remember that, although the inference or 
 reasoning, deductively considered, is false, the conclusion may 
 be true as a matter of fact. Only we have no right from the 
 nature of our premises to assert it. This indeterminate nat- 
 ure of the fact leaves it open to conjecture, hypothesis, or in- 
 ductive reasoning to infer that all the planets revolve about 
 the sun, because two or three of them do ; that is, that the 
 other planets being like the known planets in many respects, 
 will prove to be like them in this other characteristic. It is, of
 
 INDUCTIVE REASONING 307 
 
 course, not a necessary inference from the data given, but only 
 a possible or probable one, as the case may be. 
 
 The matter is sometimes put in the following manner in or- 
 der to avoid the appearance of a deductive fallacy : 
 
 A, B, C (magnets), attract iron. 
 A, B, C represent all magnets. 
 Therefore all magnets attract iron. 
 
 Here the inference is undoubtedly wider than the known 
 facts. But having used the phrase " represent all magnets," 
 we distribute the term magnets, so as either to bring the case 
 under the rules of the quantification of the predicate, and, 
 therefore, deductive reasoning, or we create one of those in- 
 tensive propositions with the quantification of one of its terms 
 which modifies the real form of the reasoning, and makes it 
 some other Figure than the apparent one, as has been ex- 
 plained. Besides, it is to assume the conclusion, if we thus 
 express the case in the premises, and so far from having our 
 knowledge increased by the inference, as it should be by In- 
 duction, it remains the same as in the premises in such cases. 
 Hence I do not think this form of expression conducive to 
 the proper representation of the inductive form of inference. 
 We must not assume as known in the premises what we are 
 to infer in the conclusion. Hence the better form for show- 
 ing the nature of the inductive inference is such as the follow- 
 ing, where nothing is assumed in the premises to anticipate or 
 necessitate the conclusion. 
 
 All magnets attract iron. 
 
 All magnets are attracting bodies. 
 
 Therefore all attracting bodies attract iron. 
 
 We may know the premises to express facts and yet they may 
 not include the conclusion drawn from them. But in this in- 
 stance the agreement or connection between the two predi- 
 cates and the same subject awakens the supposition that they, 
 the predicates, are as essentially connected with each other. 
 This is, of course, the thing to be proved. But it is possible
 
 308 ELEMENTS OF LOGIC 
 
 in the nature of the circumstances, and will be probable ac- 
 cording to the extent of our knowledge regarding the nature 
 and action of electricity. It may even reach the stage of 
 demonstration. But the first degree is that of possibility or 
 probability suggested by certain known resemblances. 
 
 In the second Figure the reasoning might be represented 
 as follows : 
 
 Magnets attract iron. 
 
 Loadstones attract iron. 
 
 Therefore loadstones are magnets. 
 
 In deductive reasoning this is a case of illicit middle. But 
 for the same reason as before it is possible that the two 
 subjects agree, although this is not necessarily the case. Their 
 agreement in the matter of attracting iron is merely the con- 
 ception of a single effect which has its cause, and as both ob- 
 jects produce the effect we may reasonably suppose that they 
 are identical in their nature. This is what we express by say- 
 ing that " Loadstones are magnets." 
 
 In this illustration we must not mistake the historical man- 
 ner in which magnets and loadstones were actually identified 
 for the inductive inference we are trying to illustrate. His- 
 torically, " magnets " was only another name for " loadstones," 
 and we came afterward to apply the same name to manufact- 
 ured articles having the same qualities. But we are here 
 supposing a mind made acquainted with the two things inde- 
 pendently of each other, and discovering that both attracted 
 iron in the same way, and hence inferring that they belonged 
 to the same class. In this case, then, we might say either that 
 the loadstone was a magnet, or the magnet a loadstone. Wo 
 should only intend by it that one of them was either possibly 
 or really more comprehensive than the other. 
 
 The difficulty with the second Figure in the inductive syl- 
 logism is that it is usually impossible to connect the two 
 subjects by the copula which signifies their identity in a class 
 relation, which is not always capable of being expressed by 
 language which is adjusted to observations already made.
 
 INDUCTIVE REASONING 309 
 
 There is a reason for this difficulty which will appear in the 
 sequel. It remains to see whether we can state the inductive 
 syllogism in the form of the first Figure. 
 
 Meteors are followed by a train of light. 
 Rapidly moving bodies are like meteors in their motion. 
 Therefore, rapidly moving bodies are like meteors in being 
 followed by a train of light (other things being equal). 
 
 The conclusion in this example is not meant to indicate an ob- 
 served fact, but for that very reason it is better calculated to 
 represent the inductive nature of the inference. It is not a 
 matter of experience that " all rapidly moving bodies " are fol- 
 lowed by a train of light, and yet we are accustomed to ex- 
 plain the absence of this effect by saying that we mean such 
 bodies as have the velocity of rueteors. Our argument is that 
 if they had as swift a motion as meteors they would display 
 the same effects, and the inference that they would do so is 
 inductive, in so far as it supposes true of all kinds of matter 
 what is observed to be true of meteoric matter and bodies. It 
 might not be true of terrestrial bodies, because of their pecu- 
 liar nature and established connection with the earth. But, 
 on the other hand, the effect would be probable in proportion 
 to the knowledge we have of the effect of friction from the air 
 upon rapidly moving matter of whatever kind, and the temper- 
 ature necessary to make it luminous, etc. 
 
 In regard to the form of this syllogism it will be noticed 
 that the middle term is quite peculiar. We could not state 
 that the subject of the minor premise was contained in the 
 class " meteors," because this would have made the argument 
 deductive, while it is our object either to " prove " their in- 
 clusion in that class, or to show that both classes belong to the 
 same general kind of objects on the ground of a common 
 quality observed and a common quality inferred, which shall be 
 as essential to the subject of the conclusion as to that of the ma- 
 jor premise. Hence we must, in the first Figure, state our mid- 
 dle term in such a way as to represent an identity between it and
 
 310 ELEMENTS OF LOGIC 
 
 the minor term which is only partial, and from which we may in- 
 fer a more complete identity inductively. Also we must state it 
 so that it represents properly the extent of our actual knowledge 
 about the two terms, so that the conclusion shall contain more 
 than our actual experience of the facts. In the minor prem- 
 ise, therefore, we cannot say that the subject is the predicate 
 or middle term, but only that it is like it in certain particulars, 
 and then we can infer its identity with the major term so as to 
 complete the conception of the identity between the two classes 
 of objects. Otherwise, as we have remarked, the argument 
 would be purely deductive. It will be interesting to remark 
 that this is the valid inductive inference, the counterpart, and 
 correspondent of the invalid deductive process in Ambiguous 
 Middle. 
 
 It will not be necessary for present purposes to represent 
 an inductive syllogism in the fourth Figure. We have in- 
 tended only to show that so far as Induction is reasoning it 
 must follow the same form as Deduction, and that it differs 
 from Deduction in the character of its conclusion, permitting 
 us to go beyond the premises with our inferences, although 
 affording nothing but a probability of truth instead of a cer- 
 t-iinty or necessity. But there is a peculiar characteristic 
 which is best illustrated in the example representing an induc- 
 tive syllogism of the first Figure, and which is our reason for 
 having given it. We have observed the peculiar form of state- 
 ment necessary to prevent the case from being a deductive 
 syllogism, and in comparing mathematical with ordinary logical 
 or deductive reasoning we remarked that the former was purely 
 quantitative and the latter both quantitative and qnali!atii-<>. 
 We have now still further to observe that inductive reasoning 
 is purely qualitative. Mathematical reasoning we found to be 
 quantitative because it dealt only with quantity, and was not 
 concerned with the qualities which constitute objects, and we 
 chose to call it Traduction because its data could be trans- 
 posed and substituted without any reference to the Figure of 
 the syllogism. Induction, on the other hand, has nothing to 
 do with quantity, but solely with the qualities of things. It
 
 INDUCTIVE REASONING 311 
 
 reasons from quality to quality, from attribute to attribute. 
 Thus in identifying electricity and magnetism we argue from 
 certain resemblances of effect to an identity of cause, although 
 in a modified form in each instance. We do not have any special 
 class or numerical quantity of instances under which the minor 
 premise can be subsumed, and hence we argue from the known 
 qualities in one or more things to the existence of the same 
 quality in other objects. The number of objects specified in the 
 premises is of no importance. Each premise may represent 
 only a single individual, and the data be sufficient for an in- 
 ductive inference of some kind. Indeed, when class terms are 
 employed they have no more value than singular terms, except 
 for strengthening our convictions about the fixity of the con- 
 nection between the subject and the predicate. We do not so 
 much use them for class or general terms as for conceptions 
 representing the presence of certain qualities, and it is from the 
 peculiar conjunction of certain qualities with each other that we 
 infer a similar conjunction where it has not yet been observed. 
 Thus we are not arguing from a class to a sub-class or indi- 
 vidual under it, on the ground of numerical inclusion in the 
 larger, but from a quality or qualities observed to the same 
 things unobserved in other objects, or in a modified form con- 
 cealing their identity. Thus Franklin, in arguing to the iden- 
 tity of lightning and electricity, reasoned from certain known 
 phenomena, the production of sparks by an electric current, to 
 the production of the same by lightning, if the proper condi- 
 tions were satisfied and his flying of the kite was only a veri- 
 fication of the inference. The number of instances in which 
 he had observed electrical phenomena and lightning had noth- 
 ing to do with his reasoning, but only in strengthening his 
 conviction regarding the permanence or uniformity of the re- 
 semblance between the two sets of phenomena. The logical 
 process was then not concerned with the mathematical aspect 
 of his conceptions, but only with their qualitative import ; 
 that is, with the resemblances between phenomena. A veri- 
 fied induction may enlarge the area or extension of general 
 concepts, but it is not done by any logical stress upon the
 
 312 ELEMENTS OF LOGIC 
 
 mathematical import of the conceptions employed. It is done 
 only by observing, or inferring, and then verifying a resem- 
 blance of qualities. After the general conception has been 
 formed, we may use its quantitative import as a basis for as- 
 sured deductive reasoning, but only the qualities of objects 
 and their connections can be the basis of induction, and hence 
 we feel justified in calling it purely qualitative reasoning. 
 
 We must be on our guard, however, about the use of the 
 terms quantitative and qualitative in describing these several 
 kinds of reasoning, because they are used here in a somewhat 
 new and modified sense. Jevons, for instance, and most writers 
 on Logic, speak of " quantitative " reasoning when they mean 
 reasoning with extensive propositions, and " qualitative " rea- 
 soning when they mean reasoning with intensive propositions. 
 The meaning I attach to the terms rather includes this than 
 differs from it essentially, because I have already laid it down 
 as a fact that all extensive propositions have a corresponding 
 intensive import, and intensive propositions an extensive im- 
 port ; so that they may be conceived either quantitatively or 
 qualitatively. But by quantitative reasoning I mean the com- 
 parison of terms on the ground of quantitative identity, or 
 quantitative relations of whole and part, and by qualitative 
 reasoning, the comparison of terms on the ground of a qualita- 
 tive identity, which never permits of additions to form quanti- 
 tative wholes ; and hence quantitative or mathematical reason- 
 ing will deal with conceptions in purely numerical relations ; 
 qualitative or inductive reasoning only with the qualitative re- 
 semblances and connections of phenomena. If, then, we can 
 apply the term Traduction to mathematical reasoning, we may 
 indicate briefly the characteristics of the three forms of infer- 
 ence. Traduction is purely quantitative, the most assured in 
 its certitude and the most free from fallacy. Induction is 
 purely qualitative, the least assured in its conclusions and the 
 least exempt from mistakes of inference. Deduction is both 
 quantitative and qualitative, and so combines the characteris- 
 tics of both Traduction and Induction. It is assured in its 
 conclusions precisely in proportion as the reasoning turns
 
 INDUCTIVE REASONING 313 
 
 upon the quantitative import of its conceptions and proposi- 
 tions, and it is exposed to fallacies precisely in proportion to 
 the lack of coincidence between the group of qualities consti- 
 tuting the individuals in a class of objects and the mathemati- 
 cal import of the class term denominating them. That is, if 
 the same term has a wider mathematical application than the 
 group of qualities it may denote there is great liability to fal- 
 lacies of all kinds in deductive reasoning, and hence when the 
 question turns upon the qualitative import of terms we see 
 the probability, or at least the possibility, of error increas- 
 ing. 
 
 3d. Kinds of Inductive Inference and its Principles. 
 The inductive conclusion does not always take the exact 
 form which has been given it in our illustrations. We had in 
 view such a statement of the process as would give a clear idea 
 of what it was in its essential characteristics, and so to sepa- 
 rate it from that class of conclusions which were the result cf 
 verification or of a surreptitious introduction into the prem- 
 ises of matter which ought first to appear in the conclusion. 
 But there is some confusion about what matters of fact should 
 belong to an inductive inference, and this is generally caused 
 by the mode of representing its syllogistic form. This con- 
 fusion, however, is not one with which logicians have dealt, or 
 about which there has been any controversy. It is rather the 
 unconscious classification under the head of inductive reason- 
 ing of two or more kinds of general conceptions, beliefs, 'or 
 principles, which are suggested by individual facts much nar- 
 rower than the truths they give rise to, that has caused the 
 confusion. They are spoken of as inductive inferences with- 
 out taking into account either the complications and condi- 
 tions that make some of them appear quite different from 
 others, or the various degrees of certitude that attach to them, 
 making some of them quite assured and others quite conjec- 
 tural. 
 
 We spoke of " the surreptitious introduction into 1 the prem- 
 ises of matter which ought first to appear in the conclusion," 
 if the inference were to be truly inductive, as characteristic of
 
 314: ELEMENTS OF LOGIC 
 
 some representations of the process. Hence it is common to 
 illustrate inductive reasoning in the following manner : 
 
 The Earth is molten under Vesuvius. 
 
 Vesuvius fairly represents the interior of the Earth. 
 
 Therefore the interior of the Earth is molten. 
 
 The only appearance of induction in this instance lies in the 
 fact that the major premise is a particular circumstance, and 
 the conclusion another supposed to be suggested by it. It 
 also appears to be a syllogism of the fourth Figure, but when 
 stated in another form it appears to be of the third Figure. 
 Thus, 
 
 The Earth under Vesuvius is molten. 
 
 The Earth under Vesuvius fairly represents the interior of 
 
 the Earth. 
 Therefore the interior of the Earth is molten. 
 
 But even this is not the completely correct form in which 
 the logical relation of the terms is actually thought, because 
 the only conclusion, inductive or deductive, which can be 
 drawn from these premises is, "That which fairly represents 
 the interior of the Earth is molten." The minor premise, as 
 we have indicated before in propositions of this kind, is an in- 
 tensive judgment which requires reduction to express the true 
 relation of the terms in it, and hence it would be, " The inte- 
 rior of the Earth is like that under Vesuvius." But this makes 
 the case one of AAA in the first Figure, and also a deductive 
 syllogism, unless we so state the resemblance as not to indi- 
 cate that it is complete, but only in a matter that supports the 
 probability of a conjunction with the predicate "molten." In 
 stating the likeness in the premises as we do, we practically 
 assume in them all that is found in the conclusion, and so 
 many of tne cases which are taken to represent inductive are 
 simply instances of deductive reasoning. The appearance of 
 induction comes only in the admitted probability of the conclu-
 
 INDUCTIVE REASONING 315 
 
 sion. But this characteristic is due solely to the probability 
 of one or perhaps both of the premises. If the premises, stated 
 in the manner of the above illustration, be proved facts, the 
 conclusion is a necessary one, and not probable at all. The 
 probability of an inductive inference is one which is the result 
 of an inference from established facts, and so is due to going 
 beyond the premises under the stimulus of certain general prin- 
 ciples which may be called the Principles of Induction. These 
 will be considered in a moment. But when we say in the mi- 
 nor premise of an inductive syllogism that certain things " rep- 
 resent " the subject of the major, or " are like " it, without 
 qualification, we assume in it what ought to be an inference in 
 the conclusion from admitted or supposed facts which do not 
 include the inference. The probability of the conclusion must 
 not be borrowed from that of the premise, but from the prin- 
 ciples which justify inferences beyond the actual data given, 
 otherwise we should practically be guilty of a petitio principii. 
 Thus if we were asked to give inductive proof for the prop- 
 osition that " the interior of the Earth is molten," we should 
 state the syllogism as follows : 
 
 The Earth immediately under Vesuvius is molten. 
 
 The interior of the Earth and that immediately under Vesu- 
 vius resemble those conditions in which openings are 
 outlets of what is contained deeper within. 
 
 Therefore the interior of the Earth resembles that immedi- 
 ately under Vesuvius in being molten. 
 
 4 
 
 Here are two known facts with such resemblances as inevi- 
 tably suggest a further resemblance, that of a quality which is 
 observed and known in one of the objects, but unobserved in 
 the other. It is also to be noted in this case that the middle 
 term is peculiar. There is no such identity or inclusion as in 
 Deduction. It is only partial, and this is the true characteris- 
 tic of inductive reasoning. The known facts of the middle 
 term must represent partial identity and resemblances, and the 
 inference must contain the total identity. It is this feature of
 
 316 ELEMENTS OF LOGIC 
 
 the inductive syllogism or reasoning ; namely, the purely qual- 
 itative, and not quantitative, nature of the comparison, that 
 lends support to the supposition that a truly form al expres- 
 sion of the process is impossible. It is certainly very "difficult 
 in many cases. But aside from these questions the important 
 feature of the inductive proof is the statement of certain 
 known and partially agreeing facts. They may wholly agree, 
 or they may only have a partial agreement, but which of the 
 two it really is must be determined by the verification or dis- 
 proof of the inductive inference. All that we are presumed to 
 know in the premises is their partial agreement or actual con- 
 nection, and we are to infer from them under the circum- 
 stances their total agreement or their necessary connection. 
 We are not to assume any data in the premises which should 
 first appear in the conclusion. 
 
 It has been necessary to dwell upon this point in order to 
 distinguish clearly between deductive probability and induc- 
 tive probability, as a condition of fixing upon truly inductive 
 inferences. It has been too common to confuse the proba- 
 bility of a conclusion with its inductive nature, and we need 
 to know exactly the marks which make the inference the one 
 or the other. In deductive reasoning, if one or both of the 
 premises be probable truths, the conclusion will be probable, 
 but it will necessarily follow as an inference from the supposed 
 data. But in inductive reasoning this is not the case. The in- 
 ference is not a necessary one, although it is always a prob- 
 able one. Its probability is independent of the question 
 whether the premises are positively known facts or only likely 
 assumptions. It comes from the peculiar nature of inductive 
 principles in connection with its extension beyond the data of 
 the premises. The difference, therefore, in this respect, be- 
 tween deductive and inductive inferences is simply this : In 
 deduction the probability of the conclusion is purely material ; 
 that is, determined by the material character of the premises 
 in this respect. In induction the probability is both formal 
 and material. But it is not due to that characteristic in the 
 premises. It is due to the manner in which the inference is
 
 INDUCTIVE REASONING 317 
 
 drawn, and this is the introduction of new matter in the con- 
 clusion. Hence inductive inferences are not determined by 
 the mark of probability alone, but by probability phis an incre- 
 ment of knowledge, or conceptions not necessarily involved in the 
 premises. This maxim must be constantly kept in mind or the 
 student will confuse with each other two entirely distinct 
 kinds of reasoning. 
 
 It will be important to examine this liability to confusion 
 because of its bearing upon the general theory of inductive 
 reasoning. There are, at least, three conceptions of the pro- 
 cess, which are not necessarily conflicting, but will be so 
 upon the supposition that each one is exclusive of the other. 
 They are : 
 
 (1.) That the inductive inference is from the particular to 
 the particular, or from one or more individual cases to an- 
 other. 
 
 (2.) That it is from the particular to the universal, or an in- 
 ference which forms a generalization from a particular instance 
 or instances. 
 
 (3.) That it is an inference from actual to necessary connec- 
 tion, or from uniformities of coexistence and sequence to 
 causal relations. 
 
 It can be shown that each of these forms of inference ulti- 
 mately result in generalization, and this is probably the reason 
 that John Stuart Mill identifies inductive reasoning with gen- 
 eralizations from particular cases. But there are some general- 
 izations which only seem to be from particular instances, and 
 hence may be confused with inductive inferences, merely be- 
 cause the data we start with happen to be individual facts and 
 the conclusion includes more than these, and perhaps all sim- 
 ilar facts, when in the meantime a deductive process has been 
 surreptitiously but unconsciously introduced. This is the case 
 when a probable conclusion in a deductive syllogism is based 
 upon an inductive conclusion as one of the premises, or when 
 our generalization simply states explicitly what was implicitly 
 involved in the conceptions arrived at by induction. Let us 
 examine the application of these principles in the example
 
 318 ELEMENTS OF LOGIC 
 
 which Mr. Fowler presents as an illustration of inductive 
 reasoning, as follows : 
 
 (a) I observe that these two bodies (though of unequal 
 weight) reach the bottom of the receiver at the same moment. 
 
 (6) This fact must be due to some cause or combination of 
 causes (Law of Universal Causation). 
 
 (c) The only cause operating in this instance is the action 
 of gravity. 
 
 (d) Therefore the fact that these two bodies reach the 
 bottom of the receiver at the same moment is due to the 
 action of gravity operating alone. 
 
 (e) But whenever the same cause or combination of causes 
 is in operation, and that only, the same effect will invariably 
 follow (Law of Uniformity of Nature). 
 
 (/) Therefore whenever these two bodies, or any other two 
 or more bodies (even though of unequal weight) are subject to 
 the action of gravity alone, they will reach the bottom of the 
 receiver at the same moment, or, in other words, will fall in 
 equal times. 
 
 The first remark here is that we have in this instance two 
 syllogisms, one ending with proposition (d) and the other 
 with proposition (f). The first can be shown to be purely de- 
 ductive, and the second to contain both a deductive and an 
 inductive conclusion. Whether Mr. Fowler intended that 
 proposition (d) should be an inductive inference I cannot say. 
 But it is certain that the only appearance of such a character 
 in the inference is the nature of proposition (a), which merely 
 states a fact or phenomenon, and of proposition (d), which in- 
 cludes the cause of the phenomenon. In the meantime, how- 
 ever, if we observe closely, all the conditions for making the 
 conclusion deductive are introduced into propositions (b) and 
 (e). Propositions (6), (e), and (d) form a complete deductive 
 syllogism, with (c) as the major and (6) as the minor premise ; 
 or the mode of statement can so be altered as to make (6) tbe 
 major and (c) the minor premise. Let us see how this is the 
 case. 
 
 Proposition (6) is not an inductive inference, nor an ob-
 
 INDUCTIVE REASONING 319 
 
 served fact, although it contains new data not found in propo- 
 sition (a). It is a deductive inference from the assumption 
 that all phenomena must have a cause, and the observed fact 
 of the two falling bodies, which is taken to be a phenomenon. 
 Now proposition (6) becomes a premise for another syllogism, 
 and it is assumed again that the action of gravity is the only 
 operating cause in the case, from which it immediately and 
 necessarily follows that this instance of falling bodies has 
 gravity for its cause. The causal influence of gravity, instead 
 of being directly inferred from the observation of a fact, is as- 
 sumed in the premise, and so makes the conclusion deductive. 
 The only proper way to state the inductive inference in 
 such cases is as follows : 
 
 I observe that two bodies of unequal weight fall through 
 
 equal spaces in equal times. 
 I observe that the action of gravity is isolated at the same 
 
 time. 
 Therefore the isolated action of gravity is the cause of the 
 
 phenomenon ; that is, of the falling through equal spaces 
 
 in equal times. 
 
 In this form of statement we merely assert the observed coinci- 
 dence of two facts, the falling of the two bodies and the isola- 
 tion of gravity, and from the coincidence we infer the cause, 
 or from their actual connection we infer their necessary con- 
 nection. This is not involved in the data known, although 
 we may have some foreign reason for making the inference. 
 But to assume that gravity is a cause in the case, and more 
 especially that it is the only cause, is to introduce in the 
 premises what we find in proposition (d], and so make the in- 
 ference deductive. 
 
 In the second syllogism the inferences seem to be some- 
 what different. They are that the same two bodies will al- 
 ivays fall in the same manner under the same conditions, and 
 that all other bodies will do so. Here we have two extensions 
 beyond the observed fact. But we must deal with only one
 
 320 ELEMENTS OF LOGIC 
 
 of them at a time. We take the first, that these two bodies 
 will always fall through equal spaces in equal times, and con- 
 sider it in relation to the premises from which it is drawn. 
 
 As before, we have a deductive instance from the very fact 
 that the universality of the case is assumed in proposition (e). 
 The universality of the connection must be suspected or 
 drawn from individual cases in order to be inductive. Propo- 
 sition (e) really states what should be the inference, and since 
 it states it as a premise the conclusion only deduces it in an- 
 other form, but does not induce it. That the same effect will 
 invariably follow the operation of the same cause, and that 
 alone, is the truth to be inferred and proved. But if it be 
 stated in one of the premises its presence in the conclusion 
 cannot be the result of an inductive inference. To make its 
 universality in time an inductive act of reasoning, we should 
 compare the same effects with each other in several different 
 times, and then from the observed fact that a difference of 
 time had exercised no influence upon the result, infer that this 
 incident alone never would do so, and that with the changes 
 of time, the effect would always be the same. But a grave 
 doubt may exist about the inductive nature of this inference, 
 because to infer that a certain event will always happen under 
 the same conditions, if we have reason to believe that those 
 conditions are its cause, or necessarily connected with it, may 
 well be claimed to be an a priori or deductive inference on the 
 ground that the universality of the causal connection is im- 
 plied in the fact of its existence at all in a particular case and 
 that difference of time is always implied in universality. If 
 this be so, then to infer that an event will ahcays occur under 
 the same circumstances that were once its cause, is only to 
 state explicitly what is implicitly involved in the idea of their 
 being the cause in the first place. 
 
