AN INTRODUCTORY LOGIC AN INTRODUCTORY LOGIC BY JAMES EDWIN CREIGHTON NEW EDITION', REVISED AND CORRECTED THE MACMILLAN COMPANY LONDON : MACMILLAN & CO., LTD. I9O6 All rights reserved COPYRIGHT, 1898, 1900, BY THE MACMILLAN COMPANY. Set up and electrotyped September, 1898. Reprinted July, 1899. New Edition, Revised and Corrected, March, 1900 ; April, October, 1901 ; July, 1902 ! February, 1903 ; February, August, 1904; January, October, 1905; September, 1906. Staek PREFACE THIS volume is intended primarily as a text-book for college students, and grew out of my lectures on Logic to undergraduate classes in Cornell University. It aims at being both practical and theoretical. In spite of the obvious deficiencies of formal Logic as a theory of the nature of thought, I am convinced that it is one of the most valuable instruments in modern education for promoting clear thinking, and for developing criti- cal habits of mind. J. S. Mill, speaking in the Auto- biography of the discipline which he received from working logical exercises, expresses the following opinion : " I am persuaded that nothing, in modern education, tends so much, when properly used, to form exact thinkers, who attach a precise meaning to words and propositions, and are not imposed on by vague, loose, or ambiguous terms." Although in treating the syllogistic Logic I have followed to a large extent the ordinary mode of presentation, I have both here, and when dealing with the Inductive Methods, endeavoured to interpret the traditional doctrines in a philosophical way, and to prepare for the theoretical discussions of the third part of the book. The advisability of attempting to include a theory of thought, or philosophy of knowledge, even in outline, Vi PREFACE in an elementary course in Logic, may at first sight appear doubtful. It seems to me, however, that this inclusion is not only justifiable, but even necessary at the present time. Psychology is no longer a ' philoso- phy of mind ' ; but, under the influence of experimental methods, has differentiated itself almost entirely from philosophy, and become a ' natural ' science. As a natural science, it is interested in the structure of the mental life, the characteristics of the elementary processes, and the laws of their combination, and not primarily in the function which ideas play in giving us knowledge. It is clear that psychology does not undertake to describe all that mind is and does. It belongs to Logic to investigate intelligence as a know- ing function, just as it is the task of Ethics to deal -with the practical or active mental functions. The practical question still remains as to whether this side of Logic can be made profitable to students who have had no previous philosophical training. I am well aware of the difficulty of the subject, but my own experience leads me to believe that the main con- ceptions of modern logical theory can be rendered intelligible even to elementary classes. Of the incom- pleteness and shortcomings of my treatment I am quite conscious ; but I have endeavoured to make the matter as simple and concrete as possible, and to illustrate it by means of familiar facts of experience, For a number of the practical questions and exer- cises, I am indebted to Professor Margaret Washburn of Wells College ; others are original, or have been collected in the course of my reading. I have also PREFACE vii taken a number of arguments from the examination papers of different universities, and from various works on Logic, especially from Jevons's Studies in Deductive Logic, from the little volume entitled Questions on Logic by Holman and Irvine (2d ed., London, 1897), and from Hibben's Inductive Logic. In writing the book, I have been under obligation to a large number of writers and books. My heaviest debt is doubtless to Bosanquet, and perhaps next in order I am under obligations to Mill, Jevons, Sigwart, and Bradley. I have also derived help from Minto's Logic, Deductive and Inductive, the chapter on ' Rea- soning' in James's Principles of Psychology, J. H. Hys- lop's Elements of Logic, and from other works to which reference is made throughout the book. My colleagues in the Sage School of Philosophy have kindly aided me from time to time with advice and encouragement, and I have also received valuable suggestions from other teachers of Logic with whom I have talked and corresponded. In particular, I wish to express my obligations to my former colleague, Pro- fessor James Seth, who read nearly all of the book in manuscript, and to Dr. Albert Lefevre, who kindly assisted me in reading the proofs. J. E. C. CORNELL UNIVERSITY, August, 1898. TABLE OF CONTENTS INTRODUCTION CHAPTER I THE STANDPOINT AND PROBLEM OF LOGIC PACK I. Definition of the Subject I 2. Relation to Psychology 4 3. Logic as a Science and an Art 8 4. The Material of Logic 13 CHAPTER II IMPORTANT STAGES IN THE DEVELOPMENT OF LOGIC 5. The Logic of the Greeks : Aristotle 1 8 6. Logic during the Middle Ages 26 7. The Logic of Bacon 28 8. Logic since the Time of Bacon 29 PART I. THE SYLLOGISM CHAPTER III THE SYLLOGISM AND ITS PARTS 9. The Nature of the Syllogism 10. The Parts of the Syllogism n. The Proposed Division of Mental Operations . . . CHAPTER IV THE VARIOUS KINDS OF TERMS 12. Singular, General, and Collective Terms . . 13. Abstract and Concrete Terms . > . , . .. . TABLE OF CONTENTS 14. Positive and Negative Terms 52 15. Absolute and Relative Terms 54 1 6. Extension and Intension of Terms 55 ' CHAPTER V DEFINITION AND DIVISION 17. Fixing the Meaning of Terms 61 18. Definition 63 19. Division . . .'.''. 71 CHAPTER VI PROPOSITIONS 20. The Nature of a Proposition 78 .21. The Quality and Quantity of Propositions . . . .80 22. Difficulties in Classification 83 23. Formal Relation of Subject and Predicate .... 85 CHAPTER VII THE INTERPRETATION OF PROPOSITIONS 24. The So-called Process of Immediate Inference ... 92 25. The Opposition of Propositions 94 26. The Obversion of Propositions 98 27. The Conversion of Propositions loo CHAPTER VIII THE SYLLOGISM 28. The Nature of Syllogistic Reasoning 105 29. The Rules of the Syllogism 108 30. The Figures of the Syllogism . ... . ."3 CHAPTER IX THE VALID MOODS AND THE REDUCTION OF FIGURES 31. The Moods of the Syllogism . . . . . . . 115 32. The Special Canons of the Four Figures . . . . 1 1 7 33. The Determination of the Valid Moods in Each of the Figures 120 34. The Mnemonic Lines 122 TABLE OF CONTENTS xi CHAPTER X ABBREVIATED AND IRREGULAR FORMS OF ARGUMENT PAGE 35. Enthymemes 126 36. Episyllogisms and Prosyllogisms . . . . . . 127 37. Sorites, or Chains of Reasoning . . . . . .129 38. Irregular Arguments . . . . . . . .132 CHAPTER XI HYPOTHETICAL AND DISJUNCTIVE ARGUMENTS 39. The Hypothetical Syllogism 136 40. Relation of Categorical and Hypothetical Arguments . .139 41. Disjunctive Arguments ' . . 145 42. The Dilemma . 148 CHAPTER XII FALLACIES OF DEDUCTIVE REASONING 43. Classification of Fallacies . . . . . . 44. Errors in Interpretation ....... 45. Formal Fallacies 46. Material Fallacies PART II. INDUCTIVE METHODS CHAPTER XIII THE PROBLEM OF INDUCTION. OBSERVATION AND EXPLANATION 47. The Problem of Induction 172 48. Observation . . . . ~ 176 49. Explanation . . 182 CHAPTER XIV METHODS OF OBSERVATION. ENUMERATION AND STATISTICS 50. Enumeration or Simple Counting . . . . . . 185 51. Statistics and Statistical Methods . . . . . . 189. 52. The Calculation of Chances . . . . . . .194 xii TABLE OF CONTENTS CHAPTER XV METHODS OF OBSERVATION. DETERMINATION OF CAUSAL RELATIONS fAGB 53. Mill's Experimental Methods 198 54. The Method of Agreement 200 55. The Method of Difference 205 CHAPTER XVI METHODS OF OBSERVATION. DETERMINATION OF CAUSAL RELATIONS (continued} 56. The Joint Method of Agreement and Difference . . . 209 57. The Method of Concomitant Variations . . . . .211 58. The. Method of Residues 213 CHAPTER XVII METHODS OF EXPLANATION. ANALOGY 59. Explanation by Analogy 219 60. Analogy as Suggestive of Explanatory Hypotheses . . 223 6l. The Incompleteness of Analogical Reasoning . . . 226 CHAPTER XVIII METHODS OF EXPLANATION. THE USE OF HYPOTHESES 62. Reasoning from Hypotheses 230 63. The Formation of Hypotheses 234 64. The Proof of an Hypothesis 237 65. Requirements of a Good Hypothesis 240 . CHAPTER XIX FALLACIES OF INDUCTIVE REASONING 66. The Source of Fallacy 67. Fallacies due to the Careless Use of Language 68. Errors of Observation 69. Mistakes in Reasoning ....... 70. Fallacies due to Individual Prepossessions . '-. .;.-/.> ' , TABLE OF CONTENTS xiii PART III. THE NATURE OF THOUGHT CHAPTER XX JUDGMENT AS THE ELEMENTARY PROCESS OF THOUGHT PAGE 71. Thinking the Process by which Knowledge grows or develops 260 72. The Law of Evolution and its Application to Logic . . 262 73' Judgment as the Starting-point ...... 266 74. Concepts and Judgment . . . . . . . 268 CHAPTER XXI THE MAIN CHARACTERISTICS OF JUDGMENT 75. The Universality of Judgments 274 76. The Necessity of Judgments 276 77. Judgment involves both Analysis and Synthesis . . . 279 78. Judgment as constructing a System of Knowledge . . 284 CHAPTER XXII THE LAWS OF THOUGHT 79. The Law of Identity 288 80. The Law of Contradiction 295 81. The Law of Excluded Middle 297 CHAPTER XXIII TYPES OF JUDGMENT 82. Judgments of Quality 300 83. Judgments of Quantity 304 84. Judgments of Causal Connection 307 85. Judgments of Individuality 315 CHAPTER XXIV THE NATURE OF INFERENCE. INDUCTION AND DEDUCTION 86. Judgment and Inference 318 87. The Nature .of Inference 324 88. Induction and Deduction ....... 329 xiv TABLE OF CONTENTS CHAPTER XXV RATIONAL AND EMPIRICAL THEORIES PAGE 89. The Point of View of Rationalism 335 90. The Doctrine of Empiricism 337 91. Reasoning from Particular ta Particular .... 340 92. Reasoning from Particulars to a Universal .... 344 QUESTIONS AND EXERCISES 348 INDEX 389 AN INTRODUCTORY LOGIC INTRODUCTION CHAPTER I THE STANDPOINT AND PROBLEM OF LOGIC i. Definition of the Subject. Logic may be defined as the science of thought, or as the science which in- vestigates the process of thinking. Every one knows, in a general way at least, what is meant by think- ing, and has noticed more or less consciously some of its peculiarities. Thinking is the intellectual act by means of which knowledge is obtained. We do not really know any fact until we think it ; that is, until the mind sets it in its proper relation to the other parts of its experience, and thus comes to understand its true meaning. We make a distinction, for example, between what has come to us through report or hearsay, and conclusions which we have reached by our own think- ing. ' I have heard,' we say, ' that A is dishonest, but I do not know it.' That is, this fact has not been reached as a result of our own thinking, and cannot therefore claim the title of knowledge. On the other hand, that the earth is round, is not a mere matter of hearsay for an educated man. It is a piece of know- ledge, because it is a conclusion which he has reached by thinking, or by putting together various facts for himself. B I 2 THE STANDPOINT AND PROBLEM OF LOGIC Logic, then, in dealing with thinking, is concerned with the process by which knowledge is obtained. In defining it as a science, we mean that it seeks to sub- stitute exact and systematic knowledge regarding the nature of thought for the popular notions to be found in everyday life. Like all the sciences, logic has to correct and supplement ordinary knowledge. It is its mission to help us to understand more exactly and completely the way in which thinking goes on, and to discover the laws which are followed in gaining knowledge. But it is also the business of a science to system- atize facts. Logic, then, cannot content itself with a mere description of this or that kind of thinking, in isolation from other ways in which we think. It must also deal with the way in which the various kinds of thinking are related. For example, we apply such terms as 'conception,' 'judgment,' 'induction,' and 'de- duction ' to different intellectual operations, and give the distinguishing characteristic in each case. But it is necessary as well to understand how these processes are related. Since all thinking has one end, the dis- covery of truth, the various intellectual operations must mutually cooperate and assist in this result. All of the logical processes, then, stand in relation to one another. They are all parts of the one intelligence, though they may well represent different stages or steps in its work of obtaining knowledge. It becomes the business of logic, then, to show us the organic structure of thought. In other words, it must furnish a comprehensive view of the way in which intelligence i. DEFINITION OF THE SUBJECT 3 acts, and the part which processes like ' conception,' 'judgment,' 'induction,' etc., play. (1) The word 'logic 1 is derived from the adjective corresponding to the Greek noun Aoyos, which signifies either a complete thought, or a word as the expression of that thought. The singular form of the adjective Aoyuo/, from which the English word is derived, was supposed to qualify either eTrtoT??/^ as applying to the theoretical science of logic, or re'xv?? as referring to the practical application of its rules and as affording guidance in the art of correct reason- ing. We shall have to raise the question in a subsequent section how far it is possible to regard logic as an art, or a system of rules which teach us how to reason correctly. (2) We have defined logic as the science of the operations and processes of thought, or as the science of thinking. It is evident, however, that this definition does not carry us very far unless we know what thinking means. And to gain a clearer idea of this com- mon term may be said to be the problem of logic. This is, however, by no means as easy a task as may at first appear. Familiar words and phrases often conceal difficulties. They are constantly repeated without reflection, and this very frequency of repetition is likely to prevent us from trying to gain any clear ideas regarding the nature of the objects which they denote. It is only when we become discontented with our knowledge regarding any subject, when doubts arise whether we really understand the meaning of the words which we use, that we attempt to make our knowledge scientific, i.e., to gain clear, definite, and systematic ideas. This can perhaps be made clearer by considering the main differences between an educated and an uneducated man. The educated man has, of course, a great deal more information than the other, and his knowledge is more definite and systematic. But a second and more important distinction is found in the attitude of mind which education begets. The educated man is desirous of knowing more, because he is sensible of his own ignorance. The uneducated man, on the other hand, supposes that he knows all about things whose names are familiar to him. He can settle puzzling theo- logical or political problems off-hand in a way which is per* 4 THE STANDPOINT AND PROBLEM OF LOGIC fectly satisfactory to himself, without study, and almost without reflection. It is clear that no intellectual salvation is possible for a man so long as he remains in this state of mind. A sense of one's own ignorance is the beginning of wisdom. Socrates, one of the great pioneers of science among the Greeks, and especially of the sciences of logic and ethics, was so firmly convinced of this that he made it the business of his life to go about the streets of Athens and con- vince those " who thought they were wise and were not wise," of their ignorance. " And because I did this," he says naively, " many of them were angry, and became my enemies." 2. Relation to Psychology. It may aid us in obtaining a clearer view of what thinking is, if we compare the general standpoint of logic with that of psychology. Both of these sciences deal with what goes on in mind or consciousness, and are thus opposed to the so-called objective sciences, which are all con- cerned with some group or field of external facts. But in spite of this agreement, there is an important dis- tinction between logic and psychology. In the first place, psychology deals with all that there is in mind. It describes pleasures and pains, acts of will, and the association of ideas, as well as what is usually called logical thinking. But logic does not differ from psy- chology simply by being less inclusive than the latter. It is true that, from the standpoint of psychology, the thought-process is merely a part of the mental content, which has to be analyzed and described like anything else which goes .on in consciousness. Thinking has doubtless for psychology peculiar marks or charac- teristics which distinguish it from other related pro- cesses like those of association; but when these have 2. RELATION TO PSYCHOLOGY 5 been found, and the psychological description of think- ing is complete, the question with which logic deals has not yet been raised. For logic, as we shall see pres- ently, adopts a different standpoint, and investigates with a different end in view. The important difference is this : In psychology we are interested in the content of consciousness for its own sake, and just as it stands. We try to find out what actually goes on in our minds, and to describe it just as we should any event which occurs in the exter- nal world. But in logic the question is not : What are mental processes ? but rather : What knowledge do they give us, and is this knowledge true or false? Logic, in other words, does not regard the way in which ideas exist, and is not interested in them for what they are, but rather in the purpose which they sub- serve in affording us knowledge of something beyond themselves. Psychology, in its description of conscious states, inquires regarding their quality, intensity, dura- tion, etc., and the ways in which they combine with each other to form complex ideas. The problem with which logic is concerned, on the other hand, has refer- ence to the value of ideas when they are taken to represent facts in the real world. In other words, the question which logic raises is not regarding the actual character of ideas as existing processes, but regarding their value or significance as pieces of knowledge. (i) The relation between logic and psychology may perhaps be illustrated by referring to that which exists between morphology and physiology. Morphology deals with the form and structure of living organisms, and physiology with the various acts and func- tions which these organisms discharge. Thus we speak of the former as the science of form or structure, and of the latter as the science of function. In the same way, psychology may be said to deal with the actual structure of mental processes, and logic with the part which they play in giving us knowledge. It must be noticed, however, that this is a distinction made for purposes of investigation, and does not denote that structure and function have nothing to do with each other. On the contrary, some knowledge of the function is often necessary in order to under- stand the structure of an organ ; and, on the other hand, it is usually true that the nature of a function only becomes completely intelligi- ble when the character of the mechanism with which it works is known. And the same holds true, I think, of the relations between psychology and logic. Although it has been found profitable when dealing with consciousness, as in the biological realm, to investigate the nature of structure and function separately, yet here, as there, the two lines of inquiry cross each other ; for it is beyond question that the knowledge we obtain by thinking is largely dependent upon the character (quality, intensity, etc. ) of the actual processes in con- sciousness. To understand the nature of a logical idea, then, it is often necessary to refer to the psychological facts and their actual mode of behaviour. And it is equally true that one cannot carry on a psychological investigation into the nature of mental processes without taking account, to some extent, of the part which they play in giving us knowledge. No psychology is able to take ideas simply as existing conscious processes to which no further meaning or importance attaches ; it is only with reference to the function they perform as knowing states that their own peculiar character can be understood. In other words, the intellectual activities and purposes of mind must be presupposed in psychology, though this science, for the most part, goes its way as if the ideas were not cognitive at all. At least this seems to be true of the 'new' or experimental psy- chology as opposed