ON THE THEORY OF LOGIC Ex Libris C. K. OGDEN ON THE THEORY OF LOGIC: AN ESSAY. BY CARVETH READ. LONDON: C. KEGAN PAUL & Co., i, PATERNOSTER SQUARE. 1878. [The Rights of Translation and of Reproduction are reserved^ SANTA BAIiBAFvA 71 PREFACE. THREE or four years ago a Travelling Scholarship was granted me by the Hibbert Trustees. One con- dition of holding it was that I should write something on some subject connected with my studies ; and I was glad to have an opportunity of writing the following Essay. Now that it is on the eve of publication I cannot help reflecting that almost every page is liable to two criticisms, (i) that it abounds with truisms, (2) that it strains after a spurious sort of originality. There is no sort of opposites which it is easier to unite than faults : but for the first of these I plead, that I have been dealing with the, most general facts, and that it would be strange if these were not sometimes also the most obvious ; and as to the second, I hope the vi Preface. reader will do me the justice to believe, that I am not blind to the difference between discovering a new truth and finding new expressions for an old one. My thanks are due to my friend Mr. E. S. Thompson, of Christ's College, for advice and suggestions upon many points of difficulty. March, 1878. TABLE OF CONTENTS. CHAPTER I. INTRODUCTION. PAGE i. General Purposes I 2. Order of Exposition I 3. Logic an ' Objective ' Science 3 4. Position of Logic among the Sciences 15 Classification of the Sciences . . . . .15 Metaphysics . . . . . . . . 16 5. Redistribution of the Contents of Scholastic Logic . . 17 Rhetoric '. . . . 18 6. Fallacies ' " 20 7. The Logical Calculus 21 8. Acknowledgments 22 CHAPTER II. OF RELATIONS. I. Definition of Logic 24 2. Of Relations in General . . . . . . . -25 3. Of Relations of Likeness and Unlikeness . . . . . 26 4. Of Relations of Succession ....... 29 5. Of Relations of Coexistence 31 6. Table of Relations 34 viii Table of Contents. CHAPTER III. OF TERMS. PAGE I. Of Terms in General 35 2. Of Feelings as Terms ........ 36 3. Of Feelings as Qualities ........ 36 4. Of Relations as Terms ........ 37 5. Of Compound Terms 39 6. Of Compound Terms in their Relations ..... 43 7. Table of Terms 45 CHAPTER IV. OF THE IMMEDIATE AND MEDIATE RELATIONSHIP OF SINGLE TERMS, ETC. PART I. Immediate Relationship of Single, Terms. I. Of Identity and Sameness 46 2. Correlatives . . . . . . . . 5 1 3. Of the Mutual Exclusion of Terms . . . . . 5 1 4. Comparison of Relations . . . . . . 53 1st. Symbols 53 2nd. Coincidence of Relat : ons . . . . . . 54 3rd. Immediate Implication ...... 54 4th. Compatibility . . ...... 55 5th. Incompatibility 55 6th. Alteruternity 56 7th. Hexagon of Comparison ...... 58 5. Qualities and Defects 58 6. Converse Relationship . . . . . . . -59 PART II. Mediate Relationship of Single Terms. I. Immediate and Mediate Implication . . . . . . 61 2. The Units ^of Mediate Relation 62 3. Psychological Digression on the Intuition of Conjunct Relations 65 4. Rule of Triterminal Correlation 69 5. Rule of Quadriterminal Correlation ..... 73 6. Proof, or Probation 76 Table of Contents. ix CHAPTER V. / t OF CLASSES. PAGE j I. Of Classes in General . ,, . ... . . . .80 2. Of the Constituencies of Classes . . . . 80 3. Of the Determinants of Classes ....... Si 4. Of Attribution and Privation ...... . . . 82 5. Table of Determinants, Attributes, and Privations ... 83 6. Positive and Counter Classes 83 7. Class and Subordinate 84 8. Complementary Classes ........ 85 9. Subdivision of Positive Classes 85 10. Natural and Artificial Classes . . . . . 89 n. Table of Classes 91 CHAPTER VI. OF THE DISCOVERY OF CLASSES. I. The Problems of Logic 92 2. Definition 94 3. Probation ........... 96 4. Causation in General ... . . ... 97 5- Objective Causation ......... 99 6. Analysis of Causation . . . . . . . .103 7. Law of Conservation 105 8. Persistence of Relations among Modes of Energy . . . 107 9. Quantitative and Qualitative Aspects of Causation . . . 109 10. Are Cause and Effect Identical ? i 10 11. Can a Cause exist before its Effect ? 112 12. Does the Effect cease with its Cause ? 113 13. Is the Effect like its Cause ? 115 14. Are there Vicarious Causes ? 117 15. Law of Causation 120 1 6. Elimination of Causal Instances . . . . . .121 17. Deduction of the Experimental Methods 123 18. Concomitant Variations . . , 128 19. Supplementary Methods 129 Table of Contents. PAGE 20. Vicariousness of Causes . . . . . . . .129 Joint Methods 132 21. Composition of Causes and Intermixture of Effects . . . 134 The Deductive Method 136 22. Probation of Classes of Substances and Individuals . . . 138 23. Probation of Coherent Coexistence . . . . X 39 24. Coexistences due to Causation . . . . . .141 25. Natural Kinds H 2 26. Superordinate Kinds ........ 144 27. Accidental Conjunctions 145 28. Classification of Laws of Coexistence ..... 146 29. Causation disguised as Coexistence 147 30. Definition and Probation . ....... 147 31. Laws 148 32. Explanation .......... 149 33. Subsumption 151 CHAPTER VII. OF THE IMMEDIATE RELATIONSHIP OF CLASSES. I. Inclusion and Exclusion -155 2. Knowledge and Reality 157 3. Designation .'.... 158 4. Qualitative and Quantitative Aspects of the Relationship of Classes 159 5. Conditions of Subsumption 160 ist Subsumption of Terms 1 60 2nd. Subsumption of Classes 161 6. Propositions concerning the Necessary Concomitance of certain Relations between the Constituents and Attributes of Classes 162 7. Unidesignate Relationship 165 8. Comparison of Unidesignate Relations of Classes . . . 166 ist. Implication ........ 166 2nd. Compatibility 1 68 Table of Contents. xi 3rd. Incompatibility 169 4th. Alteruternity . ...... . . . 170 5th. Square of Comparison . . . . 170 9. Equivalent Aspects of Unidesignate Relations . . . . 172 1st. Obverse Rela'.ionship . . . . . .172 2nd. Converse Relationship . . . . . 173 3rd. Converse of Obverse Relationship . . . .176 10. Genus and Species 176 11. Of the Qualities which appertain to a Term with regard to its Subsumption 178 Table of these Qualities . . . . . . 182 12. Propositions concerning Genus and Species . . . .183 13. Division *' . . . . . 188 14. Bidesignate Relationship . . 193 15. Deduction of Bidesignate Relations . . .' . . . 194 16. Obverse Aspect of the Relations of Genus and Species . . 197 Propositions Ampliative 198 17. Bidesignate Relationships detached from the Doctrine of Genus and Species 199 1st. Octagon of Comparison 201 2nd. Obverse Aspects 20 1 1 8. Of the Addition and Subtraction of Attributes as affecting the Relations of Classes . . . 203 ist. Abstraction and Generalization .... 203 2nd. Class and Class ....... 204 3rd. Term and Class 206 CHAPTER VIII. OF HYPOTHETICALS. I. Of Hypotheses in General ........ 208 2. Of Hypothetical Relationship . . . . . . .210 3. Of Conditionals . . . . . . . . ..211 4. Of Disjunctives ......... 213 5. Probation of Hypothetical . . . . . . . 215 xii Table of Contents. CHAPTER IX. OF THE MEDIATE RELATIONSHIP OF CLASSES. i. The Question Stated 2. Definitions. . . . . . . . . ..219 3. Possible Combinations of the Unidesignate Relations of Two Classes to a Third. . 219 4. Conditions of Mediation . . . . . . ..221 5. Axioms of Mediate Subsumption ...... 222 6. Cautions as to Mediate Subsumption . . . . . . . 226 7. Direct Moods . . . . . . . . . . 227 8. Reduction of Irregular Cases . . . . ... 229 9. Indirect Moods . 232 10. Reduction of Indirect Moods 232 11. Inconclusive Combinations 233 12. Mediation of Bidesignates ........ 235 13. Mood and Figure ......... 235 14. Mediation of Hypothetical ....... 236 15. Sorites 237 1 6. How many Terms has a Syllogism ? . . . . . 239 17. Table of the Modes of Implication ...... 244 1 8. Classification 245 CHAPTER X. OF SECONDARY RELATIONS. i. Symbols . . 250 2. Comparison of Secondary Relations . . . . . . 251 3. Mediate Relations of Secondary Relations .... 