 A complete theory of Induction would require us to dis- 
 cuss this question at length. But in an elementary treatise we 
 cannot be expectel to do so, and we must therefore be con- 
 tent to announce the conclusion we have adopted for such 
 cases. It is that all such generalizations as we have men-
 
 INDUCTIVE REASONING 321 
 
 tioned, namely, the inference that a given event will always 
 happen under the same circumstances, are deductive when 
 they are from a causal connection to its universality, but in- 
 ductive when we reason from coincidences and sequences to 
 causes. To infer a causal connection in a single instance of 
 observed coincidences is of the nature of a universal gen- 
 eralization, not because it is inferred from a single incident, 
 but because the universality of the causal relation is involved 
 in the idea of its necessity, and hence no new conception is in- 
 volved in the generalization. But to infer the causal connec- 
 tion from the mere fact of actual connection is different, and 
 so is inductive. Consequently Mr. Fowler's first inference is 
 deductive because it only states explicitly what is implied in 
 the idea of necessary connection in the observed case. 
 
 These observations show how we are to dispose of the 
 second inference in proposition (f) of Mr. Fowler's illustration. 
 We find there that he has argued from the instance of two 
 bodies of unequal weight to all bodies of equal or unequal 
 weight. Inequality of weight is shown by the observed case 
 not to be a material circumstance, but for all that we know 
 about them the different qualities of other bodies may be 
 precisely the circumstances which will prevent the formation 
 of a true or perfect middle term for the syllogism, and pre- 
 vent the expected effect in the case of experiment. These dif- 
 ferences are not included in the premises, and hence can only 
 be included with a certain degree of probability in the con- 
 clusion, according as the principles of induction determine it. 
 For this reason, therefore, we agree that the inference from 
 the specified case to the idea that all bodies will fall through 
 equal space in equal times, is inductive, because it is a gener- 
 alization involving an increment in the conclusion not contained 
 in the premises. To suppose it true of all bodies is to infer 
 it of other conditions than those in which it was first in- 
 ferred. 
 
 We are now prepared to examine the kinds of inductive in- 
 ference. They may be divided, first, into two species (1) the 
 statical and (2) the dynamical inference. By a statical induc- 
 21
 
 322 ELEMENTS OF LOGIC 
 
 tive inference we mean an inference to the actual existence of 
 certain qualities, coincidences, or sequences, from their obser- 
 vation in given cases. This class we subdivide into two subor- 
 dinate species determined by the different characteristics men- 
 tioned in the definition. There are, first, those which infer re- 
 semblances or differences, or the existence of certain qualities 
 from the partial resemblances and differences of observed 
 cases. The second class consists of those which infer the 
 repeated or continued existence of observed coincidences and 
 sequences. The two classes often coincide, so that it may be 
 difficult to decide whether the inference is to resemblances 
 and differences, or coincidences and sequences. But the one 
 or the other aspect predominates frequently enough to use 
 the difference for distinguishing different forms of the same 
 kind of reasoning. By a dynamical inductive inference we 
 mean an inference from actual to necessary connections. The 
 observed incidents may be of coincidences, or sequences, and 
 the inference in this case must not be to the mere probability 
 of their recurrence, but to the probability that one of the cir- 
 cumstances is the cause of the other. Whenever we infer, 
 therefore, a particular cause from the mere occurrence of a 
 phenomenon, we are reasoning inductively. Reasoning to the 
 existence of resembling qualities may be called attributive in- 
 ductive inference ; to coincidences and sequences, may be 
 called connective inductive inference ; and to causes may be 
 called causal inductive inference. The causal inference, how- 
 ever, may be implied along with the others, and hence its 
 distinction from them may not be absolute. But it is often 
 so prominent a feature of the inference as to serve for a cri- 
 terion of its nature. In general, the terms statical and dy- 
 namical comprehend the two characteristics determining 
 inductive inferences, namely, reasoning to unobserved facts, 
 whether qualitative or connective, and the causal nexus. The 
 following outline summarizes results : 
 
 Inductive j Statical j Attributive = Unobserved identity and differences. 
 Inference ) ' C nnectlve = Recurrent coincidences and sequences. 
 
 ( Dynamical = Causal = Necessary connections.
 
 INDUCTIVE REASONING 323 
 
 Another classification of inductive inferences can be made 
 upon the basis of the kinds of generalization involved in them. 
 Directly or indirectly, primarily or ultimately, all inductive 
 inferences result in a generalization, explicit or implicit, and 
 hence we may be able to distinguish them by this characteris- 
 tic. But it is not all generalizations that are inductive infer- 
 ences. We have seen that those of " perfect induction " are 
 not cases of reasoning at all ; so also with the universalizing of 
 the causal connection. Hence we must assign the characteris- 
 tics which mark the generalizations of inductive inferences. 
 They are (1) inferences from one particular case to another, or 
 from species to species. This results in forming or implying 
 a larger class, comprehending both particulars, and so identi- 
 fying them in a higher genus. Thus if I infer the qualitative 
 identity of magnets and loadstones, of electricity and magnet- 
 ism, of potassium and metals, from resemblances of qualities 
 or of effects produced by them, I am forming a higher genus, 
 or as in the case of potassium and the metals, widening the 
 last class. In chemistry, biology, and zoology this inference is 
 very frequent. It is very common in the classification of ani- 
 mals, and especially in investigations involving the doctrine of 
 evolution. There are, then, (2) inferences from part to whole, 
 from particular to universal, or from the individual to the 
 class, from the species to the genus. This form, however, needs 
 explanation. It might be better to define it as an inference 
 from accident to essence, or from differentia to conferentia, 
 only we must not assume that the basis of the inference ia 
 knoivn to be an accident or a difference. It is merely an ob- 
 served fact that a certain quality is present, and when we in- 
 fer that it will be found to characterize the class we infer that 
 it will be a common quality, and we thus pass from the rela- 
 tion of accident or difference to that of essence or conferentia. 
 Thus if we observe that very frequently ruminants are horned, 
 and infer tbat all ruminants are horned, we do not extend the 
 genus as a class term, but we merely increase the number of 
 qualities to be taken as the essence or conferentia of the class ; 
 we infer that what might be a mere accident is probably an es-
 
 324 ELEMENTS OF LOGIC 
 
 sential property. Or if we infer from the conjunction of a 
 single or several east .winds with rainfall that all east winds 
 must be so characterized, we infer that rainfall is an essential, 
 not an accidental, accompaniment of east winds. We do not 
 widen the genus, but we increase the conferentise. 
 
 It is important to remark the difference between these two 
 classes of generalization. The first class widens the genus, and 
 the second does not. The first does not change the conceived 
 character of the quality or relation, but only its extent, and the 
 second does so change it. A further important difference is, 
 that in the first class the resemblances between the particulars 
 are actually observed, and it is the identity of the cause or 
 quality producing them, when that identity of kind is other- 
 wise not apparent, that constitutes the object of the inference. 
 In the second class, the whole genus is known, not inferred, 
 from the individual or species ; but it is merely a sufficient 
 number of common qualities, or conferentise to determine the 
 group of individuals named. The resemblances, however, are 
 not observed, but inferred. The coincidence of the essentia 
 with a quality which might be an accident is observed, and 
 then its universal coincidence with the essentia inferred, sup- 
 posing that it is a general instead of an accidental characteris- 
 tic. This is why we may be said in some cases to pass from 
 the individual to the universal, or from the species to the 
 genus. It is only when the genus is already known by a 
 definite essentia or conferentia, that this inference can be 
 made. Otherwise we should have to depend upon " perfect 
 induction," or the addition of individual instances in order to 
 form a generalization. It may be that observation forms the 
 first generalization ; but afterward the process may be materi- 
 ally aided by the inference we have just discussed. 
 
 The two generalizations we have just considered are stati- 
 cal. When we come to the third and dynamical we have to 
 be cautious in our judgment of their inductive nature, as 
 we have already observed in our examination of Mr. Fowler's 
 example. But there is, nevertheless, a third class of generali- 
 zations that are inductive. They are (3) inferences from act-
 
 INDUCTIVE REASONING 325 
 
 ual to necessary and universal, or causal connections. The 
 basis of these inferences must always be the coincidences or 
 sequences of two or more phenomena, and the increment sup- 
 plied by the inference must be either the causal nexus or the 
 universality of the coincidence and sequence, or both. But 
 they are very much complicated. We sometimes directly in- 
 fer that a given phenomenon is the cause of another, from the 
 fact of their connection, and then we infer that it will always 
 be so. But in the last instance we may imply that the condi- 
 tions are absolutely the same for all cases, or that if they are, 
 the events will always happen in this way. If this be our 
 assumption the inference is due to a subsumption under the 
 universal law of causation, and becomes deductive, or merely 
 an explicit statement of what is involved in the idea of causa- 
 tion, as already explained. On the other hand, if we infer 
 that a particular event will always be the cause of another, we 
 may do so either on the ground that it is the known cause in 
 this instance, or that its causal relation is the result of a real 
 or an implied inductive inference. In the former case the 
 generalization is like the previous one, a mere enunciation of 
 what is implied by the idea of cause. In the latter case the 
 generalization is a probable one, but might be considered a 
 deductive inference with a probable conclusion, on the ground 
 that the supposition of the particular case being inductively 
 an instance of causality, was the minor premise of a syllogism, 
 in which the universal law of causation was the major premise. 
 The generalization could be inductive only upon the assump- 
 tion that it was only another way of making the transition 
 from observed facts of coincidence or sequence to necessary 
 connection, or that, instead of supposing tacitly or openly the 
 existence of absolutely the same conditions, we inferred that a 
 particular one of them was sufficient in the future to produce 
 the effect. This last, however, is really based on the assump- 
 tion that the event is not known as a cause in the first place, 
 but that we are still dealing with it merely as a coincidence 
 or sequence. 
 
 Let us take an illustration. I observe in one or more in-
 
 326 ELEMENTS OF LOGIC 
 
 stances that the dew falls on clear, cool nights. If I infer 
 that clear, cool nights will always cause a fall of dew, I evi- 
 dently have a generalization with an increment of conception 
 not involved in the data known, and one which, in meaning, 
 is, that all clear, cool nights will produce this effect. But 
 whether the generalization is inductive or not will depend 
 upon the question whether the inference turns upon the uni- 
 versality of the causal nexus in such cases, or the probable 
 similarity of all other clear, cool nights with the one ob- 
 served. If it turn upon the universality of the causal nexus, 
 this may be merely a statement of what is implied in suppos- 
 ing that it was the clearness and coolness of the night that 
 caused the dewfall in the observed cases, so that the inductive 
 inference will lie in the mind's implicit or explicit transition 
 from the fact of coincidence in the first place to the idea of 
 causal connection, and not in universalizing it. On the other 
 hand, if it turn upon the probable similarity in other respects 
 of all other clear and cool nights to those observed, the infer- 
 ence may be inductive, on the ground that it is not from the 
 assumption of identity in all the conditions to the effect, but 
 is an inference to the single efficiency of the clearness and 
 coolness. Of course, if we suppose at the outset that the 
 clearness and coolness of the night were the sole causes of the 
 dew falling, the generalization involving all such nights is 
 only an explicit enunciation of that idea, and not a new con- 
 ception. But usually the generalization in fact means the se- 
 lection of those qualities out of all that are present, as the 
 sole cause of the phenomenon. In this light the inference 
 might be regarded as inductive because it adds to the ob- 
 served possibility of a number of conditions the conception 
 that a particular condition is the sole one and will always be 
 the only one. 
 
 These illustrations show the limitations under which causal 
 generalizations are to be regarded as inductive, and they all 
 practically resolve themselves into the one rule that dynamical 
 inferences are inductive only when the transition is from coin- 
 cidences or sequences to causes. The mere transition from
 
 INDUCTIVE REASONING 327 
 
 individual instances to the universal in cases involving the 
 causal condition is not necessarily inductive, because it is com- 
 plicated with the assumptions about reasoning from cause to ef- 
 fect which is usually spoken of by logicians as deductive. It 
 is, therefore, necessary in such cases to find some other crite- 
 rion than the mere generalization as a mark of what the nat- 
 ure of the inference is, and this is to observe whether the case 
 involves any transition, explicit or implicit, from merely act- 
 ual to necessary connection. 
 
 The possibility of a confusion of deductive with inductive 
 inferences where the causal nexus is involved, is brought out 
 admirably by Mr. Venn in a passage of his " Empirical Logic," 
 which we quote in illustration. 
 
 " A man is bitten by a cobra. We have known or heard of 
 many other such cases, and they all proved fatal. We con- 
 clude with some confidence that XY, the present sufferer, 
 will die ; as A, B, C, the former ones, are all supposed to have 
 died. Here in these few words we have had all the requisite 
 facts put before us, and we also have the inference from them. 
 
 " Now, since we are looking, in the spirit of logicians, at the 
 existence of this belief, which we know will inevitably arise in 
 every normal mind, we proceed to exercise what Hume calls 
 ' our sifting humor,' by beginning to press a series of ques- 
 tions. We start by asking the observer why he believes in 
 the approaching death of XY ? To this question two distinct 
 answers might readily be given. Some would say off hand, 
 ' Because every one who is so bitten always dies ' ; others 
 would say, ' Because A, B, C, whom we know to have been 
 previously bitten, have all died.' When these answers are ex- 
 panded into proper shape they would stand respectively as 
 follows : 
 
 " Deductive. All men who are bitten die : the man XY is 
 bitten : therefore XY will die. 
 
 "Inductive. The men A, B, C, were bitten and died. The 
 man XY has also been bitten. Therefore XY will die." 
 
 This illustration is only to say that the inference " XY will
 
 328 ELEMENTS OF LOGIC 
 
 die " may be either inductive or deductive, according as the 
 premises do or do not include it. In the second form it is 
 assumed to be inductive because there is a transition from 
 one particular case or set of cases to another, so that the 
 known and inferred instance may ultimately be included in a 
 wider generalization than those already observed. But if this 
 individual inference be a deduction from the tacit assumption that 
 " all persons bitten by the cobra mil die," the induction origi- 
 nally was not from the particular known cases to this one, but 
 was made from the known instances to this universal assumption 
 by supposing the connection to be a necessary one in any case. 
 If anyone wishes to object to the cogency of the reasoning in 
 such cases, it must be on the ground that the assumption made 
 in the original instance was not proved, or that the inference 
 is only an inductive one. But whether it is inductive or de- 
 ductive must be determined by first deciding what are the 
 real premises in the case. 
 
 This is sufficient to recommend caution in regard to the nat- 
 ure of inferences in particular cases respecting the causal nexus 
 and generalizations involving this idea. The present treatise 
 does not require a complete exposition of the conditions and 
 limitations of induction in this respect. The student must 
 therefore be satisfied with the ordinary criterion, that causal 
 generalizations and inferences, to be inductive, must be transi- 
 tions from what is conceived as actual coincidences to neces- 
 sary ones. 
 
 The Principles of Induction are the next subject of consid- 
 eration. They are suggested by the demand to know why or 
 upon what ground we make the generalizations just discussed, 
 or inferences extending beyond the known data of the 
 premises. 
 
 The principle supposed to lie at the foundation of inductive 
 reasoning is sometimes announced as the Law of Universal 
 Causation, and sometimes the Law of the Uniformity of Nature. 
 There is an important difference between these two principles 
 which we shall have to notice, but it is not such as to affect 
 seriously the theory of induction. The latter form of state*
 
 INDUCTIVE REASONING 329 
 
 ment is adopted by those who feel doubtful about the exist- 
 ence of any such necessary connection as is implied by the no- 
 tion of " cause." The difference between the two conceptions 
 can be brought out by their definition. 
 
 The Law of Universal Causation means that every event must 
 have a cause, and every cause must have an effect.- This im- 
 plies that the mind is not satisfied with the mere existence or 
 occurrence of a phenomenon, but must know upon what it de- 
 pends, the ground for its being what it is. The principle is 
 usually spoken of as an d priori one, which means that we must 
 assume it in all our thinking and can give no proof of it. As 
 Logic is not concerned with proving that it is a priori, or with 
 the speculations centering about the use of the term, we do 
 not enter into these discussions, but must content ourselves 
 with recognizing that Logic and logical processes take the 
 law for granted. 
 
 The Law of the Uniformity of Nature means that the phe- 
 nomena of the world present a certain uniformity of qualities or 
 occurrence that enables us to conceive it as a system of them hav- 
 ing a definite unity and order. The law differs from that of 
 causation in not representing a necessary assumption or prin- 
 ciple of the mind, but in expressing the observed facts of ex- 
 perience. We must think of events as having a cause, whether 
 they occur regularly or not. But the discovery of their ac- 
 tual uniformity, whether of resemblance or occurrence, is a 
 matter of actual observation and experience. The law is 
 therefore called empirical, in contrast with the term d priori 
 describing the former law. 
 
 In regard to their relation to Induction, the two laws may 
 be said to be complementary of each other, as we shall ex- 
 plain. But it is important to remark that they are not ex- 
 clusively principles regulating the process of induction. They 
 are both equally related to deductive reasoning. Thus the 
 first law, the Law of Universal Causation, is only another 
 statement of the Law of Sufficient Reason, which we have dis- 
 cussed in the chapter on the Laws of Thought. It is true 
 that some writers would limit the Law of Sufficient Reason to
 
 330 ELEMENTS OF LOGIC 
 
 inductive processes, but the present writer sees no ground for 
 excluding the same law from a determining influence on the 
 deductive inference. The main distinction is that the law is 
 applied in a somewhat different way in the two processes. 
 But it is not important in this treatise to discuss the question, 
 nor even to take any positive attitude upon one side or the 
 other of it. It is sufficient to know the general opinion of 
 logicians,*that the law of causation is an accepted principle af- 
 fecting inductive inferences. 
 
 The manner in which the law affects induction is this : 
 There is no reason in the fact itself why we should go- beyond 
 the data of the premises. When we infer something which 
 has not been stated in the premises we must know why we are 
 so disposed to act, or upon what ground we thus go from the 
 known to the unknown, from the particular to the universal, 
 etc. The only ground or reason assignable is that there is, in 
 the circumstances involved, a phenomenon requiring explana- 
 tion, and that the most probable one in the case is to be 
 found in the inference drawn. Thus I observe a very marked 
 resemblance between certain phenomena of sound and light, 
 and I know that in the case of sound this phenomenon is con- 
 nected with its nature as vibrations. This resemblance re- 
 quires explanation because every event must be supposed to 
 have its cause. In this special instance the resemblance of 
 the phenomenon, and its known connections in the other, are 
 taken as facts justifying the inference that undulations in light 
 will account for a phenomenon which is essentially the same 
 as that which is accounted for in the same way when occur- 
 ring in the case of sound. 
 
 But the mere rationality of the demand for an explanation 
 of a resemblance, a difference, a coincidence, or a sequence, is 
 not a proof that the inference or conjecture is the right one. 
 It only explains why the mind seeks to go beyond the known 
 facts, and guarantees only that the phenomena shall have some 
 cause, but does not indicate even with a degree of probability 
 what particular cause it shall be. Another law must come in 
 at this point to determine what inference shall be the probable
 
 INDUCTIVE SEASONING 331 
 
 one. This is the Law of the Uniformity of Nature, or the act- 
 ually observed frequency with which certain phenomena have 
 been connected in our experience. This law decides the 
 greater probability of any given inference both by itself and in 
 relation to other possible inferences at the same time. The 
 occurrence of a resemblance between two bodies, say of elec- 
 trical effects in the case of electricity and magnetism, on a 
 single occasion, might justify under the law of causation the 
 supposition of their possible identity. But owing to an equal 
 possibility that the resemblance was only accidental, the infer- 
 ence would be a weak one, unless we were able to give it some 
 probability from the peculiar nature of the resemblance ; and 
 this characteristic is often a factor in such cases, but has no 
 rules for determining when it does and when it does not give 
 probability. But if we frequently, and under all sorts of vary- 
 ing circumstances, have observed this resemblance between the 
 two phenomena, the probability of their identity is greatly in- 
 creased. The law of causation determines the possibility of 
 the identity, and even the likeliness that the same effect has 
 the same cause ; but the law of the actual uniformity of nature 
 determines the probability that the cause is to be found in 
 these particular bodies. The frequency with which peculiar 
 resemblances and coincidences, differences and sequences oc- 
 cur affords a probability not only of their reoccurrence in the 
 same connection, but also of the fact that this cause will be 
 found in some of the known causal agencies in connection with 
 the objects manifesting the phenomena. 
 
 But in addition to the two general principles thus enunci- 
 ated, there are two subordinate principles or canons which are 
 determinative of the process, and which are, in a measure at 
 least, corollaries of the general law of the uniformity of nature, 
 and are mainly the means of determining the legitimacy of 
 the inductive inference. I shall call them the Principle or 
 Canon of Agreement, and the Principle or Canon of Differ- 
 ence. These are practically the same as the Methods of 
 Agreement and Difference so denominated by logicians gener- 
 ally, only I wish here to distinguish them as organs of discov-
 
 332 ELEMENTS OF LOGIC 
 
 ery, from their frequent and actual application in connection 
 with deductive assumptions, as verifications of inductive infer- 
 ences already made. This latter use of them we shall com- 
 ment upon again. 
 
 The Principle of Agreement can be briefly defined as the 
 principle which determines the probability of a given identity 
 or connection on the ground of the actual frequency of certain 
 resemblances or coincidences under varying conditions. Or 
 more simply still, the agreement of two phenomena in respect 
 of the qualities producing them, or in respect of the connection 
 in which they occur is a criterion of their cause. Thus, to il- 
 lustrate, if A, B, C, and D, E, F, resemble each other in a cer- 
 tain marked phenomenon or quality known to be characteristic 
 of each object, the probability that they are identical in their 
 nature, structure, functions, etc., is proportioned to the de- 
 gree of resemblance, the frequency of the phenomenon's occur- 
 rence, and the assurance we feel about the relation between 
 particular causes and effects. If A, B, C, and D, E, F, repre- 
 sent events associated together, the probability that they are 
 necessarily connected is proportioned to the frequency with 
 which they occur together under varying conditions. " For 
 example, bright prismatic colors are seen on bubbles, on films 
 of tar floating upon water, on thin plates of mica, as also on 
 cracks in glass, or between two pieces of glass pressed to- 
 gether. On examining all such cases they seem to agree in 
 nothing but the presence of a very thin layer or plate, and it 
 appears to make no appreciable difference of what kind of 
 matter, solid, liquid, or gaseous, the plate is made. Hence we 
 conclude that such colors are caused merely by the thinness 
 of the plates, and this conclusion is proved true by the theory 
 of the interference of light. Sir David Brewster beautifully 
 proved in a similar way that the colors seen upon mother-of- 
 pearl are not caused by the nature of the substance, but by the 
 form of the surface. He took impressions of the mother-of- 
 pearl in wax, and found that although the substance was en- 
 tirely different the colors were exactly the same. And it was 
 afterward found that if a plate of metal had a surface marked
 
 INDUCTIVE REASONING 333 
 
 by very fine grooves, it would have iridescent colors like those 
 of mother-of-pearl." 
 
 It should be remarked, respecting these examples, that they 
 also contain illustrations of the Canon of Difference, as all 
 cases of agreement actually do which have any conclusiveness 
 at all. But the important thing to be observed in them is that 
 the inference is first suggested by a number of resemblances 
 where they might not be suspected ; that is, in conjunction 
 with differences that make the resemblances significant. 
 
 The Canon of Difference lays the emphasis upon the differ- 
 ences between groups of phenomena, and so infers a causal 
 connection from the separation of certain phenomena from 
 others. It means that, if two phenomena are constantly iso- 
 lated together from other groups which remain invariable, the 
 separated phenomena may be taken as necessarily connected 
 in the relation of cause and effect ; that which is known to be 
 the antecedent being the cause, and the consequent the effect. 
 An example of this method is given in the case quoted from 
 Mr. Fowler, in which the isolated action of gravity is inferred 
 to explain the equal velocities of bodies falling through 
 equal spaces in equal times. Ordinarily, bodies of different 
 weights, where this difference is considerable say lead and 
 feathers show marked differences of velocity in falling, and it 
 was inferred that gravity did not affect all bodies equally. 
 Here was a case of differences, however, which did not take 
 into account the uniformly accompanying fact of resistance 
 from the air. But the separation of this influence and the con- 
 comitant isolation of gravity and of the falling bodies gives 
 rise to another inference, and the very opposite of the previ- 
 ous one, upon the basis of the difference between this phe- 
 nomenon and others. The inference is conclusive in propor- 
 tion to the certitude that the conditions are as we suppose 
 them. If we had not felt in isolating gravity that there could 
 be no other disturbing factor at the time, our inference would 
 have been liable to the same error as that which had been 
 based upon the actual differences of velocity in falling bodies. 
 