to the philosophies of mind. (2) It would of course be presumptuous, as well as utterly useless, for any writer to draw a hard and fast line between logic and psy- chology, and to forbid others to overstep it. In attempting to dis- 2. RELATION TO PSYCHOLOGY 7 cover the dividing line between two closely related sciences one must be guided by the procedure of those who are working in the fields which it is proposed to divide. Now, it must be admitted that by no means all of the recent writers in psychology limit the sphere of their science in the way above described ; that is, there are certain psychologists who do not confine their attention to the mere mental processes as such, but include in their investigations the fur- ther problem regarding the part which these processes play in giving us knowledge. Thus in Professor James's Principles of Psychology there is an excellent chapter on ' Reasoning' which certainly con- tains as much logical as psychological matter. In the same way, one finds problems of knowledge discussed in the psychological writings of Professor Ladd, and also, to some extent, in the recent work by Mr. Stout entitled Analytic Psychology. In spite of this, it is evident that the tendency of the ' new, 1 or laboratory psy- chology, is towards a sharper differentiation of its problems from those of logic. The ' natural science of psychology ' is interested in the conscious process as an event in time with certain defi- nitely ascertainable characteristics. It is perhaps not a matter of great moment whether the name 'psychology' be limited to this kind of inquiry, or whether philosophical inquiries regarding the nature of knowledge be also included under it. I have assumed, however, in this section, that psychology is now being differentiated from the more general inquiries regarding the nature of mind, and that it has taken for its field of investigation the nature of mental processes regarded merely as mental processes. Consider a little further the nature of the ideas with which logic deals. Every idea, as we have seen, not only exists in some definite fashion in some particular consciousness, connected with certain other ideas, and with a definite quality, intensity, etc., but it has a mean- ing or significance as a piece of knowledge. It not only is something, but it also stands for or signifies something. Now it is not with the existence, but with 8 THE STANDPOINT AND PROBLEM OF LOGIC the meaning side of ideas that logic has to do. A logical idea, or piece of knowledge, is not merely a modification of consciousness which exists in the mind of some individual at a particular time. For example, the proposition : ' The three angles of a triangle are equal to two right angles,' will give rise to a number of definite psychological processes (probably auditory or visual in character) in the mind of any individual. These processes would also probably differ in character in the case of two persons. The meaning of the propo- sition, however, is distinct from the definite processes which arise in particular minds. The proposition has a significance as an objective fact, or piece of know- ledge, outside my mind ; the psychological images or processes may differ for different persons, but the fact expressed is the same for all minds and at all times. $ 3. Logic as a Science and an Art. We have de- fined logic as the science of thought, but it has often been pointed out that there are equally strong reasons for considering it to be an art. Jevons makes the distinction between a science and an art very clear by saying that " a science teaches us to know, and an art to do." A science is interested in the discovery of facts and laws without any thought of what use may be made of this knowledge; an art, on the contrary, gives practi- cal guidance and direction for some course of action. The question before us, then, is this : Does logic merely give us knowledge about the ways in which we think, or does it also help us to think rightly ? Before we attempt to answer this question, we must 3. LOGIC AS A SCIENCE AND AN ART 9 note that practical rules of action are based upon sci- entific knowledge. An art, in other words, depends upon science, and grows in perfection with the advance of scientific knowledge. Thus medicine, as the art of healing, is founded upon the sciences of chemistry, physiology, and anatomy, and it is because of the great discoveries which have been made in these fields within recent years, that it has been able to advance with such gigantic strides. Again, the art of singing, in so far as it is an art which can be taught and learned, depends upon a knowledge of the physical and physiological laws of the vocal organs. An art, then, always pre- supposes a certain amount of science, or knowledge, and is simply the application of this knowledge to some practical purpose. In some cases the application is very obvious and direct; in others, it is much more difficult to determine ; but, in general, there is always this relation between theory and practice, between knowing and doing. From what has been already said, it will be evident that logic must first be a science before it can become an art. Its first business must be to investigate the nature of thought, and to attempt to discover the differ- ent forms which the latter assumes in the course of its development. So that we were right in defining it as primarily a science. But the further question remains: How far is it possible to apply the laws of logic after they have been discovered in such a way as to obtain directions how to reason correctly in every case ? Can we not apply our knowledge of the laws of thought in such a way as to get a complete art of reasoning, just as 10 THE STANDPOINT AND PROBLEM OF LOGIC the laws of chemistry and biology are applied in medi- cine ? It is no doubt true in logic, as everywhere, that scien- tific knowledge is capable of practical application. But I do not think that logic can be regarded as an art, in the sense that it furnishes a definite set of rules for thinking correctly. There is an important distinction in this case which must not be left out of account. The physical, and even the biological sciences, deal with things whose way of acting is perfectly definite and uniform. The character of any of the physiological functions, as, e.g., digestion, may be comparatively com- plex and difficult to determine, but it always attains its end through the use of the same means. When once its laws are understood, it is not difficult to prescribe just how the proper means may always be secured for the attainment of the desired end. But thinking has much more flexibility in its way of acting. We cannot say with the same definiteness as in the cases we have been considering, that in order to reach a certain end we must use a definite set of means. It is not possible, that is, to say : If you would learn what is true about this sub- ject, you must follow this rule and that in your thinking. Logic, it seems to me, cannot be regarded as an art like photography, or even like medicine ; for it is not possible to lay down definite rules for the guidance of thinking in every case. What we can do, is to show the method by which new truths have been discovered, and the general conditions which must always be fulfilled in reasoning correctly. And it is also possible to point out the more common errors which arise when these 3. LOGIC AS A SCIENCE AND AN ART 1 1 conditions are violated. But it is beyond the power of logic to formulate any definite set of rules for the guidance of thinking in every case. We have found that we must give up all extravagant hopes of the practical advantages to be gained from a study of logic. There is no set of rules which will make us infallible reasoners. That being admitted, the question may be raised as to the utility of the study. What will it profit us to devote ourselves to this subject? It might be a sufficient answer to point out that this question pre- supposes that knowledge has always some ulterior motive. The assumption upon which it is based is, in other words, that the prac- tical advantages arising from any study furnish the only justification for undertaking it. But it is scarcely necessary to say that this is not an attitude which any student should adopt. A student is one who prosecutes a study for its own sake, with no other motive than the desire to know. And to such a person logic should not be without interest. For as we have seen, it is an inquiry into the nature of intelligence. Its results, therefore, are not in themselves less in- teresting or less important than a knowledge of the various forms of geological formation, or of plant or animal life. " If it is re- garded as a valuable achievement," says Hegel, " to have discovered sixty odd species of parrot, a hundred and thirty-seven species of veronica, and so forth, it should surely be held a far more valuable achievement to discover the forms of reason." 1 The necessity of devoting oneself to a science quite unselfishly cannot be too strongly enjoined, nor the evils which arise when one begins a study greedy ' for quick returns of profit,' too often emphasized. Nevertheless, since the question has been raised, it would not be just to refuse altogether to speak of the particular results 1 Hegel, Werke, Bd. V., p. 139. Quoted by Bosanquet at the beginning of his work on Logic. 12 THE STANDPOINT AND PROBLEM OF LOGIC arising from a study of logic. As we have seen, we cannot hope to become infallible reasoners by its aid. It is just as true here as in any other field, however, that knowledge is power, and ignorance synonymous with weakness. For even if one resolves never to look inside a logic book, one must nevertheless have some theory, or act upon some principle it may be quite unconsciously in deciding what is true and what is false. For instance, a man may act upon the principles that those things are likely to be true which are favour- able to his own interests, or which agree with his own prejudices, or with the articles of his church or political party. Or again, he may regard his senses as the standards of truth. Mr. Bradley says that if dogs reason, they proceed upon the principle, ' what smells, exists, and what does not smell does not exist.' It is not uncommon to hear it announced : What can be perceived through the senses is true ; what cannot be sensed, or is contrary to the testimony of the senses, is an absurdity. This was the standard of truth adopted, for example, by those who attempted to overthrow the Copernican theory by declaring it to be in plain contradiction to the tes- timony of the senses. It seems evident, therefore, that intellectual beings cannot escape some kind of logical theory, whether they hold" it consciously or unconsciously. It is clear, too, that the character of this theory will determine to a great extent their thoughts and opinions. The only question which remains is whether it is better to leave this matter entirely to chance, or to attempt to gain some clear ideas regarding the nature of thinking, 4- THE MATERIAL OF LOGIC 13 and the conditions under which knowledge arises. It can scarcely be doubted that, even from a practical point of view, a true theory is better than a false one. A man who has reflected upon the nature of proof, and the principles of reasoning, is much less likely to be deceived than one who is guided unconsciously by assumptions which he has never examined. It is always an advan- tage to know exactly the nature of the result at which we are aiming, and to be perfectly clear as to our own purposes. And this is just what a study of logic aids us in attaining. It helps us to understand the structure of knowledge and conditions of proof. Moreover, it engenders the habit of criticising propositions, and ex- amining the evidence upon which they rest. Further, the importance of this study for a theory of education may well be emphasized. For education, at least so far as it undertakes to train the knowing powers of the individual, must be based upon a knowledge of the necessary laws of intelligence, and of the steps or stages which it passes through in its process of development. 4. The Material of Logic. The business of logic, as we have seen, is to discover the laws of thought and to show the differences which exist between real and imaginary knowledge. Where now shall we find the materials for this study ? Where are the facts which are to be taken as a starting point ? It is, of course, impossible to learn directly from one's own conscious- ness all that thinking is, or everything of which it is capable. For, quite apart from the difficulty of observ- ing the process of thought while it is actually going on, 14 THE STANDPOINT AND PROBLEM OF LOGIC no one can suppose that his own mind furnishes an example of all that thinking has done, or can do. It is necessary to take a broader view, arid learn how other men think. Of course, we cannot look into the con- sciousness of other men, but we can study the products and results of their thoughts. The history of the way in which truth has been discovered is of the greatest importance for logic. It must not be forgotten that thought is not a thing which can be described once for all. It is rather a living activity, which is constantly showing what it is in what it does. The history of the various sciences furnishes a record of the steps by means of which thought has built up knowledge. And, in this record, we have also a revelation of the nature of the thinking process itself, and of the stages through which it has passed in the course of its development. It is by a reflection, then, upon the nature of proposi- tions which are universally regarded as true that the laws of logic are obtained. There is always a permanent body of knowledge which no one thinks of calling in question. Both in everyday knowledge, and in the sciences, there is always found a great number of propo- sitions which appear true to everybody. And it is here that logic finds its material. Taking the facts and propo- sitions which are recognized as certain by everybody, logic examines their structure in order to learn about the nature of the intellectual processes by which they have been discovered. What principles, it asks, are involved in those pieces of knowledge, and what partic- ular acts of thought were necessary to discover them ? It is only by examining various pieces of knowledge 4- THE MATERIAL OF LOGIC 15 in this way, and attempting to trace out the conditions of their discovery, that one can learn anything new regarding the laws and character of thought. In other words, there is no way of learning about thinking ex- cept by studying what it has done. The best way of getting information about what thought can do, is to study what it has already accomplished. Every piece of knowledge, as the product of thinking, is to some extent a revelation of the nature of intelligence. But scientific knowledge by this I mean the results of the philosophical and historical sciences as well as of the so-called natural sciences^ exhibits perhaps most clearly the nature of thought. For the history of these sciences enables us to see the process of know- ledge, as it were, in the making. In tracing the history of philo- sophical and scientific ideas, we are at the same time following the laws of the development of thought. It is this fact which makes the history of philosophy and of the various sciences so instructive. It was with this object in view, to take but a single example, that Whewell wrote his famous History of the Inductive Sciences. He was interested, that is, not so much in the mere facts and names with which he dealt, as in showing the nature of thinking and the methods which had been employed in gaining a knowledge of the world. This is made very clear in the introduction to another work of Whewell from which I quote : " We may best hope to understand the nature and conditions of real knowledge by studying the nature and conditions of the most certain knowledge which we possess ; and we are most likely to learn the best methods of discov- ering truth by examining how truths, now universally recognized, have really been discovered. Now there do exist among us doc- trines of solid and acknowledged merit certainly, and truths of which the discovery has been received with universal applause. These constitute what we commonly term sciences ; and of these bodies of exact and enduring knowledge we have within our reach so large a collection that we may hope to examine them and the history of 1 6 THE STANDPOINT AND PROBLEM OF LOGIC their formation with a good prospect of deriving from the study such instruction as we need seek." 1 We have been insisting that the materials for the study of logic are to be found mainly in the records which we possess of what thinking has actually accom- plished. Our own consciousness, it was said, can supply but a very small quantity of material. To learn what thinking is, one must have as broad a survey as possible of its achievements. But there is another side to the matter. It must never be forgotten that it is the actual operations of thought with which logic is concerned. The words and proposi- tions which express the results of thinking must never be allowed to take the place of the thoughts themselves. Now, we cannot directly study the thoughts of any other individual. It is only in so far as we interpret, through our own consciousness, the records of what thinking has done, that these records are able to throw any light upon the problem of logic. So in this study, as else- where, we must find the key to the material in our own consciousness. If we are to gain any real ideas of the character of the thinking processes by means of which the sciences have been built up, we must reproduce these in our own minds. One's own consciousness must after all furnish the key which makes intel- ligible the account of the various steps which the thought of mankind has taken in building up science or knowledge. 1 Whewell, History of Scientific Ideas, 3d ed., Vol. I., p. 4. 