253 ON THE THEORY OF LOGIC. CHAPTER I. INTRODUCTION. i. General Purposes. THE purposes of this Essay are chiefly two : i . To restore to Logic the synthetic order of exposition ; 2. To sketch an outline of the Science as con- sistently as possible from the matter-of-fact point of view. While pursuing these main ends, I endeavour to present the Science in its nakedness ; on the one hand avoiding as much as possible the discussion of adjacent topics in Psychology and Metaphysics ; and on the other hand refraining from suggesting prac- tical applications : and this I do not out of a fastidious purism that fears to disfigure Logic, but because the practical bearings of the Science have recently been exhibited by writers more competent to do so. 2. Order of Exposition. During the long period in which Logic was almost entirely confined to the Deductive department, it 2 Theory of Logic. attained by the care of multitudinous expositors an admirable order and neatness of arrangement. Be- ginning with what were regarded as its most abstract elements, it moved forward by stages of increasing complication, to the Syllogism with its imposing array of Mood and Figure, and all the perplexity of Hypothetical : presenting a symmetrical whole, i KrtVjLA-*-^ cwvov*. !- ... 1*1 i J^^^^ bristling with elaborate detail and precise termi- nology, and impenetrable with mnemonics and bad verses. But since the development of Inductive Logic much of this formal excellence has been lost. The new doctrine, instead of being incorporated with the old, has merely been added to it. It is true that Mill explained to some extent the natural connection of the different parts of the Science, but he did not reorganise the whole accordingly. And Prof. Bain, though pointing out what the natural course of expo- sition would be, prefers to adopt another.* Thus the orderly succession of topics according to depend- ence and complexity is lost ; and probably many still think that by the intrusion of Induction into the Science, its unity has been destroyed. I hope it may not prove so. My excuses for deviating from the example of authorities to whom I owe much are, that it is peculiarly anomalous for a Science, so old and fundamental among Sciences as Logic, not to con- form to the plainest principles of scientific exposition ; that to those who can really grasp the subject, the * Logic, Introduction, 55 ; cf. MilL Logic, B. II. ch. i. 3. Introduction. 3 synthetic order is the easiest to follow and remember ; that the example of coherence, precision, economy, and method, which a Science so expounded presents, has a good influence on the minds of most people, especially Englishmen ; and that although there is an incipient tradition in favour of a different course, it cannot yet be too late to mend, in as much as the history of Logic in the future is likely to be very much longer than it has been in the past. Returning, therefore, to the example of the older Logicians, I have endeavoured to mould in accordance with it the more copious materials of the modern Science : beginning with simpler elements and more general truths, interpolating topics formerly neglected, modifying to some extent the arrangement of the parts always recognized, omitting what now seems extraneous, and carrying the synthesis to a stage of greater definiteness. At the same time I have taken a point of view, which I am not aware that any pre- vious writer on Logic has taken and consistently maintained ; and which for want of a better expres- sion, I have called the matter-of-fact point of view. 3. Logic an < Objective' Science. This has been done at the instigation of certain passages in the works of Mr. H. Spencer, particularly Principles of Psychology, Part VI., ch. 8 ; where it is announced, and I think proved, that Logic is an B 2 4 Theory of Logic. Objective Science, or Science of objective existence, " a Science that formulates the most general laws of correlation among existences considered as objective:" language which I could almost adopt, if allowed to give a special explanation of the meaning of the word objective. Modern Logicians have been roughly divisible into two schools : the Conceptualists who regard Logic as the Science or Art of Thought, that is, of certain Mental operations or products ; and the Nominalists, who hold it to be concerned primarily with the use of language in thinking or reasoning. It is seldom however that an adherent of either view has con- sistently maintained his position. The Nominalist has continually to consider the reference of language to things or thoughts ; and only a few Conceptualists have had the hardihood to pretend to exclude from Logic all that concerns the relation of thought to things. The thing, or matter-of-fact, is apt to con- front every Logician before long, whatever theory he starts with ; and so there have been some writers who held more or less clearly, that Logic is a Science of things. Those who show a strong leaning this way are called Materialists, because they seem to take more interest in the matter of any statement or pro- cess of thought than in its form ; but its metaphysical associations make the name very misleading. Logical materialism, to use the name for once, has naturally been a note of those who have done anything to introduction* 5 advance the theory of Induction ; but here again no one has been consistent. Among recent writers the most materialistic are, I suppose, Mill and Prof. Bain; and a word or two on their positions may throw some light on this Essay. Mill defines Logic to be " the Science of the opera- tions of the Understanding which are subservient to the estimation of evidence : both the process itself of! advancing from known truths to unknown, and all; other intellectual operations in so far as auxiliary to this." * Such a definition prepares one for the statement in the Examination of Hamilton f that " Logic is not a science distinct from and co-ordinate with Psychology. So far as it is a science at all, it is J a part or branch of Psychology Its theoretic grounds are wholly borrowed from Psychology." Accordingly in the Logic we find chapters on Infer- ence^ on the Functions of the Syllogism, Evidence of the Law of Causation, and on Abstraction or the Forma- tion of Concepts ; all which (with others) contain more or less Psychological speculation. On the other hand, at the opening of the chapter on the Import of Propositions, we read : " An inquiry into the Nature of Propositions must have one of two objects : to analyse the state of mind called belief, or to analyse what is believed Logic, according to the con- ception here formed of it, has no concern with the act * System of Logic ; Introduction, 7. t Ch. xx. p. 445 (jrd edit.}. 6 Theory of Logic. of judging or believing ; the consideration of that act, as a phenomenon of the mind, belongs to another science." The other Science is presumably Psycho- logy; although it seems strange to speak of that Science, of which Logic is said to be a branch, as "another Science." But it is more important to ob- serve that Logic is here said to have no concern with the act of judging; though, surely, the act of judging is an " operation of the Understanding subservient to the estimation of evidence." In the Examination (p. 447), however, we read : " He (Hamilton) says : ' Logic considers Thought not as the operation of thinking, but as its product ; it does not treat of Con- ception, Judgment, and Reasoning ; but of Concepts, Judgments, and Reasonings.' Let me begin by saying that I give my entire adhesion to this distinction." This passage agrees with that from the chapter on the Import of Propositions ; but how does it agree with the definition of Logic; and how with the existence in the System of a chapter on the Formation of Conceptions ? I cannot reconcile these statements (and still others might be adduced of a similar kind) ; but the above definition of Logic is expressly given as only provi- sional, on the ground that a complete definition can- not be framed until the Science is further advanced : so that in the meanwhile there is some room for vacilla- tion. I am sorry to say, however, that the definition first given (and no other is offered) is asserted to be " at all events a correct definition of the subject of Introduction. 