 The Principle of Difference is complementary to that of
 
 334 ELEMENTS OF LOGIG 
 
 Agreement, and is often, if not always, found in connection 
 with it, except that the inference does not turn upon the 
 agreements, but upon the differences. But the differences are 
 noticeable directly in proportion to the invariability of other 
 phenomena without those differences, and hence such circum- 
 stances are a great indirect help to the inference. In the 
 example quoted to illustrate the Principle of Agreement, it 
 was certain differences between the instances that made the 
 inference drawn a possible one. The differences between the 
 natures of the various substances whila.the iridescence was a 
 common quality showed very distinctly that the peculiar kind 
 of substance had nothing to do with the effect, and this nega- 
 tive inference had as important a part in determining the in- 
 ference as the agreement of the various substances in reflecting 
 light. So, in the example quoted from Mr. Fowler, the agree- 
 ment between the retarding influence of the air and the de- 
 creased velocity of the lighter body, and the uniformity of 
 opposite effects without the isolation of gravity, are as neces- 
 sary to the inductive inference as the variation from the usual 
 order when gravity is isolated. But the explicit ground of 
 the inference depends in the one instance upon the agreement, 
 and in the other upon the difference, between the phenomena. 
 It must be remarked regarding the Principle of Difference, 
 that it is most frequently applicable in cases of experiment, 
 and experiment belongs more properly to verification rather 
 than discovery, or to confirming the inference rather than first 
 suggesting it. This is not exclusively the case, however. The 
 same remark is not so true of the Principle of Agreement, 
 which is more frequently the incident of discovery, although 
 it can be made, and often is, the instrument of verification and 
 experiment. It is when the two principles are combined with 
 observation and experiment artificially applied, and with de- 
 ductive principles assumed or proved, that they become means 
 of verification rather than the means of originating the induc- 
 tive inference ; and it is this fact which makes it so difficult, if 
 not impossible, to distinguish as accurately as is desirable be- 
 tween pure inductive reasoning and scientific method or verifi-
 
 INDUCTIVE REASONING 335 
 
 cation. They are regulative of inductive inferences pure and 
 simple, when the reasoning is suggested by them alone, and not 
 complicated with assumed or known facts or principles in- 
 volving an application of the Laws of Identity and Contradic- 
 tion. As principles, however, they are only modifications of 
 the Uniformity of Nature.* 
 
 * For general references on the nature of induction, the following 
 works may be consulted : Mills: Logic, Bk. III., Chaps I. to V. inclu- 
 sive. Hamilton : Lectures on Logic, Lects. XVII. and XXXIII. Venn : 
 Empirical Logic, Chaps. XIV. and XV. Jevons : Principles of Science, 
 Bk. I., Chap. VII. ; Bk. II , Chap. XI. (the whole book should be care- 
 fully read). Fowler ; Inductive Logic, Chap. I. Ueberweg : System of 
 Logic and History of Logical Doctrine, Sections 127, 128, and 129, pp. 
 476-490.
 
 CHAPTER XXin. 
 
 4 
 
 SCIENTIFIC METHOD 
 
 I. THE NATURE OF SCIENTIFIC METHOD.- We have 
 hinted at the distinction between inductive reasoning and 
 scientific method, and perhaps we have implied that the latter 
 is limited to the verification of the former. So far as scien- 
 tific method is identified only with what is usually called "In- 
 ductive Method," this is perhaps the case. But there is a 
 broader use of the conception, which involves also the applica- 
 tion of deductive principles, which, besides being distin- 
 guished from purely inductive reasoning, may appear in com- 
 bination with the inductive, or be a deductive process solely. 
 This broader use of the conception gives rise to two subordi- 
 nate forms of logical procedure. They are usually called : 
 
 (1) The Method of Discovery. 
 
 (2) The Method of Instruction. 
 
 The method of discovery is occupied with the acquisition of 
 knowledge, and is usually identified with Induction or the In- 
 ductive Method, as it is generally called. The method of in- 
 struction is occupied with the communication of knowledge. 
 But the name purports to include more than the mere im- 
 parting of truth once discovered, and hence it is meant to be 
 identified with what is called the Deductive Method, which 
 aims to prove as well as to impart truth. But the two 
 methods are not wholly independent of each other, as the se- 
 quel of the present discussion is intended to show. They 
 may be combined in certain stages, both of discovery and in- 
 struction. This requires us to recognize the divisions only 
 for provisional purposes, and to indicate processes which do 
 not exactly coincide. In certain features the two methods are
 
 SCIENTIFIC METHOD 337 
 
 entirely distinct from each other. It is when they are both 
 implicated in the discovery and establishment of the same 
 truth, that they together form the perfect application of scien- 
 tific method. We shall call them respectively the Inductive 
 and the Deductive Methods, and examine the deductive first 
 in order. 
 
 //. THE DEDUCTIVE HfETHOD.The deductive method 
 has to do with the communication and proof of existing 
 knowledge. The knowledge may exist as a positive acquisi- 
 tion, or be positively known, only by the person imparting it, or 
 proving it, and so involve something of discovery to the per- 
 son receiving it. But this fact does not prevent the mental 
 processes of the receiver from being deductive as well as 
 those of the imparter. The object of the method is to assure 
 the truth of the matter concerned, to give it more than a 
 probable value. The method comprehends three distinct pro- 
 cesses : Definition, Division, and Probation. 
 
 I st. Definition. Definition is the process of making clear 
 all the conceptions entering into the thesis or proposition to 
 be proved. It unfolds their intension. The first thing, of 
 course, is to know what is to be established, and the method 
 of proving it begins with the definition of the elements of the 
 judgment or thesis enunciated for proof. The nature and 
 principles of definition have already been discussed, and need 
 not be repeated here. The process itself does for the concep- 
 tions of a proposition what proof does for the proposition it- 
 self. 
 
 2d. Division. Division is the process of rendering our 
 conceptions more definite in respect to their relations to each 
 other. It is complementary to Definition, and so unfolds the 
 extension of a notion. It is to some extent implied in Defini- 
 tion, but can go beyond that by showing the exhaustive na- 
 ture of our ideas, and the amount of truth involved in the 
 thesis to be proved. This subject has also been duly treated 
 in its proper place, and requires no further mention. 
 
 To illustrate what is meant by Definition and Division, as 
 aids to instruction and proof, suppose it is required to estab-
 
 338 ELEMENTS OF LOGIC 
 
 lish the truth of the proposition, " Governments are useful." 
 I must first define what " Government " is, and what " utility " 
 is. In so doing I indicate the characteristics constituting 
 them, conferential and differential. But this does not tell me 
 how much is involved in the proof of the thesis. I must di- 
 vide the two terms into their species ; " government," for in- 
 stance, into monarchic, oligarchic, and democratic, or other 
 such species as the convenience of the argument may require. 
 This will show more distinctly what is involved in proving the 
 main thesis. So also if "utility "-be divided into species, 
 such as economic, artistic, scientific, etc. The two processes, 
 definition and division, bring out more clearly than one of 
 them alone, the conferentia and differentia, genus and species, 
 or qualitative and quantitative aspects of our conceptions. 
 
 3d. Probation. Probation is the process of proof; or the 
 statement of certain truths which render the thesis a neces- 
 sary conclusion from them. The thesis is the proposition to 
 be proved. The truths which prove it are the known facts and 
 principles which may constitute the premises, and the thesis 
 will be the conclusion. These determining truths may be 
 axioms, postulates, proved propositions, or any truth which 
 the person to whom the probation is made may accept. Their 
 acceptance is the condition of their proving anything. We 
 must observe, therefore, that probation is a material as well as 
 a formal process. It is always syllogistic, but, as Hamilton 
 observes, the converse is not necessarily true ; namely, that 
 all syllogistic reasoning is probation. The object in probation 
 is to prove the material truth of the thesis, and not to con- 
 duct merely a formal process of reasoning. We must there- 
 fore enunciate some facts or principles accepted by the per- 
 son to whom the probation is made, and then bring the 
 thesis under it in such a way that it will be a necessary con- 
 sequence of what is already admitted. In this way we 
 give absolute certainty to our proposition, provided we avoid 
 the usual fallacies in syllogistic reasoning. 
 
 To illustrate, suppose I am required to prove that the sum 
 of the angles in a triangle is equal to two right angles. I
 
 SCIENTIFIC METHOD 339 
 
 must first define the terms involved, namely, " angle," " right 
 angle," "triangle," etc., and then I must announce the funda- 
 mental principles that are either assumed or admitted in the 
 process of proof. There may be such axioms and postulates 
 as that things equal to the same thing are equal to each 
 other, all right angles are equal to each other, and all tri- 
 angles are essentially identical in their properties, etc. If then 
 I can show, either by observation or proof, that the sum of the 
 angles of the triangle is equal to some quantity which is known 
 or admitted immediately to be equal to two right angles, the 
 major premise involved in the axiom mentioned insures the 
 conclusion I am required to draw. The proposition may not 
 carry with it its own evidence, but the truth of some prior 
 proposition or propositions may prove it. The following dia- 
 gram and argument illustrates the whole method. The the- 
 
 Fio. 31. 
 
 sis to be proved is that a + b + c = 2 right angles. It is as- 
 sumed that c and d + e are right angles and equal to each 
 other. Hence c + d + e = 2 right angles. By construction 
 or previous proof, b is taken to be equal to d and a to e, so 
 that a + b = d + e, a right angle ; a + b + c, therefore, equals 
 c + d + e, and by assumption c + d + e 2 right angles. 
 Hence whoever admits the previous steps and conditions must 
 admit that a + b + c 2 right angles. 
 
 The process would be quite similar to this in any other 
 proposition, such as " civil law is necessary to the preserva- 
 tion of order." If this statement did not evince its own 
 truth, we should be obliged to announce some acceptable 
 truth which carried the given conclusion with it. Thus we 
 might show that restrictions of individual liberty were neces- 
 sary to this end, and this assertion might be admitted with- 
 out proof, but on its own transparency. Then, if the defini- 
 tion of civil law showed it to be the only restriction adapted to
 
 340 ELEMENTS OF LOGIC 
 
 such an end, the truth of our proposition must follow as an 
 inference, whether the person to whom the proof was directed 
 could fully see the contents and meaning of the conclusion or 
 not. 
 
 It is not always essential that the deductive method should 
 take an explicit syllogistic form, because one of the premises 
 may be so apparent that the conclusion would follow r upon the 
 enunciation of a single conditioning truth, or it may follow as 
 an immediate inference from the conditioning proposition. 
 Usually, however, a syllogism is implied. In disproof, which 
 involves the same general method, we state or assume the con- 
 tradictory of the thesis, as in probation we state the condition- 
 ing proposition. These serve to measure the truth of a given 
 assertion by its agreement or its contradiction with truths al- 
 ready known or accepted. No guessing, or probabilities, based 
 upon the actual uniformities of nature, enter into the case, ex- 
 cept as they might have originally determined the proposition 
 which is assumed as proof. The proof depends upon the ac- 
 ceptance of the principles, and not upon the method by which 
 the}* were originally suggested. We take up the inductive 
 method next. 
 
 ///. THE INDUCTIVE ME77IOD.-A.iter what has been said 
 about the meaning of the term " induction," it might seem ob- 
 jectionable to use the term for describing a method which is 
 to be sharply distinguished in some of its features from the 
 process of reasoning going by that name. But as long as we 
 are careful to distinguish between induction or inductive rea- 
 soning as a mode of inference, and the process usually called 
 " the inductive method" and comprehending the principles of 
 verification, there will be no serious reason for rejecting the 
 traditional use of the term ; especially as the method is mainly, 
 if not wholly, occupied with the acquisition of knowledge. It 
 is often identified with scientific method, which, as we have 
 explained, is really a combination of inductive and deductive 
 processes in many of the cases assumed to be purely inductive. 
 Because of this fact and general usage, we shall treat, in this 
 section, of " the inductive method " as usually conceived, and
 
 SCIENTIFIC METHOD 341 
 
 without trying to push the distinction between inductive rea- 
 soning and the complications of induction and deduction in the 
 process of verification into any new departures of phraseology. 
 Were we discussing the subject for any other than elementary 
 purposes, we might be tempted b} T innovations. But they are 
 not necessary here, and hence we shall employ the expression to 
 include all the steps involved in the passage from facts to general 
 ideas and their verification. The process so named includes an 
 exposition of the preliminary methods or inductive inferences, 
 the four inductive methods and the verification therein in- 
 volved, along with the frequent application of deductive princi- 
 ples which give greater assurance to truths first suggested by 
 induction. In other words, we use the expression here to in- 
 dicate all the processes involved in the acquisition and verifica- 
 tion of knowledge that is not deduced solely from previous 
 conceptions. These processes may be reduced to two general 
 classes, somewhat merging into each other in actual experi- 
 ence. They may be called Acquisition and Verification, pri- 
 mary and secondary conditions, or subsidiary and confirmatory 
 conditions of new inferential knowledge. Their import and 
 the extent of their application will appear on examination. It 
 must be observed that we are here dealing only with new in- 
 ferential truth, and not with what may be acquired by non- 
 logical processes. In scientific method, however, we shall be 
 permitted to mention the conditions or data upon which such 
 inferences are based. Some of them are included under the 
 subsidiary processes to induction, as they are called by writers 
 generally. 
 
 I st. Acquisition, or Primary Stages of Ascertaining New 
 Facts and Truths. These are so called because they are the 
 first conditions of the truths which turn out to be acquired by 
 a method which is not deductive, or which does not represent 
 in the first stages of its conception anj* derivation from estab- 
 lished knowledge. In the acquisition of knowledge in general, 
 there are of course more processes involved than mere infer- 
 ence or reasoning. Indeed, the first and most important 
 means of attaining such are entirely prior to any form of
 
 342 ELEMENTS OF LOGIC 
 
 reasoning whatever. They are the common processes of Per- 
 ception and Introspection, by which we simply observe facts 
 of experience, and perhaps group them together into classes 
 by comparison and generalization. But these are processes 
 whose analysis and investigation belong more properly to Psy- 
 chology. Yet it is not possible to understand their relation to 
 the properly logical processes of the mind unless we mention 
 them, and indicate the manner in which they either condition 
 the inductive inference, or confirm it when made. We there- 
 fore begin with a brief consideration of these processes as fur- 
 nishing the data of scientific method. 
 
 1. OBSERVATION. Observation is simply the perception of 
 facts. It denotes to watch for, and merely involves what is 
 contained in perception and introspection. The data from 
 which to make inferences must first be known, and observation 
 is the means of ascertaining them. The amount of our knowl- 
 edge is determined by its accuracy. Those who are good ob- 
 servers will discover more data for inductive inferences than 
 those who are not. Observation may be of two kinds, spon- 
 taneous or casual, and voluntary or rational. The former re- 
 quires no distinct effort on the part of the person observing ; 
 the latter requires an act'of attention and is frequently, if not 
 always, employed to verify some expected or anticipated phe- 
 nomenon. The progress of science depends much more upon 
 rational than upon casual observation. 
 
 Experiment is generally mentioned as a subsidiary process 
 to induction. This is often the case in regard to particular 
 kinds of induction, but it is not a condition of all inductions, 
 nor of the process in its earliest forms. It is, however, a very 
 important means to increasing the data for inductive infer- 
 ences. It is not an instrument which can be employed inde- 
 pendently of observation, but is only a help to that process. 
 It is the means both of varying and multiplying observations. 
 It is the artificial production or repetition of natural phe- 
 nomena for the purpose of insuring the accuracy of first ob- 
 servations or of correcting them. It is therefore one of the 
 incidents of observation in its highest forms. " Mr. Mill dis-
 
 SCIENTIFIC METHOD 343 
 
 tinguishes between the two processes by saying that in obser- 
 vation we find our instance in nature ; in experiments we 
 make it by an artificial arrangement of circumstances. When, 
 as in astronomy, we endeavor to ascertain causes by simply 
 watching their effects, we observe ; when, as in our laboratories, 
 we interfere arbitrarily with the causes or circumstances of a 
 phenomenon, we are said to experiment" Experiment may 
 also be used as a means of verification as well as acquisition, 
 and so also can observation. But of these functions we shall 
 speak in the proper place. It is sufficient to remark of the two 
 that they occupy, as subsidiary processes, the position in in- 
 ductive method that is held by definition in deductive method. 
 
 There are two rules for regulating the use of observation as 
 an instrument of scientific discovery, in so far as it is merely 
 a subsidiary process to inductive inferences. They are : 
 
 (a) The observations should be precise. 
 
 (6) They should be relevant or material. 
 
 The first rule enjoins accuracy, and the second the disposition 
 to seek and see what is essential and important. 
 
 2. CLASSIFICATION. This process does the same for induc- 
 tive methods that Division does for the deductive, and it is 
 also the reverse of Division. It is the process of reducing 
 phenomena to systematic groups on the ground of their re- 
 semblances and invariable connections. It is the means of 
 keeping together all relevant and material circumstances in 
 connection with objects and phenomena, and hence is of great 
 help in determining the extent of the inferences we draw on 
 particular occasions, as well as assigning the limits within 
 which they can be drawn. It is the reduction of phenomena to 
 their classes or genera, as Division reduces genera to species. 
 But the process need not be dwelt upon at length in this 
 treatise. We turn to the third and more important process. 
 
 3. HYPOTHESIS, OB THE FORMING OF HYPOTHESES. An hypo- 
 thesis may be briefly defined as a supposition. It is an infer- 
 ence from certain facts to others, or to the cause of those 
 which are matters of observation. This account of it identi- 
 fies it with the inductive inference, and this is our intention.
 
 344 ELEMENTS OF LOGIC 
 
 Some would distinguish it from "induction." But the dis- 
 tinction, when examined, only turns out to be one between a 
 supposition expressing a mere possibility and one expressing 
 a probability of such a degree as to set aside the legitimacy of 
 other suppositions, at least for the time. An hypothesis simply 
 supposes, in its primary stage, a possible fact or cause to ex- 
 plain phenomena, and may turn out to be tenable or not, ac- 
 cording as further observation may determine. But an "in- 
 duction " is supposed to represent a firmer mental allegiance 
 than hypothesis, and so something more approaching to valid- 
 ity. But we insist that in their nature they are the same, and 
 that at most they can differ only in degree. We, of course, use 
 the term " induction " in this comparison as denoting a par- 
 ticular kind of inference suggested by observed facts, and in- 
 cluding more than they contain. Hypothesis we use as a con- 
 venient term for representing this function, of whatever degree, 
 in the application of scientific method, and so indicating the 
 mental act which must be preliminary to all verification, and 
 intermediate between the observation of facts and the proof 
 of whatever guesses, surmises, or probable inferences we may 
 make. 
 
 The term is sometimes compared with the term theory, and 
 a distinction drawn between them. An hypothesis is called a 
 supposition, meaning that it represents a mere possibility, and 
 a theory is called a verified hypothesis. But this distinction 
 is only an arbitrary one, except as it implies the difference 
 between the lowest and the higher degrees of probability in 
 various inferences. The fact is, also, that the terms are often 
 used interchangeably. We speak indifferently of the "Dar- 
 winian hypothesis," and the " Darwinian theory ; " the " nebu- 
 lar hypothesis," and the " nebular theory." Also we say " un- 
 dulatory theory " of light and sound, but never " hypothesis," 
 although the conception is precisely that of a conjecture or 
 supposition awaiting satisfactory verification. On the other 
 hand, the term theory is sometimes used to comprehend a 
 body of ascertained truth, or a number of related ideas 
 brought together to represent a system of truths. Thus we
 
 SCIENTIFIC METHOD 345 
 
 say "the theory of the universe," "the theory of the solar 
 eclipse," "the theory of heat," "the theory of equations." 
 It is this conception which has given rise to the notion that 
 a theory is a verified hypothesis. But their actual inter- 
 changeability in many crucial and important instances, and 
 the fact that they both represent an inductive process hav- 
 ing indistinguishable degrees of probability, are sufficiently 
 cogent arguments for making them identical in all essential 
 features, and so using one of them for describing the whole 
 process incident to the attainment of new conceptions which 
 require verification. An hypothesis, then, we regard as a sup- 
 position or inference from given data to their cause, their 
 principle of unity, or their identity. Although the term is 
 usually employed to denote a very low degree of probabil- 
 ity in the supposition, this does not mate the act of form- 
 ing it different in kind from more assured inductive infer- 
 ences, and so we used it merely to denote the stage of thought 
 immediately antecedent to verification. 
 
 To illustrate : When Franklin observed certain resemblances 
 between lightning and electricity, his supposition that they 
 were identical was an hypothesis. His experiment with the 
 kite was of the nature of a verification. The supposition that 
 the sun's heat is supplied from the falling of meteors into the 
 sun, is an hypothesis. So also is the vulcanic theory of the 
 molten character of the earth's centre. Some such inference 
 or supposition from given facts must first be made before any 
 steps can be taken to give it greater probability or positive 
 proof. In some cases, of course, the circumstances give so 
 great a probability to the first inference that it scarcely re- 
 quires verification. But the inference is nevertheless an hy- 
 pothesis, if we are to take the term in its broadest sense, to 
 denote what is less than a positively ascertained certainty. 
 
 The formation of hypotheses must be subject to certain con- 
 ditions affecting the right to make them. It is important to 
 observe that there can be no absolute rules regulating their 
 legitimacy, because nothing can determine their positive va- 
 lidity or invalidity except the process of verification. Hence
 
 346 ELEMENTS OF LOGIC 
 
 there must be a wide range of liberty in forming them. 
 Nevertheless, there are certain proximate rules regulating 
 their formation and determining the degree of presumption 
 in their favor, which ought to be observed by everyone enter- 
 ing into scientific study. Some of the more important of 
 them may be briefly discussed. 
 
 (a) An hypothesis should not be inconsistent with known facts 
 and causes, but should be required by new phenomena that are 
 not explained by existing suppositions. 
 
 We have no right to introduce any new hypothesis when the 
 facts, forces, or causes already known are admittedly capable 
 of explaining the phenomena under consideration. If there 
 be anything new and unaccounted for, there can be no objec- 
 tion to new hypotheses ; or if the known causes account for 
 only a part of the effect, the new supposition is admissible. 
 But we must in all such cases be assured that the old suppo- 
 sitions are not adequate to the effect, and that there are new 
 phenomena to be explained. Thus, for example, if the law of 
 supply and demand be a fully accepted one as regulative of 
 prices, we should be careful about introducing a^y other in- 
 fluence as determinative of them. Or, if the material re- 
 sources of a country are known to be especially rich and fer- 
 tile, it will not be legitimate to assume that its prosperity is 
 due to government interference with industrial action, until we 
 have shown phenomena which cannot be accounted for by any 
 other means. For example, we often see protection appealed 
 to as the cause of industrial wealth where the impression con- 
 veyed is that this influence is the sole cause of it. But al- 
 though it may be a factor, as long as other causes are known 
 to be active it is illegitimate to introduce protection as the 
 only supposition required to account for the effect. We must 
 first be assured of an unexplained increment. Or, to give an' 
 other illustration, if it be known that petrifactions actually 
 occur, and that animal and vegetable life may leave traces of 
 their existence in deposits of sand and rock during the pres- 
 ent period of the earth's development, it is unnecessary to 
 suppose the existence of some " materia pinguis," fatty mat-
 
 SCIENTIFIC METHOD 347 
 
 ter, or " lapidifying juice " to account for the traces of fossils 
 which do not represent present forms of organic life. It is 
 easier to assume that nature is uniform where precisely the 
 same kinds of effects are discernible, and in this case the 
 known fact of death or the extinction of life, in many instances, 
 has only to be added to the suppositive fossil deposits in past 
 ages in order to account for the difference of genera and species 
 which is observed. The instance only proves that we must 
 first exhaust the explanatory power of known causes before 
 resorting to new ones. 
 
 There are several rules which may be regarded as mere cor- 
 ollaries of the one just discussed, or at least as closely related 
 to it. Such as that an hypothesis should be as simple and 
 as free from complexity as possible ; or, that it should not be 
 arbitrary, but should be relevant to the phenomenon and ac- 
 cording to actual experience. An illustration of unnecessary 
 complexity is the persistence of some astronomers in the theory 
 of epicycles and eccentrics after the simpler hypothesis of Co- 
 pernicus had been proposed. An example of arbitrary as- 
 sumptions is that of the Italian philosopher, mentioned by 
 Fowler, who sought to reconcile the Aristotelian and Platonic 
 theory of the rotundity of heavenly bodies, with the supposi- 
 tion of Galileo, proved by observations of the telescope, that 
 the moon was of a rough surface, by imagining that the hollow 
 parts were filled with transparent crystal, which would permit 
 the same appearances of light and shadow as those we observe, 
 and would observe without this substance. We may reconcile 
 facts by a new hypothesis, but we should not undertake to 
 reconcile theories or suppositions when the simpler of the two 
 may explain the facts or cover all that is known of them. The 
 difficulty, however, in all such cases is to distinguish between 
 the facts and the hypotheses, because suppositions which are 
 believed for a long time, and which rendered a large body of 
 phenomena intelligible, come to be considered as facts, and 
 so give rise to attempts to reconcile them with new theo- 
 ries apparently in conflict with them. What is needed is a 
 criterion to distinguish between facts and hypotheses, or
 
 348 ELEMENTS OF LOGIC 
 
 verified and unverified theories. We turn next to the second 
 rule. 
 