4- THE MATERIAL OF LOGIC I/ References The following references may be given in connection with I and 2 : C. Sigwart, Logic, Vol. I., General Introduction. F. H. Bradley, The Principles of Logic, pp. i-io. B. Bosanquet, Logic, Vol. I., Introduction. H. L. Mansel, Prolegomena Logica, Chap. I. R. Adamson, The first part of the article ' Logic ' in the Encyclo- pedia Britannica. D. G. Ritchie, The Relation of Logic to Psychology, Philos. Review, Vol. V., pp. 585-600, Vol. VI., pp. 1-17. CHAPTER II IMPORTANT STAGES IN THE DEVELOPMENT OF LOGIC 5. The Logic of the Greeks : Aristotle. In the fourth and fifth centuries before Christ, a great interest in debate and public controversy sprang up in Athens. There were several reasons for this. In the first place, the Athenians of this period were a very acute and intel- lectual people ; they therefore required some outlet for their mental activities. The various sciences of nature which occupy so much of the thought of the modern world did not exist at that time, nor did the interest exist which was necessary to create them. For although the Greeks of this period had the greatest love and rever- ence for nature, their interest in natural objects was rather like that of the poet and the artist, than that of the modern man of science ; in other words, they were content to enjoy the beauty of natural objects, and to take delight in the harmonies of sound and color which their senses presented to them. They had no desire to pull things to pieces to see how they are made, or to discover the laws according to which they act, and so their mental energy and mental acuteness found its chief outlet in argumentative controversy, and public debating became one of their favourite diversions. The Athenians of those days used to argue, from the pure love of argument, wherever they met, in the market- 18 5. THE LOGIC OF THE GREEKS 19 place, in the groves and gardens, and at their meals and banquets. There was in addition, however, a very practical reason why it was necessary and desirable for one to be able to argue well. A man of property in Athens was constantly exposed to lawsuits, and was obliged to be his own lawyer and defend his cause by pleading before the judges. It was of the utmost practical importance, then, that he should be able to state his cause well, and should be master of all the arts by which the judges would be likely to be influenced. Under these circumstances, it is not difficult to under- stand why the art of public speaking came to be regarded in Athens as a necessary part of education. And, in response to this demand, there arose a class of teachers called Sophists, who made it their business to instruct young men in all the practical affairs of life, and especially in the art of public speaking, or rhetoric, as it was called. The Sophists do not seem to have made it their object to teach truth to their pupils, or to inculcate in them a love and reverence for truth; they rather sought to make those whom they taught clever men of the world. In teaching the art of argu- mentation or public speaking they did not seek to point out the methods by which true conclusions could be reached, but rather taught the arts by which the judges could be persuaded, and tricks for the discomfiture of one's adversary. The rhetoric of the Sophists, in other words, was not a science of reasoning, but an art of persuasion and of controversy. It was not necessary to have any real knowledge of the subject under dis- 2O DEVELOPMENT OF LOGIC cussion in order to argue well, but only to be well versed in all the arts of persuasion, and quick to take advantage of the omissions of an opponent. The theory on which the teaching of the Sophists was based is usually known as scepticism. The Sophists, that is, had come to the conclusion that it is impossible to find any fixed standard of truth. Looking at the diversity of individual opinions and of individual feelings, they declared that knowledge or truth as something objective, or the same for all, is an illusion. Only individual opinions exist; there is no standard by reference to which these opinions may be measured. It is impossible, then, to distinguish false opinions from true. Indeed, the words ' truth ' and ' falsehood ' can have no real meaning ; each indi- vidual must be the measure of truth for himself. Moreover, in the opinion of the Sophists, the same state of things exists with regard to our moral ideas. There is no standard of right and wrong, just as there is no standard of truth and falsehood. Each man has the right to choose what he regards as most advantageous f or s_ himself. The traditional rules of morality have no authority over the individual, nor is it possible to discover any rules of morality which are binding on all men. It is the part of wisdom to con- sult one's own interest in acting, and to seek to secure one's own advantage. Moral distinctions, like logical distinctions, are purely relative and individual. Socrates was the great opponent of the ethical scepti- cism of the Sophists. They had concluded, from the diversity of individual opinion on moral questions, that 5- THE LOGIC OF THE GREEKS 21 there is no real or absolute distinction between right and wrong. Socrates, however, was convinced that, if one ex- amined more carefully the nature of the judgments which men pass on matters of right and wrong, one would find common elements or ideas. It is possible, he believed, to find a fixed standard, both in matters of theory and in' matters of practice. This common element, however, is not to be discovered in sensation, nor in feelings of pleasure and pain; these are purely individual, and can never serve as a universal standard. But beneath the diversity of sensation and feelings there is the thought, or concept, which is common to all men. When rational beings come to understand each other, they must agree as to the nature of the fundamental virtues, justice, temperance, courage, etc. It is true that few men have thought about these matters, and are able to express their meaning clearly ; but every man, as a rational being, carries these fundamental notions in his mind. Now, in order to refute the moral scepticism of the Sophists (and it was this side of their teaching which Socrates especially opposed), it is necessary that the ethical notions, or concepts, which are implicit in the minds of men shall be drawn out and carefully defined. How is this to be accom- plished? Socrates did not undertake to teach men what ideas they should hold regarding the nature of any of the virtues ; he rather made them partners in an investigation, and by means of skilful questions tried to assist them' in discovering the real nature of goodness for themselves. Another point to be noticed is that the definition of the various virtues was reached 22 DEVELOPMENT OF LOGIC as a result of comparing the views of a number of individuals. In this way, by comparing the opinions of many men, of different professions, and of different grades of society, he was able to separate what was merely individual and relative in these opinions, from what was unchanging and absolute. Plato, the disciple of Socrates, continued the work of his master. He did not confine his attention wholly to the moral conceptions, but showed that the Socratic method could also be used to refute the intellectual scep- ticism of the Sophists. In other words, he proved that in the concept, or thought, as opposed to sensation, a standard of truth is to be found, as well as a standard of morality. Knowledge arises from thinking, and it is possible to compare our thoughts, however impossi- ble it may be to find any basis of comparison in our sensations. Plato's disciple, Aristotle, is of great importance in the history of logic. He undertook a thorough investi- gation of the process of reasoning, and sought to show what conditions and principles are necessarily involved in reaching certainty. Aristotle was thus the founder of logic, as well as of psychology, zoology, and a number of other sciences. His most important logical works are the Categories, De Interpretation, Prior Analytics, Posterior Analytics, Topics, and the Sophistical Elenchns, a treatise on Fallacies. These writings came after- wards to be known as the Organon (or scientific instru- ment) of Aristotle. They contained, in the first place, what we call theory of knowledge (a discussion of the structure of knowledge, and of the scientific principles s- THE LOGIC OF THE GREEKS 23 upon which it rests), which formed an essential part of Aristotle's philosophical system. But they also fur- nished the practical application of these principles. In his doctrine of the syllogism, which is found mainly in the Prior Analytics, he showed what are the only valid forms of reasoning, and thus furnished the pattern or type to which all proofs must conform. He also classi- fied, in his work on Fallacies, the various species of false reasoning ; and showed how false arguments could be refuted and exposed by the principles which he had discovered. The form to which Aristotle maintained that all true reasoning can be reduced was as follows : All men are mortal, Socrates is a man, Therefore Socrates is mortal. This is called a Syllogism, and it is made up of three propositions. The first two propositions are called Premises, and the last the Conclusion. Every piece of reasoning, all proof, can be reduced to this form. Of course, the propositions which make up the syllogism do not always stand in this order, and sometimes one of them may be omitted. Thus in the argument : ' he ought to be supported by the state, for he is an old soldier,' the conclusion stands first, and one premise is wanting entirely. It is easy to see, however, that the real argument when properly arranged is equivalent to this : All old soldiers ought to be supported by the state, He is an old soldier, Therefore he ought to be supported by the state. Now the part of Aristotle's logic which was best 24 DEVELOPMENT OF LOGIC worked out, was a theory of proof or demonstration by means of the syllogism. Here he showed clearly the various ways in which different kinds of propositions could be combined as premises to yield valid conclu- sions, and proved that no conclusion could be drawn from other combinations. This part of the Aristotelian logic has c-ome down to us almost unchanged, and is the subject of Part I. of the present volume. It will be noticed that, in the doctrine of the syllogism, Aristotle was dealing with that kind of reasoning which undertakes to demonstrate the truth of some fact, by showing its relation to a general principle which every one admits. In other words, this part of his work may be called the logic of proof or demonstra- tion. Aristotle was at one time of his life a teacher of rhetoric, and he seemed always to have aimed at putting this art of reasoning on a scientific basis. That is, for the rules of thumb and questionable artifices of the Sophists, he wished to substitute general laws and methods of procedure which were based upon a study of the principles and operations of reason. By com- plying with the rules which he laid down, an argument will necessarily gain the assent of every rational being. But we do not employ our reason merely in order to demonstrate to ourselves or to others what we already know. We seek to discover new facts and truths by its aid. In other words, we not only wish to prove what is already known, but also to discover new facts, and we need a logic of Discovery, as well as a logic of Proof. This distinction between proof and discovery corre- sponds in general to that between Deduction and In- 5. THE LOGIC OF THE GREEKS 2$ duction. Deduction is the process of showing how particular facts follow from some general principle which everybody admits, while Induction shows the methods by which general laws are obtained from an observation of particular facts. Now Aristotle, as we have seen, furnished a very complete theory of Deduction, or method of proof. But he did not treat of Induction, or the method of passing from particular facts to gen- eral laws, with anything like the same completeness. Moreover, what he did write on this subject received no attention for many centuries. Aristotle was himself a great scientific observer, and may well be regarded as the father of the natural history sciences. But, in his logical writings, his main object seems to have been to present a true theory of argumentation, as opposed to the false theories of the Sophists. Science, too, was only in its beginning when Aristotle wrote, and it was impossible for him to foretell the methods of discovery which it has actually employed. After Aristotle's death (322 B.C.), and after the loss of Athenian independence, there was a great decline of interest in matters of mere theory which had no direct application to the practical affairs of life. The Stoic school did make some slight additions to logical theory, but like their opponents, the Epicureans, they regarded practice, the art of living well, as. the supreme wisdom of life. The Romans, who derived their knowledge of Greek philosophy largely from the Stoics, were also in- terested in the practical advantages of logic, rather than in its theoretical side. It was the possibility of apply- ing the laws of logic to rhetoric and public speaking 26 DEVELOPMENT OF LOGIC which especially interested Cicero, who was the first to make Latin paraphrases and adaptations of Greek logic in his rhetorical works. 6. Logic during the Middle Ages. For more than seven hundred years, during the Middle Ages, the Greek language and literature was almost unknown in Western Europe. During this time, almost the only sources of information regarding logic were Latin translations of Aristotle's Categories, and of an Introduction to the same work by Porphyry, who lived 232-303 A.D. Both of these translations were made by Boethius (470-525), who is best known as the author of The Consolations of Philosophy. Even when scholars again became acquainted with the original works of Aristotle, in the latter part of the Middle Ages, they did not really understand their true significance. They took the husk, one may say, and neglected the kernel. They adopted the Aristotelian logic as an external and arbitrary set of rules for the guidance of thinking, and neglected entirely 'the sci- entific theory upon which these rules were based. A great deal of ingenuity was also shown in subdividing and analyzing all possible kinds of argument, and giv- ing the particular rule for each case. This process of making distinctions was carried so far that scholastic logic became extremely cumbersome and artificial. Its pretensions, however, rapidly increased ; it claimed to furnish a complete instrument of knowledge, and a sure standard for discriminating between truth and false- hood. It is not very difficult to understand why this set of logical rules 6. LOGIC DURING THE MIDDLE AGES 27 seemed so satisfactory to the age of Scholasticism. The men of this period had no desire to increase their knowledge ; they supposed that they were already in possession of everything which was worth knowing. Their only object was to weave this knowledge into a system, to show the connection and interdependence of all its parts, and thus to put it beyond the possibility of attack. And for this purpose, the school logic was admirably adapted ; it was always possible to bring every case which could arise under one or other of its rules. There is no doubt that the Aristotelian logic had a real value of its own, and that it exercised a very important influence upon Western civilization, even in the form in which it was taught by the Schoolmen; but there is, of course, nothing complete or final about it. Its main purpose, as we have already seen, was to furnish a method by means of which the knowledge we already possess may be so arranged as to be absolutely convincing. But the centre of intellectual interest has changed since mediaeval times. We are not content merely to exhibit the certainty and demonstrative char- acter of the knowledge which we already have, but we feel that there is a great deal of importance still to be discovered. So that, in modern times, one may say the desire to make discoveries, and so add to the general stock of knowledge, has taken the place of the medi- aeval ideal of showing that the traditional doctrines taught by the church are absolutely certain and con- vincing. And when men became conscious of the importance of gaining new knowledge, and especially knowledge about nature, they at once saw the neces- sity for a new logic, or doctrine of method, to aid them in the undertaking. 28 DEVELOPMENT OF LOGIC 7. The Logic of Bacon. All the great thinkers of the sixteenth and seventeenth centuries saw clearly that the school logic is simply a method of showing the certainty of the knowledge we already possess, and does not aid us at all in making new discoveries. A new method, they all declared, was an absolute neces- sity. The new point of view was put most clearly and eloquently by the famous Francis Bacon (1561-1626), at one time Lord Chancellor of England. Bacon called his work on logic the Novum Organum, thus contrast- ing it with the Organon, or logical treatises of Aristotle. An alternative title of the work is, True Suggestions for the Interpretation of Nature. Bacon begins this work by showing the advantages to be gained from a know- ledge of nature. It is man's true business, he tells us, to be the minister and interpreter of nature, for it is only by becoming acquainted with the laws of nature that we are ever able to take advantage of them for our own ends. " Knowledge and human power are synonymous, since ignorance of the cause prevents us from taking advantage of the effect." The discovery of the laws of nature, which is therefore of so great practical impor- tance, cannot be left to chance, but must be guided by a scientific method. And it is such a method which Bacon endeavours to supply in the Novum Organum. The method which Bacon proposed seems to us very simple. If we would gain new knowledge regarding nature, he says, and regarding natural laws, we must go to nature herself and observe her ways of acting. Facts about nature cannot be discovered from logical propositions, or from syllogisms ; if we would know the 8. LOGIC SINCE THE TIME OF BACON 29 law of any class of phenomena, we must observe the par- ticular facts carefully and systematically. It will often be necessary, also, to put pointed questions to nature by such experiments as will force her to give us the information we want. Knowledge, then, must begin with observation of particular facts ; and only after we have made a great number of particular observations, and have carefully classified and arranged them, taking account of all the negative cases, are we able to discover in them the general law. No hypotheses or guesses are to be made ; but we must wait until the tabulations of the particular phenomena reveal the general ' form ' or principle which belong to them all. It will be frequently necessary to refer to Bacon's work in what follows. At present, it is sufficient to note that Bacon showed that a knowledge of nature cannot be attained through general propositions and logical arguments, but that it is necessary to begin with the observation of particular facts. He empha- sized, also, the importance of systematic observation and carefully planned experiments, and showed that knowledge must begin with facts of perception. This is the method of induction, and Bacon is usually said to have been the founder of the inductive sciences of nature. 8. Logic since the Time of Bacon. Another and quite different method of extending knowledge was pro- posed by the great Frenchman, Descartes (1596-1650), who took mathematics as the type to which all know- ledge should conform. That is, he supposed that the 30 DEVELOPMENT OF LOGIC true method of extending knowledge was to begin with general principles, whose truth could not be doubted, and to reason from them to the necessary character of particular facts. Descartes and his followers thought that it was possible to discover certain axiomatic propo- sitions from which all truth could be derived through reason. They thus emphasized Deduction rather than Induction, and reasoning rather than observation and experiment. The spirit of Bacon's teaching was, how- ever, continued in England by John Locke, in the Essay Concerning Human Understanding (1690). Dur- ing the next centuries, philosophical thinkers were divided into two great schools, Rationalists, or those who agreed in the main with Descartes, and Empiricists, or Sensationalists, who followed the teachings of Bacon and Locke. Although the natural sciences made great advances during the seventeenth and eighteenth centuries, there seems to have been no effort made to analyze and describe the methods which were actually being em- ployed. In England, at least, it seems to have been assumed that all discoveries were made by the use of the rules and methods of Bacon. One of the first writers to attempt to explain the method used by the natural sciences was Sir John Herschel (1792-1871). His work, Discourse on the Study of Natural Philosophy, was published in 1832. A little later, and with the same object in view, William Whewell (1794-1866), afterwards Master of Trinity College, Cambridge, un- dertook his History of the Inductive Sciences, which was followed some time after by the Philosophy of the 8. LOGIC SINCE THE TIME OF BACON 3 1 Inductive Sciences, The man, however, who did most towards putting the study of logic on a new basis was John Stuart Mill (1806-1873), the first edition of whose Logic appeared in 1843. We shall have frequent occa- sion to refer to this work in future discussions. It is sufficient to say here that Mill continues the empirical tradition of the earlier English writers in his general philosophical position. Mill's book gave a great im- pulse to the study of logic. Before it was published, writers on the subject had confined their attention almost exclusively to the syllogistic or deductive rea- soning. Mill, however, emphasized strongly the impor- tance of induction ; indeed, he regarded induction as the only means of arriving at new truth, deduction being merely a means of systematizing and arranging what we already know. Though few logicians of the present day adopt this extreme view, the importance of inductive methods of reasoning, and the necessity of studying them, have now become generally recognized. Most modern writers on logic devote a considerable amount of attention to induction. The reader will find that Part II. of the present volume deals with this subject. There is still another side of logic which has been developed in the English-speaking world since the time of Mill, though it is a direct continuation of the move- ment started in Germany by Kant more than a hun- dred years ago. The so-called ' modern ' logic has laid aside the formalism and paradoxical mode of expression adopted by Hegel, but the fundamental conceptions with which it works