7 these volumes : " for that I must dispute. It appears to me that the subject of those immortal volumes is not the operations of the mind, but primarily the Laws of Nature and their Proof. And the satisfactory proof of a Law of Nature consists always according to that work, in bringing" it within the sweep of some highest Law, which itself rests upon constant and un- contradicted experience. The highest Laws are the Axiom of the Syllogism, the Law of Causation with its derivative Canons of Experiment, the theory of Probabilities, and perhaps the doctrine of Kinds ; all of which are plainly conceived by Mill to be Laws of Nature. Then in the First and Fourth Books there is much discussion of matters subsidiary to the dis- covery and proof of Laws, such as Names and Naming, Definition, Classification, &c. ; and here again facts and the order of Nature are the chief concern. I grant that all this is interspersed with Psychological disser- tations in answer to such questions as, If axioms are based on particular experiences, whence the feeling of their certainty ? What is the true process of Infer- ence ? of Abstraction ? Is Volition an efficient Cause ? &c, : and the immense value of these passages I would be the last to question. But they form a comparatively small portion of the book; and I venture to think that, regarded merely as a treatise on Logic, the book would be nearly as complete without them. Of course the writer who maintains that names and propositions refer not to ideas, but to things, is free from the least 8 Theory of Logic. taint of Conceptualism : as little is he a Nominalist, Although his position was not perfectly clear to him- self, Mill was in reality a matter-of-fact Logician. In Prof. Bain's great work I am not aware that Logic is anywhere, strictly speaking, defined ; but it is described, its scope (as viewed in that work) is stated, and it is divided. In the Introduction^ i , we read : " Logic may be briefly described as a body of doctrines and rules having reference to Truth ; " and " the Truth of things, no matter what the subject be." And this, I suppose, is as much as to say that Logic is concerned with matter-of-fact in general (no matter what the subject be) ; or, in other words, with the most general laws of the correlation of phenomena. However, in 2, we read : " Logic under every view involves frequent references to the laws and workings of the mind ; " and so indeed throughout the work we find these frequent references ; though an advance has been made on Mill's practice, as it appears to me, by collecting very many of them into the Introduction. But I can hardly admit that Logic really involves these references to the workings of the mind. It is true that some of the principal doctrines of Logic have been attained by the help of Psychology ; but those doctrines once reached, the Psychological ladder may be kicked away. The doctrine of Rela- tivity, for instance, fundamental in Logic, was first demonstrated in Psychology ; but being demonstrated, or rather accepted, it is no longer a peculiarly Introduction. 9 Psychological doctrine ; for it is true not of the sub- jective order of phenomena only, but equally of the objective order ; and it is in its universality, as prevailing in both orders, that it is, I conceive, funda- mental in Logic. To be sure Logic is Science, and Science is knowledge, and in every act of knowledge (with some qualification in the case of psychological knowledge) Object and Subject are inseparable co- efficients. But this is no more true of Logic than of the other Sciences. The Laws of Nature contemplated in Logic are in one aspect cognitions, but so are the Axioms of Mathematics ; so are the Laws of Chemis- try : and an account of any Law of Nature may be given from the subjective side. But there is a Science in which the nature of all cognitions is investigated once for all ; and in no case, except Logic, is it deemed necessary to interrupt the course of a special Science, in order to give an account of the cognitions involved. What is present everywhere, once recog- nized, may be everywhere suppressed. The sub- jective element is present everywhere ; and having been recognized in Psychology, may in all the other Sciences be overlooked. Indeed we may call it a postulate of the Abstract and Objective Sciences, that the subjective element may be neglected : we write, Such is the course of Nature ; not, Thus it appears to us. The passion of British philosophers for psychological explanations and foundations, is perhaps due to the somewhat exclusive cultivation of i o Theory of Logic. that Science which has always characterized British Philosophy. Prof. Bain's position that Logic involves frequent references to the laws of the mind, is supported by citing the custom of Logicians. The custom must be admitted, but it does not guarantee its own pro- priety; it is only a sign of the imperfect state of Logic and adjacent Sciences. Metaphysics, Logic, Psychology, Rhetoric, &c., growing up together, and to some extent mutually dependent, have become very much tangled. Indeed, formerly the tangle was much worse than now : now it is at last possible to undo it ; and the second object of this Essay is to clearly extricate Logic. I think we shall gain by it : though it must be allowed that hitherto the inter- mixture of Logic with other Sciences has had some good results : and as for the Psychological discussions in Prof. Bain's work, they appear to me to be always just and instructive. The real theme of that work, however, is, like that of Mill's, the Laws of Nature. And I do not think I can be wrong in claiming Prof. Bain as a Logician in whose view Logic is a Science of matter-of-fact.* But the writer who has expressed this view most distinctly is Mr. Spencer, and I cannot do better than quote the passage : " A distinction exists which, in consequence of its highly abstract nature, is not easily perceived, * Cf. Logic, Appendix B. Introduction. 1 1 between the Science of Logic and an account of the process of Reasoning". The distinction is in brief, this, that Logic formulates the most general laws of correlation among existences considered as objective; while an account of the theory of Reasoning, formu- lates the most general laws of correlation among the ideas corresponding to these existences. The one contemplates in its propositions, certain connections predicated, which are necessarily involved with certain other, connections given : regarding all these con- nections as existing in the Non-Ego not, it may be, under the form in which we know them, but in some form. The other contemplates the process in the Ego by which these necessities of connection come to be recognized."* This passage points out clearly the nature of the error committed by those who regard the Theory of Reasoning, which is a part of Psychology, as an essential part of Logic. But it does not bring out quite all that I mean by saying that Logic deals with matter-of-fact ; for it includes in Logic some things that are not matter-of-fact, and excludes some things that are. To explain, let us first inquire What are " existences considered as objective r " The Object is rightly opposed to two other kinds of existence, real or supposititious ; namely, the Subject, and the Noumenon. The Metaphysical universe is usually divided, I conceive, into Phenomena and * Psychology, Part VI. ch. viii. IT 2 Theory of Logic. Noumena; and Phenomena are again classed as Subjective and Objective: and if Noumena are also sometimes similarly subdivided, the Noumenal Object or Subject is, or always ought to be, expressly qualified as Noumenal or Transcendent. Thus exist- ences considered as merely objective should always be Phenomenal, and Phenomena are existences in the form in which we know them. According to Mr. Spencer, however, the connections regarded by Logic, or some of them, exist " in the Non-Ego, not it maybe in the form in which we know them;" that is, I suppose, are Noumenal. I understand, then, that Mr. Spencer in this passage (as, I think, very often in the Metaphysical portions of his writings) means to include in the Object, or among existences con- sidered as objective, not only objective Phenomena, but also Noumena, or at least some Noumena. Now whether the Noumenon be a reality or an illusion this is not the place to discuss, but probably most philosophers will admit that it is not a matter-of-fact ; and, therefore, I do not include any connections that may exist in it within the scope of Logic. For who can tell whether relations of Likeness, Coexistence, and Succession, or anything parallel to these familiar entities, obtain in that untrodden realm ? Again, it is clear that among existences considered as objective, Mr. Spencer does not include the Subject ; for " the most general laws of correlation among existences considered as objective," are treated as Introduction. 