 (6) An hypothesis should permit the application of deductive 
 reasoning, or of the inference to consequences which are capable 
 of comparison with the results of observation. 
 
 This is the first of Jevons's rules, and seems to have been 
 suggested to him by the observations of Hobbes and Boyle. 
 It differs from the first rule we have formulated only in the 
 requirement that the hypothesis be capable of deductive in- 
 ferences consistent with the facts of observation, and so is re- 
 garded by Jevons as merely another form of the rule which 
 he regards as the sole test of the legitimacy of the hypothe- 
 sis, namely, conformity with observed facts. The simplicity of 
 this last mode of statement, however, is so great that illegiti- 
 mate hypotheses might appear admissible for lack of the defi- 
 niteness which is necessary to make the rule useful. 
 
 The importance of this rule is, that an hypothesis will not 
 be capable of verification unless deductive inferences can be 
 drawn from it, and the results compared with experience and 
 observation. The theory of the transparent crystal filling the 
 hollow spaces of the moon's surface, which was mentioned 
 above, is of this nature. Perhaps the old theory of Phlogis- 
 ton is of the same kind, since no deductions could be made 
 from it. Jevons states the case in the following manner : 
 "We can only infer what would happen under supposed con- 
 ditions by applying the knowledge of nature we possess to 
 those conditions. Hence, as Boscovitch truly said, we are to 
 understand by hypotheses ' not fictions altogether arbitrary, 
 but suppositions comformable to experience or analogy.' It 
 follows that every hypothesis worthy of consideration must 
 suggest some likeness, analogy, or common law, acting in two 
 or more things. If in order to explain certain facts a, a', a", 
 etc., we invent a cause A, then we must in some degree ap- 
 peal to experience as to the mode in which A will act. As the 
 laws of nature are not known to the mind intuitively, we 
 must point out some other cause, B, which supplies the requi- 
 site notions, and all we do is to invent a fourth term to an
 
 SCIENTIFIC METHOD 349 
 
 analogy. As B is to its effects b, b', b", etc., so is A to its 
 effects a, a', a", etc. When we attempt to explain the passage 
 of light and heat radiations through space unoccupied by 
 matter, we imagine the existence of the so-called ether. But 
 if this ether were wholly different from anything else known 
 to us, we should in vain try to reason about it. We must ap- 
 ply to it at least the laws of motion, that is, we must so far 
 liken it to matter. And as, when applying those laws to the 
 elastic medium air, we are able to infer the phenomena of 
 sound, so by arguing in a similar manner concerning ether, we 
 are able to infer the existence of light phenomena correspond- 
 ing to what do occur. All that we do is to take an elastic 
 substance, increase its elasticity immensely, and denude it of 
 gravity and some other properties of matter, but we must re- 
 tain sufficient likeness to matter to allow of deductive calcula- 
 tions." 
 
 These two main rules are sufficient for all practical pur- 
 poses. Most other rules are either corollaries of the second, 
 or combinations of this and the corollaries of the first rule. 
 Hence there is no need of multiplying them unnecessarily. 
 
 It is important to remark, however, that conformity to 
 these rules does not verify or prove an hypothesis. It only 
 shows their right to be entertained as representing a possi- 
 bility or probability. They are legitimately made when they 
 explain a given number of facts, but may be set aside by a 
 better one which supersedes the place of the old one on ac- 
 count of superior simplicity and comprehension, or superior 
 deductive capacity in connection with observed facts. 
 
 Hypotheses are sometimes advanced for the purpose of de- 
 ductions which shall be a reductio ad absurdum. These cor- 
 respond to the negative proof of deductive methods, and as- 
 sume a positive hypothesis already in mind, and in this way 
 conform to the rules we have laid down. 
 
 2d. Verification. Verification is the process of testing 
 hypotheses and inferences to see whether they have been cor- 
 rect or not. After a supposition has been made, or an in- 
 ference drawn that something is possibly or probably true,
 
 350 ELEMENTS OF LOGIC 
 
 the mind naturally looks for other evidence to substantiate 
 them. This evidence may come in the repeated occurrence 
 of the original phenomenon, showing that it is probably not a 
 mere accident of circumstances, or it may come in its being a 
 necessary deduction from some known fact or law, but was 
 not perceived until the event was better known. The verifi- 
 cation, then, is intended to give greater assurance than the 
 hypothesis can be supposed to have on the first occasion of 
 its suggestion to the mind. When Newton first thought of 
 the law of universal gravitation, he was not content with its 
 power to explain the single phenomenon which suggested it, 
 but he saw that certain other facts must follow from it, or 
 must inevitably be associated with it. He therefore set about 
 a mathematical calculation to see if the result coincided with 
 what ought to occur in the case, and seeing that it did not, he 
 gave up his theory until new data, some ten years later, were 
 discovered regarding the true distance of the moon from the 
 earth. He then resumed his calculations, and found the re- 
 sult to coincide with his hypothesis, and this was regarded as 
 a verification of it. And so, when the hypothesis that all 
 gases might be compressed into liquids or solids was ad- 
 vanced, it was at least a partial verification of it to have suc- 
 ceeded in compressing hydrogen into a liquid under a low 
 temperature and high pressure. 
 
 The process of Aerification may be of two kinds. The first 
 is that which increases the probability of an hypothesis or in- 
 ference by the four inductive methods ; the second is that which 
 establishes the certitude of it by deductive or by observational 
 principles. The one insensibly passes into the other, as we 
 shall see when considering certain peculiarities of the method 
 of difference ; or the two may be combined in such a way that 
 no absolute line of distinction can be maintained, and we can 
 only rely upon extreme instances of their application for mak- 
 ing the distinction at all. The manner in which the two 
 methods may be connected, consciously or unconsciously, will 
 be remarked in its place. We must take the several forms of 
 verification according to their simplicity, and not according to
 
 SCIENTIFIC METHOD 351 
 
 the degree of certitude supplied by them. These we reduce to 
 four : Observation, Experiment, Inductive Methods, and Deduc- 
 tion, and shall treat them in that order. 
 
 1. OBSERVATION. Observation in verification is the same 
 process that it is in acquisition, only it is applied with some- 
 what different presumptions. When an inference or hypothe- 
 sis has once been made, observation may come in to ascertain 
 whether it is correct or not, and according as the expectation 
 is realized or not, will the verification be. Thus, when the 
 Copernican theory of planetary motion around the sun was 
 proposed and the explanation of the phases of the moon was 
 accepted, it was argued that Mercury and Venus ought to ex- 
 hibit similar phases. This was admitted by the advocates of 
 the theory, and when Galileo turned his telescope upon them 
 the inference was verified by the discovery of these phases. 
 Here the observation of a fact was the condition of proving 
 the hypothesis. Whenever the inference concerns the exist- 
 ence of some physical fact beyond the control or production 
 of experiment, observation is the only resource for verification, 
 and the assumption made is, that the realization of expectation 
 cannot be due to chance. The effect of finding the inference 
 turn out true is like that of seeing a prediction realized. It is 
 a proof of the assumptions on which it is made. If I have in- 
 ferred from observations in a particular region that ague is 
 due to miasma from swampy soil, it will be something of a 
 verification of my hypothesis to observe that the same disease is 
 associated with like conditions in other places, and my certi- 
 tude regarding it will depend upon the number of my subse- 
 quent observations, and the varied circumstances under which 
 the disease occurs. 
 
 The certainty which verification by observation gives varies 
 with the nature of the conditions under which it is made. If 
 we are assured of the simple and isolated nature of the con- 
 ditions, it may amount to positive proof. But if the combina- 
 tion of the conditions is great, the observations may only in- 
 crease the probability of our hypothesis, and this varies 
 between the lowest and the highest degree, which last in most
 
 352 ELEMENTS OF LOGIC 
 
 cases cannot be distinguished from demonstration or certainty, 
 so far as the feelings of the mind are concerned. 
 
 2. EXPERIMENT. As already remarked, experiment is only 
 a modified method of observation. But in the process of veri- 
 fication it is much more valuable than observation. It is the 
 artificial reproduction of phenomena under conditions that 
 isolate them more successfully than we can expect to find 
 them, or at least be assured of, in nature. Besides, it enables 
 us to vary the circumstances under which the effect may 
 occur. Such was the effect of Sir David Brewster's experi- 
 ment showing that iridescence was due to the form and not 
 the nature of a substance. By taking a wax impression of 
 mother-of-pearl, as before mentioned, he produced the effect 
 where there could be no condition affecting the result but the 
 form of the surface, and hence whatever doubt might have ex- 
 isted in regard to the observation of natural instances was 
 here removed by a decisive experiment. In observing the oc- 
 currence of phenomena in nature there may be so many con- 
 ditions associated with their production that a doubt may 
 often exist regarding the conclusiveness of the verification by 
 observation, because the effect may be due to other than the 
 inferred cause which has not yet been separated from the 
 others. But experiment eliminates many possible conditions 
 of the phenomenon, and so gives greater assurance to our in- 
 ferences. This certitude may vary from all degrees of prob- 
 ability to positive certitude, the degree depending wholly 
 upon the amount of isolation effected in the conditions of the 
 phenomenon. It is the great means of verifying conjectures 
 in the physical sciences, and has more force than any other 
 method. 
 
 It should be remarked, however, that both observation and 
 experiment, or modes of verification, are very likely to be as- 
 sociated either with the inductive methods of Agreement and 
 Difference, or with inductive principles, as verifying instru- 
 ments. Indeed, it may be said that observation can hardly be 
 separated from them. But the two processes will verify an 
 hypothesis in proportion to the extent of their connection
 
 SCIENTIFIC METHOD 353 
 
 with the above-mentioned methods and principles. If obser- 
 vation enlarges the number of instances illustrating the 
 method of agreement, it increases the probability of the infer- 
 ence, and if it detects a case or several cases of difference, the 
 probability may amount to a proof. Experiment will do the 
 same. It will enlarge the area of application for the method 
 of agreement, and more easily effects an application of the 
 method of difference, so that its verifying power is greater 
 than that of observation. And, of course, if any deductive as- 
 sumptions, or known facts and principles, are illustrated by 
 the results of the two processes, the verification becomes a de- 
 monstrated certainty. Usually some deductive assumption is 
 made, whether of a probable or an assured truth, before ob- 
 servation and experiment are applied. Hence we must always 
 calculate whether our assurance or verification is due more to 
 the methods of observation and experiment than to the realiza- 
 tion of assumptions already made under the inspiration of 
 agreement and difference or deductive truths. 
 
 3. INDUCTIVE METHODS. We have already explained the 
 methods of agreement and difference as principles of induction, 
 or grounds of the inductive inference and hypothesis. It re- 
 mains to show that the same principles may be used for veri- 
 fication, in that they may be the means or conditions of 
 increasing the probability of our inductive reasoning. They 
 renew the inference under circumstances which increase its co- 
 gency and probability. They are, however, quite as frequently 
 the source of the inductive inference or hypothesis as the 
 means of verification. They act as verifying agencies, when 
 we can assume that the re-occurrence of a phenomenon in 
 accordance with the conditions involved in these methods ful- 
 fils the expectation expressed by our hypothesis, and so the 
 original inference is confirmed and strengthened by the mul- 
 tiplication of the instances involving the effect. The verifica- 
 tion consists more in the number or quantity than in the 
 nature or quality of the instances, because the fact of fre- 
 quency is a better indication of a law of nature. Although we 
 speak a great deal of the uniformity of nature, there is as much 
 23
 
 354: ELEMENTS OF LOGIC 
 
 variety at the same time, and it is this variety that acts as a 
 disturbing factor to our expectations and calculations. Be- 
 sides, this variety and diversity indicate a constant change of 
 causal agencies, so that we may never be assured that an event 
 will have the same connections a second time. But if these 
 connections are repeated under a variety of circumstances and 
 other changes, the supposition suggested by a peculiar inci- 
 dent will be confirmed because of an approximation thus made 
 to a law of nature. This is the Method of Agreement. But 
 if with the isolation of a particular cause we discover a simul- 
 taneous isolation of a given phenomenon, we may infer their 
 causal connection. But the verification of this inference by 
 the Method of Difference, as it was suggested by the same 
 method, may come in one of two ways yet to be noticed. We 
 take up the methods separately. 
 
 (a) Method of Agreement. This has been sufficiently de- 
 fined. To understand its application as a source of verifica- 
 tion, we have only to recall the illustration of iridescence, 
 where the common fact was this phenomenon in connection 
 with a variety of substances. By increasing this variety of 
 conditions and retaining the common fact of iridescence, I 
 increase the probability that the cause, which was supposed 
 after a few observations to be the form of surface on the 
 bodies, is the true one. The highest degree of probability in 
 this case will be reached where the observations pass over to 
 the method of difference, which we shall notice presently. 
 
 Again, I find that after taking a particular kind of food I 
 am ill. But I have taken it along with other kinds at the 
 same time, and my conjecture that the effect may be due to 
 the particular kind supposed will be very weak, until I have 
 repeated the circumstances frequently enough to confirm my 
 belief in the supposed cause. This is greatly strengthened by 
 changes of general condition climate, temperature, air, and 
 food which show that this presence or absence is not materi- 
 al to the result. Here again is an instance where the hypoth- 
 esis was first suggested by the principle of agreement, and 
 then verified or confirmed by the same, on an extension of the
 
 SCIENTIFIC METHOD 355 
 
 instances to which it was applicable ; and it is also noticeable 
 that the verification increases in probability or cogency with 
 the approximation of the case to one involving the method of 
 difference. This approximation to the method of difference 
 consists in the constant elimination of conditions which might 
 possibly be the cause of the phenomenon in question ; that is, 
 the elimination of what appears by this separation not to be 
 the cause, until the phenomenon and its causes are left in 
 complete isolation, when the inference becomes distinctly veri- 
 fied by the fact. 
 
 (&) Method of Difference. Mr. Mill states the Canon of 
 Difference in the following language : " If an instance in which 
 the phenomenon under investigation occurs, and an instance 
 in which it does not occur, have every circumstance in com- 
 mon save one, that one occurring only in the former ; the cir- 
 cumstance in which alone the two instances differ is the effect 
 or the cause, or an indispensable part of the cause, of the 
 phenomenon." 
 
 A clear conception of it may be afforded by comparing it 
 with the method of agreement. In the method of agreement, 
 everything may vary except the phenomena in question ; in the 
 method of difference, nothing may vary ; that is, everything 
 may be the same, except the phenomena in question. In the 
 latter case the probability of the inference is always regarded 
 as greater than in the former. The reason for this is that with 
 the method of agreement it is assumed that several causes may 
 produce the same effect, and hence the probability that any 
 one of them is the true one is diminished. Besides, we are 
 arguing from effect to cause. On the other hand, with the 
 method of difference, when everything is common except the 
 phenomena in question, we have instances in which causes are 
 present and yet the effect to be accounted for does not follow, 
 but something else connected with another isolated circum- 
 stance. If these common qualities were causes of the phe- 
 nomenon, it must accompany them ; but as it does not accom- 
 pany them they cannot be its causes. Hence, since the common 
 qualities are not the causes, we have no other choice but to
 
 356 ELEMENTS OF LOOIO 
 
 accept the antecedent which has been isolated with the phe- 
 nomenon as its cause. The probability is here proportioned 
 to the assurance we feel that the phenomena have really been 
 isolated. Besides, a part of the argument is from cause to 
 effect, and deductive. That is deductive which proves what is 
 not the cause, and the remainder, even if regarded as induc- 
 tive, has its assurance based upon the probabilities of complete 
 isolation in the phenomena concerned. 
 
 A passage from Jevons will furnish illustrations of the 
 method of difference. " Thus we can clearly prove that friction 
 is one cause of heat, because when two sticks are rubbed to- 
 gether they become heated ; when not rubbed together they 
 do not become heated. Sir Humphry Davy showed that even 
 two pieces of ice when rubbed together in a vacuum produce 
 heat, as shown by their melting, and thus completely demon- 
 strate that the friction is the source and cause of the heat. AVe 
 prove that air is the cause of sound being communicated to 
 our ears by striking a bell in the receiver of an air-pump, as 
 Hawksbee first did in 1715, and then observing when the re- 
 ceiver is full of air we hear the bell ; when it contains little or 
 no air we do not hear the bell We learn that sodium, or any 
 of its compounds, produces a spectrum having a bright-yellow 
 double line, by noticing that there is no such line in the spec- 
 trum of light when sodium is not present, but that if the small- 
 est quantity of sodium be thrown into the flame or other source 
 of light, the bright yellow line instantly appears. Oxygen is 
 the cause of respiration and life, because if an animal be put 
 into a jar full of atmospheric air, from which the oxygen has 
 been withdrawn, it soon becomes suffocated." 
 
 These are instances where the method of difference is applied, 
 and it will be apparent on examination that the inference 
 turns upon the elimination of the two phenomena with each 
 other while all else remains the same. But now it requires to 
 be shown how the method is one of verification as well as of 
 suggesting the inductive inference. The inductive inference 
 is from the fact to its cause ; the verification is an increase 
 of its probability or its proof. How does the method of dif-
 
 SCIENTIFIC METHOD 357 
 
 ference effect this at the same time that it is the source of 
 the inference ? 
 
 In answer to this question it is proper to remark first, that 
 some logicians, notably Mr. Venn, regard the method of differ- 
 ence as properly one of verification, and not of suggestion. 
 But this mere expression of opinion is not a proof of it, nor 
 have adequate reasons been given for it. The first inci- 
 dent of its verifying power, however, would be the repetition 
 of the observations and experiments involving its application, 
 so as to show that the phenomena concerned were not acci- 
 dental, and in this way the inference would be strengthened. 
 But usually the inference seems so cogent that it requires no 
 such repetition to reinforce it. The mind seems convinced 
 positively by a single instance. The inference and its verifi- 
 cation seem to occur at once. The reason for this is the exist- 
 ence of certain assumptions that are an invariable part of the 
 method of difference. In the first place, there is the deduc- 
 tive argument involved, which- we have observed, and which 
 shows what is not the cause. The separation of the common 
 qualities from the phenomenon to be explained settles this 
 fact. In the second place, the assumption made whenever the 
 method of difference is strictly applied and which is, that there 
 are no other than the isolated circumstances to be taken into ac- 
 count, leaves no room for doubt, and verifies the case by an- 
 other implied deductive syllogism with this assumption as its 
 major premise. Of course this may be doubtful, but this does 
 not alter the form of the argument or the method of verification, 
 and hence we see that the whole certainty, or the degree of 
 probability attaching to the inference, is determined by the as- 
 surance we feel that there are no other circumstances to be 
 considered in the case. These are the verifying agencies in 
 the method of difference, and they are probably associated 
 with it in all its forms and applications. 
 
 There are three other methods which are simply modifica- 
 tions of one or the other of the methods just considered, and 
 which may be briefly considered. 
 
 (c) Method of Concomitant Variations. This is the method of
 
 353 ELEMENTS OF LOGIC 
 
 difference applied in circumstances where the variations of one 
 phenomenon are constant and simultaneous with another. 
 Thus, all other things being equal or remaining the same, the 
 variations of the altitude of the mercury in a thermometer 
 along with a corresponding variation of temperature, lead to 
 the inference that the temperature is the cause, and the veri- 
 fication by the same method is as before. We may repeat the 
 observation or experiment, or use the reasoning which is in- 
 cident to the method of difference in order to add more prob- 
 ability to our inference. In so far as the method is one of dif- 
 ference it represents more of verification than suggestion or 
 acquisition. 
 
 (d) Method of Residues. This is again another application 
 of the method of difference. If we subtract or eliminate from 
 any group of phenomena what we know is due to a given 
 cause, the residue will be the effect of the remaining condi- 
 tions. Thus, if we suspect the impurity of a mass of gold, we 
 have only to determine its specific gravity and compare this 
 with what we know of the specific gravity of pure gold. The 
 existence of , difference proves the case. We can thus deter- 
 mine what influence the sun has on the tides by subtracting 
 the influence of the moon from the total effect. 
 
 (e) Joint Method of Agreement and Difference. " This meth- 
 od consists in a double employment of the method of agree- 
 ment and a comparison of the results thus obtained, the com- 
 parison assimilating it to the method of difference. We, first 
 of all, compare cases in which the phenomenon occurs, and, 
 so far as we can ascertain, find them to agree in the posses- 
 sion of only one other circumstance. But, though we may 
 not be justified in regarding this inference as certain, we may 
 increase our assurance by proceeding to compare cases in 
 which the phenomenon does not occur," The assurance may 
 not be complete in this case as in the method of difference, 
 but it is greater than in the method of agreement. We quote 
 Fowler's illustration of the method : 
 
 " A very thin sheet of light proceeding from incandescent 
 hydrogen is passed through a prism, and it is invariably found
 
 SCIENTIFIC METHOD 359 
 
 (with one exception) that in the spectrum thus obtained there 
 are, in proportion to the intensity of the light, one, two, or 
 more bright lines occupying precisely the same relative posi- 
 tion. Moreover, very thin sheets of white light proceeding 
 from various incandescent substances are passed through in- 
 candescent hydrogen, and the emergent light is then separated 
 into its constituent elements by a prism. In the spectra thus 
 obtained it is found that there are invariably (with the above- 
 mentioned exception) dark lines occupying exactly the same 
 positions in the spectrum as the lines above mentioned. Hence 
 it is inferred, by the method of agreement, that a sheet of light, 
 whether it proceed directly from incandescent hydrogen itself, 
 or be transmitted through it from some other incandescent 
 substance, will invariably (allowing for the exceptional case) 
 produce these lines. But, if we try the same experiments with 
 any other element than incandescent hydrogen, although we 
 may obtain bright or dark lines, we never find these lines oc- 
 cupying the same positions in the spectrum as the lines in 
 question." 
 
 The principles of verification involved in the joint method 
 of agreement and difference are those of the two methods 
 themselves, and require no illustration at length. We turn to 
 the last and most conclusive methods of verifying hypotheses. 
 
 4. DEDUCTION. This method of verification is simply the 
 application of deductive reasoning to show that the conclu- 
 sion of the inductive process may be a necessary deduction 
 from some other known fact, law, or principle. We have seen 
 what place it has in the method of difference, but it may be 
 applied in many instances to the inferences obtained by the 
 method of agreement, and it may possibly be a contingent 
 factor in all forms of verification. But we shall not insist 
 upon this possibility. It suffices to find some cases where the 
 proof of an inference originating inductively may become de- 
 ductive and so absolutely assuring it. 
 
 This deductive method of verification merely consists in 
 comparing an hypothesis or inductive inference with a known 
 truth, and ascertaining whether they agree or not. If they do
 
 360 ELEMENTS OF LOGIC 
 
 not agree, but contradict each other, we have positive proof by 
 the principles of opposition that the inference is false. If they 
 do agree, and the inference bears all the marks of being identi- 
 cal with the known truth, it must be true. Whenever it is 
 possible the method is always resorted to. The certitude it 
 possesses depends wholly upon the certitude of the general 
 principle by which the particular inference is tried. But in 
 all cases where a conclusion by induction can be confirmed by 
 its agreement with some other known truth or principle, its 
 probability is very greatly increased, and if the incidents are 
 of the proper kind the verification may be demonstrative. 
 
 The most conclusive illustrations of verification by the de- 
 ductive method are those where the hypothesis accords with 
 the results of mathematical calculation. Thus the conformity 
 of Newton's theory of gravitation with the mathematical cal- 
 culations made in regard to planetary motion was universally 
 taken as a proof of the theory. Then again, Newton's Bi- 
 nomial Theorem was at first an inductive inference from a 
 few observed instances, and it was afterward proved deduc- 
 tively by later mathematicians ; that is, it was shown by ad- 
 mitted mathematical principles that the co-efficients and ex- 
 ponents inductively inferred by Newton must be the ones 
 assumed. Galileo inferred from some observations and ex- 
 periments that the area circumscribed by a cycloidal curve is 
 three times that of the generating circle or wheel. But he 
 had no means of proving it, although he tried to do so exper- 
 imentally " by cutting out cycloids in pasteboard, and then 
 comparing the areas of the curve and the generating circle by 
 weighing them. In every trial the curve seemed to be rather 
 less than three times the circle," so that he began to suspect 
 the correctness of his supposition. But afterward Torricelli 
 showed by mathematical calculations that this ratio must be 
 true. 
 
 The method has applications outside of mathematics. If 
 we know any general principle we may compare it with an as- 
 sumed or inferred fact and test its validity. Thus if I infer 
 from a few observations of the fact of potatoes driving out
 
 SCIENTIFIC METHOD 361 
 
 wheat as a food, that the cheaper food has a tendency to sup- 
 plant the more costly, I may deduce this fact from Gresham's 
 law regarding the influence of cheaper upon better currencies, 
 when I know or assume, of course, that money is only a com- 
 modity like all other things. Mr. Fowler mentions two in- 
 stances of the combination of induction and deduction which 
 are to the point. 
 