are essentially the same as those 32 DEVELOPMENT OF LOGIC employed by the latter in his Wissenschaft der Logik (1816-1818). It has been within the last twenty years that the results of German idealism the doctrines of Kant, Fichte, Schelling, and Hegel have become naturalized in England and America. And largely as a consequence of these teachings, a new conception of the nature of thought has grown up, and given rise to investigations which may be said to have created a 'modern' logic that is fairly entitled to rank beside its sister science, the ' new ' psychology. The Aristotelian doctrine of the syllogism is a purely formal science. In the form in which it is represented in ordinary text-books, it might perhaps be more prop- erly described as the art of arranging our knowledge in such a way as to compel assent. The ' matter ' with which thought is supposed to work is supplied to it in form of concepts and judgments. The problem which formal logic has to solve is to define and classify the various kinds of concepts with which thought operates, and to determine the various relations in which these stand when combined into judgments. Similarly, it has to show what combinations of judgments can be employed as premises leading to valid conclusions in the syllogism. The criterion of truth employed in these investigations is the principle of non-contradiction or consistency. Inconsistent combinations of concepts, that is, are ruled out ; but so far as the doctrine of the syllogism goes, anything is true which is not self- contradictory. Now, without questioning the practical value of its canons, it is obvious that formal or syllogistic logic does 8. LOGIC SINCE THE TIME OF BACON 33 not take any account of many of the processes of every- day thought, and that its rules go but a little way in helping us to distinguish the true from the false. For, in the first place, to think is not merely to combine and arrange ideas already in our possession. This might enable us to render clearer and more definite what we already know, but would never enable us to gain new knowledge. The real movement of thought as op- posed to its merely formal procedure consists in the formation of new ideas and new knowledge through actual contact with the world of experience. A com- plete account of the intellectual process, then, must deal with the relation of the mind to objects; it must investigate the various activities by means of which thought interprets the world and builds up the various sciences of nature and of man. The recognition of the importance of induction, and of the necessity of studying the methods of the induc- tive sciences which was brought about by Whewell, Mill, and others, was a step in the right direction, for it called attention to a kind of thinking which occupies a large place in our intellectual life, and also gave rise to a truer conception of the nature of thought itself. But even Mill did not reach the idea which guides modern logicians, that thought or intelligence is one from beginning to end, and that the various logical processes are all parts of one whole, or rather ways in which intelligence operates in different circumstances, or at different stages of its development. He still treats of logical processes, like conception, judgment, and reasoning, as if they were quite separate from D 34 DEVELOPMENT OF LOGIC each other ; and, as has already been noticed, in his zeal for induction, he fails completely to do justice to syllogistic reasoning. As opposed to the division of mind into separate faculties, the thought by which modern logic is domi- nated is that of the unity and continuity of all intel- lectual life. Thought is regarded as an organic, living function or activity, which remains identical with itself throughout all its developing forms and phases. The problem, accordingly, which logic must set before itself is to show the unity and interrelation of all of the intellectual processes. No one of the steps or stages in this process can be completely understood when viewed by itself : each is what it is only in and through its connection with the whole of which it forms a part. No hard and fast boundary lines are to be drawn be- tween the different stages of the reasoning process, but it must be shown that the whole nature of intelligence is involved more or less explicitly at each step. So far only the broad outlines of this theory have been filled in ; but the conception of an organism whose parts are developing in mutual relation and inter- dependence, promises to be as fruitful when applied to logic as it has already shown itself to be in the other sciences. Besides the ordinary histories of philosophy the reader may con- sult for the history of logic : Prantl, Geschichte der Logik im Abend- lande, 4 vols., Leipsic, 1855-1870; which extends, however, only to the dose of the mediaeval period. Harms, Geschichte der Logik, Berlin, 1881. Ueberweg, System der Logik, 4th ed., 1874; Eng. trans, of 3d ed., London, 1874. Adamson, article 'Logic,' in tha 8. LOGIC SINCE THE TIME OF BACON 35 Encyl. Brit., gth ed. Sir William Hamilton's Lectures on Logic, also contain much historical information. Among modern works on logic, the following may be mentioned : J. S. Mill, A System of Logic, London, ist ed., 1843; Qth ed., 1875. W. S. Jevons, The Principles of Science, London, 1874; 2d ed., 1877. Also, by the same author, Studies in Deductive Logic, 1880 ; and Pure Logic, 1890. H. Lotze, Logik, 1874; Eng. trans., Lon- don, 1 88 1 and 1888. W. Wundt, Logik, 2d ed., 1896. C. Sigwart, Logik, 2d ed., 1889-1893 ; Eng. trans., London and New York, 1895. The newer development of logic is well represented by F. H. Brad- ley, The Principles of Logic, London, 1886. B. Bosanquet, Logic, or the Morphology of Knowledge, London, 1888 ; and The Essentials of Logic, London and New York, 1895. L. T. Hobhouse, The Theory of Knowledge, London, 1 896, may also be mentioned in the same group of writers, although he has been, perhaps, more influenced by Mill than by any other writer. The following works, among others, have proved useful as text- books : W. S. Jevons, Elementary Lessons in Logic, London and New York, 1870. A. Bain, Logic, Deductive and Inductive, New York, 1883. J. H. Hyslop, The Elements of Logic, New York, 1892. W. Minto, Logic Inductive and Deductive, New York, 1894. J. G. Hibben, Inductive Logic, New York, 1 896. PART I. THE SYLLOGISM CHAPTER III THE SYLLOGISM AND ITS PARTS 9. The Nature of the Syllogism. The theory ol the syllogism, as has been already stated ( 5), was first worked out by Aristotle. And it stands to-day in almost the same form in which he left it. A few additions have been made at different points, but these do not affect materially the main doctrine. In deal- ing with the nature of the syllogism, we shall first try to understand its general aim and purpose, or the results which it seeks to bring about. We shall then have to analyze it into the parts of which it is com- posed, and to examine and classify the nature of these elements. Finally, it will be necessary to discover what rules must be observed in order to obtain valid conclusions, and to point out the conditions which most commonly give rise to error or fallacy. In the first place, it is to be noticed that syllogistic logic deals with the results of thinking, rather than with the nature of the thought-process. Its object is not to give an account of the way in which thinking goes on, but to show how the ideas and thoughts which we already possess may be combined so as to compel 36 9. THE NATURE OF THE SYLLOGISM 37 assent. The ideas which it uses as material are fixed by having been expressed in language. Indeed, it is largely with words, as the expression of thoughts, that syllogistic logic deals. Many of the discussions with which it is occupied have reference to the meanings of words and propositions ; and the rules which it fur- nishes may be taken as directions for putting together propositions in such a way as to lead to a valid conclu- sion. Nevertheless, it is important to remember that these rules are not arbitrary and external, but find their justification in the nature of thought. Indeed, the theory of the syllogism, when rightly understood, may be said to reveal the fundamental characteristics of the process of intelligence. For it brings together facts in such a way as to make evident their relation and dependence. It connects a judgment with the grounds or reasons which support it, and is thus a process of systematization. In order to understand the signifi- cance of the rules of syllogistic logic, then, it will frequently be necessary to look beyond words and propositions to the act of thought whose result they express. A great deal has been written regarding the princi- ples, or laws of thought, which are employed in syllo- gistic reasoning. , It seems better, however, to postpone the definite consideration of this subject until the student has learned more about the various kinds of syllogisms, and has had some practice in working examples. In dealing with the nature and principles of thought in the third part of this book, it will be necessary to discuss this question at length. Even at the present stage of 38 THE SYLLOGISM AND ITS PARTS our inquiry, however, it is important to notice that syl- logistic reasoning presupposes certain simple and fun- damental principles of thought whose nature we shall have to examine hereafter. In particular, the regular syllogism is founded on a principle which we may call the law of Identity, or the law of Contradiction, according as it is stated affirmatively or negatively. Stated affirm- atively, this so-called ' law ' simply expresses the fact that every term and idea which we use in our reason- ings must remain what it is. A is A, or has the same value and meaning wherever employed. The law of Contradiction expresses the same thing in negative language. A cannot be both B and not B. If any term is taken to be the same as another in one connec- tion, it must always be taken to be so ; if it is different, this relation must everywhere be maintained. .The data or materials which are employed in the syllogism are ideas whose meaning is supposed to be perma- nently fixed, and expressed in words which have been carefully defined. It would be impossible to reason, or to determine the relation of our ideas, if their mean- ing were to change without notice, or if the words by means of which they are expressed were used now in one sense, and now in another. It is of course true that our ideas regarding the nature of things change from time to time. And, as is evident from one's own experience, as well as from the history of language, a corresponding change takes place in the meaning of words. But the assumption upon which syllogistic reasoning proceeds, is that the ideas which are to be compared are fixed for the mean time, and that the io. THE PARTS OF A SYLLOGISM 39 words by which they are expressed are used in the same sense throughout the course of the argument. In this kind of reasoning, then, just as in geometry, it is essential that the terms which enter into the argu- ment be clearly and precisely defined, and that when thus determined they shall be taken as fixed and un- changeable until further notice is given. It is quite possible that all the requirements of the syllogism may be met without its conclusions being true of reality. In other words, an argument may be formally true, but really false. It is not difficult to understand why this may happen. The syllogism ac- cepts the ideas and judgments which it compares with- out criticism. These data are of course the product of previous acts of thinking. But in proceeding to ar- range them in syllogistic form, we do not inquire whether or not they are true ; i.e. adequate to express the nature of the things for which they stand. For the purposes of the syllogism it is only essential that their meanings be clearly understood, and that these meanings be regarded as fixed and permanent. io. The Parts of a Syllogism. The syllogism may be said to express a single comprehensive act of thought. We may define inference as a judgment which has been expanded so as to exhibit the reasons by which it is supported. In the syllogism The geranium has five pointed sepals, This plant has not five sepals, Therefore it is not a geranium. we may say that we have the judgment, 'this plant is 40 THE SYLLOGISM AND ITS PARTS not a geranium,' supported by the propositions which precede it, and that the whole syllogism taken together expresses a single thought, which is complete and self- sufficient. It is possible, however, even when one is dealing directly with the process of thinking, to dis- tinguish in it different subordinate steps, various stages which serve as resting places, in the course of its passage to the complete and comprehensive form represented by "the syllogism. But it is usual, in dealing with the syllogism, to take a more external view of its nature, and to regard it primarily as made up of words and propositions. In this sense, a syllogism can, of course, be divided into parts. In the first place, it is composed of three propositions. In the example given above the two propositions which stand first are called the premises, since they furnish the grounds or reasons for the propo- sition which stands last, and which is known as the conclusion. However, it is not true that we always find the two premises and the conclusion arranged in this regular order in syllogistic arguments. Oftentimes the conclusion is given first. Frequently, too, one of the premises is not expressed, and has to be supplied in order to complete the argument. Thus the statement, 'he must be more than sixteen years of age, for he attends the university,' is an incomplete syllogism. The conclusion, as will be readily seen, stands first. There is also only one premise expressed. To put this statement in the regular syllogistic form we have to supply the missing premise and arrange it as fol- lows : io. THE PARTS OF A SYLLOGISM 4! All students of the university are more than sixteen years of age, He is a student of the university, Therefore he is more than sixteen years of age. When one premise of an argument is lacking, the name of enthymeme is applied to it. When an argument is defective in this way, it must be remembered that the missing proposition is to be regarded as in consciousness, though not expressed. It is of great importance to form the habit of making clear to oneself the premises by which any conclusion claims to be supported. In this way groundless assumptions are often brought to light, and the weakness of an argument exposed. Whenever words like 'therefore,' 'for,' 'because,' 'it follows,' etc., are used in their proper signification, it is possible to find an argument composed of two premises and a con- clusion. But one must not allow oneself to be imposed upon by the mere words, but must insist on understand- ing exactly what are the premises in the case, and how the conclusion follows from them. It is possible to carry the division of a syllogism still further. Every logical proposition may be divided into two terms, and a copula or connecting link. The terms, which are the extremes of the proposition, are named the subject and the predicate. Thus in the proposition, ' the fields are covered with snow,' ' the fields ' is the subject, 'are,' the copula, and, 'covered with snow,' the predicate. To reduce a proposition to the logical form in which it is most conveniently treated, it is neces- sary to express it in such a way that the two terms are united by some part of the verb 'to be,' preferably 'is' or ' are.' Thus the sentence, ' No plant can grow with- 42 THE SYLLOGISM AND ITS PARTS out light and heat,' would be expressed as a logical proposition in the following, or some similar, form : ' No plant is an organism which can grow without light and heat.' ' Men have strong passions,' may be written, ' Men are beings having strong passions.' It is always well to reduce a sentence to some such form, by substi- tuting for the verb of predication some part of the verb ' to be.' The analysis of the syllogism gives us the divisions under which it is convenient to treat this part of logic. We shall accordingly deal (i) with Terms, (2) with Propositions, and (3) with the Syllogism as a whole. These divisions, however, are only made for the sake of convenience in treatment. It must not be forgotten that a term is a. part of a proposition. To understand the nature of a term, it is necessary to consider the part which it plays in the judgment which the propo- sition expresses. In other words, the function of the term, rather than the form of the word or words em- ployed, must be considered. It is, of course, true that we naturally and commonly use certain word forms to express certain kinds of ideas, just as in the grammati- cal sentence the different 'parts of speech' nouns, verbs, etc., have each a definite and comparatively permanent function. But even in the sentence, it is the part which the word in its grammatical function plays, rather than its form, which determines whether it is to be classified as a noun or an adjective, a preposition or a conjunction. In dealing separately with terms, as we propose to do in the next chapter, we shall be occupied to a large extent with the form of words in which cer- II. PROPOSED DIVISION OF MENTAL OPERATIONS 43 tain kinds of ideas are usually expressed. But, as the same word or group of words may be used for different purposes, it will be necessary, in order to understand the meaning of terms, to refer frequently to the various ways in which they are used in a proposition. The same difficulty exists when propositions are con- sidered by themselves, the relation to the complete argument of which they form a part being thus ig- nored. In this case, however, the results of the isola- tion are not so apparent, for a proposition forms, in a certain sense, a whole by itself. It is the expression of a judgment which, as we shall see later, is the unitary process of thought. It has thus a significance of its own, as expressing a more or less complete and inde- pendent act of thought. Nevertheless, it must not be forgotten that its independence and completeness are only partial and relative. A single proposition cannot stand alone. Taken strictly by itself, a proposition is only a fragment. In order to make it intelligible, it must be brought into relation with the other proposi- tions which state the grounds or reasons upon which it rests, or the conclusion which it helps to support. The logical nature of a proposition will, therefore, de- pend upon its function in an argument, and in treating of propositions this fact must not be forgotten. ii. The Proposed Division of Mental Operations. It is frequently stated in text-books on logic that corre- sponding to the division into Terms, Propositions, and Syllogisms, there must be a division of the different kinds of thought, or of operations of the mind. These differ- 44 THE SYLLOGISM AND ITS PARTS ent operations are usually called Simple Apprehension, Judgment, and Reasoning. " The first of these, Simple Apprehension, is the act of mind by which we merely become aware of something, or have a notion, idea, or impression of it brought into the mind. The adjective simple means apart from other things, and apprehension, the taking hold by the mind. Thus the name or term ' iron ' instantaneously makes the mind think of a very strong and very useful metal, but does not tell us any- thing about it, or compare it with anything else." 1 Judgment, the account continues, is an entirely dif- ferent action of mind, and comes later than Simple Apprehension. It consists in comparing two notions or ideas derived from simple apprehension in order to ascertain whether they agree or differ. In order to judge, we must have two notions or ideas ready in the mind. The judgment results from comparing these, and affirming, that they agree or do not agree. In the sarne way, having already made judgments, we can combine them into arguments or processes of reasoning by a new and still different activity of mind. Apprehension, judgment, and reasoning are thus sup- posed to be separate and distinct mental operations. It is true that the later forms employ as their mate- rial the finished products of the earlier. But from this point of view, apprehension, judgment, and reasoning simply succeed one another. The real unity which belongs to these operations as forms of intelligence is not set forth. 