1 3 equivalent to " certain connections regarded as exist- ing in the Non-Ego." But the Subject is a matter-of- fact, and I wish to include it (in a sense to be presently explained) within the scope of Logic. This is why I can accept the description of Logic as an Objective Science only on condition of being allowed to give a special explanation of the word " objective," as here used. Strictly speaking, Object and Subject are mutually exclusive, that is, so far as the nature of the matter will admit ; but as the Object is something contrasted with the Subject, so within the Subject itself some phenomena may be contrasted with others still more subjective. This happens in all psychological analysis ; in which the Subject is often said to be made the object of study ; and by putting a special strain upon the words, certain states of the Subject might then be said to be con- sidered as objective. But far be it from me to contri- bute to confusion : and, therefore, I will not describe Logic as an Objective Science. It is, I hold, neither an Objective, nor a Subjective Science, nor partly one and partly the other, but is raised above the distinc- tion of Subject and Object, a universal Science, formulating the most general laws of correlation among existences whether objective or subjective. But now it may perhaps seem that, according to this account of Logic, it must include the Theory ot Reasoning, which was lately excluded. Not at all . ran account of the process of reasoning formulates the 14 Theory of Logic. most general laws of correlation among ideas corres- ponding with certain other existences, and (I may add) regarded as corresponding with certain other exist- ences. Whereas Logic deals with ideas and their correlations as such, and not merely as corresponding with certain other existences. This distinction, it will be observed, is similar to that drawn by Mr. Spencer between Psychology and Biology.* That Logic may, nay, must so deal with subjective phenomena is obvious ; for some at least of the rela- tions which obtain in the Object, obtain also in the Subject Likeness, Succession, and in some degree Coexistence : and so far as similar relations obtain among phenomena of both orders, the science of those relations is the same. Logic, then, can only be described as to its matter by calling it a Science of universal matter-of-fact, I know no short name for it : Realistic and all cognate words are excluded by historical considerations, but this is a clumsy ex- pression, and it is better to describe it according to uts form. Logic is an Abstract Science ; and the absence of any other generic name for Logic is a reason for confining the name, Abstract, as Mr. Spencer does, to Logic and Mathematics. For, of course, Mathematics, like Logic, is neither an objec- tive nor a subjective Science, but indifferent to this distinction. For in as much as quantitative relations of Number, Intension, Pretension (subjective Exten- * Principles of Psychology, 53. Introduction. 1 5 sion, though it exists, is too indefinite to be measur- able) occur among subjective phenomena, Mathe- matics treats of them, at the same time with similar relations in the Object. 4. Position of Logic among the Sciences. With the qualifications above indicated, the position of Logic among the Sciences appears to be that which has been assigned to it by Mr. Spencer. And indeed so far as (with only a superficial knowledge of most Sciences) I may presume to judge, the whole classifi- cation of the Sciences given by him seems to me just and admirable. But it is a pity that in that classifi- cation the place of Subjective Psychology is not ex- pressly marked ; and so I propose, by recasting the general Table given in the Classification of the Sciences, p. 12,* both to find a place for that Science, and to show the place of the Abstract Sciences as conceived in this Essay. Thus it will be seen how the Abstract are the only universal Sciences. SCIENCE. Abstract, dealing with Relations in general, whether their Terms be Objective, or Subjective. Abstract-Concrete, treating of phenomena in their elements. Concrete, treating of phenomena in their totalities. Spencer, Essays, vol. iii. 1 6 Theory of Logic. In this classification there is still no place for Metaphysics, or First Philosophy, and there seem to be reasons why Metaphysics cannot be classed among Sciences. i. Since no accepted body of doctrine yet exists which can be called Metaphysics, we cannot be sure whether such a body of doctrine, if ever it should exist, would have sufficient unity to be called a Science. 2. It is probable that Meta- physics will never be a body of dogmas, as the Sciences are, but rather a place of criticisms. I hardly expect indeed that it will ever be a Science at all, as Mr. Lewes does ; though I hope that by some such method as he has elucidated,* we may one day have a criticism of Axioms and instinctive beliefs, to which most well-formed minds will be able to assent : such a criticism rendered as systematic as possible might be aptly called Metaphysics ; and so that wandering word find rest at last. That there should be a special place for such criticism, instead of leaving to the several Sciences the criticism of their own Axioms, is very desirable. For the mental attitudes of Science and Criticism are strongly contrasted, and the aptitudes for them are very different. Moreover it seems that the chief focus of Metaphysical criticism is the union or mutual im- plication of Object and Subject : whereas the special Sciences assume the differentiation of Object and Subject ; except the Abstract Sciences, and these * Problems of Life and Mind, Introduction, Part I. ch. iii. Introduction. \ 7 merely neglect it. And the existence of a competent Metaphysics of this nature would be a great relief to the special Sciences ; particularly to Psychology, and to Logic, as we shall more than once have occasion to notice. I understand a discussion to be critical, when the question stated does not admit of a complete solution, and the possibility of a solution is inquired into, or an approximate solution is sought by investigating the origin of the question, and of the rival solutions of it, besides balancing the arguments for the rival solutions. Kant's Transcendental Dialectic, appears to me to furnish a better model for a system of Meta- physics than any other work I am acquainted with. The Classification of the Sciences would itself, I suppose, be an outlying topic of Metaphysics, or First Philosophy. 5. Redistribution of the Contents of Scholastic Logic, We have seen how heterogeneous were the con- tents of the Scholastic Logic : including the Science of the use of language in Reasoning ; the Theory of Reasoning itself; occasional discussions in Meta physics ; and expressly or by implication, some of the most general laws of the correlation of phenomena. These ingredients we have to redistribute : and, first, only the last named portion of Scholastic Logic was really Logical. As for the metaphysical discussions : 1 8 Theory of Logic. pending the constitution and general acceptance of some body of metaphysical criticism, its questions must be dealt with to some extent in the special sciences, wherever intelligence requires it. The Theory of Reasoning is of course absorbed by Psy- chology. And the Science of the use of language in Reasoning, I propose to cede to Rhetoric. For, surely, it is anomalous that there should be one Science which treats of the use : of language in discourse generally; whilst the use of language in a particular kind of discourse, and that the most important, is dealt with in another Science. If Logic deals with the use of language in reasoning ; of what does Rhetoric treat ? Is it conversant with i AA^UVC . the use of language in obfuscation ? I fear it is commonly thought so. I believe an 'impression prevails, that if you wish to state your case plainly and fairly, and avoid misleading yourself and others, you may perhaps derive some assistance from Logic ; but that if you want to overcolour your facts, and make the worse appear the better reason, you had better apply your mind to the study of Rhetoric. Rhetoric, in short, according to this view, is the art of so using language as to " minify the great and magnify the little." The prevalence of such notions explains the neglect of Rhetoric in modern times, and why so few influential minds have given it their attention. It was impossible that such notions should not arise, while the use of language as the Introduction. 