 " In the science of Political Economy, Bicardo's theory of 
 rent, when stated in the slightly modified form that ' the rent 
 of land represents the pecuniary value of the advantages 
 which such land possesses over the least valuable land in cul- 
 tivation,' is an easy deduction from two principles which are 
 supplied by every one's experience ; namely, (1) that land 
 varies in value, and (2) that there is some land either so bad 
 or so disadvantageously situated as to be not worth the culti- 
 vating." 
 
 Professor Cairn es' work on the Slave Power furnishes a re- 
 markable example of the successful application of the deduc- 
 tive method to the determination of economical questions. 
 " The economical effects of slavery are thus traced. We learn 
 from observation and induction that slave labor is subject to 
 certain characteristic defects : it is given reluctantry ; it is un- 
 skilful ; and, lastly, it is wanting in versatility. As a conse- 
 quence of these characteristics, it can only be employed with 
 profit when it is possible to organize it on a large scale. It re- 
 quires constant supervision, and this for small numbers or for 
 dispersed workmen would be too costly to be remunerative. 
 The slaves must, consequently, be worked in large gangs. 
 Now, there are only four products which repay this mode of 
 cultivation, namely, cotton, sugar, tobacco, and rice. Hence a 
 country in which slave labor prevails is practically restricted 
 to these four products, for it is another characteristic of slave 
 labor, under its modern form, that free labor cannot exist side 
 by side with it. But besides restricting cultivation to these 
 four products, some or all of which have a peculiar tendency 
 to exhaust the soil, slave labor, from its want of versatility, 
 imposes a still further restriction. ' The difficulty of teaching
 
 362 ELEMENTS OF LOGIC 
 
 the slave anything is so great the result of compulsory igno- 
 rance in which he is kept, combined with want of intelligent 
 interest in his work that the only chance of rendering his 
 labor profitable is, when he has once learned a lesson, to keep 
 him to that lesson for life. Accordingly, where agricultural 
 operations are carried on by slaves, the business of each gang 
 is always restricted to the raising of a single product. What- 
 ever crop be best suited to the character of the soil and the 
 nature of slave industry, whether cotton, tobacco, sugar, or 
 rice, that crop is cultivated, and that crop only. Rotation of 
 crops is thus precluded by the conditions of the case. The 
 soil is tasked again and again to yield the same product, and 
 the inevitable result follows. After a short series of years its 
 fertility is completely exhausted, the planter abandons the 
 ground which he has rendered worthless, and passes on to seek 
 in new soils for that fertility under which alone the agencies 
 at his disposal can be profitably employed.' Thus from the 
 characteristics of slave labor may be deduced the economical 
 effect of exhaustion of the soil on which it prevails, and the 
 consequent necessity of constantly seeking to extend the area 
 of cultivation. From the peculiar character of the crops which 
 can alone~be successfully raised by slave labor may be explained 
 the former prevalence of slavery in the southern, and its ab- 
 sence in the northern States of the American Union ; and from 
 the necessity, of constantly seeking fertile virgin soil for the 
 employment of slave labor may be explained the former policy 
 of the southern States, which was invariably endeavoring to 
 bring newly constituted States under the dominion of slave 
 institutions." 
 
 The confirmation of an inductive hypothesis by deductive 
 reasoning from accepted principles is very common, and it 
 varies in degrees of assurance according to the assurance we 
 feel about the principles or premises upon which it depends. 
 As we have already explained, it may still be deductive, al- 
 though the premise may only be probably true, provided the 
 verification is reasoned out deductively. But we require in 
 such cases to be cautious about mistaking another instance of
 
 SCIENTIFIC METHOD 363 
 
 comparison by agreement for a deductive premise. We may 
 take as proof in some cases merely another instance of the 
 same kind, and so only increase the probability of our inference 
 by enlarging the area of the induction. Hence we must be 
 careful, in attempts at deductive verification, to secure a 
 principle or a group of facts under which the particular case 
 or cases involved in the hypothesis may be placed. The in- 
 stances, however, where this method is applicable are mostly 
 in Physics, or Mechanics, Astronomy, and the Mathematico 
 physical sciences in general. They are more rare in what may 
 be called the moral sciences. 
 
 IV. FALLACIES AND ERRORS IN INDUCTIONS. Errors 
 in application of scientific method, when it is purely inductive, 
 are not so easily detected as in deduction. There are no 
 fixed rules for determining the degrees of probability attaching 
 to inductive inferences, and hence the errors in them are 
 mostly discoverable only after some contradictory truth has 
 been proved. But it is known that we are liable to mistakes 
 in the process, and hence, in order to avoid them we are re- 
 quired to observe certain conditions as precautions against 
 error, or to keep in mind the sources of error that may appear 
 after the attempts at verification. These sources of error may 
 be divided into two kinds : first, Errors of Observation, and 
 second, Fallacies of Inductive Inference. The latter is an 
 error connected with reasoning. 
 
 I st. Errors of Observation. These are called errors rather 
 than fallacies because they are not due to any process of 
 reasoning, but to mistakes in the acquisition of facts. They 
 give rise to an error in the conclusion by falsifying the data 
 from which the inference is drawn. This may occur in two 
 ways, and against these contingencies the scientific student 
 should perpetually be on his guard. There are no rules, how- 
 ever, to determine when they may occur. But the two 
 sources of error in observation are as follows : 
 
 1. NON-OBSERVATION. This is simply the failure to observe 
 the facts which determine the nature of the inference or the 
 right to make it. We may fail to observe all the facts, or we
 
 364 ELEMENTS OF LOGIC 
 
 may fail to observe those which are essential to a legitimate 
 inductive inference. The failure may be due to various causes 
 with which it is hardly the business of Logic to deal at length, 
 or farther than to refer in general to the admitted sources of 
 mistake in perceptions ; but it is an important fact in scien- 
 tific method that we are liable to such errors in the acquisi- 
 tion of our data as vitiate the conclusion, by bringing it into 
 conflict with non-observed facts. This non-observation may 
 be of some of the cases or instances involved in, or necessary 
 to, the inductive inference, or of incidents in connection with 
 instances that are observed. We may not be able always to 
 avoid these errors, but we should be on our guard against 
 them. 
 
 2. MAL-OBSEBVATION. Mai - Observation is not a failure to 
 see facts, but is mistaken or distorted observation. It may 
 be either sensuous or intellectual. HI observation by the 
 senses may be due to various causes, such as defect of the 
 organs, indistinctness of the impressions, indirectness of vision 
 in the case of eyesight, etc., so that we may often mistake a 
 thing for what it is not. HI observation from intellectual 
 causes may be due to mistaking an inference for a fact, to pre- 
 conceived ideas distorting the appearance of those which are 
 facts, to the concentration of attention which may prevent the 
 distinct perception of all that is in the indirect field of con- 
 sciousness, etc. Some of the last may actually be a source of 
 error to the senses at the same time. And there may be cases 
 where there are tendencies in the very mental and physical 
 organism to modify the impressions and perceptions which 
 consciousness must receive. The mistaking of an inference 
 for a fact of observation is a very frequent error under this 
 head. Mr. Fowler mentions the instance of the objections at 
 first made to the Copernican system of astronomy. People 
 claimed that the theory was contrary to what they saw, when 
 the truth was that what they claimed to see was only an infer- 
 ence from certain facts. Illusions often exhibit the same phe- 
 nomenon. We misinterpret the data of sense and take the 
 result as a fact.
 
 SCIENTIFIC METHOD 
 
 2d. Fallacies of Inductive Inference. It is difficult to 
 treat of fallacies in this field, because there can be no absolute 
 rules laid down to determine the limits of correct inductive in- 
 ference ; that is, to determine when we may and when we may 
 not go beyond the data of the premises. Perhaps the only 
 rules applicable here are those which determine the limits of 
 legitimate hypothesis. But these have no formal character, 
 and hence must be capable only of material application and 
 limitaUons. In deductive reasoning the fallacies are mostly 
 fallacies of inference which can be determined by rules. In 
 inductive reasoning what is called an error of inference is 
 never, perhaps, formal, but must be material, and hence there 
 is no way of determining them until after the process of veri- 
 fications has been applied and a result established contrary to 
 our supposition. But if errors do not arise from the mode of 
 our inference, they may arise from the assumptions we make 
 in the interpretation of phenomena. Hence there may be 
 what we may call presumptive fallacies in the inductive pro- 
 cess, which vitiate the condusiueness of the inference ; that is, 
 its probability or certainty, although it may be admissible as 
 representing a conceivable case awaiting distinct verification. 
 Of these fallacies it is sufficient to recognize two general 
 kinds : first, the mistaking an inductive for a deductive infer- 
 ence ; and second, the mistaking of partial for total causes. 
 
 1. CONFUSION OF INDUCTIVE WITH DEDUCTIVE INFERENCE. We 
 may often mistake the degree of proof we have for an asser- 
 tion. We may have arrived at a truth inductively, and then 
 assume that the process was one of positive proof, or deductive. 
 Or in answering the demand for proof of a proposition, we 
 may quote instances or facts which are only the incidents from 
 which the inductive inference was drawn. They undoubtedly 
 support the inference as an inductive one, but if we assume 
 that they prove it, we assume that we have given deductive 
 reasons, when we have only referred to facts. In deduction 
 this would be called a petitio principii. Thus, if I inferred that 
 the moon influenced the weather, or that frost was caused by 
 cool calm nights, and supposed that my belief was confirmed
 
 366 ELEMENTS OF LOGIC 
 
 or proved by the incidents of my observations and experience, 
 I would be committing this fallacy. It is a very frequent er- 
 ror, and in no phenomena is it more frequent than in the con- 
 fusion of coincidence or sequence with the causal connection. 
 This is the non causa pro causa, or post hoc, ergo propter hoc 
 fallacy already discussed. We have a right to suppose it pos- 
 sible or probable, but not as proved by the circumstances. 
 Perhaps, however, these should not be called inductive falla- 
 cies, so much as fallacies which accompany inductive reason- 
 ing. Their association with it justifies the consideration of 
 them in this connection. 
 
 2. CONFUSION OF PAKTIAL AND TOTAL CAUSES. This is merely 
 the presumptive error of supposing that what may be one of 
 the causes is the only one. Thus we may infer from a set of 
 observations that the moist air of a given region is unhealthy, 
 when it may be the moist air in conjunction with the tempera- 
 ture, or it may be due to influences of temperature as well as 
 moisture. Or again, the effect may be as much determined 
 by the character of the individual as by the circumstances in 
 which he is placed. The error here, however, is not in infer- 
 ring a cause, but in assuming that the discovered cause is the 
 only or the whole cause of the phenomenon, and hence in the 
 application of scientific method we need to be as much on our 
 guard against this as against the preceding fallacy. 
 
 This fallacy may take several forms, and there are perhaps 
 others of an allied nature. But we do not consider it impor- 
 tant to discuss any of them at length in this elementary work. 
 The careful treatment of them belongs to special treatises on 
 scientific method.* 
 
 * For a more complete discussion of this subject the student may con- 
 sult the following references : Jevons : Principles of Science ; Fowler : 
 Inductive Logic ; Mill: Logic, Books III., IV., and VI. ; Hamilton: Lec- 
 tures on Logic, Lects. XXIV., XXV., and XXVI. ; Venn: Empirical 
 Logic Chapters XIV.-XVIIL, and Chapter XXIV.
 
 PKACTICAL QUESTIONS AND PEOBLEMS. 
 
 CHAPTER I. 
 
 1. What is the distinction between "science " and " art," and how ap- 
 ply it to Logic ? 
 
 2. Examine the merits of the several definitions of Logic. 
 
 3. Define the logical use of the term " thought." 
 
 4. What is Sir William Hamilton's account of " thought ? " 
 
 5. What is the meaning of the term "law" and such expressions as 
 "laws of nature " and " laws of thought ? " 
 
 6. Illustrate the use of the term " law " in the sciences. 
 
 7. What are the meanings of the term "form" in common usage and 
 in Logic ? Also the term " matter ? " 
 
 8. What is the relation of Logic to the other sciences ? Especially to 
 Psychology ? 
 
 9. What are the divisions of Logic ? How define the meaning of each 
 subdivision ? 
 
 CHAPTER II. 
 
 1. What are the elements of logical doctrine ? From what points of 
 view are they to be regarded, and why ? 
 
 2. Define "terms," "propositions," and "syllogisms.'' 
 
 3. What are the " formal " elements of logical doctrine, and how de- 
 fine them ? 
 
 4. What ambiguity is found in the word " conception ? " 
 
 5. What is the difference belween a " term '' and a "concept ? '' 
 
 6. What is the definition of " concept," and to what ambiguities is the 
 term exposed ? 
 
 7. What is a "percept," and how are concepts formed ? 
 
 8. What is meant by Perception or Apprehension ? 
 
 9. What is meant by "attribute " or " individual '' and "class wholes ?" 
 
 10. Explain the difference between mathematical and metaphysical 
 or logical concepts. 
 
 11. What is the use of Denomination in Logic ? 
 
 12. Define Judgment and compare the process with that of Conception. 
 
 13. Define Reasoning and compare it with Conception ?
 
 368 ELEMENTS OF LOGIC 
 
 CHAPTEH III. 
 
 1. What are categorematic and syncategorematic terms ? 
 
 2. Define and illustrate singular terms. When will terms ordinarily 
 singular become general ? 
 
 3. Define and illustrate general terms. 
 
 4. Distinguish between distributive and collective terms. 
 
 5. Examine the following propositions and state the distributive and 
 collective use of the terms : 
 
 () The inhabitants of Germany constitute a nation. 
 (&) " All men find their own in all men's good, 
 
 And all men join in noble brotherhood." Tennyson. 
 (e) All standing armies are dangerous to the state. 
 
 (d) N"on omnis moriar (i.e., I shall not all die). 
 
 (e) All the men cannot lift this' weight. 
 (/) All of the regiment was put to flight. 
 
 6. Define and distinguish concrete and abstract terms. 
 
 7. Give illustrations of pure concrete and pure abstract terms, and show 
 why terms cannot be classified according to the distinction between con- 
 crete and abstract. 
 
 8. Indicate in the following list the concrete and the abstract terms, 
 and those also, if any, which may be both : 
 
 Act. 
 
 Action. 
 
 Agency. 
 
 Agent. 
 
 Beauty. 
 
 Man. Wisdom. Introduction. Stone. 
 
 9. Do abstract terms admit of being plural ? 
 
 10. What disposal should be made of general terms in the classification 
 of concrete and abstract concepts ? 
 
 11. What are the common uses of "concrete'' and "abstract." and 
 why? 
 
 12. State the views of Wundt on concrete and abstract terms. 
 
 13. In the following list of terms show how each term may be taken as 
 concrete and abstract, according to its meaning : 
 
 Individuality. Personality. Equivocation. Government. 
 
 Society. Science. Philosophy. Institution. 
 
 14. Define and distinguish positive, negative, and privative terms, 
 naming the marks of the negative. 
 
 15. Define and illustrate nego-positive terms, giving the reasons for 
 recognizing such a class. 
 
 Ability. 
 
 Plato. 
 
 Production. 
 
 Presidency. 
 
 Solitude. 
 
 Warmth. 
 
 Timeliness. 
 
 Dexterity. 
 
 Science. 
 
 Virtue. 
 
 Government. 
 
 Art. 
 
 Excellence. 
 
 Library. 
 
 Truth.
 
 PRACTICAL QUESTIONS AND PROBLEMS 369 
 
 16. What is meant by infinitated conceptions ? 
 
 17. How can any term be considered as the negative of all others but 
 its synonym ? 
 
 18. Illustrate absolute and relative terms. 
 
 CHAPTER IV. 
 
 1. What is meant by the ambiguity of terms ? 
 
 2. Define and illustrate uuivocal terms. 
 
 3. Distinguish the three kinds of equivocal terms. 
 
 4. Distinguish the three causes by which the third and most important 
 class of ambiguous terms has been produced. 
 
 5. Explain the ambiguity of the following terms, referring each to its 
 cause : 
 
 Chair. 
 Man. 
 
 Paper. 
 
 Stock. 
 
 Minister. 
 Count. 
 
 Volume. 
 Scale. 
 
 Earth 
 Law. 
 
 Country. 
 Sensation. 
 
 Air. 
 
 Glass. 
 
 Period. 
 Clerk. 
 
 Feeling. 
 Kind. 
 
 Art. 
 Bolt. 
 
 Bill. 
 
 Peer. 
 
 Order. 
 
 State. 
 
 Star. 
 
 Table. 
 
 Sense. 
 
 Wood. 
 
 Service. 
 
 End. 
 
 Term. 
 
 Letter. 
 Commons. 
 Mount. 
 
 Ball. 
 Interest. 
 Church. 
 Currency. 
 
 Bull. 
 Pole. 
 House. 
 
 Fault. 
 
 Subject. 
 Age. 
 Virtue. 
 Lace. 
 
 Class. 
 Can. 
 Light. 
 Dip. 
 
 6. Explain and illustrate what is meant by the generalization and 
 specialization of terms, and how they give rise to ambiguous ideas. 
 
 7. How are the laws of generalization and specialization applicable to 
 concrete and abstract terms ? 
 
 CHAPTER V. 
 
 1. What terms are used to express the same meaning as the terms in- 
 tension and extension? 
 
 2. Define and distinguish what is meant by the intension and extension 
 of concepts. 
 
 3. Can abstract terms have both intension and extension ? 
 
 4. What is the relation between intension and extension, and how may 
 it be formulated ? 
 
 5. Between what class of terms can a comparison of extension be 
 made ? 
 
 C. When can a comparison between them not be drawn, and what is 
 peculiar in this respect regarding affirmative and negative propositions? 
 
 7. What can be said about the nature and accuracy of the law express- 
 ing the ratio or relation between intension and extension ? 
 24
 
 370 ELEMENTS OF LOGIC 
 
 8. How can the relation be expressed by symbols ? 
 
 9. What arguments can be produced for the substantial truth of the 
 law? 
 
 10. What is meant by denotative, connotative, and non-coimotative 
 terms ? 
 
 11. Examine Mill's doctrine upon this point, and state what modifi- 
 cations are commended by Keynes and Fowler ? 
 
 12. Select from the following list the terms which belong to the same 
 classes, and arrange them in the order both of the greatest intension and 
 the greatest extension : 
 
 Emperor. 
 
 Person. Animal. 
 
 Ruler. 
 
 Teacher. 
 
 Horse. Dissenter. 
 
 Organized substance. 
 
 Baptist. 
 
 Heavenly body. Individual. 
 
 Lawyer. 
 
 Timber. 
 
 Christian. Jupiter. 
 
 Alexander. 
 
 Planet. 
 
 Matter. Quadruped. 
 
 Napoleon III. 
 
 Mammalian. 
 
 Solicitor. Being. 
 
 Episcopalian. 
 
 Frenchman. 
 
 Man. Word. 
 
 Greek. 
 
 13. Select and arrange the following, with the use of triangular sym- 
 bols, so as to show both the order of intension and extension: 
 
 Animal. 
 
 Machine. 
 
 President. 
 
 Engine. 
 
 Tree. 
 
 Vertebrate. 
 
 Manufactures. 
 
 Locomotive. 
 
 Gladstone. 
 
 Man. 
 
 Statesman. 
 
 Englishman. 
 
 Substance. 
 
 Element. 
 
 Vegetable. 
 
 Iron. 
 
 Lily. 
 
 Metal 
 
 Steel. 
 
 Flower. 
 
 Organism. Being. Lincoln. American. 
 
 CHAPTER VI. 
 
 1. What are the five predicables ? 
 
 2. Define and illustrate the term " property." What other terms are its 
 equivalents ? 
 
 3. What is meant by essential property ? Illustrate. 
 
 4. Define and illustrate what is meant by non-essential properties or 
 " accidents." 
 
 5. What is meant by universal and contingent accidents ? Also by the 
 term " peculiar " property ? 
 
 6. What is meant by " differentia" or difference ? 
 
 7. What is the meaning of the term conferentia, and what is the reason 
 for using it ? 
 
 8. Define " genus" and "species," illustrating them and showing how 
 they are purely relative in their meaning, with two exceptions. 
 
 9. What two meanings has the term " genus ? " 
 
 10. What is meant by suinmum genus and infima species?
 
 PRACTICAL QUESTIONS AXD PROBLEMS 371 
 
 11. What is the analysis of concepts ? 
 
 12. What is logical division ? Illustrate and show what is meant by 
 the f '(indumentum divisionis. 
 
 13. Apply division as far as possible to the following list of concepts : 
 
 Animal. Government. Matter. 
 
 Vegetables. Man. Vertebrate. 
 
 Stones. Book. School. 
 
 Trees. Science. Poetry. 
 
 Races. Furniture. MetaL 
 
 14. What is meant in division by the terms super-ordinate, subordinate, 
 and co-ordinate ? 
 
 15. What is dichotomy and trichotomy? Illustrate the "tree of Por- 
 phyry." 
 
 1C. Define and illustrate "partition." What is meant by mathematical 
 and logical partition ? 
 
 17. Analyze the following concepts by partition : 
 
 Metal Picture. Cathedral. Knowledge. Ink. 
 
 Iron. Stone. Religion. Money. Book. 
 
 Plant. House. Literature. Government. Ice. 
 
 German. Diamond. Virtue. Production. Wheat. 
 
 18. Name and explain the several kinds of definition. 
 
 19. Compare the processes of definition and division. 
 
 20. What are the rules for correct definition ? 
 
 21. Give a logical definition of the following terms : 
 
 Biped. 
 
 Spirit. 
 
 Desk. 
 
 Action. 
 
 Proposition. 
 
 Reptile. 
 
 Water. 
 
 House. 
 
 Book. 
 
 Literature. 
 
 Nation. 
 
 Ability. 
 
 Agent. 
 
 Religion. 
 
 Spectacle. 
 
 Purity. 
 
 Honor. 
 
 Imagination. 
 
 Club. 
 
 Gravitation. 
 
 Diet. 
 
 Success. 
 
 Republican. 
 
 Money. 
 
 Race. 
 
 22. Criticise the following definitions : 
 
 (a) A member of the solar system is anything over which the sun has 
 continued influence. 
 
 (b) A chair is an object upon which men sit. 
 
 (c) An animal is a being which increases in size. 
 
 (d) Death is the opposite of life. 
 
 (e) A king is one who exercises regal functions. 
 
 (/) A gentleman is a man having no visible means of subsistence. 
 (g) Tin is a metal lighter than gold. 
 (A) Government is an association of men. 
 
 (i) Science is the study of phenomena with a view to a scientific knowl- 
 edge of them. 
 
 (j) Man is a bundle of habits.
 
 372 ELEMENTS OF LOGIC 
 
 23. What can be said of the definition of faith in the eleventh chapter 
 of the Epistle to the Hebrews ? 
 
 24. Give examples of indefinable words and explain why they are inde- 
 finable. 
 
 CHAPTER VII. 
 
 1. What is the definition of a judgment or proposition ? 
 
 2. What is meant by subject and predicate logically considered. 
 
 3. What are the divisions of judgments, and the difficulties suggested 
 by some of the current divisions ? Define and illustrate each. 
 
 4. How reduce disjunctive propositions to hypothetical ? 
 
 5. What is meant by the quality of propositions ? 
 
 6. What is meant by the division of propositions according to quantity? 
 
 I. Show how the five forms of propositions according to quantity can 
 be reduced to two. 
 
 8. Define and illustrate analytic and synthetic judgments. 
 
 9. Explain the difficulty of drawing an absolute line of distinction 
 between analytic and synthetic judgments. 
 
 10. What are tautologous, pure, and modal propositions ? 
 
 II. What are the three sources of ambiguity in propositions ? 
 
 12. Define what is meant by Inverted and Duplex propositions. 
 
 13. Illustrate and explain each of the kinds of duplex propositions. 
 
 14. State the complementary of the following duplex propositions: 
 
 (a) None but the wise can be virtuous. 
 
 (6) All persons except criminals and foreigners are not allowed to vote. 
 
 (e) Only bipeds have hands. 
 
 (d) Man alone is not obedient to his instincts. 
 
 (e) Few persons are as strong to resist temptation as they should be. 
 (/) Only those substances which are not subject to gravity are imma- 
 terial. 
 
 15. Explain the quantity of exclusive propositions. 
 
 CHAPTER VIII. 
 
 1. Explain the meaning of "judgments of extension" and "judg- 
 ments of intension." 
 
 2. How show the possibility of expressing a mathematical relation be- 
 tween the subject and the predicate in at least the extensive judgments ? 
 
 3. How show that intensive judgments have an extensive or quantita- 
 tive, as well as an intensive or qualitative import ? 
 
 4. What is the difference between the quantity of extension and quan- 
 tity of intension expressed by propositions ? 
 
 5. Symbolize propositions A, E, I, and O, by Euler's diagrams in every
 
 PRACTICAL QUESTIONS AND PROBLEMS 373 
 
 possible form, and reduce them to the simplest expression or representa- 
 tion. 
 
 6. How symbolize the quantity of intension in judgments ? 
 
 7. Define what is meant by the distribution of the subject and predi- 
 cate, and give the rules for it. 
 
 CHAPTER IX. 
 