1 Jevons, Lessons on Logic, pp. n, 12. II. PROPOSED DIVISION OF MENTAL OPERATIONS 45 The whole of Part III. of the present book may be regarded as an argument against this point of view. We shall there endeavour to show that thinking is not a process of externally joining on part to part, but consists in a development or expansion of knowledge from within. And, in particular, we shall try to ex- hibit the essential unity of intellectual processes by whatever name they may be called, and at whatever stage of development they may be found. Without anticipating too far our future discussions, we may point out that the primary process of thought is not ' Simple Apprehension,' but Judgment. In other words, it is impossible to apprehend or passively receive ideas. 'To get an idea/ or to understand the meaning of a term, is only possible when the mind judges or inter- prets things for itself. To have an idea or concept of anything, then, is to be able to judge more or less clearly and confidently regarding it. I have an idea of 'iron' when I judge that it is 'black' and 'heavy' and 'malleable.' And the more complete and exact we can make our judgments, the better is the idea or appre- hension which we obtain of the thing in question. In- telligence or thought must not be regarded as at first merely receptive. It does not begin by laying hold of separate ideas or terms, and afterwards call in judg- ment as a new kind of process to bring the former into relation. But it is from the first a systematizing and relating activity which proceeds from the less perfect to the more perfect form of judgment (cf. 73, 74). CHAPTER IV THE VARIOUS KINDS OF TERMS 12. Singular, General, and Collective Terms. A logical term, as we have already seen, is an element of a proposition. In dealing with terms apart from prop- ositions, we shall be concerned mainly with different classes of words and the meanings which they usually express. It will be impossible, however, to fix ths meanings of terms absolutely without reference to the way in which they are used in propositions. The first division which we have to notice is that into Singular or Individual, General, and Collective terms. (i) A Singular or Individual term is one which can be applied in the same sense to but a single thing. The main purpose of Singular terms is to refer to, or identify, some individual object. Proper names are all singular. It is true that proper names are some- times used to denote a class of objects, as, e.g., 'a Daniel,' ' a Mephistopheles.' But when thus employed they lose their real character as proper names. That is, their function is no longer merely to identify certain individuals by naming them, but to describe them by mentioning certain qualities or characteristics which they are supposed to possess. But the ordinary pur- pose in using a proper name is to indicate some indi- vidual to whom the name belongs. In this sense, then, proper names are Singular. 46 12. SINGULAR, GENERAL, AND COLLECTIVE TERMS 47 In addition, any word or group of words which is applied to a single thing may be regarded as singular. And by 'single thing,' we mean anything which is thought of as one, as well as objects which are per- ceived through the senses. Thus, 'the waterfall just below the bridge,' ' the centre of the earth,' are singu- lar terms, and so also are words like 'justice,' 'good- ness,' 'the chief end of man.' It is perhaps more doubtful whether we should call terms such as ' white- ness,' ' sweetness,' singular, since we speak of differ- ent degrees and kinds of whiteness and sweetness. The question would have to be decided in every case by reference to the way in which the terms are em- ployed in propositions. (2) A General term is a name which applies to a whole group of objects. It is not limited, like the sin- gular name, to a single thing, but applies to a number of different things. All class names like ' metal,' ' man,' ' works on logic,' are of this character. The general name belongs to each and every individual of a whole class. Thus iron, gold, silver, etc., are ' metals ' ; and A, B, and C, ' men.' (3) A Collective term, on the other hand, is a name applied to a number of individuals when taken together and treated as a whole, as 'an army,' 'an audience.' It is important to distinguish carefully between general and collective terms. A general term is a name which applies equally to each individual of the group ; or, in other words, it is used of the individuals distributively. A collective name belongs to the whole, but not to the separate parts of the whole. Thus we say that 'sol- 48 THE VARIOUS KINDS OF TERMS dier ' is a general name, and is used distributively of each man in a regiment. ' Regiment,' however, is a collective name, for it applies only to the whole group, and not to the individual soldiers. Ambiguity sometimes arises from the fact that the English word 'all' is used in both of these senses. That is, it may rnean ' all taken together/ or ' each and every.' Thus we can say : ' All the angles of a tri- angle are less than two right angles ' ; and ' all the angles of a triangle are equal to two right angles.' In the former sentence, the word ' all ' is used distribu- tively ; in the latter, collectively. In Latin two different words are used : cuncti expresses the collective sense of ' all,' and omnes its distributive signification. It is worth noticing in this connection that it is the use which is made of terms, rather than the form of the words composing them, which determines their logical character. Thus terms which are collective in one connection may be general in another. ' Regi- ment,' for example, is a collective term with reference to the soldiers which compose it, but general when used as a common term for a number of similar divisions of an army. The same is also true of terms like ' grove,' 'mob/ 'class/ etc. Again, collective terms may be very properly regarded as singular when the proposition in which they are used emphasizes the unity and solidarity of the group. A proper name is sometimes applied to a collection of in- dividuals that are permanently united or that have acted together on some historic occasion, as, for example, ' The Fifth Cavalry regi- ment/ < The Charge of the Six Hundred.' 13. Abstract and Concrete Terms. Terms are fur- ther divided into abstract and concrete terms. The word ' abstract ' is often used popularly to describe anything which is difficult to understand. Etymologi- 13. ABSTRACT AND CONCRETE TERMS 49 cally, it signifies drawn off, separated (abstraho, to draw off, take away). We may distinguish two senses in which the word is used, both, however, being derived from its etymological signification. (i) A term is called abstract when it refers to some object which cannot be directly perceived through the senses, and concrete when such perception is possible. Thus 'a beech tree,' ' a tall man,' 'a sweet taste,' being names of things which can be perceived, are concrete. Words like 'sweetness,' 'hardness,' etc., have no objects of sense directly corresponding to them, and are for this reason called abstract. The same is true of terms like 'individuality,' 'equality,' 'justice,' etc. These words represent objects of thought, rather than ob- jects of sense. There may be cases or instances of 'equality,' 'justice,' etc., which fall under our percep- tion, but the real object to which these words corre- spond is not a thing which can be perceived through the senses at all. Their reality is conceptual, or for thought, not something directly revealed through the senses. It is important to notice that there are degrees of abstractness in terms, according as the objects for which they stand are nearer to, or further removed from ordinary sense-perception. All general or class names are abstract. One cannot point to a single object, to which the term 'metal,' for example, or the term ' man 1 corresponds. But although such terms have no direct sensuous object, yet we feel that they stand nearer to sense-perception, and are therefore tess abstract, than words like 'animal,' 'inorganic substance.' These terms, again, are perhaps less abstract than 'energy,' or 'spirit,' or even than singular terms like 'justice,' 'the ground of the universe,' etc. 50 THE VARIOUS KINDS OF TERMS (2) Again, the word ' abstract ' is applied to any ob- ject which is treated apart from the whole to which it belongs. Thus it would be an abstraction to attempt to represent the nature of a leaf in complete isolation from the plant to which it belongs, or to consider the nature of a man without regard to the social institu- tions family, church, state, etc. of which he is a member. Of course, it is essential when dealing with a complex whole to analyze it into its parts, and to under- stand just what is the nature of each part when taken by itself. But in order to comprehend fully the nature of the parts, it is necessary to restore them to their proper setting, and to see their relation to the concrete whole. In this sense of the word, then, ' abstract ' applies to what is taken out of its proper setting, broken off, and considered apart from the things to which it is organically related. Concrete, on the other hand, means what is whole and complete, a system of things which mutually support and explain one another. Since science has to analyze things into their elements, and to investigate and describe these elements in detail, it is impossible entirely to avoid abstraction. But it is necessary, in order to completely understand the nature of a complex object, that the abstractions of analysis shall be corrected. In other words, the concrete rela- tions in which things stand must not be ignored in investigating them. The conception of evolution in recent times has done much to render the biological sciences more concrete in the sense in which we are now using the term. For it has substituted, for the old method of treating each species of plant and animal as 13- ABSTRACT AND CONCRETE TERMS 51 distinct and separate, ' cut off from each other as if by a hatchet,' the view that all organic beings are members of one family, and can be properly understood only in their relations to one another. It is interesting to notice that, from this point of view, sense- perception is more abstract than thought. For the senses represent things in isolation from each other. Each thing is known in sense- perception as a separate individual, occupying its own space and time, and in this way, cut off from its fellows. It is the business of thought, on the other hand, to discover the relations between things, and the principles according to which they are united. Thinking thus overcomes the abstract point of view of sense-perception by showing that what appear to the latter as separate objects are really closely and necessarily connected as members of a com- mon unity or system. Each science takes as its province certain facts which resemble one another, but which nevertheless appear to sense-perception to be quite independent. It attempts by thinking to bring these facts into relation, to show that they are all cases of some law, that there is a common principle which unites them as parts of a whole or system. The law of gravitation, for example, expresses the unity which thought has discovered in things which appear to sense-perception as different as the falling of an apple, the movements of the heavenly bodies, and the ebb and flow of the tides. Scientific knowledge, then, is more con- crete than the facts which we learn from ordinary sense-percep- tion, because it brings to light real unity and connection in facts which appear to be entirely isolated and independent from the latter point of view. In employing the terms 'Abstract' and 'Concrete' it is of the utmost importance to distinguish the two sig- nifications of the words. From one point of view, as we have seen, all thought terms are abstract, as opposed to words which refer directly to objects of sense-perception. 52 THE VARIOUS KINDS OF TERMS In another sense, ' abstract ' denotes what is partial and incomplete, what is taken by itself and out of relation to the system of things to which it belongs. And since the real connection and relations of things are not given by perception, but have to be discovered by thought, the knowledge which the latter yields is more concrete, in this latter sense of the term, than that afforded by the former. 14. Positive and Negative Terms. The distinction between Positive and Negative terms is very obvious. Positive terms express the existence of some quality, or group of qualities, in the objects which they denote; as, e.g., ' happy,' ' good/ ' equality,' ' organism,' etc. A Neg- ative term, on the other hand, indicates the absence of qualities or properties in some object; 'bad,' 'un- happy,' 'inorganic,' 'injustice,' for example, are negative terms. Negative terms are often formed from positive by means of the affix, less, as in ' hopeless/ or by means of certain prefixes, of which the more common are un, in, dis, a, anti. Words which are positive in form are, how- ever, often negative in meaning, and are used as the contradictories of other terms. Thus 'ignorant' is generally regarded as the negative of ' learned/ ' dark- ness ' is the negative of ' light/ etc. It is not always possible, however, to find a separate word to express the exact opposite of every positive term. Words are used primarily to express the presence of qualities, and the negative idea may not be referred to so frequently as to require a separate word to express it. Thus there is no independent term to express the opposite of ' trans- H. POSITIVE AND NEGATIVE TERMS 53 ferable,' but by employing ' not ' as a negative prefix we obtain ' not-transferable.' It is always advisable when we wish to limit a term strictly to its negative application to employ not or non as a prefix. Words which are negative in form frequently have a more or less definite positive signification. Jevons points out that words like 'unloosed' and ' invaluable, 1 though negative in form, have a positive meaning. But, in addition, terms like 'unhappy,' 'immoral,' do not merely indicate the absence of positive qualities, but also express some positive properties of the objects to which they are applied. We speak of a person ' being positively unhappy ' ; and we employ 'non-moral' to express the simple negative relation rather than ' immoral.' On the other hand, there are certain terms which are positive in form that express the absence of qualities or attributes. Words like 'blind,' 'dumb,' 'maimed,' 'orphaned,' may be given as examples. These are often called Privative terms, rather than Negative, the distinction being that they refer to qualities or attributes which the objects to which they are applied naturally and usually have, but of which they have been deprived, or which they have never possessed. Thus ' blind,' as applied to a man, implies that he has lost or is desti- tute of the ability to see which naturally belongs to a human being. Again, other terms seem to be positive and negative solely in relation to each other. ' Element ' and ' compound ' are related as negatives or contradictories. It is difficult, however, to say which term is in itself negative or positive. It is important to notice the distinction between the relation in which positive and negative terms stand to each other, and that expressed by words which have to do with opposite extremes of something which pos- sesses quality or degree. Positive and negative terms are mutually contradictory. An element is what is not a compound, ' dishonest ' is the contradictory of ' honest,' 54 THE VARIOUS KINDS OF TERMS and as contradictories there is no middle ground be- tween them. What is not an element, is a non-element or a compound. Opposite or contrary terms, on the other hand, express a great difference of degree in the objects to which they refer. Thus 'foolish' is the op- posite of ' wise,' ' cold ' the opposite of ' hot,' and ' bitter ' of 'sweet.' But there is always the possibility of a middle ground between opposites. We cannot say that a man must be either wise or foolish, a taste either sweet or bitter. The logical contradictory of ' wise ' is 'not-wise,' of 'bitter,' is 'not-bitter,' etc. Opposite or contrary terms, then, must be carefully distinguished from contradictories. 15. Absolute and Relative Terms. Another classi- fication of terms, which is usually given by logicians, is that into absolute and relative terms. An absolute term is one which refers to an object which exists by itself, and has an intelligible meaning when taken alone. Thus, 'tree,' 'house,' 'the State of New York,' are ex amples of absolute terms. A relative term, on the con- trary, is a name which only derives a meaning from its relation to something else. The term ' parent,' for ex- ample, cannot be thought of except in relation to 'child.' Similarly, 'teacher' is relative to 'pupil,' and 'cause' to 'effect.' Relative terms usually go in pairs and are known as Correlatives. Adjectives, as well as nouns, may be related in this way. The presence of one quality or characteristic in a thing frequently implies the presence of others. Thus, ignorance and super- stition, sympathy and tolerance, are necessary correla- 16. EXTENSION AND INTENSION OF TERMS 55 tives, because the one involves the other, or is invariably connected with it. It is of course true that no finite thing is completely absolute or independent of other things. The nature of everything is largely determined by the nature of the other things with which it stands in relation. A tree, for example, is relative to the seed from which it sprang, the soil in which it grew, the sunshine, rain, etc., which accompanied its growth. All finite things have a beginning and an end, and are also influenced throughout the whole period of their lives by the action of other things. They are therefore not com- pletely absolute or independent. It is, however, possible to make a distinction between words which are the names of things that are comparatively independent, and may for ordinary purposes be con- sidered by themselves, and those which have only a meaning when regarded as correlatives. 