19 vehicle of truth was discussed by another Science. It was impossible that the modern nations, so anxious about truth, as to develop the experi- mental Sciences, should bestow much thought upon a study which, at best, appeared to aim at nothing better than ornamentation. So long as it bears that appearance, Rhetoric can never prosper : language is the instrument of truth, as the epigram to the contrary bears witness; and truth brooks no rival interest. But I see no reason why care, accuracy, and elegance in the use of language should not be united in one discipline. Professor Bain * in his English Composition and Rhetoric, under Exposition and Persuasion, expounds some parts of the Scho- lastic Logic : why should it not all be expounded there, so far as it is concerned with Names, Proposi- tions, and Arguments ? By giving to Rhetoric such a core of necessary matter, it would certainly be rescued from neglect ; and the remainder of its sub- stance, serviceable to beauty and perfection, would secure the regards of many more students ; who must be edified accordingly. By being entangled with Logic, a hardier Science, Rhetoric has been robbed of its own, and stunted in its growth ; if given more room, it may perhaps flourish again. But it is not a Science that can be * Cf. Whately : Rhetoric ; Part I. c. ii. Campbell : Rhetoric ; Bk. I. cc. iv vi. The close affinity of the first Book of Aristotle's Rhetoric to Logic is also obvious. C 3, 2O Theory of Logic. altogether separated from others ; it is not a funda- mental Science. Language is a mediator between thought and fact, and the science of the use of language must depend upon Psychology and Logic. Upon the principles of Logic will depend, for instance, all that part of Scholastic Logic, which we propose to cede to Rhetoric. The principles of Consistency, which belong to Rhetoric, represent certain aspects of the constancy of nature, which are laws of Logic : were nature inconsistent (so to speak) we should be under no obligation not to be so ; since inconsistent statements might then both be true. The import of Names and Propositions, the processes of Obversion, Conversion, &c., as con- cerned with language, are all explained by reference to corresponding logical principles, which will appear in subsequent chapters. 6. Fallacies. Logic, as I try to regard it, has little or nothing to do with Fallacies. It is no doubt quite possible to commit Fallacies in expounding Logic, or in in- terpreting the exposition ; but in the actual corre- lations of phenomena, in matter-of-fact, there can lurk no Fallacies. Fallacy is a kind of Error; it is incident to the correspondence of Subject and Object, and arises when that correspondence is im- perfect. The proper place to treat of Fallacies, Introduction. 2 1 therefore, would seem to be the Science which inves- tigates the means of furthering the correspondence, that is, the Science of Education, especially of the Intellect. There at least what Prof. Bain calls the "fallacious tendencies of the mind," would be most suitably corrected. But by far the greater portion of what are usually called Fallacies, must be handed over to Rhetoric. To Rhetoric naturally belong all Fallacies occasioned by the use of language, whether in private meditation, or in the communication of ideas ; and whether the misrepresentation which essentially constitutes the Fallacy, prevail in the mind of the thinker himself, or be one which he wishes to make prevail in the minds of those whom he addresses. Thus the Table of Fallacies in Whately's Logic might be transferred whole to a treatise on Rhetoric. Indeed many Fallacies, all those which may be called devices of sophistry, are plainly such as ought never to have been men- tioned in Logic. Petitio principii, ignoratio elenchi, argumentum ad hominem, &c. ; these are tricks of the hustings ; and to treat of them in Logic shows with what an arrogant and grasping spirit the Logician has invaded the province of Rhetoric. 7. The Logical Calculus. In this Essay everything proceeds, somewhat as in Euclid, by a comparison of intuitions, and I only 22 Theory of Logic. need mention once or twice, in passing, the systems of Logical calculation, developed by Boole, Prof. Jevons, &c. Not that I underrate the advantages of a Calculus : probably by its means conclusions may be reached, which few, or no one could prove without it ; and certainly, once mastered, it saves effort even in less complicated and protracted trains of reasoning. Still it is not the Science of Logic, but a machinery con- structed on Logical principles, and related to Logic, as the Rules of Arithmetic are related to the Science of Number. And it would be a great mistake, I think, to substitute a drill in the Calculus, for an. explanation of the Science, as a means of Education. For in using the Calculus we lose to a great extent that discipline of the power of abstract intuition, which is the great benefit of Logical studies. It would be sad indeed if the study of Logic should sink into that state in which the study of elementary Mathematics still almost everywhere grovels : there are at this moment half a million children in the country, having rules and formulae drummed and brayed into their ears, unmitigated by one note of Science. But of this there is little danger. 8. Acknowledgments. In the foregoing pages, I have noticed certain opinions of Mill, Prof. Bain, and Mr. Spencer, in order to point out how I differed from them : it was Introduction. 23 necessary to distinguish their views from my own, because they are the writers with whom I feel myself in closest agreement. And I now hasten to add, that if there is anything of value in the ensuing pages, it is probably derived from their works. The idea of the whole and the substance of parts, are derived chiefly from Mr. Spencer ; most of the remainder is founded on the writings of Prof. Bain and Mill. As to quotations and references, when I might have used either Prof. Bain or Mill, I have generally preferred the work of" Prof. Bain, in as much as he has made several improvements in the modes of statement adopted by his great forerunner. After writing the last chapter, I found that De Morgan in his First Notions of Logic had anticipated to a great extent my treatment of the Syllogism, or Mediate Relation of Classes ; and I was able to make some improve- ments from hints supplied by him. Lesser obliga- tions will be acknowledged as they occur. CHAPTER II. OF RELATIONS. i. Definition of Logic. LOGIC has been defined by Mr. Spencer as " an Abstract Science, treating of the Laws of Relations that are qualitative ; or that are specified in their natures as relations of coincidence or proximity in Time and Space, but not necessarily in their terms : the nature and amount of which are indifferent." * And this definition with two slight qualifications I am willing to accept : first, Logic cannot altogether ignore Relations that are quantitative ; secondly,, besides Relations of Contiguity in Time and Space, those of Likeness and Unlikeness must continually be considered. The Likeness and Unlikeness of Terms lies at the foundation of the Logic of Classes ; which was nearly the whole of the Scholastic Logic : as Mr. Spencer has elsewhere described it " a science of the relations implied in the inclusions, exclusions, and overlapping of classes." * * Classification of the Sciences, Table I. + Study of Sociology, ch. ix. Of Relations. 25 2. Of Relations in General. A Relation cannot be defined, for we know of nothing more elementary. The only way of bringing it to light is by contrasting it with its co-ordinate abstraction, the Term. Every Relation lies between, or connects, or ties two Terms, and no more. All Terms are connected and tied by Relations. We may be helped to realize these notions by the figure of two balls tied together with a string. The world consists of related Terms or terminated Relations. This seems to be the end of all analysis, whether of the Object or Subject.* The ultimate modes f of Relation are 1. Likeness and Unlikeness. 2. Succession and Nonsuccession. 3. Coexistence and Noncoexistence. And it must be observed that although in each of these couples, one name has a negative prefix, the Relation signified thereby is not less real than the other. Negation is an artifice of language : in nature there is only contrast and incompatibility. J Likeness has only a single contrast, Unlikeness : but Succession * Bain : Logic, Appendix C. t Bain : Logic, Appendix C ; and Bk. I. ch. iii. 16, 17. J Kant : Versuch den Begriff der Negativen Grb'ssen c. Bain : Logic, Bk. I. ch. i. 12. 