 1. Explain the meaning of opposition. 
 
 2. What is meant by the terms contrary, contradictory, subalterns, 
 subalternans, subalternate, and sub-contrary ? 
 
 3. If we assume any one of the four propositions, A, E, I, and 0, to be 
 false, what follows in regard to the others ? 
 
 4. How must we express the opposite of any singular proposition ? 
 
 5. How treat the relation between the two propositions, "Daniel Web- 
 ster was an American," and " Daniel Webster was not an Englishman ? " 
 
 6. How are propositions best proved or disproved, and what difficulty 
 do universal propositions present in disputation ? 
 
 7. Select pairs of the following propositions and arrange them so as to 
 show all the various relations of opposition illustrated by them : 
 
 (a) All men are mortal. 
 
 (b) Some men are not mortal, 
 (e) No men are mortal. 
 
 (d) Some men are mortal. 
 
 (e) Most men are mortal. 
 (/) All men are not mortal. 
 (<7) Not all men are mortal. 
 (h) Only men are mortal, 
 (z) Few men are not mortal. 
 (j) Only mortals are men. 
 
 (k) All men except a few are mortal. 
 
 8. Examine the relation between the following propositions and exam- 
 ples of assertion, and state what is implied by a given assertion against 
 an opponent. 
 
 (rt) One man asserts that all men are wise, and another that they are all 
 ignorant. 
 
 (b) Apples are a species of fruit, but they are not an ordinary vege- 
 table. 
 
 (c) Free-trade lowers prices and protection raises them. 
 
 (d) The leaders of one party assert that Mr. A will be elected presi- 
 dent, and the leaders of the other that Mr. B will be elected president. 
 
 (e) Mr. X asserts that not a nail was made in this country before 1861.
 
 374 ELEMENTS OF LOGIC 
 
 So far is this statement of X from being true that in 1856 there were 
 2,645 nail machines in operation in this country with an output of 86,462 
 tons, and in 1859 as many as 4,686,207 pounds of nails were exported. 
 
 (/) " Will the educated woman marry ? So queried one of our alum- 
 nae in a recent magazine article. The review roll of our alumnae shows 
 that of 76 ladies who graduated in our classes, 32 have already married." 
 
 (g) "Great efforts are made to show that a general glut of the market 
 is not a source of evil, and that the appreciation of gold has been the 
 cause of the extraordinary fall in prices. To those who hold this view I 
 would piit the question why ivory and whalebone have not fallen in 
 price, but, on the contrary, have steadily risen in price during the last 
 decade." Cotter Morrison, Sen-ice of Man, Preface, pp. x.-xi. 
 
 (K) " An import duty is not a tax, and yet raises the price of articles to 
 the consumer. 
 
 " The object of raising prices to the consumer is to enable the domestic 
 manufacturer to pay higher wages to his workmen. 
 
 "The duty does not raise prices to the consumer, but, on the contrary, 
 lowers them. 
 
 "The manufacturers are enabled to pay higher wages in spite of this 
 fall in prices. " 
 
 CHAPTER X. 
 
 1. What is the meaning of inference? Of immediate inference? Of 
 mediate inference ? 
 
 2. Define and illustrate Conversion. 
 
 3. What are the rules for Conversion, and what is meant by the con- 
 vertend and the converse ? 
 
 4. How many kinds of Conversion are there ? How are they variously 
 applied to propositions A, E, I, and O ? 
 
 5. What is true of the conversion of definitions, and of singular propo- 
 sitions with a singular predicate, and why ? 
 
 6. What is Conversion by Negation, to what proposition is it applied, 
 and how ? Show whether it is a legitimate form of conversion or not. 
 
 7. Explain and illustrate the process of Obversion. and apply it to the 
 four propositions. 
 
 8. Explain and illustrate Contraversion or Contraposition, and show 
 why it cannot be applied to proposition I. 
 
 9. Explain in what sense Contraversion is a process of immediate infer- 
 ence, if it be so at all. 
 
 10. Define and illustrate the process of Inversion, and show to what 
 propositions it is applicable, and. to what propositions it is not applicable. 
 
 11. What is meant by Inference by Contribution, and what are its di- 
 visions ? Illustrate each.
 
 PRACTICAL QUESTIONS AND PROBLEMS 375 
 
 12. When will Inference by Added Determinants and by Complex Con- 
 ception be invalid ? 
 
 18. State the logical process by which we pass from each of the follow- 
 ing propositions to the succeeding one : 
 
 (a) All metals are elements. 
 
 (b) No metals are non-elements. 
 
 (c) No non-elements are metals. 
 
 (d) All non-elements are not-metals. 
 
 (e) All metals are elements. 
 (/) Some elements are metals. 
 (g) Some metals are elements, 
 (/i) No metals are elements. 
 
 14. Convert the following propositions : 
 
 (1.) Every man is a biped. 
 
 (2.) No triangle has one side equal to the sum of the other two. 
 
 (3.) Some books are dictionaries. 
 
 (4.) " Every consciousness of relation is not cognition." 
 
 (5.) Vegetables only are deciduous. 
 
 (6.) A stitch in time saves nine. 
 
 (7.) Perfect happiness is impossible. 
 
 (8.) Few are acquainted with themselves. 
 
 (9.) No one is free who does not control himself. 
 
 (10.) Good orators are not always good statesmen. 
 
 (11.) Some inorganic substances do not contain carbon. 
 
 (12.) All men are not born equal. 
 
 (13.) Only the brave deserve the fair. 
 
 (14.) No one is a hero to his valet. 
 
 (15.) He jests at scars who never felt a wound. 
 
 (16.) Uneasy lies the head that wears a crown. 
 
 (17.) Better late than never. 
 
 (18. ) A certain man had a fig tree. 
 
 (19.) Familiarity breeds contempt. 
 
 (20.) Every mistake is not culpable. 
 
 (21.) I shall not all die (non omnis inoriar). 
 
 (22.) Not many of the metals are brittle. 
 
 (23.) Great is Diana of the Ephesians. 
 
 (24.) Talents are often misused. 
 
 (25.) Romulus and Remus were twins. 
 
 (26.) Some books are to be read only in part. 
 
 (27.) Nothing is praiseworthy but virtue. 
 
 (28. ) Two blacks will not make a white. 
 
 (29.) Not one of the Greeks at Thermopylae escaped. 
 
 (30.) No one is always happy.
 
 376 ELEMENTS OF LOGIC? 
 
 V 
 
 (31.) Metals are all good conductors of heat. 
 
 (32.) There is none good but one. 
 
 (33.) All that glitters is not gold. 
 
 (34.) He can't be wrong whose life is in the right. 
 
 15. State the relation between the following propositions as indicated 
 by the figures in parentheses at the end of each proposition r 
 
 (1.) Good men are wise. 
 
 (2.) Unwise men are not good (1.). 
 
 (3.) Some wise men are good (1.). 
 
 (4.) No good men are unwise (1.) (2.). 
 
 (5.) Some unwise men are not good (2.) (4.). 
 
 (6.) Some good men are wise (1.) (3.). 
 
 (7.) No good men are wise (1.) (4.) (6.) (3.). 
 
 (8.) Some good men are not wise (1.) (3.) (6.) (7.). 
 
 (9.) No unwise men are good (1.) (2.) (4.) (5.) (8.). 
 
 (10.) No wise men are good (1.) 2.) (6.) (7.) (8.). 
 
 16. What is the logical relation, if any, between the two following prop- 
 ositions : "A false balance is an abomination to the Lord, but a just 
 weight is his delight." 
 
 17. What can be inferred by Obversion, Conversion, and Contraversion 
 from the following proposition : " The angles at the base of an isosceles 
 triangle are equal." 
 
 18. State the Contraverse and Obverse of propositions (1.) (5.) (6.) (7.) 
 (9.) (15.) (19.) (23.) (81.) and (33.) under question 14. 
 
 19. Can we logically infer that because heat expands bodies, therefore 
 cold contracts them ? 
 
 20. State the relation between the following three propositions: " The 
 voluntary muscles are all striped, and the unstriped are all involuntary, 
 but a few of the involuntary muscles are striped." 
 
 CHAPTER XI. 
 
 (1.) What is meant by a middle term ? Explain what is meant by ma- 
 jor and minor terms, and show how each of the three terms can be dis- 
 tinguished in a syllogism. 
 
 (2.) State the rules of the syllogism. 
 
 (3.) Define what is meant by a formal fallacy, and state what three 
 kinds of them occur. 
 
 (4.) How can we symbolize by diagrams the several kinds of formal 
 fallacy. 
 
 (5.) In the following syllogisms and reasonings point out the major,
 
 PRACTICAL QUESTIONS AND PROBLEMS 377 
 
 middle, and minor terms, with the corresponding premises and conclu- 
 sion : 
 
 (a) All men are fallible. (5) Platinum is a metal. 
 
 All kings are men. All metals are heavy. 
 
 .*. All kings are fallible. , .*. Platinum is heavy. 
 
 (e) Cattle are ruminants. (d) Iron is a metal. 
 
 Horses are not ruminants. Metals are substances. 
 
 . '. Cattle are not horses. . '. Some substances are iron. 
 
 (e) He is wise, because he knows what his interest is and whoever 
 knows his interest is wise. 
 
 CHAPTER XII. 
 
 1. Define what is meant by the Mood of a syllogism. Illustrate and 
 show how many combinations are possible with the four propositions A, 
 E, I, O. 
 
 2. Mark those moods which are invalid and state the reason. 
 
 3. What kind of a conclusion can be drawn from the following prem- 
 ises, AA, EA, IA, AE, OA, El. 
 
 4. Define what is meant by the Figure of a syllogism, and illustrate 
 each Figure. 
 
 5. What is meant by a weakened conclusion ? 
 
 6. What peculiar value attaches to each of the first three Figures of 
 syllogism ? 
 
 7. Show in what figures the following premises give valid conclusions, 
 AA, AE, AI, EA, OA, El, AO. 
 
 8. Show that O cannot stand as a premise in the first Figure, as major 
 premise in the second Figure, and as minor premise in the third Figure. 
 
 9. What fallacies would be committed by having A as a conclusion in 
 any Figure but the first ? 
 
 10. Why can we prove only negative propositions in the second Fig- 
 ure ? 
 
 11. What fallacy is committed if the minor premise of the first Figure 
 be negative ? 
 
 12. If one premise be 0, what must the other be ? 
 
 13. What are the Moods and Figures of the following syllogisms ? Name 
 the valid and the invalid Moods. 
 
 (a) Some M's are P's (&) All P's are M's. ' 
 
 No S's are M's. No M's are S's. 
 
 . . Some P's are not S's. . . No P's are S's. 
 
 (c) All S's are M's. (d) No M's are P's. 
 No P's are M's. All M's are S's. 
 
 .. Some S's are not P's, .'. Some S's are not P's.
 
 378 ELEMENTS OF LOGIC 
 
 (e) All feathered animals are vertebrates. 
 
 No reptiles are feathered animals. 
 .'. Some reptiles are not vertebrates. 
 (/) All vices are reprehensible. 
 
 Emulation is not reprehensible. 
 
 /. Emulation is not a vice. . 
 
 (g) All men are rational beings. 
 
 All Caucasians are rational beings 
 . '. All Caucasians are men. 
 (7i) All vices are reprehensible. 
 
 Emulation is not a vice. 
 . *. Emulation is not reprehensible, 
 (t) Only citizens are voters. 
 
 A, B, C are voters. 
 . \ A, B, C are citizens. 
 
 14. Deduce conclusions, stating Moods and Figures, from the following 
 premises : 
 
 (a) All planets are heavenly bodies. 
 
 No planets are self-luminous. 
 (6) All Europeans are Caucasians. 
 
 All Caucasians are white. 
 
 (c) All lions are carnivorous animals. 
 
 No carnivorous animals are devoid of claws. 
 
 (d) Some animals are quadrupeds. 
 All quadrupeds are vertebrates. 
 
 - (e) Oak-trees are not evergreen. 
 Pine-trees are evergreen. 
 
 15. Invent examples showing true conclusions with false premises. 
 
 CHAPTER XIII. 
 
 1. Explain the mnemonic lines which represent the valid Moods and 
 Figures, and the reduction of the last three Figures to the first. 
 
 2. Construct syllogisms in Camenes, Cesare, Ferison, Fesapo, Camestres, 
 and Datisi. and reduce them to the corresponding Moods in the first Figure. 
 
 3. What is the difference between Direct and Indirect Reduction. 
 
 4 Why cannot Baroko and Bokardo be reduced directly ? Show how 
 they may be reduced indirectly. 
 
 5. Apply indirect reduction to Cesare and Camenes. 
 
 CHAPTER XIV. 
 
 1. Define Prosyllogism, Episyllogism, Enthymeme, Epicheirema, and 
 Sorites. Illustrate each.
 
 PRACTICAL QUESTIONS AND PROBLEMS 379 
 
 2. What is the difference between the regressive and the progressive 
 Sorites ? 
 
 3. Upon what is the distinction between the three orders of Enthymeme 
 based ? 
 
 4. Produce an example of syllogism in which there are two Prosyllo- 
 gisms. 
 
 5. Complete the following syllogisms : 
 
 () Europeans are Caucasians because they are white. 
 
 (b) Since he was directed to deliver the message, and did not, I am at 
 liberty to do as I please. 
 
 (c) We cannot know what is false because knowledge cannot be decep- 
 tive. 
 
 (d) A is B, because C is B 
 E is A, because G is A 
 
 .-. EisB. 
 
 (e) A manor cannot begin at this day, because a court-baron cannot now 
 be founded. 
 
 CHAPTER XV. 
 
 1. What is hypothetical reasoning ? Illustrate. State the signs of it and 
 define what is meant by the antecedent and the consequent. 
 
 2. What are the valid and the invalid forms of hypothetical reasoning ? 
 What is meant by modus ponens and modus fattens ? 
 
 3. How can hypothetical syllogisms be reduced to the categorical ? 
 
 4. What moods and figures of the categorical syllogism do the modus 
 ponens and modus tollens of the hypothetical syllogism belong ? 
 
 5. What formal fallacies are committed by the invalid forms of hypo- 
 thetical syllogism ? Prove by reduction to the categorical. 
 
 6. Examine the following instances of hypothetical reasoning : 
 
 (a) Rain has fallen, if the ground is wet ; but the ground is not wet ; 
 therefore rain has not fallen. 
 
 (b) If rain has fallen, the ground is wet ; but rain has not fallen ; there- 
 fore the ground is not wet. 
 
 (c) The ground is wet if rain has fallen ; the ground is wet ; therefore 
 rain has fallen. t 
 
 (d) If the ground is wet, rain has fallen ; but rain has fallen ; therefore 
 the ground is wet. 
 
 (e) If a man cannot make progress toward perfection, he must be a 
 brute ; but no man is a brute ; therefore every man is capable of such 
 progress. 
 
 (/) If two and two may make five in some other planet, Mill's opinion 
 about the matter is correct ; but they do not make five in any place and 
 kence Mill is wrong.
 
 380 ELEMENTS OF LOGIC 
 
 CHAPTER XVI. 
 
 1. Define and illustrate Disjunctive Reasoning. 
 
 2. Upon what does incomplete disjunction depend ? 
 
 3. Classify the forms of disjunctive syllogism, and show how they may 
 be reduced to either the hypothetical or the categorical form. 
 
 4. To what fallacy is disjunctive reasoning incident ? 
 
 5. Examine the following cases of reasoning: 
 
 (a) Criminals are either good or bad. 
 
 They are bad. 
 . *. They are not good. 
 (Z>) The weather will be either clear or warm. 
 
 It will not be warm. 
 . '. Therefore it will be clear. 
 
 (c) Aristotle was either very talented or very industrious. 
 He was very industrious. 
 
 .'. He was not very talented. 
 
 (d) If the government enacts such a law it must either adopt socialism 
 or go into bankruptcy. But it will not enact such a law, and hence there 
 is no danger of either socialism or bankruptcy. 
 
 (e) If pain is severe it will be brief, and if it last long it will be slight ; 
 it is either severe or it lasts long, and therefore will be either brief or 
 slight. 
 
 (/) If capital punishment involves cruelty to its victims it ought to 
 be abolished in favor of some other penalty ; if it does nc good for society 
 it should also be abolished. But it either involves cruelty to its victims 
 or does no good to society, and hence it ought to be abolished. 
 
 CHAPTER XVII. 
 
 1. Define the term "fallacy " and explain what is meant by formal and 
 material fallacies. 
 
 2. Explain what is meant by the fallacies of Amphibology and of Ac- 
 cent. 
 
 3. Give outline form of the classification of fallacies. 
 
 4. Illustrate the formal fallacies and also those of Amphibology and 
 Accent. 
 
 CHAPTER XVIII. 
 
 1. What are the grounds upon which a twofold division of material 
 fallacies may rest ? 
 
 2. Explain what is meant by the following terms : petitio principii, non
 
 PRACTICAL QUESTIONS AND PROBLEMS 381 
 
 sequitur, non causa pro causa, or post hoe, ergopropter 7ioc, circulus in pro- 
 lando, assumptio non probata, and ignoratio denclii. 
 
 3. Explain the fallacies of Quantity and Quality, or those of Composi- 
 tion and Division and of Accident. 
 
 CHAPTER XIX. 
 
 1. What is meant by the quantification of the predicate ? 
 
 2. What effect upon the number of propositions to be considered by 
 Logic is produced by quantifying the predicate explicitly ? 
 
 3. How does the quantification of the predicate effect the process of 
 conversion ? 
 
 4. What additional rule must be added to the rules of the syllogism if 
 we accept the doctrine of the quantification of the predicate ? 
 
 5. What is peculiar about definitions and exclusive propositions in re- 
 lation to this doctrine ? 
 
 CHAPTER XX. 
 
 1. What is the nature of mathematical propositions ? 
 
 2. What effect do they produce upon the Figures of the syllogism ? 
 
 3. Write out the list of valid Maods when propositions are mathemati- 
 cal, and show why each one is valid that is not valid in ordinary reason- 
 ing. 
 
 4. How does mathematical reasoning simplify the symbolic representa- 
 tion of the syllogism ? 
 
 5. What characteristic of conceptions appears in mathematical reason- 
 ing, and what is excluded ? 
 
 6. What is meant by Traduction, or traductive reasoning ? 
 
 7. How can you treat syllogisms which have been called irregular by 
 Jevons and others ? Invent instances and illustrate. How are they re- 
 lated to the principles of mathematical reasoning ? 
 
 CHAPTER XXI. 
 
 1. What is the general nature of the "Laws of Thought" and what 
 are their divisions ? 
 
 2. Define and illustrate the laws of Identity, Contradiction, Excluded 
 Middle, and Sufficient Reason. 
 
 3. Enumerate the secondary laws of thought. 
 
 4. How are the primary laws related to the principles enunciated in 
 Formal Logic ?
 
 382 ELEMENTS OF LOGIC 
 
 CHAPTER XXII. 
 
 1. What views have been taken regarding the nature of inductive rea- 
 soning ? 
 
 2. What two recognized forms of induction are to be considered in the 
 definition, and how are they to be treated ? 
 
 3. What is the difference between generalization by enumeration and 
 inductive inference ? 
 
 4. What is the essential conception involved in the idea of induction 
 since the time of Bacon, and how did it arise ? 
 
 5. Illustrate Inductive reasoning and compare it with the deductive. 
 
 6. State the form of the inductive syllogism, illustrate and compare it 
 with the deductive. 
 
 7. What are the reasons for regarding inductive reasoning as purely 
 qualitative, and deductive as mainly, if not wholly, quantitative f 
 
 8. What is meant by the division of inductive inferences into statical 
 and dynamical ? 
 
 9. What are the so-called "principles" of induction? Define their 
 meaning and relation to inductive reasoning. 
 
 10. What is meant by the principles or canons of Agreement and Dif- 
 ference, and their functions in inductive reasoning ? 
 
 CHAPTER XXIII. 
 
 1. Explain the meaning of scientific method, also what is meant by the 
 methods of discovery and of instruction. 
 
 2. Define and illustrate Deductive Method. State its divisions and ex- 
 plain their relation to the method. 
 
 8. Explain what is meant by the Inductive Method as compared with 
 inductive reasoning. 
 
 4. Define and illustrate the primary processes of ascertaining new 
 knowledge. 
 
 5. What are the rules to be observed in forming hypotheses ? 
 
 6. Explain the process of verification, stating its relation to inductive 
 inference. 
 
 7. What are the two or more kinds of verification ? Explain those 
 which seem also to be conditions or factors of the inductive inference, 
 namely, Observation and Experiment. 
 
 8. What are the Inductive Methods, and how do they serve as verify- 
 ing processes. 
 
 9. Show how deduction may enter as a process of verification. 
 
 10. Define and illustrate what are called the fallacies of induction. 
 Why distinguish between errors of observation and errors of inference ?
 
 PEACTICAL EXERCISES 
 
 DEDUCTIVE 
 
 THE student is expected to examine the following arguments ; to state 
 the mood and figure of the syllogism where necessary ; to complete im- 
 perfect syllogisms ; to indicate the instances of valid and invalid reason- 
 ing ; and if invalid, to state whether the fallacy is formal or material, 
 and what the particular fallacy is. He should also be prepared to make 
 any resolution of the propositions and syllogisms which the rules of Logic 
 would enable him to do. 
 
 1. None but animals are quadrupeds. 
 Horses are quadrupeds. 
 Therefore horses are animals. 
 
 2. Personal deformity is an affliction of nature. 
 Disgrace is not an affliction of nature. 
 Therefore personal deformity is not a disgrace. 
 
 3. All roses are beautiful. 
 Lilies are not roses. 
 Therefore lilies are not beautiful. 
 
 4. All paper is useful ; and all that is useful is a source of comfort to 
 men ; therefore all paper is a source of comfort to men. 
 
 5. Some statesmen are also authors ; for such are Burke, Macaulay. 
 .Gladstone, Lord Russell, etc. 
 
 6. Some philosophers are logicians. 
 
 No logicians are ignorant of the works of Aristotle. 
 Therefore some philosophers are not ignorant of the works of Aris- 
 totle. 
 
 7. No persons destitute of imagination are true poets. 
 Some persons destitute of imagination are good logicians. 
 Therefore some true poets are not good logicians. 
 
 8. This explosion must have been occasioned by gunpowder ; for noth- 
 ing else would have possessed sufficient force. 
 
 9. If Caesar was a tyrant, he deserved to die. 
 Caesar was not a tyrant. 
 
 Therefore he did not deserve to die. 
 
 10. Good is the object of moral approbation. The highest good is, 
 therefore, the ultimate object of such approbation, the end of action.
 
 384 ELEMENTS OF LOGIC 
 
 11. Every one desires his own good. 
 
 Justice and temperance are every one's good. 
 Therefore every one desires to be just and temperate. 
 
 12. " But it is doubtful yet whether Caesar will come forth to-day or not. 
 For he is superstitious grown of late." 
 
 13. Every man should be moderate ; for excess will cause disease. 
 
 14. All Parisians are Frenchmen. 
 No Chinese are Parisians. 
 
 Therefore some Chinese are not Frenchmen. 
 
 15. Some men are not virtuous. 
 All Americans are men. 
 
 Some Americans are not virtuous. 
 
 16. Blessed are the merciful ; for they shall obtain mercy. 
 
 17. As almost all the organs of the body have a known use, the spleen 
 must have some use. 
 
 18. Some of the inhabitants of the globe are more civilized than others. 
 No savages are more civilized than other races. 
 
 Some savages are not inhabitants of the globe. 
 
 19. Cogito, ergo sum (I think, therefore I am). 
 
 20. He must be a Mohammedan, for all Mohammedans hold these opin- 
 ions. 
 
 21. He must be a Christian, for only Christians hold these opinions. 
 
 22. Logic is either a science or an art. 
 It is a science. 
 
 Therefore it is not an art. 
 
 23. No idle person can be a successful writer of history ; therefore, 
 Hume, Macaulay, Hallam, and Grote must have been industrious. 
 
 24. Who spareth the rod hateth his child ; the parent who loveth his 
 child therefore spareth not the rod. 
 
 25. The coronation took place either at Paris, Berlin, or Vienna ; it did 
 not occur at Paris or Berlin, and consequently must have occurred at 
 Vienna. 
 
 26. Every moral man obeys the law ; every citizen does not do so, and 
 therefore is not moral. 
 
 27. Rational beings are accountable for their conduct ; brutes, not be- 
 ing rational, are therefore free from responsibility. 
 
 28. All valid syllogisms have three terms. 
 This syllogism has three terms. 
 
 This syllogism is therefore valid. 
 
 29. All syllogisms are valid that have three terms. 
 This syllogism has three terms. 
 
 Therefore this syllogism is valid. 
 
 30. Comets are heavy matter ; for otherwise they would not obey the 
 law of gravitation.
 
 PRACTICAL EXERCISES 385 
 
 31. A charitable man has no merit in relieving distress, because he 
 merely does what is pleasing to himself. 
 
 32. None but savages were in America when it was discovered. 
 The Hottentots were savages. 
 
 Therefore they were in America when it was discovered. 
 
 33. None but despots possess absolute power. 
 The Czav of Russia is a despot. 
 Therefore he possesses absolute power. 
 
 34. Bacon was a great philosopher and statesman, and as he was also 
 a lawyer we may infer that any lawyer may be a great philosopher and 
 statesman. 
 
 35. Mathematical studies undoubtedly improve the reasoning powers ; 
 but as Logic is not a mathematical study we may conclude that it does 
 not improve our reasoning powers. 
 