1 6. Extension and Intension of Terms. In the foregoing sections of this chapter we have explained the nature of the various kinds of terms with which logic deals. It is now necessary to notice two different purposes for which terms are employed. In the first place, terms are used to refer to things, to name and identify them. Thus ' man ' refers to the different individual men, John Smith, Thomas Brown, etc., as well as to the various classes of men, Caucasians, Indians, Mongolians, etc. As denoting or naming ob- jects, whether these be individual things or classes of things, terms are said to be employed in Extension. But words are also used to describe as well as to name. That is, they represent the qualities or attributes be- longing to things for which they stand. They are not bare names without signification, but as the expression 56 THE VARIOUS KINDS OF TERMS of ideas they stand for certain qualities or character- istics which things are judged to possess. ' Man,' for example, is not merely a name which may be applied to individual human beings or races of men, but it implies that the objects so named have certain qualities, such as animal life, reason, and the power of com- municating with their fellows. When words are used in this way to define or describe things, rather than merely to name them, they are said to be employed in Intension. The terms ' Denotation ' and ' Connotation ' were used by Mill instead of Extension and Intension, respectively, and have been adopted pretty generally since his time. To 'denote,' is to point out or specify the objects for which a term stands ; and to 'connote' is to take account of the attributes or qualities which a name implies. The words 'breadth,' and 'comprehension,' are also sometimes used as synonymous with Extension, and ' depth,' or ' content,' instead of Intension. The terms to be remembered, however, are Extension or Denotation, and Intension or Connotation. It is useful to accustom ourselves to distinguish these two functions or uses of a term, to notice, that is, the things or classes of things to which the name applies, and also to reflect upon the signification, or ways of judg- ing about these things, for which the name stands. The Extension of a term, as has been said, indicates the objects to which a name applies, and the Intension the qualities or attributes which it signifies. From the point of view of extension, therefore, ' planet ' may be defined by mentioning the names of the various planets, Mer- cury, Venus, the Earth, Mars, etc. Similarly, a term like 'carnivora' might be given in extension by nam- 1 6. EXTENSION AND INTENSION OF TERMS 57 ing seals, bears, weasels, dogs, wolves, cats, lions, etc. Usually, however, we define from the point of view of intension, that is, by stating the qualities or character- istics for which the term stands. Thus we give the intensive meaning of ' planet,' as a heavenly body which revolves in an elliptical orbit round the sun. ' Car- nivora,' defined from the same point of view, are mam- malian vertebrates which feed upon flesh. It is not unusual, however, to supplement an intensive definition by turning to extension and enumerating examples. Thus we might add to the definition of 'carnivora' just given, the words, ' as lions, tigers, dogs, etc.' It is sometimes said that the intension and extension of terms vary inversely. This is simply an attempt to give a mathematical form of statement to the fact that the more a term is defined, or limited, by the addition of attributes, the fewer are the objects to which it applies. ' As the intension of a term is increased its extension is diminished, and vice -versa,' is the form in which the relation is often stated. For example, let us begin with some class-name like 'animal,' which has a great extension, and add a new attribute, 'rational.' We get ' rational animal ' = man. This term now applies to a much smaller number of individuals than 'animal.' The extension of the former term has been diminished, that is, by increasing the intenJ\i.. If we add to 'man' still another attribute like 'vinte,' we .again lessen the num- ber of individuals to which the term applies. In gen- eral, then, it can be seen that the extension of a term is lessened as it is made more definite by the addition of new attributes. And, conversely, by stripping off 58 THE VARIOUS KINDS OF TERMS attributes, by ' decreasing the intension,' the number of individuals to which a term applies is increased. There is, however, no exact ratio between the increase or decrease of intension and the corresponding change in extension. Indeed, the extension of a class may increase greatly without any loss of intension on the part of the term by which the idea is expressed. Thus the meaning or intension of the term ' man ' has not lost, but rather gained, during the last hundred years by the increase of population throughout the world. Extension and intension, according to the view just given, represent two different uses or functions of terms. Every term denotes some object or group of objects more or less directly, and at the same time connotes or signifies certain qualities or attributes. Sometimes the one purpose, sometimes the other, is the predominant one. Proper names, for example, are used primarily to denote or mark out things, and do not directly qualify or describe them. In the proposition, 'these animals are all vertebrates,' the predicate term ' verte- brates ' is employed less as a name of a number of animals, than as a description of their qualities. Never- theless, in both these cases the terms employed have the double function of naming or denoting objects, and of connoting qualities. Mill, however, and certaVpother logicians who follow him, make a distinction betftsim connotative and non- connotative terms. " A non-connotative term is one which signifies a subject only, or an attribute only. A connotative term is one which denotes a subject, and implies an attribute. By a subject is here meant any- 16. EXTENSION AND INTENSION OF TERMS 59 thing which possesses attributes. Thus ' John,' or ' Lon- don,' or ' England ' are names which signify a subject only. 'Whiteness,' 'length,' 'virtue,' signify an attribute only. None of these names, therefore, are connotative. But 'white,' 'long,' 'virtuous,' are -connotative. The word ' white ' connotes all white things, as snow, paper, the foam of the sea, etc., and implies or, as it was termed by the schoolmen, connotes the attribute whiteness. . . . All concrete general names are connotative. The word 'man,' for example, denotes Peter, James, John, and an indefinite number of other individuals, of whom, taken as a class, it is the name. But it is applied to them because they possess, and to signify that they possess, certain attributes." 1 There is no real ground, I think, for such an abso- lute distinction between connotative and non-connota- tive terms. When we consider the use or function of terms, we find that they are never used merely to name things, or merely to connote attributes, though in cer- tain cases the former purpose is the primary one, and in other cases the latter object is more prominent. Even when proper names are employed, the qualities or characteristics of the objects named are indirectly im- plied. The very fact that a proper name is given to an object implies that it possesses a certain definitely marked individuality. And a proper name when used intelligently carries with it some still more definite im- formation regarding the qualities of the thing to which it is applied, as, for example, whether it is a name of a person, an animal, or a place. 1 Mill, System of Logic, Bk. I. Ch. II. 5. 6O THE VARIOUS KINDS OF TERMS The reader may consult, in connection with this chapter : J. S. Mill, Logic, Bk. I. Ch. II. F. H. Bradley, The Principles of Logic, pp. 155-173. B. Bosanquet, Logic, Vol. I., pp. 46-71. " " The Essentials of Logic, Lecture V. CHAPTER V DEFINITION AND DIVISION 17. Fixing the Meaning of Terms. --We have al- ready referred to the necessity of definitely fixing the meaning of the terms which we employ in reasoning. In ordinary life, words are frequently used in a loose and shifting way, without any clear conception of the qualities or properties which they connote, or of the objects to which they apply. Logic demands, in the first place, that we shall have clear and definite ideas corresponding to our words, and that the signifi- cation and scope of the latter shall be carefully deter- mined. But this is a demand to which little attention is paid in the ordinary affairs of life. To define our terms in explicit language, or even to make clear to ourselves the ideas and things for which they stand, is by no means a natural or a universal mode of proced- ure, but something which requires a distinct, conscious effort. Bacon, Hobbes, Locke, Hume, and nearly all of the older philosophical writers have warned us against the abuse of words. The whole matter has been expressed very clearly by Locke, from whom I quote the follow- ing passage : " For he that should well consider the errors and obscurity, the mistakes and confusion, that are spread 61 62 DEFINITION AND DIVISION in the world by an ill use of words will find some reason to doubt whether language, as it has been employed, has contributed more to the improvement or hindrance of knowledge amongst mankind. How many are there, that when they would think on things fix their thoughts only on words, especially when they would apply their minds to moral matters ; and who then can wonder if the result of such contemplations and reasonings, whilst the ideas they annex to them are very confused and very unsteady, or perhaps none at all ; who can wonder, I say, that such thoughts and reasonings end in nothing but obscurity and mistake, without any clear judgment or knowledge ? " This inconvenience in an ill use of words men suffer in their own private meditations ; but much more manifest are the discords which follow from it in con- versation, discourse, and arguments with others. For language being the great conduit whereby men convey their discoveries, reasonings, and knowledge from one to another; he that makes an ill use of it, though he does not corrupt the fountains of knowledge which are in things themselves ; yet he does, as much as in him lies, break or stop the pipes whereby it is distributed to the public use and advantage of mankind." l The remedy for the obscurities and confusions of words is to be found in clear and distinct ideas. We must endeavour to go behind the words and realize clearly and distinctly in consciousness the ideas for which they stand. Now the means which logic re- 1 Essay concerning Human Understanding, Bk. III. Ch. XI. i8. DEFINITION 63 commends for the attainment of this end is definition. The first requirement of logical reasoning is that terms shall be accurately defined. There are, however, two ways in which the meaning of a term may be defined or explained. Every term, as we have already seen ( 1 6), may be regarded either from the point of view of intension, or from that of extension. To define in the narrower sense is to explain from the standpoint of intension, to state the attributes or qualities which are connoted by the term. The process of explaining terms with reference to the objects, or classes of objects, for which they stand is known as Division. We may include, then, under the general term definition, (i) In- tensive definition, or definition in the narrower sense, and (2) Extensive definition or division. 1 8. Definition. To define a term is to state its connotation, or to enumerate the attributes which it implies. Thus we define a parallelogram as a quadri- lateral figure whose opposite sides are parallel. A distinction is often made between verbal and real defi- nition. When we merely wish to explain the mean- ing in which we intend to employ some term, we have verbal definition. But when it is the purpose of our assertion to state the real nature or essential character- istics of some object, the proposition employed is said to constitute a real definition. This distinction, though not without importance, cannot, I think, be regarded as ultimate. For we never define a word or term for its own sake merely, but in order to understand the nature of the objects to which it refers. Indeed, a mere word, 64 DEFINITION AND DIVISION apart from the things for which it stands, has no inter- est for us. In denning a term, then, we are always attempting to explicate or explain, more or less directly, the nature of a thing, or our idea about a thing. Nevertheless, there is an advantage in distinguishing propositions whose immediate purpose is to expound the meaning of a word, from those which assert some- thing directly of an object. ' Monarchy consists in the authority of one man over others,' may be regarded as a verbal definition, because the purpose of the propo- sition is simply to explain the meaning of the subject term. On the other hand, ' iron is malleable ' is a real definition (though not a complete one), because it does not primarily refer to the signification of the word ' iron,' but to the real object to which the name is ap- plied. In this connection, it is interesting to notice that a proposition which amounts to nothing more then a verbal definition, is some- times put forward as if it were an assertion which contained some real knowledge. The solemn commonplaces in which ignorant per- sons delight are often of this character. ' A republic is a govern- ment by the people, 1 ' a just man will do what is right, 1 ' if it rains, the ground will be wet, 1 may serve as examples. The mistake in such cases consists in supposing that these assertions are anything more than verbal. There are two points of view from which the subject of definition may be considered. We might either discuss the best method of obtaining real definitions of the nature of things^ or might confine our attention to the requirements which a good definition has to fulfil. A person's ability to define either a term, or the thing 1 8. DEFINITION 65 for which the term stands, depends, however, upon the possession of clear and distinct ideas on the subject. The problem, then, as to the best method of finding definitions, resolves itself into an inquiry concerning the means to be used in obtaining and classifying our ideas in general ; and the answer to this question, so far as an answer can be given, must be found in the theory of logic as a whole. In our treatment of the subject we shall, therefore, confine our attention mainly to a consideration of the requirements of a logical definition, and the rules which must be observed in stating it in language. Before entering upon the subject, however, it is in- teresting to refer briefly to the method proposed by Socrates for obtaining definitions. Socrates, as we have already seen ( 5), was the first to emphasize the necessity of defining and fixing the meaning of familiar terms. He found that, though the people of Athens were constantly using terms like 'good,' 'beau- tiful,' 'justice,' and 'temperance,' none of them, not even those with the greatest reputation for wisdom, were able to give any clear and consistent statement of what these terms implied. Socrates himself did not profess to be wiser than the rest, but he had a genuine spirit of inquiry, and made it the business of his life to try to arrive at clear conceptions, especially with regard to certain fundamental ethical virtues, like justice, and temperance, and wisdom, which he regarded as of the utmost practical importance. It was by means of con- versation with others that he sought to gain clear ideas regarding the nature of these virtues. By a r 66 DEFINITION AND DIVISION series of questions and answers, by comparison of any definition proposed with particular facts which are admitted, he led his interlocutors to expose and refute the inadequacies of their earlier statements. In the Republic, for example, the question is regarding the nature of justice. The first definition suggested is, that it is just 'to speak the truth, and to restore to each man his own.' But supposing that a man were out of his mind and demanded his weapons which had been placed in the hands of a friend, would the friend be an unjust man if he refused to return the weapons, or abstained from telling the whole truth ? Evidently not. The definition is then modified to read, ' It is just to give to each man what is his due.' Socrates then questions further, What is due to each man ? What is due to a friend, and what to an enemy ? This leads to the further modification that 'justice means doing good to our friends and harm to our enemies.' By referring again to particular instances and familiar analogies, Socrates leads the person maintaining this definition to admit that to injure a person is to make him less virtuous, and therefore less just. But how can justice render the character of another less just than it was before ? The idea is absurd ; therefore the definition has to be abandoned, and a fresh start made. This method of proceeding by means of question and answer, and thus compelling a speaker to admit par- ticular facts which refute the general thesis which he is maintaining, is called Dialectic. This was the means by which Socrates constantly strove to advance to consis- tent and adequate definitions. Apart from the dialectical 1 8. DEFINITION 6/ and dramatic form which the Socratic argument took, the method employed is essentially that of induction. For the definition, or conception, is derived from a com- parison of particular instances, both positive and nega- tive. By a consideration of individual cases, Socrates sought to obtain a definition which would be a complete and adequate expression of the nature of all the individ- uals which share in the class name. Aristotle says that it is to Socrates we owe the method of induction and logical definitions. Clear and distinct conceptions, for- mulated in exact definitions, constituted the scientific goal for Socrates, and the inductive procedure of ob- serving and classifying particular instances was the means which he employed for reaching this goal. The second question has reference to the formulation of a definition in language. Suppose that we already possess a clear conception of the meaning of the terms to be defined, what are the conditions which a logical definition must fulfil ? The answer to this question is usually given in logical text-books by means of a set of rules for definition. Before stating these rules, how- ever, it is necessary to explain the meaning of the terms 'genus,' 'species,' and 'differentia,' which will be fre- quently employed throughout the remainder of this chapter. These terms, together with ' property ' and 'accident,' constitute what the older logicians call the predicables, and to which a great deal of importance was supposed to belong. It will only be necessary, however, for us to consider briefly the signification of the first three terms. 68 DEFINITION AND DIVISION In logic, any term may be regarded as a genus which contains two or more subordinate classes or species. A species, on the other hand, is simply a subdivision or subordinate class of some larger whole. Thus ' metal ' is a genus with reference to iron, gold, silver, etc., which are its species. ' Rectilinear figure ' is the genus to which belong the various species, triangle, quadri- lateral, pentagon, etc. The differentia of any term is made up of the qualities or characteristics which dis- tinguish it from other terms, from the genus to which it belongs, as well as from the species which are co- ordinate with it. Thus the logical differentia of a triangle, is the property of having three sides, the dif- ferentia of man, is that which distinguishes him from other animals, whether this be the power of speech and reason, or some other characteristic either physical or mental. The use of the terms 'genus' and 'species' in logic is entirely relative. That is, any term may be considered either as a species or a genus, according as it is regarded as forming a part of some more comprehensive class, or as itself including other classes. Thus man, for example, is a species of the genus ' animal ' ; but the same term also may be regarded as a genus including various species of men, Caucasians, Negroes, Mongolians, etc. In the same way, ' animal ' may be considered a species of the still more comprehensive class 'organized being,' and this latter term again as a species of the genus ' material being.' A still higher or more comprehensive term which includes as its species material and spiritual beings alike is 'being.' Since this term includes every- i8. DEFINITION 69 thing which exists, and can therefore never be included in any more ' general class, it is sometimes called the highest genus ' (summum genus}. On the other hand, we might proceed downwards until we come to a class which did not admit of division into any subordinate classes. Such -a term is called in logic the lowest species (infima species}. It is important to notice that the terms ' genus ' and ' species ' have not the same signification in logic as in the natural sciences. In classifying objects in natural history, we use the terms 'variety,' ' species, ' genus, 1 ' family,' and ' order,' to denote varying degrees of relationship between certain groups or classes of objects. These terms, as thus employed, also indicate certain relatively fixed divi- sions, or permanent ways of grouping the various forms of plant and animal life. But in logic the terms ' genus' and 'species' are em- ployed to indicate the relationship between any higher and lower class whatsoever. Moreover, as we have seen, any term (excepting only the highest genus and the lowest species) may be regarded from different standpoints, as either a genus or a species. We shall now proceed to state the requirements of a logical definition : (l) A definition should state the essential attributes of the thing to be defined. This is done by stating the genus to which the object belongs, and also the pecul- iar marks or qualities by means of which it is distin- guished from other members of the same class. Or as the rule is usually stated : A logical definition should give the next or proximate genus, and the dif- ferentia of the species to be defined. Thus we define a triangle as a rectilinear figure (genus), having three sides (differentia) ; and man as an animal (genus), which has the power of speech and reason (differentia). 