26 Theory of Logic. is contrasted indefinitely with Nonsuccession, defi- nitely with Coexistence ; and Coexistence is con- trasted indefinitely with Noncoexistence, definitely with Succession.* Likeness precedes Succession and Coexistence in the order of exposition because it is involved in them ; for they are Unlikeness and Like- ness with respect to Time. And Succession precedes Coexistence, because it is simpler, and according to Psychological Theory, prior in experience; and because we shall find that coexistences often result from successions (Causation), but we have not to notice any cases in which succession results merely from coexistence. If we call Likeness, Coexistence, and Succession, Positive ; Unlikeness, Noncoexistence, and Nonsuc- cession, may be called Counter Relations. 3. Of Relations of Likeness and Unlikeness. The Likeness and Unlikeness of phenomena is the fundamental fact of nature. From the Cosmological point of view, that phenomena are alike and unlike is the reason why identification and discrimination are the ultimate faculties of mind : from the Psycho- logical point of view, they are two expressions of the same fact. Ultimate Relations themselves are alike and unlike; else they could not be classified as above. * Spencer : Psychology, Part VI. ch. viii. Of Relations. 2 7 But Likeness and Unlikeness prevail amongst phenomena in various degrees, from the vaguest and most superficial resemblances and contrasts to exact Likeness and Unlikeness with respect to Quantity. Relations of Likeness and Unlikeness are thus either 1. Quantitative, or 2. Qualitative. The Quantitative division, comprising Relations of Equality and Inequality of amount in respect of Number, Intensity, Time, Space, is the matter of Mathematics.* And several recent works on Logic have given some account of the methods of Mathe- matics : but no such task falls within the design of the present Essay ; which aims hardly at all at being practical, but mainly at pure Science ; treats not of how Relations are dealt with, but of Relations them- selves; and therefore, since Quantitative Relations are treated of in Mathematics, so far as possible only of Qualitative Relations : though some of the discussions are so abstract as to be almost equally applicable to Relations of both orders. For indeed it is obvious that, if there are any truths concerning Relations in general, they must be common to Logic and Mathe- matics ; being the contents of that generic Abstract Science of which these sciences are co-ordinate species. The Qualitative division of Relations comprises * Spencer : Classification of the Sciences, Table I. &c. 28 Theory of Logic. ' Relations of Likeness and Unlikeness in respect of Quality merely, or Nature ; of Likeness and Unlike- I ness in respect of Time, that is, Simultaneity and Succession; and Likeness in Time, with adjacency in Space vaguely implied, or Coexistence ; with the indefinite Relations of Nonsuccession and Non- coexistence. But I must add that when Relations of Succession and Coexistence are definitely mea- sured, they become subject to Mathematical rather than Logical treatment. Strictly, it is only when Succession and Coexistence are considered as such and apart from measurement of Time and Space, that they belong to Logic. For Mathematical treat- ment, on account of its greater definiteness and immense resources, has the preference whenever applicable. Merely Qualitative Relations of Likeness and Un- likeness may again be generally distinguished from one another, as 1. Definite, or 2. Indefinite. It is with Definite Likeness or Unlikeness that Logic has to do ; since it is only so far as these Relations are definite, that Laws of phenomena can be estab- lished : wherefore, too, it is with such Relations that Reason is conversant ; since only so far as there are Laws can there be safe inferences. Indefinite Like- ness and Unlikeness, on the other hand, belong to ... A - fc /~\s r> r i' Uf jKelations. u -KYUT > . axc J \k Fancy, and often furnish matter to poetry and wit ; as 20 (li- when the " flying fiend " is compared with a fleet of ships, or a cloud is said to be like a weasel, or like a whale. But such vague and transitory resemblances afford no footing to Science. The most definite Relations of Qualitative Likeness may sometimes be called Equal ; though there is a tendency to confine that name to Quantitative Rela- tions. 4. Of Relations of Succession. Relations of Succession are either 1. Inconstant, or 2. Constant. An eclipse of the sun may or may not be followed by a disastrous battle ; it is always followed by dark- ness upon earth : the former Succession of events is classed as Incoherent or Inconstant, the latter as Coherent or Constant. In the infinite movement of the world from moment to moment, the Incoherent Successions are of course incalculably more numerous than the Coherent ; since all events of the second moment follow each event of the first ; while on the recurrence of any event of the first moment, only one or a few events of the second moment would recur. But Coherent Successions afford most scope for Science, or generalized knowledge, and are those which are chiefly treated of in Logic. 30 Theory of Logic. Constant Successions are said to involve Causation, and this may happen either directly or indirectly. A Relation of Direct Causation is called a Relation of Cause and Effect; such is the Relation between sunrise and daylight upon earth. A Coherent Succession by Indirect Causation may obtain, or seem to obtain, between two Part-Effects of a single Cause ; as between day and night over the same hemisphere, the Joint-Effect of a planet's rotation in the sunshine. This indeed may be viewed as two Effects of the continued action of a Cause ; and it must be admitted that an unexcep- tionable example of this Relation is hard to find. The flash and report of a gun seem to make a case in point ; but here distance of the observer is a condition of the succession of the Part-Effects. The difficult subject of Causation will be discussed at greater length in Chapter VI., and all relevant remarks else- where, I should wish to be interpreted in the sense of fuller discussion. A Coherent Succession by Indirect Causation may be called a Relation of Coeffectionally Coherent Suc- cession. A Coherent Succession by Direct Causation may be called a Relation of Efficiently Coherent Succession, or a Relation of Efficient Coherence, or simply a Causal Relation ; and a series of events so related may be called a Causal Series. Amongst Causal Series, again, we may distinguish from the others, those which consist of several events Of Relations. 31 that happen again and again in a very similar order.* Such repetitive Series make up the lives of plants and animals : they may be called Cyclical ; all other Causal Series, Acyclical. It is possible that all Causal Series are in the long run Cyclical : this is the famous speculation that in the infinite lapse of Time the World repeats itself. But it is enough if the above distinction be real in experience. The indefmiteness of Relations of Nonsuccession prevents their being similarly classified : they can only be contrasted with each and every sort of Succession. 5. Of Relations of Coexistence. Relations of Coexistence, like Relations of Succes- sion, are either 1. Inconstant, or 2. Constant. Inconstant or Incoherent Coexistence is the Rela- tion of all things in the world to one another at any moment of the world, in so far as they cannot be expected to recur continually in the same Relation. Thus, a book on the table, and a tree in the garden, though not without a certain Coherence in the system of the World, would usually be said to stand to one another in a Relation of Incoherent Coexistence. Bain : Mental Science, Bk, II. ch. L 46. 