 36. If a man cannot obey the law he must be either a mere machine or 
 a demon ; but no man is either of these, and hence he must be able to 
 obey the law. 
 
 37. Whatever tends to withdraw the mind from pursuits of a low nat- 
 ure deserves to be promoted ; classical learning does this, since it gives 
 us a taste for intellectual enjoyments ; therefore it deserves to be pro- 
 moted. 
 
 38. Alexander the Great was the son of King Philip, and therefore 
 King Philip was the father of Alexander the Great. 
 
 39. He that withholdeth corn, the people shall curse him. But bless- 
 ing shall be upon the head of him that selleth it. Proverbs of Solomon. 
 
 40. If virtue is involuntary, vice is involuntary. 
 Vice is voluntary. 
 
 Therefore virtue is voluntary. 
 
 41. All civilized people are inhabitants of the earth. Few Indians are 
 civilized, and therefore few Indians are inhabitants of the earth. 
 
 42. To improve is to change, and to be perfect is to have changed often. 
 What hope can we entertain of those who oppose change ? 
 
 43. The Germans are a mtion. Bismarck, Stein, Kant, and Hegel 
 were Germans, and hence must have been a nation. 
 
 44. The right should be enforced by law. Hence as the exercise of the 
 suffrage is a right, it should be enforced by law. 
 
 45. Napoleon was not a great emperor ; for though he would have been 
 great had he succeeded in retaining his power, he did not do so. 
 
 46. Nothing is better than wisdom ; dry bread is better than nothing ; 
 therefore dry bread is better than wisdom. 
 
 47. Knowledge is of no use to any one in preventing him from com- 
 mitting crimes ; for we hear every day of frauds and forgeries which 
 
 25
 
 386 ELEMENTS OF LOGIC 
 
 never would have been committed had the person not learned to read and 
 write. 
 
 48. The end of punishment is either the protection of society or the 
 reformation of the individual. Capital punishment ought, therefore, to 
 be abolished, because it neither prevents crimes of violence, nor protects 
 society, nor does it reform the individual. 
 
 49. Wealth is value ; value is purchasing power ; purchasing power is 
 the product of labor, and the product of labor is property ; therefore 
 wealth is property. 
 
 50. Every rule has exceptions ; this is a rule, and therefore has excep- 
 tions ; therefore there are some rules that have no exceptions. 
 
 51. All who think this man innocent think he should not be punished ; 
 you think he should not be punished ; therefore you think him innocent. 
 
 52. All who think this man innocent think he should not be punished ; 
 you think he should be punished ; therefore you do not think him inno- 
 cent. 
 
 53. Haste makes waste, and waste makes want. A man, therefore, 
 never loses by delay. 
 
 54. All equilateral triangles are equiangular, and therefore all equian* 
 gular triangles are equilateral. 
 
 55. Only the virtuous are truly noble ; some who are called noble are 
 not virtuous; therefore some who are called noble are not truly noble. 
 
 56. For those who are bent on cultivating their minds by diligent study 
 the incitement of academic honors is unnecessary ; and it is ineffectual 
 for the idle and such as are indifferent to mental improvement ; therefore 
 the incitement of academic honors is either unnecessary or ineffectual. 
 
 57. Logic as it was cultivated by the schoolmen proved a fruitless study ; 
 therefore Logic, as it is cultivated at the present day must be a fruitless 
 (study likewise. 
 
 58. Repentance is a good quality ; wicked men abound in repentance, 
 and therefore abound in what is good. 
 
 59. Warm countries alone produce wine. Spain is a warm country, and 
 therefore produces wine. 
 
 60. It is an intensely cold climate that is sufficient to freeze mercury ; 
 the climate of Siberia is sufficient to freeze it, and hence must be intensely 
 cold. 
 
 61. No designing person ought to be trusted ; engravers are by profes- 
 sion designers ; therefore they ought not to be trusted. 
 
 62. I will not do this act because it is unjust ; I know it is unjust be- 
 cause my conscience tells me so, and my conscience tells me so because 
 the act is wrong. 
 
 63. Is a stone a body ? Yes. Then is not an animal a body ? Yes. 
 Are you an animal ? I think so. Ergo, you are a stone, being a body. 
 Lucian.
 
 PRACTICAL EXERCISES 387 
 
 64. If ye were Abraham's children ye would do the works of Abraham. 
 John viii. 39. 
 
 65. He that is of God heareth God's words ; ye therefore hear them not, 
 because you are not of God. John viii. 47. 
 
 66. His imbecility of character might have been inferred from his 
 proneness to favorites ; for all weak princes have this failing. 
 
 67. He is brave who conquers his passions , he who resists temptation 
 conquers his passions ; so that he who resists temptation is brave. 
 
 68. Suicide is not always to be condemned ; for it is but voluntary 
 death, and this has been gladly embraced by many of the greatest heroes 
 of antiquity. 
 
 69 All that glitters is not gold ; tinsel glitters and is therefore not gold. 
 
 70. Meat and drink are the necessaries of life. The revenues of the 
 king were spent on meat and drink, and were therefore spent on the nec- 
 essaries of life. 
 
 71. He who calls you a man speaks truly ; he who calls you a fool calls 
 you a man ; therefore he who calls you a fool speaks truly. 
 
 72. Theft is a crime , theft was encouraged by the laws of Sparta ; 
 therefore the laws of Sparta encouraged crime. 
 
 73. Since all gold is a metal, the most rare of all masses of gold must be 
 the most rare of all the metals. 
 
 74. Nothing but the express train carries the mail, and as the last train 
 was an express it must have carried the mail. 
 
 75. Protective laws should be abolished, for they are injurious if they 
 produce scarcity, and they are useless if they do not. 
 
 76. The Quaker asserts that if men were true Christians" and acted xipon 
 their religious principles there would be no need of armies ; hence he 
 draws the conclusion that a military force is useless, and being useless is 
 pernicious. 
 
 77. Detention of property implies at least possession ; for detention is 
 natural possession. 
 
 78. "Profit" is interpreted or defined to be "advantage;" to take 
 profit, then, is to take advantage , it is wrong to take advantage of one's 
 neighbor , therefore it is wrong to take profit. 
 
 79. Peel's remission of taxes was beneficial ; the taxes remitted by Peel 
 were indirect, and therefore the remission of indirect taxes is beneficial. 
 
 80. Some poisons are vegetable , no poisons are useful drugs, and 
 therefore some useful drugs are not vegetable. 
 
 81. Whosoever intentionally kills another should suffer death ; a soldier 
 therefore who kills his enemy should suffer death. 
 
 82. Few towns in the country have 500,000 inhabitants, and since all 
 such towns ought to have three representatives in Congress, it is evident 
 that few towns ought to have three representatives. 
 
 83. If Bacon's opinion be right it is improper to stock a new colony with
 
 388 ELEMENTS OF LOGIC 
 
 criminals from prison ; but this course we must allow to be proper if the 
 method of colonizing New South Wales be a wise one. If this be wise, 
 therefore Bacon's opinion is not right. 
 
 84. The people of the country are suffering from famine, and as A, B, C 
 are people of the country, they must be suffering from famine. 
 
 85. You are not what I am ; I am a man ; therefore you are not a man. 
 
 86. Gold and silver are wealth ; and therefore the diminution of the 
 gold and silver of a country by exportation is the diminution of the wealth 
 of the country. 
 
 87. The holder of some shares in a lottery is sure to gain a 'prize, and 
 as I am the holder of some shares in a lottery I am sure to gain a prize. 
 
 88. A monopoly of the sugar-refining business is beneficial to sugar re- 
 finers ; and of the corn trade to corn growers ; and of the silk manufact- 
 ure to the silk-weavers ; of labor to the laborers. Now all these classes 
 of men make up the whole community. Therefore a system of restric- 
 tions upon competition is beneficial to the community. 
 
 89. Over-credulous persons should never be believed; and as- the ancient 
 historians were in many instances over-credulous they ought never to be 
 believed. 
 
 90. That is unfortunate ; you insolently assert that you are a Darwinian, 
 while the truth is that you are a poet. 
 
 91. Every incident in the narrative is probable, and hence the narrative 
 may be believed since it is probable. 
 
 92. If a substance is solid it possesses elasticity, and so also it does if it 
 be liquid or gaseous ; but all substances are either solid, liquid, or gaseous ; 
 therefore all substances possess elasticity. 
 
 93. Who is most hungry eats most ; who eats least is most hungry ; 
 therefore who eats least eats most. 
 
 94. If the Elixir of life is of any value those who take it will improve 
 in health ; now my friend who has been taking it has improved in health, 
 and therefore the Elixir "is of value as a curative agent. 
 
 95. The policy of protection was immediately followed by a great in- 
 crease in the prosperity and wealth of the country, and hence we may 
 infer that the result was due to its connection with the enactment of the 
 protective law. In reply, however, we are told that before the passage of 
 the law the loss by fire in Chicago in one year was $200,000,000, but was 
 only $3,000,000 for the year after its passage, so great was the effect of 
 this act. 
 
 96. What produces intoxication should be prohibited ; the use of intox- 
 icating liquors causes intoxication ; therefore the use of spirituous liquors 
 should be prohibited. 
 
 97. When we hear that all the righteous people are happy, it is hard to 
 avoid exclaiming, What ! are all the unhappy persons we see thought to 
 be unrighteous ?
 
 PRACTICAL EXERCISES 389 
 
 93. Italy is a Catholic country and abounds in beggars ; France is also 
 a Catholic country, and therefore abounds in beggars. 
 
 99. The Latin word "virtus" originally meant "manliness;" hence 
 the virtue of manliness or courage is the highest virtue and type of all 
 other virtues. 
 
 100. If it be fated that you recover from your present disease, you will 
 recover, whether you call in a doctor or not ; again, if it be fated that you 
 do not recover from your present disease, you will not recover, whether 
 you call in a doctor or not. But one or the other of these contradictories 
 is fated, and therefore it can be of no service to call in a doctor. 
 
 101. This person may reasonably be supposed to have committed the 
 theft, for he can give no satisfactory account of himself on the night of 
 the alleged offence ; moreover he is a person of bad character, and, being 
 poor, is naturally liable to a temptation to steal. 
 
 102. All the trees in the park make a thick shade; this oak-tree is one 
 of them and therefore makes a thick shade. 
 
 103. All visible bodies shine by their own or by reflected light. The 
 moon does not shine by its own ; therefore it shines by reflected light ; 
 but the sun shines by its own light ; therefore it cannot shine by reflected 
 light. 
 
 104. The two propositions, "Aristotle is living," and "Aristotle is 
 dead," are both intelligible propositions ; they are both of them true or 
 both of them false, because all intelligible propositions must be either 
 true or false. 
 
 105. How can anyone maintain that pain is always an evil who admits 
 that remorse involves pain, and yet may sometimes be a real good ? 
 
 106. I am charged with absenteeism from my post, and on that ground 
 I am accused of ignorance in regard to the proper duties of my office. 
 But my accuser himself, who was my predecessor in the same office, was 
 not in the country, of which he was the ruler, longer than five days. 
 
 107. Every law is either useless or it occasions hurt to some person ; 
 now a law that is useless ought to be abolished ; and so ought every law 
 that occasions hurt ; therefore every law ought to be abolished. 
 
 108. What fallacies are implied or charged against Mr. Spencer in the 
 following criticism? 
 
 "Mr. Spencer's distinction between objects and relations is far from 
 satisfactory; and even if it were a true distinction, I do not see that any 
 adequate classification of knowledge could be based upon it, because there 
 is no science within the circle of knowledge that does not deal both with 
 objects and relations." Knight : Essays in Philosophy. 
 
 109. Does a grain of millet, when dropped on the floor, make sound ? 
 No. Does a bushel of millet make sound under the same circumstances ? 
 Yes. Is there not a determinate proportion between the bushel and the 
 grain V There is. There must therefore be the same proportion between
 
 390 ELEMENTS OF LOGIC 
 
 the sonorousness of the two. If one grain be not sonorous, neither can 
 ten thousand grains be so. 
 
 110. What you say is that virtue is the power of attaining good ? Yes. 
 And you would say that goods are such as health aud wealth, and the 
 possession of gold and silver, and having office and honor in the state 
 these are what you call goods ? Yes, all these. Then, according to 
 Meno, who is the hereditary friend of the great king, virtue is the power 
 of getting silver and gold. Plato's Dialogues : Meno. 
 
 111. Injustice is more profitable than justice because those who do un- 
 just acts gain more than the just. 
 
 112. As for saying that without God man cannot have moral sentiments, 
 or, in other words, cannot distinguish between vice and virtue, it is as if 
 one said that without the idea of God man would not feel the necessity 
 of eating and drinking. John Morley. 
 
 113. I am offered a sum of money to assist this person in gaining the 
 office he desires ; to assist a person is to do him good, and no rule of 
 morality forbids the doing of good ; therefore no rule of morality forbids 
 me to receive the sum of money for assisting this person. 
 
 114. Ruminant animals are those which have cloven feet, and they 
 usually have horns ; the extinct animal which left this foot-print had a 
 cloven foot ; therefore it was a ruminant animal aud had horns. Again, 
 as no beasts of prey are ruminant animals, it cannot have been a beast of 
 prey. 
 
 115. Without order there is no living in public society, because the 
 want thereof is the mother of confusion, whereupon division of neces- 
 sity followeth, and out of division destruction. Hooker: Ecclesiastical 
 Polity. 
 
 116. The man who does any kind of work in a careless, bungling, or 
 superficial way is not acting as a reasonable, being ; for the first demand 
 of reason, as the truthful faculty in the world of action, is to realize its 
 idea completely and thoroughly, and this no hasty and superficial handi- 
 work will pretend to do. 
 
 117. Happiness signifies a gratified state of all the faculties. The grati- 
 fication of a faculty is produced by its exercise. To be agreable that exer- 
 cise must be proportionate to the power of the faculty ; if it is insufficient 
 discontent arises and its excess produces weariness. Hence to have com- 
 plete felicity is to have all the faculties exerted in the ratio of their sev- 
 eral developments. 
 
 118. We must either gratify our vicious propensities or resist them ; 
 the former course will involve us in sin and misery ; the latter requires 
 self-denial. Therefore we must either fall into sin and misery, or practise 
 self-denial. 
 
 119. Every moral aim requires the rational means of attaining it ; these 
 means are the establishment of laws ; and as happiness is the moral aim of
 
 PRACTICAL EXERCISES 391 
 
 man it follows that the attainment of it requires the establishment of 
 laws. 
 
 120. He that can swim needs not despair to fly ; for to swim is to fly in 
 a grosser fluid, and to fly is to swim in a subtler fluid. 
 
 121. What fallacy was the humorist afraid of who said that he would 
 not admit that two and two make four until he knew what use was to be 
 made of the assertion. 
 
 122. The Good is a state of consciousness ; for the Good is a possible ob- 
 ject of knowledge ; but all objects of knowledge are states of consciousness. 
 Hence the Good is a state of consciousness. 
 
 123. The Good is pleasure ; for the Good results from the due perform- 
 ance of function ; but the Good is a state of consciousness ; therefore the 
 Good is the state of consciousness which results from the due performance 
 of function. 
 
 124. Riches are for spending, and spending for honor and good actions; 
 therefore extraordinary expense must be limited by the worth of the oc- 
 casion. 
 
 125. The several species of brutes being created to prey upon one an- 
 other proves that the human species were intended to prey upon them. 
 
 126. If any objection can be urged to justify a change of established 
 laws, no laws could be reasonably maintained ; but some laws can be 
 reasonably maintained ; therefore no objection that can be urged will 
 justify a change of established laws. 
 
 127. You are inconsistent with yourself, for you told me yesterday that 
 there was a presumption of this man's guilt, and now when I say that 1 
 may presume his guilt, you contradict me. 
 
 128. The more correct the logic, the more certainly the conclusion will 
 be wrong if the premises are false ; therefore where the premises are 
 wholly uncertain the best logician is the least safe guide. 
 
 129. If our rulers could be trusted always to look to the best interests 
 of their subjects monarchy would be the best form of government ; but 
 they cannot be trusted ; therefore monarchy is not the best form of 
 government. 
 
 130. He who bears arms at the command of the magistrate does what 
 is lawful for a Christian ; the Swiss in the French service, and the British 
 in the American service, bore arms at the command of the magistrate ; 
 therefore they did what was lawful for a Christian. 
 
 131. A man that hath no virtue in himself envieth virtue in others; for 
 men's minds will either feed upon their own good or upon others evil, and 
 who wanteth the one will prey upon the other. 
 
 132. The object of war is durable peace ; therefore soldiers are the best 
 peacemakers. 
 
 133. Confidence in promises is essential to human intercourse and com- 
 merce ; for without it the greatest part of our conduct would proceed upou.
 
 392 ELEMENTS OF LOGIC 
 
 chance. But there could be no confidence in promises if men were not 
 obliged to perform them ; the obligation, therefore, to perform promises 
 is essential to the same ends aiid in the same degree. 
 
 134. If the majority of those who use public-houses are prepared to 
 close them legislation is unnecessary ; but if they are not prepared for 
 such a measure, then to force it on them by outside pressure is both 
 dangerous and unjust. 
 
 135. He who believes himself to be always in the right in his opinion 
 lays claim to infallibility ; you always believe yourself to be in the right 
 in your opinion ; therefore you lay claim to infallibility. 
 
 136. If we never find skins except as integuments of animals we may 
 safely conclude that animals cannot exist without skins. If color can- 
 not exist by itself, it follows that neither can anything that is colored ex- 
 ist without color. So, if language without thought is unreal, thought 
 without language must also be so. 
 
 137. If the light is not refracted near the surface of the moon there 
 cannot be any twilight ; but if the moon has no atmosphere light is not 
 refracted near its surface ; therefore if the moon has no atmosphere 
 there cannot be any twilight. 
 
 138. No soldiers should be brought into the field who are nt well qual- 
 ified to perform their duty ; none but veterans are well qualified to per- 
 form their part ; therefore none but veterans should be brought into the 
 field. 
 
 139. The minimum visibtte is the least magnitude which can be seen ; 
 no part of it alone is visible, and yet all parts of it must affect the mind 
 in order that it may be visible ; therefore every part of it must affect the 
 mind without being visible. 
 
 140. Improbable events happen almost every day, but what happens 
 almost every day is a very probable event ; therefore improbable events 
 are very probable events. 
 
 141. What fallacies are implied against an opponent in the following 
 statement : " Each of its links is in fact unsound. And even though no 
 flaw were visible in them, still the conclusion is demonstrably false." 
 
 142. " Now that which does not make a man worse, how can it make 
 a man's life worse ? But neither through ignorance, nor having the 
 knowledge but not the power to guard against or correct these things, is 
 it possible that the nature of the universe has overlooked them ; nor is it 
 possible that it has made so great a mistake, either through want of power 
 or want of skill, that good and evil should happen indiscriminately to the 
 good and the bad. But death certainly, and life, honor and dishonor, 
 pain and pleasure all these things happen equally to good men and bad, 
 being things which make us neither better nor worse. Therefore they 
 are neither good nor evil." Marcus Aurdim. 
 
 143. Since there is no harm or evil to the elements themselves in their
 
 PRACTICAL EXERCISES 393 
 
 continual changes into one another, a man should have no apprehension 
 about the dissolution of all elements. For it is according to nature, and 
 nothing is evil that is according to nature. Marcus Aurelim. 
 
 144. Form a syllogism and show under what conditions any one of the 
 following three fallacies may be found non sequitur, petitio principii, 
 and equivocation. 
 
 145. What fallacy is charged to the defenders of Charles the Second by 
 Macaulay in the following statements : " We charge him with having 
 broken his coronation oath, and we are told that he kept his marriage 
 vow ! We accuse him of having given up his people to the merciless in- 
 flictions of the most hot-headed and hard-hearted of prelates, and the de- 
 fence is that he took his little son on his knee and kissed him ! We cen- 
 sure him for having violated the articles of the Petition of Right, after 
 having for good and valuable consideration promised to observe them, 
 and we are informed that he was accustomed to hear prayers at six o'clock 
 iu the morning." 
 
 146. "Don't you think the possession of gold is good? Yes, said 
 Ctesippus, and the more the better. And to have money everywhere and 
 always is a good ? Certainly, a great good, he said. And you admit that 
 gold is a good ? I have admitted that, he replied. And ought not a man 
 then to have gold everywhere and always, and as much as possible in 
 himself, and may not he be deemed the" happiest of men who has three 
 talents of gold in his stomach, and a talent in his head, and a stater of 
 gold in either eye." Plata's Dialogues : Euthydemus. 
 
 147. " If we are to test the truth of materialism by its outcome for 
 well-being, we can hold it only by showing that the supreme end of man 
 is to develop a body, and that materialism is especially useful in promot- 
 ing the interests of tlie animal nature. The normal brain is that which 
 takes care of itself, and the test of truth is self-preservation. Moral aims 
 and scientific truth, so far as they have no physical value, must be voted 
 not merely worthless, but delusion ; for the test of truth is physical pres- 
 ervation. Hence the inhabitant of the sty would be the prince of ma- 
 terialistic philosophers ; he is not troubled by delusion and he preserves 
 himself." 
 
 143. If sin by itself confers the right and imposes the duty of punish- 
 ment, there must be the right to inflict either a definite punishment or 
 an infinite amount. If the latter, it is obvious that the state will always 
 have the right to inflict any quantity of punishment it pleases upon any 
 of its citizens at any time, since all have sinned and incurred thereby an 
 unlimited liability to punishment. If, on the other hand, wrong-doing 
 confers a right to inflict a merely limited amount of punishment, it will 
 not be possible to determine the amount outside of utilitarian considera- 
 tions, since moral gniit cannot be measured in terms of physical pain. 
 But it is apparent that the right to inflict an infinite punishment with-
 
 394 ELEMENTS OF LOGIC 
 
 out distinction of the crimes in regard to consequences is absurd, as also 
 the infliction of a definite amount without regard to its utility, and hence 
 sin by itself and independently of the advantage to society is not punisha- 
 ble. 
 
 149. Our tariff is found fault with because it does not make men inde- 
 pendent and virtuous, besides giving them the opportunity to become 
 prosperous. It is said to be responsible for the over production which 
 has characterized some branches of manufacture. The same evil occurs, 
 and more frequently, in Lancashire under Free Trade. 
 
 150. The usefulness of government has been established by a long ex- 
 perience in the enactment and enforcing of laws against such acts as in- 
 jure social order, and if anything be needed to establish the benefits of 
 despotic governments it can be found in the power and practice exercised 
 by them to punish crimes against life and property. Therefore we may 
 infer their usefulness as forms of government. 
 
 151. If a debater affirm a proposition in the major premise which is 
 true only in an abstract sense ; that is, of the genus or conferentia, and 
 the minor premise expresses what is true of unessential properties, what 
 fallacies would be implied, first, by indicating this difference ; second, by 
 disputing the universality of the major premise, if the argument was in 
 the first Figure ; and third, by disputing the truth of either premise. 
 
 152. " Five years ago a first-class pair of nickel-plated steel skates, 
 with the necessary clamps to fasten them to the boot or shoe, cost $15. 
 To-day precisely the same article, and with an equal finish and complete- 
 ness, can be obtained for $4. Three years ago a second grade of nickel- 
 plated steel skates cost $4. The same article can be produced to-day for 
 $1.50. The decline of seventy per cent, in five years, and of sixty percent, 
 in three years, shows just how protection cheapens prices." Miltcaukcf 
 Evening Wisconsin. 
 
 153. If the earth were of equal density throughout, it would be abouf 
 2 times as dense as water ; but it is about 5 times as dense ; therefore 
 the earth must be of unequal density. 
 
 154. " ' By open discrimination, or by secret rates, drawbacks, and re- 
 bates, a few railway managers may subject to their will every business in 
 Avhich transportation is a large element of cost, as absolutely as any Ori- 
 ental despot ever controlled the property of his subjects. No civilized 
 community has ever known a body of rulers with such power to distribute 
 at pleasure, among its mercantile classes, prosperty or adversity, wealth 
 or ruin. That this is no abstract or remote danger to society is plain to 
 any man who will look at the condition of trade and of mercantile morals 
 in the United States to-day.' How vivid! But how absurd ! how un- 
 true ! Our commercial morals are equal to the highest in the world." 
 Kirkman : Railway Rates and Government Control. 
 
 155. "Not only the effects are good, but the agent sees beforehand
 
 PRACTICAL EXERCISES 395 
 
 that they will be so. This may make the action indeed (done from an- 
 tipathy) a perfectly right action ; but it does not make antipathy a right 
 ground of action. For the same sentiment of antipathy, if implicitly de- 
 ferred to, may be, and very frequently is, productive of the very worst 
 effects. Antipathy, therefore, can never be a right ground of action." 
 Bentham : Principles of Morals and Legislation. 
 
 156. "Mr. Gladstone, however, commits himself to the principle that 
 4 all protection is morally bad.' If this has been his belief ever since he 
 became an advocate of free trade, his conscience must have received 
 many and severe wounds, as session after session, while Chancellor of the 
 Exchequer, he carried through Parliament a bounty may I not say a di- 
 rect protection ? of 180,000 to a line of steamers running between Eng- 
 land and the United States a protection that began six years before free- 
 trade was proclaimed, and was continued nearly twenty years after. " Mr. 
 Blaine, in the North American Review for January, 1890. 
 