70 DEFINITION AND DIVISION (2) A definition should not contain the name to be defined, nor any word which is directly synonymous with it. If, for example, we were to define justice as the way of acting justly, or life as the sum of vital pro- cesses, we should be guilty of a violation of this rule. (3) The definition should be exactly equivalent to the class of objects defined, that is, it must be neither too broad nor too narrow. In other words, the definition must take account of the whole class and nothing but the class. ' A sensation is an elementary state of con- sciousness,' for example, is too broad a definition, since it applies equally to affective and conative elementary processes. On the other hand, the definition of gov- ernment as 'an institution created by the people for the protection of their lives and liberties,' is too nar- row. For it takes no account of absolute forms of government which do not depend upon the will of the people. Both of these cases may be regarded as a failure to give the true differentia of the class to be defined, and hence as violations of the first rule. (4) A definition should not be expressed in obscure, figurative, or ambiguous language. The reasons for this rule are at once evident. Any lack of clearness or definiteness in a definition renders it useless as an explanation. Sometimes the words used in defining may be less familiar than the term to be explained (ignotum per ignotius\ The definition which was once given of the word 'net' as 'a reticulated texture with large interstices or meshes,' may serve as an example. (5) A definition should, whenever possible, be affirma- tive rather than negative. A definition, that is, should 19. DIVISION 71 state what a term implies rather than what it does not imply. Sometimes, however, the purpose of a defini- tion may be best attained by a negative statement of what is excluded by the meaning of the term. Thus, for example, we may define a spiritual being as a being which is not material, that is, unlike a material body made up of parts extended in space. A logical definition, as has been said, requires us to mention the proximate genus or next higher class to which the species to be defined belongs, and also the specific or characteristic differences which dis- tinguish it from other species. Now it is clear that there are certain cases in which these conditions cannot be fulfilled. In the first place, no logical definition can be given of the highest genus, be- cause there is no more general class to which it can be referred. And again, although it is possible to give the differentia of any species such as ' man ' or ' metal,' it is not possible to state indi- vidual characteristics by means of a logical definition. An indi- vidual thing may be perceived, and its various properties pointed out. But it is never possible to state in a logical definition wherein the individuality of a particular thing consists. The uniqueness of a particular object cannot be summed up in a -general definition, but must be learned through perception. We may perhaps say that the highest genus is above, and the individual thing below, the sphere of logical definition. There are, moreover, other terms such as 'space,' 'time, 1 'life,' 'thought,' which are not readily referred to any higher class, and for which therefore logical definitions cannot be given. These terms are sometimes said to denote objects which are sui generis, or of their own class. 19. Division. We have already spoken of divi- sion as a process of defining a term from the point of view of extension. This is to enumerate the objects or classes of objects which the term denotes. This 72 DEFINITION AND DIVISION enumeration must, however, be guided by certain prin ciples which we have now to consider. It is usual to begin this subject by speaking of Di- chotomy, or the division of a term into two parts (S/%a re/jiveiv, to cut in two). This is a purely formal process, and is based on the so-called law of Excluded Middle, which is regarded as one of the fundamental laws of thought. This law may be stated as follows: There is no middle ground between contradictories. Any term, a, is either b or not-. A triangle is either equilateral or not-equilateral. Of two contradictory predicates, one or the other must belong to every possible subject. Now it is clear that this is a purely formal principle of division. Some positive knowledge of the particular facts involved is always necessary, in order to enable one to determine what things do stand in this relation of logical opposition. The logical law, in other words, does not help us at all in deciding what may be re- garded as not-a in any particular case. It is not, there- fore, a means of increasing our knowledge, but merely a principle of order and arrangement. This fact, obvi- ous as it seems, was not understood by the Schoolmen who busied themselves with logic in the latter part of the Middle Ages. They clung firmly to the belief that it was possible to discover the nature of particular facts by purely formal operations of this kind. Accordingly, they spent a great deal of time in classifying and arrang- ing terms as contradictions, contraries, etc. This work was doubtless of much service in fixing the meaning of terms, and in preventing confusion in their employment. But it was a purely verbal investigation, and of course ig. DIVISION 73 could not lead to any discoveries regarding the nature of things. Moreover, it must be noticed that we do not always get propositions to which any meaning can be attached by uniting subjects and predicates in this way. If the law of Dichotomy is not guided by knowledge of the particular facts, it will give absurd propositions like, 'virtue is either square or not-square/ 'iron is either pious or not-pious.' Unmeaning propositions of this kind being left out of account, however, we may proceed to divide everything according to this principle. All geometrical figures are either rectilinear or not-rec- tilinear ; all rectilinear figures either triangular or not- triangular ; all triangles, equilateral or not-equilateral, etc. This method of division may be represented thus : Substance Material non-material I r^ j Organic not-organic I I mineral not-mineral gold not-gold If it were desirable, the terms 'non-material,' 'organic,' and 'not-mineral' might also be further subdivided in the same way. Now it is not difficult to see that the practical use of this principle will depend upon our ability to find some positive value for the negative not-a. That is, to make the law of more than formal value, we must know what 74 DEFINITION AND DIVISION concrete term excludes a, or is its logical contradictory. And knowledge of this kind comes, as already said, only from experience of the particular facts. The strictly logical contradictory of a is always not-a ; of wise, not- wise, of cold, not-cold, etc. Mistakes frequently arise in stating contradictories in a positive form. The difficulty is that terms are chosen which are not true logical con- tradictories. Thus, if we say that every man is either wise or foolish, our terms are not contradictory, for a middle ground between them is possible. The same would be true of divisions like, 'large or small,' 'rich or poor,' 'saint or sinner,' 'idle or diligent.' In general, it is safe to scrutinize all dichotomic divisions very sharply to see that the alternatives are really contra- dictories. The method of dichotomy depends, as we have seen, upon the law of Excluded Middle. But there is also another process called Division in logic, which is per- haps better known by its less technical name of Classi- fication. In classification, there is no necessary limit to the number of classes or divisions which may be ob- tained. In this respect, it of course differs fundamentally from the twofold division which we have been exam- ining. Furthermore, a classification is always made according to some principle which is retained through- out the whole process. Any common characteristic of the group of individuals to be divided may be taken as a principle of classification. If, however, the characteristic chosen is merely an external and accidental one, the classification based upon it will be regarded as artificial, and made for some special or temporary purposes, 19- DIVISION 75 Thus we might divide all flowering plants according to the color of the flowers, or the persons in any company according to the pattern of their shoes. A classification which proceeds upon such surface distinctions has, of course, no real or scientific value. It does not attempt to discover fundamental or deep-lying resemblances be- tween the individuals with which it deals. A scientific or natural classification, on the other hand, has for its purpose the discovery of real likeness or resem- blance. It seeks to find and group together, the things which are related in some essential point. Consequently, it selects as its principle of division some property which appears to be a real mark of individuality, and to be connected with changes in other properties. Such a real principle of natural classification is rarely found by comparison of merely one property or set of prop- erties in the things to be compared. To classify accord- ing to a single property may be a convenient method of giving names to any group of individuals, and of arranging them in such a way as to be useful to the student. It does not, however, give any adequate idea of the properties and true relations of the individuals compared. A really scientific, or natural, classification must be based upon a study and comparison of all the discoverable properties of the different individuals to be classified. It is only in this way that their real resemblance and affinities can be brought to light. (i) The classification of plants proposed by the famous Swedish botanist, Karl Linnaeus (1707-1778), was based upon the comparison of a single feature : the structure of the sexual organs of plants. This method proved of the greatest convenience in indexing plants in a 76 DEFINITION AND DIVISION convenient way into genera and species so that they could be named and described. Yet since the classification adopted was based upon a single property or feature of the plant, it was considered (even by Linnaeus himself) as merely artificial. Of course it is not so obvi- ously artificial as the examples of what we may perhaps call merely accidental or trivial classification given above. But Linnnsus's system did not aim at setting forth the true relations of plants, and it was not based upon any systematic study of all their properties. It is useful merely as a stepping-stone to the real study of plants which is presupposed in natural classification. Certain rules for division are usually given in con- nection with the treatment of this subject. It is not, of course, supposed that by their help one can properly divide any subject without special knowledge. The purpose of these rules is rather to warn against the logical errors to which one is most liable in the process of division. (1) Every division is made on the ground of differ- ences in some attribute (or attributes) common to all the members of the whole to be divided. (2) Every division must be based on a single prin- ciple or ground (fundamentum divisionis). (3) The constituent species (or groups into which the whole is divided) must not overlap, but must be mutually exclusive. (4) The division must be exhaustive, i.e., the con- stituent species must be equal, when added together, to the genus. The first rule requires no remark. It simply states that it is only possible to divide any whole on the basis of differences in something which is common to all its parts. The second rule warns against changing the i 9 . DIVISION 77 principle of division while the process is being carried out. This law would be violated, if, for example, one were to divide mankind into Caucasians, Negroes, Mon- golians, Europeans, Australians, and Americans. The principle of division which was first adopted in this example was obviously that of the color of the skin. But this principle was not carried through, and another principle, that of geographical distribution, was substi- tuted for it. In dividing one must be clearly conscious of the principle which one is using, and keep a firm hold of it until the division is completed. The example which we have just given also violates the third rule. For not all of the groups, European, Caucasian, etc., exclude one another. Similarly, it would not be good logic to divide animals into vertebrates, mammals, in- sects, birds, molluscs, and fishes. The fourth rule simply insists that the division must be complete. The whole must be completely included in its divisions. It would not be a complete division to say that books may be divided into folios, quartos, and duodecimos ; or vertebrates into mammals and birds. For in neither of these examples are the divisions enumerated equal to the whole class. References J. S. Mill, Logic, Bk. I. Chs. VII. and VIII. W. Minto, Logic Inductive and Deductive, Pt. II. pp. 82-130. C. Sigwart, Logic, Vol. I. 42-44. J. H. Hyslop, The Elements of Logic, Ch. VI. CHAPTER VI PROPOSITIONS 20. The Nature of a Proposition. A proposition is the expression in words of an act of judgment. It is composed, as we have already seen, of two terms, a subject and a predicate, connected by a copula. From the point of view of formal logic the predicate is affirmed (or denied) of the subject. When we come to consider the nature of judgment (cf. especially 74, 77), we shall find reasons for questioning whether this analy- sis of the proposition can be taken as furnishing a cor- rect account of what actually takes place in judgment. When we judge, we do not begin with words or terms which are not yet judgments, and then pass on to judg- ment by joining together the former in an external way. The conclusions which we shall have to adopt are, that terms represent ways of judging, that the simplest act of thought is already a judgment, and that thinking develops by advancing from incomplete to more com- plete and comprehensive judgments. The theory of the syllogism is, however, worked out on the view of the proposition already indicated. This is sufficiently accurate for practical purposes, and is not likely to lead to any serious mistakes so long as we remember that it is the proposition, rather than the actual nature of judgment, with which we are dealing. 78 20. THE NATURE OF A PROPOSITION 79 The logical proposition, as the expression of an act of thought, corresponds to the grammatical sentence. Not every sentence, however, is a logical proposition. Sen- tences which express a wish or an interrogation do not directly enter into the process of argument at all, and may therefore be neglected for the present The same is true of exclamatory sentences. Again, even indicative sentences frequently require to be rewritten in order to reduce them to the form of a logical proposition, which demands two terms and a copula. The sentence, ' the sun shines,' must, therefore, for purposes of logical treatment, be reduced to, 'the sun is a body which shines.' ' On the hillside deep lies the snow ' is ex- pressed as a logical proposition in some such form as this : ' The snow is a covering lying deep on the hill- side.' It is very important to change the grammatical sentence to the regular form of a proposition before attempting to treat it logically. The most general division of propositions is that which classifies them as Categorical and Conditional. A categorical proposition asserts directly, and without any condition. The predicate is either affirmed or de- nied unconditionally of the subject. 'A is B,' 'this room is not cold,' ' New York is the largest city in America,' are examples of categorical propositions. Conditional propositions, on the other hand, make a statement which is not immediately and directly true> but only claims to be true under a condition ; as, e.g., 'we shall go to-morrow, if it does not rain.' 'It will either rain or snow to-morrow,' is also a conditional prop- osition ; for neither rain nor snow are asserted directly 80 PROPOSITIONS and absolutely, but in each case the appearance of the one is dependent upon the non-appearance of the other. The first of these conditional propositions is known as a Hypothetical, and the latter as a Disjunctive proposi- tion ; but for the present we shall deal only with cate- gorical propositions, and with the form of syllogistic argument to which they give rise. After we have com- pleted the account of the categorical syllogism, however, it will be necessary to return to a consideration of conditional propositions, and to the class of arguments in which they are employed. 21. The Quality and Quantity of Propositions. We shall now consider the various kinds of categorical prop- ositions. Such propositions are classified with regard to quality and quantity. From the standpoint of quality, propositions are either affirmative or negative. An affirmative proposition is one in which an agreement is affirmed between the subject and predicate, or in which the predicate is asserted of the subject. The proposi- tion, ' snow is white,' for example, indicates such an agreement between the subject and predicate, and is therefore affirmative in quality. A negative proposition indicates a lack of agreement or harmony between the subject and predicate. The predicate does not belong to the subject, but all relation or connection between the two is denied. 'The room is not cold,' 'the trees are not yet in full leaf,' are examples of negative propositions. The quantity of a proposition is determined by the extension of the subject. When the proposition refers to all of the individuals denoted by the subject, it is said 21. THE QUALITY AND QUANTITY OF PROPOSITIONS 8 1 to be universal in quantity. When, on the other hand, the proposition affirms that the predicate belongs only to a part of the subject, it is said to be particular. For example, ' all metals are elements ' is a universal propo- sition, because the assertion is made of the subject in its widest or fullest extent ; ' some metals are white ' is a particular proposition, because reference is made to only a part of the subject 'metal.' We divide propositions, then, with regard to quantity, into Universal and Particular propositions. Universal propositions are often indicated by adjectives like 'all,' ' the whole,' ' every,' etc. It frequently happens, how- ever, that no such mark of universality is present. A scientific law is usually stated without any explicit statement of its quantity, though from its very nature it is meant to be universal. Thus we say, 'the planets revolve around the sun,'' 'comets are subject to the law of gravitation.' Propositions which have a singular or an individual name as subject are often called Individual propositions, as, e.g., 'the earth is a planet,' 'knowledge is power.' But since it is impossible to limit a singular subject, individual propositions are to be regarded as universal. They belong, that is, to the class of propo- sitions which employ the subject term in its complete extent. Another class, called Indefinite or Indesignate propo- sitions, has sometimes been proposed. This class is usually said to include propositions in which the form of the words does not give any indication whether the predicate is used of the whole, or only of a part of the Subject.- ' Men are to be trusted,' ' animals are capable o2 PROPOSITIONS of self-movement,' may serve as examples. This classi- fication may be useful in illustrating the evil of making indefinite or ambiguous statements. Otherwise there is nothing to be learned from it. A really indefinite proposition has no place in an argument, and logic rightfully refuses to deal with it. The first demand of logic is that our statements shall be clear and precise. A proposition is not necessarily indefinite, however, because it has no qualifying words like 'all' or 'some.' It is the meaning of a proposition as a whole, rather than the form of its subject, which renders it definite or indefinite. Where, on the other hand, it is really im- possible to decide whether the proposition is universal or particular, logic forbids us to proceed with the argument until this point has been made clear. Particular propositions are usually preceded by some word or phrase which shows that the subject is limited in the extent of its application. The logical sign of particular propositions is 'some,' but other qualifying words and phrases, such as 'the greatest part,' 'nearly all/ ' several,' ' a small number,' etc., also indicate par- ticularity. Here again, however, it is the meaning of the proposition, rather than its form, which is to be considered. ' All metals are not white,' for example, is a particular proposition, although introduced by 'all,' since it is clearly equivalent to 'some metals are not white.' 