32 Theory of Logic. Such Relations are only within narrow limits a subject of Science, or generalized knowledge ; though to deal with them (as by measurement) may be part of the object of Applied Science. They will, of course, not be confounded with Relations of Position in the abstract which belong to Geometry. Constant Coexistence is the Relation of entities at any moment, in so far as they, or similar entities, may be expected to recur continually in the same Relation. Constant Coexistence is either 1 . Coeffectional, or 2. Specific. Coeffectional Coherent Coexistence is the Relation of coexistent Part-Effects of the same Cause : such is the Coexistence of night and day over opposite hemispheres of the earth. And possibly all Co- herent Coexistence is ultimately Coeffectional. Specifically-coherent Coexistence is the Relation of qualities or parts in a member of a Natural Kind. It is either 1. Essential, or 2. Integral. Essentially Coherent Coexistence is the Relation- ship of qualities in a substantial group, as of the colour, specific-gravity, &c., of gold. And Essential Coexistence does not involve Relations of Position, Of Relations. 33 since qualities appear to subsist in mutual inter- fusion. Hence the formula of the Logical Calculus,* ABC = ACB = BCA = &c. Integrally Coherent Coexistence is the Relationship of separable parts of a whole, as of elementary sub- stances in a chemical compound, or of members in an organized body. Integral Coexistence is ultimately reducible to Essential Coexistence together with Relations of Position. The Coexistence of motions or events may be called Simultaneity. As with Relations of Nonsuccession, the indefinite- ness of Relations of Noncoexistence, or Nonsimul- taneity renders it impossible to classify them ; but they are contrasted with each and every sort of Coexistence and Simultaneity. * Jevon's Principles of Science, vol. i. p. 41 (ist ed.) 34 Theory of Logic. Likeness and Unlikeness. 6. Table of Relations. [-in Quantity [-Equality. -Inequality. ^-Quality merely r-Connature. <-Nonconnature. r Succession plnconstant. and Non-succession -Constant r Coeffectional. ^-Efficient pA cyclical l-Cyclical. -Simultaneity, plnconetant Nonsimultaneity, and (space sometimes vaguely implied) Coexistence and Noncoexistence. -Constant i-Coeffectional. ^Specific r-Essential. ^Integral. CHAPTER III. OF TERMS. i . Of Terms in General. THE Term can be defined no more than the Rela- tion. In trying to elucidate the notion of Relation- ality we have already done what we can to elucidate the notion of Terminality. Terms are contrasted with Relations as being entities related. All Terms are tied together in couples by Relations. A Rela- tion of two Terms seems to be the unit of existence. No Term without a fellow : no pair of Terms without a Relation : no Relation without two Terms : no Relation with more than two Terms. But every Term enters into many Relations : is indeed related in some way to every other Term. Terms are either 1. Simple, or 2. Compound. And simple Terms are either 1. Feelings, or 2. Relations. D 2 36 Theory of Logic. Within our present consciousness Feelings are ulti- mate Terms, although there is a Psychological hypo- thesis* that no known Feeling is really a simple or ultimate experience. Feelings being related, a Rela- tion of two Feelings may be itself related to another Relation of two Feelings ; as when the Coexistence of two qualities in one animal is like the Coexistence of similar qualities in another animal of the same kind : hence Relations may themselves be Terms and may then be called Terminal Relations. 2. Of Feelings as Terms. How Feelings are Terms hardly needs pointing out. One Feeling of warmth is like another Feeling of warmth, and unlike a sound. Certain combinations of sound are simultaneous with, or succeeded by, a sense of pain : and so on. Pure Feelings, as such, belong to the Subject : but Feelings regarded as in Essentially Coherent Co- existence with other Feelings are called Qualities ; and, as Qualities, they are either Subjective or Objective, 3, Feelings as Qualities. Jt is as Qualities that Feelings are most important in Logic, and especially as Qualities of the Object ; * Spencer : Psychology, Part II. ch. i. Of Terms. 37 for although many theorems of Logic hold good of the Subject, it is in the greater definiteness and co- herence of the Object that they are best studied. Since Feelings are Terms, of course Qualities are ; for Qualities are only Feelings that terminate par- ticular Relations. To terminate Relations of Essen- tial Coexistence is the nature of Qualities. They may also be related by Coherent Succession, as when in melting ice, the qualities of a liquid succeed those of a solid : and they may be like or unlike, as with colour in snow and May-blossoms, or in snow and poppies. 4. Of Relations as Terms. Relations themselves may be Terms of all kinds of Relations : indeed all Relations are Terms. That Relations may terminate Relations of Likeness and Unlikeness, that is, may be like or unlike one another and unlike Terms, is implied in their classi- fication among themselves, and in distinction from Terms. Relations are alike in their Relationality and unlike Terms : Relations of Likeness are alike, and unlike Relations of Unlikeness, Succession, and Coexistence ; and so on. Similarly all kinds of Relations may terminate Relations of Succession, and do so when they are implicated in Causal Series. Take a billiard-ball rolling about upon the table. At three successive 38 Theory of Logic. moments, if we make the moments short enough, the motion of the ball in the first moment is like (or imperceptibly unlike) its motion in the second ; and its motion in the second moment is like its motion in the third : thus two Relations of Likeness succeed one another. And since these Relations of Likeness coincide with Relations of Succession, the Relations of Succession likewise succeed one another. At the same time the motion of the ball is being converted into vibrations, which coeffectionally coexist; and these coexistent vibrations from moment to moment succeed one another. And all kinds of Relations may terminate Rela- tions of Coexistence, and are commonly implicated in Specific Coexistence. In the organization of an animal, in so far as it is symmetrical, we have the Coexistence of Relations of Likeness : many changes, too, involving Relations of Succession, go on simul- taneously, or coexist, in corresponding members as the effects of common causes : and Relations of the Integral Coexistence of parts constantly coexist with Essential Coexistence of qualities. Lastly, Relations of Succession may coexist with Relations of Coexistence ; and the Relations of Co- existence thus arising between Relations of Succes- sion and Coexistence, may again be related by Coherent Succession. Thus, whilst ice is melting, there are changes of consistency and specific-gravity, which coexist with the coexistence of unaltered Of Terms. 39 weight and chemical constitution ; and as the process of melting continues, moment by moment such co- existent Relations of Relations coherently succeed one another. Further complexities of Relationality the reader will follow out for himself. 5. Of Compound Terms. A Compound Term is a definite Group of Qualities; and such a Group may occur in the Subject as an Idea, or in the Object as a Thing or Event : but it is better studied in the Object. To be treated as Terms such Groups of Qualities must have some coherence ; and for Logical purposes they may perhaps be best classified according to those Relations of their parts which give them coherence. And since Relations of Likeness do not give coherence to Terms, we have only to consider how Terms may subsist by the co- herence of Qualities in Succession, or Coexistence, or both. i. As to Succession. We may suppose a Com- pound Term to consist of two Simple Terms related as Cause and Effect : but such a case is unexampled ; for Simple Terms are abstractions, and Causation is of the concrete. True, a Relation of Cause and Effect is sometimes said to be a Relation of two events or changes ; and a change is itself a Relation of Succes- sion, which if not compounded is a Simple Term. 40 Theory of Logic. But in the first place, a Relation of change is in reality always compound ; and moreover a succes- sion of two changes is not the whole Relation of Cause and Effect, as we shall see in Chapter VI. 2. As to Coexistence. We may suppose a Com- pound Term to subsist by the Coexistence of Simple Qualities : and such are most of the concrete phe- nomena of the Inorganic World. A piece of iron, a stone, a house these are instances of Groups of Qualities cohering by Coexistence. Such we may call Substances. The type of this simplest kind of Compound Term is a chemical element : a house is an outlying example. 