 DEDUCTIVE AND INDUCTIVE 
 
 Examine the following arguments, stating whether they are deductive 
 or inductive ; if deductive, show whether they are valid or invalid, and 
 why ; if inductive, show what Method of Induction is involved. 
 
 1. Two of the wealthiest men of the West are said to have been mes- 
 senger-boys. It pays to go slowly, after all. 
 
 2. Geometry contemplates figures. Figure is the termination of mag- 
 nitude ; but extension in the abstract has no definite determinate magni- 
 tude. Whence it follows clearly that it can have no figure, and conse- 
 quently is not the object of Geometry, whose object is commonly said to 
 be abstract extension. 
 
 3. The newly discovered painting must be a Rubens ; for the concep- 
 tion, the drawing, the tone and the tints are precisely those seen in the 
 authentic works of that master. 
 
 4. In nine counties, in which the population is from 100 to 150 per 
 square mile, the births to 100 marriages are 396 ; in sixteen counties, 
 with a population of 150 to 200 per square mile, the births are 390 to 100 
 marriages. Therefore the number of births per marriage is inversely re- 
 lated to the density of population, and contradicts Malthus's theory of the 
 law of population. 
 
 5. l< Cramming" for examination is detrimental rather than otherwise ; 
 for I have noticed that no matter what the subject is, I invariably write 
 a poor paper when I " cram," and a good one when I do not. 
 
 6. The great famine in Ireland began in 1845, and increased until it 
 reached a climax in 1848. During this time agrarian crime increased
 
 396 ELEMENTS OF LOGIC 
 
 very rapidly until in 1848 it was more than three times as great as in 
 1845. After this time it decreased with the return of better crops, until 
 in 1851 it was only fifty per cent, more than it was in 1845. It is evident 
 from this that a close relation of cause and effect exists between famine 
 and agrarian crime. 
 
 7. Pitt did not bribe the Irish parliament in 1800, when he so lavishly 
 bestowed peerages on its members. For he bestowed honors on only 
 forty of his followers, while in 1779, Lord North bestowed thirty peerages 
 in one day, and in 1832, Lord Grey got the king's consent to the creation 
 of a hundred. 
 
 8. "Suppose we have a southward velocity amounting, let us say, to 
 
 3 feet per second, and simultaneously an eastward velocity amounting to 
 
 4 feet per second, then we know by kinematics how to construct the 
 single velocity which is the resultant of these two. All we have to do is 
 to draw a line of length 3 southward and from its extremity a line of 
 length 4 to the eastward, and then complete the triangle. In a geometri- 
 cal sense, therefore, a velocity of 3 southward and a velocity of 4 east- 
 ward will be equivalent to a velocity which, if you calculate what the 
 third side of that triangle will be, is represented by 5 on that scale. ''- 
 Recent Adcances in Physical Science. 
 
 9. On May 27, 1875, a remarkable shower of small pieces of hay oc- 
 curred at Monkstown, near Dublin. They appeared floating slowly down 
 from a great height. A similar shower occurred a few days earlier in 
 Denbighshire. From this and many similar facts we may conclude that 
 the distribution of organisms of the same species over continents and 
 islands separated by the ocean has been effected by the agency of natural 
 forces. 
 
 10. The influence of heat in changing the level of the ground upon 
 which the Temple of Jupiter Serapis stands might be inferred from 
 several circumstances. In the first place, there are numerous hot springs 
 in the vicinity, and when we reflect on the dates of the principal oscilla- 
 tions of level this conclusion is made much more probable. Thus before 
 the Christian era, when Vesuvius was regarded as a spent volcano, the 
 ground upon which the temple stood was several feet above water. But 
 after the eruption of Vesuvius in 79 B.C. the temple was sinking. Subse- 
 quently Vesuvius became dormant and the foundations of the temple 
 began rising. Again Vesuvius became active, and has remained so ever 
 since. During this time the temple has been subsiding again, so far as 
 we know its history. 
 
 11. "I have two pendulums with very massive bobs suspended from 
 them, and have carefully made these two pendulums as nearly as possi- 
 ble the same. Both pendulums are now at rest, but suppose I set one to 
 vibrate, leaving the other at rest, you will notice, if you watch the second 
 for a short time that it begins to vibrate in its turn, and as time goes on
 
 PRACTICAL EXERCISES 397 
 
 it swings through larger and larger arcs of vibration till at last the first 
 pendulum is brought to rest. Now this is quite obviously a case of trans- 
 ferrence of energy from one pendulum to the other, effected, you will see, 
 through the wooden structure." Recent Advances in Physical Science. 
 
 12. Why should any but professional moralists trouble themselves with 
 the solution of moral difficulties ? For, as we resort to a physician in 
 case of any physical disease, so, in case of any moral doubt or any moral 
 disorganization, it seems natural that we should rely on the judgment of 
 some man specially skilled in the treatment of such subjects. 
 
 13. Take a bottle of soda-water, slightly warmer than a given tempera- 
 ture registered by the thermopile; and mark the deflection it causes. Then 
 cut the string which holds it, and the cork will be driven out by the 
 elastic force of the carbonic acid gas. The gas performs its work, and in 
 so doing it consumes heat, and the deflection of the thermopile shows 
 that the bottle is cooler than before, heat having been lost in the pro- 
 cess. 
 
 14. The occurrence of the Aurora Borealis under different meteorolog- 
 ical conditions is invariably accompanied by magnetic disturbances and 
 by the appearance of sun spots, and hence we infer that a causal connec- 
 tion exists between them and the sun spots. 
 
 15. It has been found that linnets when shut up and educated with 
 singing larks the skylark, woodlark, or titlark will adhere entirely to 
 the songs of these larks instead of the natural song of the linnets. We 
 may infer, therefore, that birds learn to sing by imitation, and that their 
 songs are no more innate than language is in man. 
 
 16. An enemy has a keener perception than a friend ; for, as Plato 
 says, the "lover is blind as respects the loved one," and hatred is both 
 curious and gossipy. Hiero was twitted by one of his enemies for the 
 foulness of his breath ; so he went home and said to his wife : " How is 
 this ? You never told me of it." But she, being pure and innocent, re- 
 plied : " I thought all men's breath was like that." Thus perceptible 
 and material things, and things that are plain to everybody, are sooner 
 learned from enemies than from friends. Plutarch's Morals. 
 
 17. A man cannot really be injured by his brethren, for no act of theirs 
 can make him bad, and he must not be angry with them nor hate them ; 
 for we are made for co-operation, like feet, like hands, like eyelids, like 
 the rows of the upper and lower teeth. To act against one another, then, 
 is contrary to nature ; and it is acting against one another to be vexed 
 and to turn away. ThougJits of Marcus Aurelius. 
 
 18. As an evidence of the remote antiquity of highly civilized man we 
 have the following facts : On one of the remote islands of the Pacific- 
 Easter Island two thousand miles from South America, two thousand 
 miles from the Marquesas, and more than one thousand miles from the 
 Gambier Islands, are found hundreds of gigantic stone images, uow
 
 398 ELEMENTS OF LOGIC 
 
 mostly in ruins. They are often forty feet high, while some seem to 
 have been much larger, the crowns on their heads, cut out of a red stone, 
 being sometimes ten feet in diameter, while even the head and neck of 
 one is said to have been twenty feet high. The island containing these 
 remarkable works has an area of about thirty square miles, and as the 
 smallest image is about eight feet high, weighing four tons, and as the 
 largest must weigh over a hundred tons or much more, their existence 
 implies a large population, abundance of food, and an established gov- 
 ernment which so small an island could not supply. 
 
 19. We observe very frequently that very poor handwriting character- 
 izes the manuscripts of able men, while the best handwriting is as fre- 
 quent with those who do little mental work when compared with those 
 whose penmanship is poor. We may, therefore, infer that poor penman- 
 ship is caused by the influence of severe mental occupation. 
 
 20. It has been shown by observation that overdriven cattle, if killed 
 before recovery from their fatigue, become rigid and putrefy in a sur- 
 prisingly short time. A similar fact has been observed in the case of ani- 
 mals hunted to death, cocks killed during or shortly after a fight, and 
 soldiers slain in battle. The contrary is remarked when the muscular ex- 
 ercise has not been great or excessive. Hence we may infer that cada- 
 veric rigidity depends upon a more or less unirritable condition of the 
 muscles immediately before death.
 
 ABSOLUTE terms, 47-49 
 
 'Abstract terms, 36-44, 65 ; relation to 
 fallacy of accident, 235 
 
 Abstraction, 26, 40 
 
 Accent, fallacy of, 220, 221 
 
 Accidens, 82 
 
 Accident, 82-86 ; fallacies of, 231-240 
 
 Accidentia, 88 
 
 Acquisition, 341 
 
 Affirmative propositions, 109 
 
 Agreement, principle of, 331, 332 ; 
 method of, 354 
 
 All, logical signification of, 111 
 
 Alternative, 109, 212 
 
 Abignity of terms, 50-67 ; of proposi- 
 tions, 115-121 
 
 Ambiguous middle, 226, 239 
 
 Ampliative propositions. 112 
 
 Analytic propositions, 112 
 
 Antecedent, 108, 204 
 
 Antithesis, 169 
 
 Any, logical signification of, 111 
 
 Argumentum ad rem, 250 ; ad judi- 
 cium, 250 ; ad populum, 251 ; ad 
 hominem, 251 ; ad ignorantiam, 251 ; 
 ad verecundiam, 251 
 
 Aristotle, 248, 264, 267 
 
 Attributive terms, 37 
 
 BACON, 298 
 Bain, 32 
 
 Barbara, Celarent, etc., 190 
 Begging the question, 240 
 Benjamin Franklin, 306 
 Bentham, Jeremy, 243 
 Bentham, George, 262 
 Berkeley, 250 
 Brewster, Sir David, 332 
 26 
 
 CATEGOREMATIC terms, 31 
 
 Categorical propositions, 107 ; reason- 
 ing, 204 
 
 Categories, 82 
 
 Cause, fallacy of false, 255 
 
 Circulus in definiendo, 104 
 
 Circulus in probando, 241, 244 
 
 Classification, 343 ; of fallacies, 219-227 
 
 Cognition, 22 
 
 Collective terms, 34, 35 
 
 Comparison, 23 
 
 Complimentary propositions, 116 
 
 Complex syllogisms, 197 
 
 Composition, fallacy of, 228 
 
 Comprehension. See Intension 
 
 Concepts and conceptions defined, 20 ; 
 formation of, 21-27; kinds of, 25- 
 26, 31-49 ; denomination of, 26 ; in- 
 tension and extension of, 68-81 ; 
 analysis of, 94 ; infinitated, 165 
 
 Conception, 5, 16, 19-27, 28 
 
 Conclusion, 29, 171 
 
 Conclusion, weakened, 184 
 
 Concomitant variations, method of, 357 
 
 Concrete terms, 36-44 ; relation to fal- 
 lacy of accident, 235 
 
 Conditional propositions, 107 
 
 Conferentia, 87, 88, 91, 102, 126, 127, 
 234, 235, 278, 323 
 
 Confusion of fallacies, 256-261 
 
 Connotation of terms, 79 
 
 Connotative, meaning of, 79-81 
 
 Consequent, 108, 204 ; fallacy of false, 
 253 
 
 Contradiction, law of, 291 
 
 Contradictory, 142 
 
 Contraposition. See Contraversion, 
 note on, 163
 
 400 
 
 INDEX 
 
 Contraries, 142, 144 
 
 Contraversion, 161-166 
 
 Contribution, 168 
 
 Converse, 156 
 
 Converse fallacy of accident, 231 
 
 Conversion, 155-160; simple, 156,264; 
 per accidens or limited, 156 ; rules 
 for, 1 50 ; by negation, 159 ; table of, 
 160 
 
 Convertend, 156 
 
 Co-ordinate species, 96 
 
 DEDUCTION, 279, 300, 301, 327, 358 
 
 Deductive reasoning, 15, 279, 295, 327, 
 328 
 
 Deductive method, 337 
 
 Definite propositions, 110; quantity, 
 267 
 
 Definition, 100-104; kinds of, 100; 
 logical, 101 ; rules for, 103 ; in meth- 
 od, 337 
 
 Definition and division, 82-104 
 
 De Morgan, 233, 238, 243, 247, 252, 255, 
 262 
 
 Denomination of concepts, 26 
 
 Denotation and connotation of terms, 
 78 
 
 Denotative, meaning of, 79-81 
 
 Descriptive definition, 100 
 
 Destructive dilemma, 217 ; hypotheti- 
 cal reasoning, 205 
 
 Desynonymization of terms, 60 
 
 Difference, principle of, 331, 333 ; 
 method of, 355 
 
 Differentia, 82, 86, 90 
 
 Differential or specific accident, fallacy 
 of, 231 
 
 Differentiation of terms, 60 
 
 Dilemma, 217; constructive, 217; de- 
 structive, 217 
 
 Dilemmatic propositions, 108 
 
 Direct reduction, 190 
 
 Disjunction, incomplete, 214 
 
 Disjunctive propositions, 107 ; syllo- 
 gisms, 212-218 
 
 Distribution of terms, 138 ; rules for, 
 140 
 
 Distributive terms, 34, 35 
 
 Division, definition and, 82-104; logi- 
 cal, 94 ; rules for, 95 ; dichotomous, 
 97 ; trichotomous, 97 
 
 Divisionis, f undamentum, 95 
 
 Dr. Johnson, 250 
 
 Duplex propositions, 1 16 
 
 Doctrine, elements of logical, 16-30 
 
 EXTHYMEME, 198 
 
 Epicheirema, 200 
 
 Episyllogism, 198 
 
 Equivocal terms, 51 
 
 Equivocation, fallacies of, 225, 228- 
 240 
 
 Essence, 86 
 
 Essentia, 85, 88 
 
 Essential properties, 84 ; propositions, 
 112 
 
 Ethics, 14 
 
 Etymological definition, 100 
 
 Euler, 130 
 
 Exclamatory propositions, 107 
 
 Exceptive propositions, 120 
 
 Excluded middle, law of, 145 
 
 Exclusive propositions, 118; quantity 
 of, 119 
 
 Experiment, 342, 352 
 
 Explicative propositions, 112 
 
 Extension of concepts, 68-81 ; relation 
 between intension and, 72 ; law of, 
 and intension, 73 
 
 Extension, judgments of, 123 ; sym- 
 bolized, 130, 131 
 
 FALLACIA conseqnentis, 253 
 
 Fallacies, classification of, 219-227; 
 hermeneutic, 220; formal or logical, 
 223 ; material, 225, 228-261 ; of am- 
 phibology, 220 ; of accent, 221 ; of 
 four terms, 223 ; of illicit middle, 
 224 ; of illicit major, 224 ; of illicit 
 minor, 224 ; of particular premises, 
 225 ; of negative premises, 225 ; of 
 equivocation, 226, 228 ; of composi- 
 tion and division, 228 ; of accident, 
 231 ; of presumption, 240 ; of petitio 
 principii, 240 ; of non sequitur, 253 ;
 
 INDEX 
 
 401 
 
 of false cause, 255 ; confusion of, 
 
 256-261 ; inductive, 363 
 Fallacy, definition of, 219 
 False cause, fallacy of, 255 
 False consequent, 253 
 False propositions, 144 
 Figures of the syllogism, 183 
 Form and matter, 9, 1 6 
 Formal fallacies. 223 
 Formal logic, 1, 13, 15 
 Fowler, 79, 81, 318, 321, 324, 364 
 Fundamentum divisionis, 95 
 
 GENERAL terms, 34-36 ; mathemati- 
 cal generals, 25, 75, 236 ; logical gen- 
 erals, 25, 101, 236 ; propositions, 110 
 
 Generalization of terms, 57 ; of obser- 
 vations, 290 
 
 Genus, 82, 86, 88 ; mathematical, 93 ; 
 logical, 93; contrasted with differen- 
 tia, 91 , 92 ; summum, 90 
 
 Grammatical division of propositions, 
 107 
 
 HAMILTON, 1, 4, 73, 95, 154, 262, 266, 
 267, 270, 271, 275, 285, 291, 338 
 
 Hypothesis, 343 
 
 Hypothetical propositions, 107 ; rea- 
 soning, 204-211 
 
 IDENTITY, law of, 145 
 
 Ignoratio elenchi, 149, 153, 241-253 
 
 Illicit process of major term, 1 74, 177, 
 224 ; of middle term, 174, 177, 224 ; 
 of minor term, 174, 178, 224 
 
 Immediate inference, 154-170 
 
 Imperfect induction, 295 
 
 Incomplete disjunction, 214 ; syllo- 
 gisms, 197 
 
 Indefinite propositions, 110 ; quantity, 
 267 
 
 Indirect reduction, 193 
 
 Individual terms, 20, 32 
 
 Induction, nature of, 295-313 ; perfect, 
 295, 296, 300-302, 320, 327; princi- 
 ples of, 328 ; fallacies of, 3(33-366 
 
 Inductive logic, 15, 295 ; method, 298, 
 340, 353; inference, 313, 321-323; 
 syllogism, form of, o06 
 
 Inference, nature of, 154 ; immediate, 
 155 ; mediate, 154, 171-189 ; by 
 privative conception, 160 ; by con- 
 tribution, 168; deductive, 15, 279, 
 295, 327, 328; inductive, 313, 321- 
 325. 
 
 Infima species, 90 
 
 Infinitated conceptions, 46, 165. 
 
 Intension of terms or concepts, 68- 
 81 ; relation between, and exten- 
 sion, 72 ; law of, and extension, 73 
 
 Introspection, 342 
 
 Inversion, 156 
 
 Inverted propositions, 1 1 5 
 
 Irrelevant conclusions, 245 
 
 S 
 
 JEVONS, 37, 50, 63, 04, 90, 96, 103, 159, 
 162, 193, 213, 220, 232, 239, 244, 263, 
 264 note, 265 note, 280, 312, 348 
 356 
 
 Johnson, Dr., 250 
 
 Joint method of agreement and dif- 
 ference, 358 
 
 Judgment, 5, 16, 28-29 ; intensive and 
 extensive, 123 
 
 KaTriyopcv/icvov, 106 
 
 Keynes, 13, 32, 37, 38, 79, 81, 187/193, 
 213, 70 
 
 LANGUAGE, 27 
 
 Law, conception of, 6-9 ; of thought, 
 6-9; of intension and extension, 
 73; of contradiction, 145, 291; of 
 excluded middle, 145, 292 ; of iden- 
 tity, 145, 291 ; of sufficient reason, 
 292, 329; of universal causation, 
 318, 329 ; of the uniformity of nat- 
 ure, 318, 329, 331 
 Laws of thought, 6-9, 290-294 
 Limitation, conversion by, 156, 157 
 Logic, definition of, 1 ; relation to 
 the other sciences, 12 ; divisions of, 
 15; inductive and deductive, 15 
 Logical concepts, 26; 276, 277 
 Logical definition, 101 ; generals, 75, 
 
 101, 236 
 
 Logical doctrine, formal elements of, 
 18
 
 402 
 
 INDEX 
 
 MAJOB term, 171 ; illicit process of, 
 174, 224 
 
 Mai-observation, 364 
 
 "Margarita Philosophica," 233 
 
 Matter, form and, 9 
 
 Material fallacies, 225, 228-261 
 
 Mathematical concepts, 25, 26, 276, 
 277 
 
 Mathematical generals, 75, 236 ; rea- 
 soning, 275-289 
 
 Mediate reasoning, 171-189 
 
 Method, scientific, 300 ; inductive, 
 298, 340, 353; deductive, 337; of 
 agreement, 297, 354; of difference, 
 297, 355 ; of concomitant varia- 
 tions, 357; of joint agreement fend 
 difference, 358 ; of residues, 358 
 
 Middle term, 171, 174-180, 224 
 
 Mill, J. S., 37, 41, 70, 79, 80, 81, 203, 
 247, 250, 317, 355 
 
 Minor term, 171 ; illicit process of, 
 174-180, 224 
 
 Mnemonic lines, 190 
 
 Miscellaneous propositions, 114 
 
 Modal propositions, 114 
 
 Modus ponendo tollens, 216 
 
 Modus ponens, 205 
 
 Modus tollendo ponens, 216 
 
 Modus tollens, 205 
 
 Moods of the syllogism, 181-189 
 
 NEGATION, conversion by, 159 
 Negative propositions, 109 ; distribu- 
 tion of, 140 
 
 Negative terms, 44 ; signs of, 45 
 Nego-positive terms, 44-46 
 Newton, 304, 350, 360 
 Non causa pro causa, 255 
 Non-observation, 363 
 Non sequitnr, 253, 856 
 Notion, 21 
 
 OBSERVATION, 342, 351 ; errors of, 363 
 
 Obversion, 160 
 
 Opposition, 141-153 ; application of 
 
 the principles of, 144 ; rules for, 144 ; 
 
 square of, 144 
 Or, signification of, 109, 212 
 
 PARTICULAR propositions, 109, 110; 
 signs of, 112 
 
 Partition, 99 
 
 Partitive propositions, 116 
 
 Per accidens, conversio, 156 
 
 Percepts, 22 
 
 Petitio argument!, 241 
 
 Petitio principii, fallacy of, 240-245 
 
 Plurative propositions, 110 
 
 Porphyry, tree of, 97 
 
 Positive terms, 44 
 
 Post hoc, ergo propter hoc fallacy, 255 
 j Predicables, 82-94 
 
 Predicate, 106; distribution of, 138; 
 relation between subject and, 122- 
 140. See also Quantification of pred- 
 icate 
 
 Preindesignate terms, 267 
 
 Premises, 29, 171 
 
 Primary laws of thought, 290 
 
 Privative conception, 44 
 
 Probation, 338 
 
 Proof. See Probation 
 
 Proper names, 31-33 
 
 Property, 82 ; kinds of. 82-85 ; table, 
 88 ; propositions, 105-121 ; defined, 
 17 ; divisions of, 197, 121 
 
 Prosyllogism, 198 
 
 Psychology, 13 
 
 Pure propositions, 114, 277, 310-312 
 
 QUALITATIVE reasoning, 277, 310-312 
 Quality of propositions, 109 
 Quantification of the predicate, 262- 
 
 274 
 Quantity of tsrms. See Extension ; of 
 
 propositions, 109 
 Quanto-qualitative reasoning, 277 
 Quaternio terminorum, 223 
 
 REASON, law of sufficient, 329 
 Reasoning, 5, 16, 29-30, 171-180; 
 forms of, 197-203 ; hypothetical, 
 204-211 ; disjunctive, 212-218 ; math- 
 ematical and other, 275-289 ; quan- 
 titative, 277 ; qualitative, 277 
 Reduction of the moods and figures, 
 190-196; of hypothetical syllogisms 
 to categorical, 209
 
 INDEX 
 
 403 
 
 Reisch, 233 
 Relative terms, 47-49 
 Residues, method of, 358 
 Rules of the syllogism, 173 
 
 SCIENCE AND ART, 1-2 
 
 Sciences, mathematical and metaphys- 
 ical, 25 
 
 Scientific method, 300-366 
 
 Secondary laws of thought, 293 
 
 Simple accident, fallacy of, 231 
 
 Simple conversion, 158 
 
 Singular propositions, 110 
 
 Singular terms, 31-33 
 
 Socrates, 296 
 
 Sorites, 201 
 
 Specialization, 59-63 
 
 Species, 82, 88 ; contrasted with genus, 
 91-92 ; and fallacy of accident, 231, 
 335 
 
 Specific accident, fallacy of, 231 
 
 Subaltern propositions, 143 
 
 Subalternans, 143 
 
 Subalternate, 143 
 
 Subcontrary propositions, 143 
 
 Subject and predicate, 106 ; distribu- 
 tion of, 138 ; relation between, 122- 
 140 
 
 Subordinate species, 96 
 
 Sufficient reason, law of, 329 
 
 Summum genus, 90 
 
 Superordinate conceptions, 96 
 
 Syllogism, 18, 171-180 ; forms of, 197- 
 203; hypothetical, 204-211 ; disjunc- 
 tive, 212-218 ; inductive, 306 
 
 Syncategorematic terms, 31 
 
 Synthesis of percepts, 20 
 
 Synthetic propositions, 112 
 
 TAUTOL'OGOUS propositions, 114 
 Terms, 31-49; ambiguity of, 50-67; 
 
 intension and extension of, 68-81 ; 
 
 denotation and connotation of, 79 ; 
 
 distribution of, 1 38 
 Terminorum, quaternio, 223 
 Theory, 344 
 Thompson, 1, 266 
 Thought, definition of, 3-6 ; nature of 
 
 its laws, 6-9 ; laws of, 290-294 
 Traduction, 279, 310, 312 
 Tree of porphyry, 97 
 Truistic propositions, 114 
 
 UEBERWEG, 1, 134 
 
 Universal propositions, 109, 110 ; 
 
 signs of, 111 ; distribution of, 140 
 Univocal terms, 51 
 
 bnoKttjjitvov, 106 
 Verification, 341, 349 
 
 WATTS, DR., 1 
 Whately, 1, 233, 245, 247 
 Weakened conclusion, 184 
 
 THE END.