'Every mark of weakness is not a disgrace,' again, is a particular proposition, and signifies that ' not all, or some marks of weakness are not disgraceful.' The words ' few ' and ' a few ' require special atten- tion. The latter, as in the proposition, ' a few persons 22. DIFFICULTIES IN CLASSIFICATION 83 have spoken to me about it,' is equivalent to 'some,' and introduces a particular affirmative proposition. ' Few,' on the other hand, is negative in character. Thus, ' few were saved from the shipwreck ' implies that only a few were saved, or that the greater number did not escape, and the proposition is therefore to be con- sidered as a particular negative. Propositions, then, are classified as affirmative and negative in Quality, universal and particular in Quantity. When these classi- fications are combined, we get four kinds of propositions, to symbolize which the vowels A, E, I, O are employed. A and I, the vowels contained in affirmo, stand for affirmative propositions ; E and O, the vowels in nego, for negative propositions. This may be represented as follows : Affirmative: All S is P A Negative: No S is P. E Affirmative : Some S is P. I Negative : Some S is not P. O We shall henceforth use A, E, I, and O to represent respectively a universal affirmative, a universal negative, a particular affirmative, and a particular negative propo- sition. In dealing with propositions logically, the first step is to reduce them to one or other of these four types. This can be accomplished readily by noticing the distinctions previously laid down. There are, how- ever, certain grammatical forms and sentences which present some difficulty, and it may therefore be useful to consider them separately. 22. Difficulties in Classification. In the first place, we may notice that in ordinary language the terms Universal < Particular < 84 PROPOSITIONS of a proposition are frequently inverted, or its parts separated in such a way that it requires attention to determine its true logical order. In the proposition, 'now came still evening on,' for example, the subject ' still evening ' stands between two portions of the predicate. As a logical proposition, the sentence would have to be expressed in some such form as the follow- ing : ' Still evening is the time which now came on.' Similarly, we should have to write an inverted sentence like, ' deep lies the snow on the mountain/ as ' the snow is something which lies deep on the mountain.' If a subject is qualified by a relative clause, the verb of the latter must not be confused with the main asser- tion of the proposition. Take the sentence, ' he is brave who conquers his passions.' Here it is evident that the relative clause describes or qualifies ' he.' Logically, then, the proposition is of the form A, and is to be written, ' he who conquers his passions is brave.' The reader will notice that all propositions which begin with pronouns like 'he who,' 'whoever,' etc., are universal in quantity, since they mean all who belong to the class in question. (i) We have reduced grammatical sentences to logical propo- sitions by changing the form in such a way as to have two terms united by ' is ' or ' are ' as the copula. Such a proposition, however, does not express time, but simply the relation existing between subject and predicate. When the grammatical sentence does involve a reference to time, and especially to past or future time, the reduction to logical form is somewhat awkward. Perhaps the best method is to throw the verb expressing time into the predi- cate. Thus 'the steamer will sail to-morrow' = 'the steamer is a vessel which will sail to-morrow ' ; ' we waited for you two hours 23. RELATION OF SUBJECT AND PREDICATE 85 yesterday ' = ' we are persons who waited for you two hours yes- terday.' (2) Exclusive propositions exclude all individuals or classes except those mentioned by the use of some such word as ' except,' ' none but,' ' only.' ' None but the guilty fear the judge ' ; ' only citizens can hold property'; 'no admittance except on business.' These propositions may all be reduced to the form E by writing 'no' before the negative si the subject term. Thus 'none but the guilty fear the judge ' = < i-^'cttu^ J r\3\ / V*^%M -6-< A Contraries Sub-Contraries FIG. 5. 96 THE INTERPRETATION OF PROPOSITIONS A and E are known as contraries ; I and O as sub* contraries ; A and O, I and E, as contradictories ; A and I, E and O, are subalterns. The relations of these propositions may now be summed up in the following statements : (1) Of contrary propositions, one is false if the other is true, but both may be false. (2) Of contradictory propositions, one is true and the other necessarily false. (3) If a universal proposition is true, the particular which stands under it is also true ; but if the universal is false, the particular may or may not be true. (4) If a particular proposition is true, the correspond- ing universal may or may not be true ; but if the par- ticular is false, the universal must be false. (5) Subcontrary propositions may both be true; but if one is false, the other is necessarily true. The knowledge that any one of these propositions is either true or false enables us to determine the truth or falsity of at least some of the others. For example, if A is true, E is false, O is false, and I is true. If A is false, E is doubtful, O is true, and I doubtful. If I is true, E is false, A is doubtful, and O doubtful. If I is false, E is true, A is false, and O true. Similarly we are also able to determine what follows when we suppose that E and O are either false or true. It ought to be carefully noted that when we affirm the truth of the particular proposition I, we do not deny the truth of the universal proposition A. The proposition, 'some students are fond of recre- ation,' for example, does not exclude the truth of ' all students are 25. THE OPPOSITION OF PROPOSITIONS 97 fond of recreation. 1 Similarly, the truth of O does not exclude the corresponding proposition in E : the statement, 'some men are not generous,' for example, does not interfere with the truth of the uni- versal proposition, ' no men are generous.' A particular proposition, in other words, asserts something of a limited part of a subject ; it neither affirms nor denies anything of the same term taken universally. The reader will remember that propositions which have the name of some singular or individual thing as subject, have been classified as universal. ' New York is the largest city in America,' ' charity is not the only virtue,' are examples of such propositions. Now it is at once evident that in cases of this kind there are no cor- responding particular propositions. What has just been said regarding the relation of universal and particular propositions, applies therefore only to propositions which have a general term or name as subject. Moreover, we must notice that when A and E propositions have a singular or individual name as subject, the relations between them are somewhat different from those just stated. A and E, we said, are contrary, but not contra- dictory propositions. By that it was implied that al- though we can proceed from the truth of the one to the falsity of the other, it is not possible to go in a converse direction, from falsity to truth. We cannot conclude, for example, from the falsity of the proposition that ' all men are selfish ' the truth of the corresponding negative proposition, 'no men are selfish.' 'With contra- dictory propositions, however, we can go from a denial to an affirmation. Now the point to be observed, with regard to propositions with a singular term as subject, 98 THE INTERPRETATION OF PROPOSITIONS is that although only contraries in form, they have yet the force of contradictories. ' Socrates is wise ' (A), and ' Socrates is not wise ' (E), are contradictory as well as contrary, propositions. 26. The Obversion of Propositions. The terms ' Ob- version ' and '^Equipollence ' were formerly used to denote any process by which the form of a proposition is changed without an alteration in meaning being involved. The name ' Obversion ' is, however, now gen- erally employed to describe the change which a propo- sition undergoes in passing from the affirmative to the negative, or from the negative to the affirmative form while still retaining its original meaning. Every fact is capable of expression either in the form of an affirmative or of a negative proposition. Whether the affirmative or negative form is chosen in any par- ticular case, is partly a matter of convenience. It is also determined largely by the psychological interest of the moment, i.e., by the purpose which we have in view in making the assertion. When, for example, we wish to repel some suggestion which may have occurred to us, or to deny something which our companions appear to believe, we naturally choose the negative form of statement. But the meaning of the proposition is the same whether we say, ' all men are fallible,' or, ' no men are infallible.' Similarly, we can say, 'not one of the crew escaped,' or, 'all of the crew perished.' Obversion, then, is the process of substituting for any affirmative proposition its equivalent in negative form, or of expressing the meaning of a negative prop- 26. THE OBVERSION OF PROPOSITIONS 99 osition as an affirmative. To obtain the obverse of proposition A, we proceed on the principle that two negatives are equal to an affirmative. Instead of 'all animals digest food,' we may write, ' no animals are beings that do not digest food'; for, 'every man has his own troubles,' ' there are no men who have not their own troubles.' Instead of affirming the predicate of the subject, the obverse of A takes the negative of the original predicate and denies it universally. Proposition I may be obverted in the same way, though it yields a particular, instead of a universal negative proposition. Thus the obverse of, 'some of the houses are comfortable,' is ' some of the houses are not not-comfortable,' i.e., uncomfortable. We deny the negative predicate in the obverse proposition, instead of affirming the positive. We obtain the obverse of the propositions E and O by changing the negation contained in them to its equivalent affirmation. This is done by attaching the negative to the predicate, and then affirming it of the subject. For example, to obtain the obverse of, ' no one who was present can forget the scene,' we first write the proposition in logical form, ' no one who was present is a person who can forget the scene.' Now the negative of the predicate term, ' a person who can forget the scene,' is, ' a person who can not forget the scene.' Affirming this universally we get, ' all persons who were present are persons who cannot forget the scene.' As an exam- ple of how the obverse of O is obtained, we may take the proposition, 'some metals are not white.' Now if we change the quality of the proposition by attaching the 100 THE INTERPRETATION OF PROPOSITIONS negative to the predicate, we obtain ' some metals are not- white.' That is, instead of denying, we affirm the neg- ative of the original predicate. When the predicate is made up of several words, it is important that the logical contradictory of the whole term be taken. For example, in the proposition, 'some men are not fond of work/ the predicate fully expressed is, ' persons who are fond of work.' Now the negative or contradictory term corre- sponding to this is, ' persons who are not fond of work.' The obverse of the original proposition therefore is, 'some men are persons who are not fond of work.' 27. The Conversion of Propositions. To convert a proposition is to transpose its subject and predicate so that each shall occupy the place previously held by the other. Thus the proposition, 'no men are infallible,' is converted by writing it, 'no infallible beings are men.' The original proposition is called the convertend, and the proposition obtained by conversion the converse. By conversion, then, a new proposition is derived directly from an old one. It is for this reason that conversion is usually ranked as a process of immediate inference. But, as we have already seen, the process of interpreta- tion which results in conversion seems to fall wholly within the proposition. In other words, it makes clear what is involved in the original proposition, but does not lead to any new fact with which the latter is connected. We therefore reached the conclusion that it might more .properly be regarded as a process of formal interpreta- tion, than as one which involves real inference. It is evident that in proceeding to convert propositions 2^. THE CONVERSION OF PROPOSITIONS IOI it will be necessary to notice whether the predicate of the convertend, or proposition to be converted, is dis- tributed or undistributed, otherwise we should not know what extension to apply to this term when used as the subject of the converse proposition. The rules usually given to limit the process of conversion are as follows : (1) No term must be distributed in the converse prop- osition which was not distributed in the convertend. (2) The quality of the converse proposition must remain the same as the quality of the convertend. The reason for the first rule is at once evident from what has been already said. The second rule is not one which is always observed. Of course, the meaning of a proposition must not be altered by changing the qual- ity simply or directly. But, in converting by Contrapo- sition, as we shall see later, it is first necessary to obtain the equivalent of the convertend by obversion, and this necessarily involves a change of quality. There are three kinds of conversion usually recog- nized : {a) Simple Conversion ; () Conversion by Limi- tation or per accidens ; (c) Conversion by Contraposition. (a) By Simple Conversion is meant the direct trans- position of the subject and predicate without any other change in the form of the proposition. Both propositions E and I can be converted in this way. Thus the converse of, ' none of the books on this shelf are novels,' is another proposition in E, ' no novels are books on this shelf.' From ' some dicotyledons are exogens ' we obtain by conversion another particular affirmative proposition, ' some exogens are dicotyledons.' IO2 THE INTERPRETATION OF PROPOSITIONS (b) Conversion by Limitation or per accidens is applied to proposition A. In this process A loses its univer- sality, and yields as a result only proposition I. To illustrate this mode of conversion we may take the propo- sition, 'brown hematite is an iron ore.' As we already know, the term 'an iron ore,' being the predicate of proposition A, is undistributed. When used as the sub- ject of a new proposition, therefore, it must be limited by the adjective 'some.' We thus obtain the converse proposition, ' some iron ore is brown hematite.' Simi- larly, the converse of the proposition, ' all sensations are mental .processes,' is ' some mental processes are sensa- tions.' When proposition A is converted by limitation, then, it yields proposition I as a result. And it is evident that the proposition has really lost something in the process. For it is impossible by converting again to obtain anything more than a particular proposition. It is, however, sometimes possible to convert proposition A without limiting the predicate. In formal definitions, for example, the subject and the predicate are of equal extent, and may be transposed simply without any limitation of the latter. Thus the converse of, 'an equilateral triangle is a plane figure having three equal sides,' is 'a plane figure having three equal sides is an equilateral triangle.' (c) In Conversion by Contraposition the negative or contradictory of the original predicate is taken as the subject of the converse proposition. This method of conversion is usually applied only to propositions A and O. When applied to A, it means that from a proposition 27. THE CONVERSION OF PROPOSITIONS 103 in the form, All B is C, we are able to assert something of what is not C. If we know, for example, that ' all the planets are bodies revolving around the sun,' we can obtain by contraposition the proposition, ' no bodies which do not revolve around the sun are planets.' The rule for contraposition is, first obvert, and then convert simply. Thus, the obverse of, 'aluminium is a white metal,' is the proposition in E, 'aluminium is not a metal which is not white ; ' and converting this simply, we get as the contrapositive of the proposition from which we started, 'no metal which is not white is alu- minium.' Proposition O can be converted only by contraposi- tion. If we were to convert simply, as, e.g., ' some metals are not white,' 'some white things are not metals,' we should fall into error ; for the term ' metal ' is distributed in the converse proposition without having been distributed in the convertend. To obtain the converse of O by contraposition, the rule given above, first obvert and then convert simply, applies once more. The obverse of the proposition in O, ' some men who make loud professions are not to be trusted,' is the equivalent in I, 'some men who make loud professions are persons not to be trusted.' Con- verting this simply, we obtain the contrapositive, ' some persons not to be trusted are men who make loud pro- fessions.' For the sake of convenience we may sum up the treatment of Conversion as follows : IO4 THE INTERPRETATION OF PROPOSITIONS Proposition A is converted (i) by Limitation, and (2) by Contra position. All S is P. (A) (1) Converting by Limitation, Some P is S. (I) i.) Obversion yields, No S is (2) Converting by Contraposition not-P. (E) ii.) The Simple Converse of this is, No not-P is S. (E) Proposition I is converted Simply. Some S is P. (I) Converting Simply, Some P is S. (I) Proposition E is converted Simply. No S is P. (E) Converting Simply, No P is S. (E) Proposition E may also be converted by Contraposition, but the result is the same as the Contrapositive of O. Thus for example : No S is P. (E) f i.) Obversion yields, All S is not- 1P. (A) v ' 11.) Converting this by Limitation, Some not-P is S. (I) Proposition O is converted by Contraposition. Some S is not P. (O) f i.) Obversion yields, Some S is Converting by Contraposition \ .. 11.) The Simple Converse of this 1 is, Some not-P is S. (I) References B. Bosanquet, Logic, Vol. I. pp. 310-319. W. Minto, Logic Inductive and Deductive, Pt. III. pp. 130-166. J. H. Hyslop, The Elements of Logic, Ch. X. CHAPTER VIII THE SYLLOGISM 28. The Nature of Syllogistic Reasoning. The syl- logism, as we have already seen ( 10), presents a con- clusion together with the reasons by means of which it is supported. A single proposition taken by itself is dogmatic : it merely asserts without stating the grounds upon which it rests. The syllogism, on the other hand, justifies its conclusion by showing the premises from which it has been derived. It thus appeals to the reason of all men, and compels their assent. To do this, it is of course necessary that the truth of the premises to which appeal is made should be granted. If the premises are disputed or doubtful, the argument is pushed a step further back, and it is first necessary to show the grounds upon which these premises rest. The assumption of syllogistic reasoning and, indeed, of all reasoning whatsoever is that it is possible to reach propositions which every one will accept. There are certain facts, we say, well known and established, and these can always be appealed to in support of our conclusions. In syllogistic reasoning, then, we exhibit the interdependence of propositions ; i.e., we show how the truth of some new proposition, or some proposition not regarded as beyond question, follows necessarily 105 106 THE SYLLOGISM from other propositions whose truth every one will admit. The question which arises in connection with the syllogism, therefore, is this : Under what conditions do propositions which are accepted as true contain or imply a new proposition as a conclusion? Or we may put the question in this form : In what ways may the four logical propositions, A, E, I, O, be combined so as to yield valid conclusions ? We pointed out in a previous chapter that a syllogism has always two premises. It is, however, impossible to obtain a conclusion by combining any two propositions at random, as e.g., All A is B. No X is Y. It is evident that any two propositions will not yield a conclusion by being taken together. In order to serve as premises for a syllogism, propositions must fulfil certain conditions, and stand in certain definite relations to each other. To determine some of the most apparent of these conditions, let us examine the argument : All mammals are vertebrates, The whale is a mammal, Therefore the whale is a vertebrate. "Vv^r-