3. Are there any Compound Terms whose in- tegrity depends on both Succession and Coexistence ? Plainly there are : the qualities and parts of an organized body are interrelated both by Succession and Coexistence: its coherently coexistent qualities at one stage of growth are coherently succeeded at another stage of growth by other qualities also coherently coexisting. But hitherto we have only considered the general Relations of constant Succession and Coexistence : how far may this classification be extended by taking account of more special modes of Relation ? First, are any sub-classes of Compound Terms subsisting by Coexistence, to be distinguished according as the constitutive Relation is Coeffectional, Integral, or Essential ? Of Terms. 41 i. We shall see in Chapter VI. that it is an aim of Science to show that all Coherent Coexistence is Co- effectional, but certainly this cannot at present be accomplished. Mixed Relations of Simple Coex- istence and Coeffection are perhaps the com- monest, as far as our knowledge reaches ; and therefore we make no subdivision at this point. And, though obvious, it may be worth observing that Coeffectional Relations of Coexistence, obtaining amongst the parts or qualities of a Substance, do not introduce into it any Relation of Succession; for though they savour of Causation the Efficient Rela- tion itself is not involved in them. If, for instance, the Relations of the qualities of gold among them- selves were shown to be Coeffectional, that would not introduce a Relation of Succession among those qualities ; but would only prove them to be small part-effects of some vast and ancient case of Causation. ii. Nor do we subdivide the class of Compound Terms dependent on Coexistence, on account of Integral Relationship. For, in the first place, all Compound Terms (with the hypothetical exception of physically simple atoms) involve both Integral and Essential Relations. And, secondly, as before re- marked, Integral Parts are themselves resolvable into Essential Coexistence of Qualities with Rela- tions of Position. Secondly, how are Compound Terms, subsisting 42 Theory of Logic. by both Succession and Coexistence, affected by the constitutive Relations of Succession being Coeffec- tional, Acyclical, or Cylical ? i. Coeffectional Relations of Succession, when known to be such, are of subordinate importance in comparison with the Efficient Relations in which they are involved; and hence establish no inde- pendent Terms. ii. Relations of Acyclical Succession, however, may be regarded as giving coherence to, and esta- blishing, independent Compound Terms. In every case of Cause and Effect, the set of coexisting circumstances making up the Cause, and the set of coexisting circumstances making up the Effect, are bound together by the Efficient Relation into a complex whole, which we may name a Causal Instance. iii. And Relations of Cyclical Succession among Coherent Coexistences, also establish Compound Terms; namely, organised bodies, which we have already described as subsisting by both Succession and Coexistence : and these we may call Individuals. But the name Individual cannot be consistently con- fined to organic bodies; we must extend it to all bodies that exhibit a cycle of evolution ; as, for instance, a planet. We find then that there is only one kind of Com- pound Terms, subsisting chiefly by Coexistence, namely, Substances ; but two kinds subsisting by Of Terms. 43 both Coexistence and Succession, namely, Causal Instances and Individuals. Both Causal Instances and Individuals involve Causation : but in the former case it is Acyclical, occurring as an incident in the general weaving of Nature, and liable to be dissipated in ever new directions according to circumstances ; in the latter it is Cyclical, caught (as it were) in a vortex, and revolved in a crowd of similar cases, through approximately similar changes in similar times. Both involve Coexistence : but the Coexist- ence involved in Causal Instances, though partly Essential, since Causation is of the concrete, needs not be Coherent throughout, but generally involves the concurrence of separable circumstances ; whereas the Coexistence involved in the nature of Individuals is throughout Integral and Essential, and indeed in their case the coherence due to Coexistence is liable to be mistaken for the whole. 6. Compound Terms in their Relations. Compound Terms in Relations of Succession and Coexistence present Logic with no new phenomena. A Coherent Succession of Substances would be re- solvable into Substances and Causation ; a Coherent Succession of Causal Instances only yields more Causal Instances ; the Causation or Succession of Individuals by generation only produces more In- dividuals. Similarly, the Coexistence of Substances 44 Theory of Logic. is a more compound Substance; the Coexistence of Causal Instances is a more complicated Causal Instance. But as the Likeness and Unlikeness of Terms in general is the fact that gives existence to Classes, so the Likeness and Unlikeness of Compound Terms gives rise to those Classes which are based upon many Attributes ; and these phenomena furnish Logic with some of its most important matter. Four of the remaining chapters of this Essay will be entirely occupied with the consideration of Classes ; and it is Classes of Compound Terms which require most consideration. And I may here observe that Compound Terms are tied with Compound Relations. A Relation of two Groups of Qualities is not a single Relation, nor a coincidence * of single Relations, but manifold, ac- cording to the multitude of the Qualities which con- stitute the terminal Groups. Compound Terms, in fact, are not tied together with a thread, but with a hawser made of many threads. And if for conveni- ence we sometimes speak of such Terms and Rela- tions as wholes, we always reserve the right to decompose them into their elements whenever intel- ligence requires it. We shall find, for instance, that, in respect of some qualities, a Compound Term may be related to others by Likeness ; but, in respect of other qualities, to the same Terms by Unlikeness. * Cf. ch. iv. 4. Of Terms. 45 7. Table of Terms. We may again give the results of the Chapter in the form of a Table. Terms r-Simple r-Feelings or Qualities. -Relations. - Compound -Substances. -Causal Instances. -Individuals. CHAPTER IV. OF THE IMMEDIATE AND MEDIATE RELATIONSHIP OF SINGLE TERMS, &c. PART I. IMMEDIATE RELATIONSHIP OF SINGLE TERMS. i. Of Identity and Sameness. WE must try to distinguish between Identity, Sameness and Similarity. All these Relations are species of Likeness. Identity (generally) and Same- ness both differ from Similarity in this, that they carry Likeness to the degree of indistinguishableness ; whereas in Similarity there is still some perceptible Unlikeness. Sameness is exact Likeness, which may be either of Quality, Quantity (Equality), or Position. And where these three modes of Sameness, together with Continuous Existence, unite in one entity (in which case the time of its existence is not marked by any perceivable changes in the entity itself) here we predicate Identity. However, this is only the most perfect Identity, and rarely or never to be met with. Many entities, Immediate Relationship of Single Terms, 47 especially the more Compound Terms, are (as we shall see) called identical notwithstanding alterations of both Quantity and Quality. And changes of Position are admissible in an identical thing, if they are such as may be rationally accounted for. But changes of Quantity and Quality in a thing con- sidered as identical are usually slow, and leave to it its indistinguishableness from moment to moment ; and all its changes must be according to Nature, and such that its Continuous Existence as a possible object of unbroken observation remains inferable : whence Hume says that Identity depends upon Causation ; * and Locke, that an identical thing can have but one beginning, f Sameness is, indeed, often made synonymous with Identity ; but it is as often confounded with Simi- larity : so that it may be a gain to both the fulness and precision of our vocabulary if we discriminate all the three. A Feeling has, strictly speaking, no Identity, for it has no Position ; or if it be called identical so long as it persists in consciousness without change or inter- ruption (which, on account of the intermittent nature of consciousness, cannot be very long) this is by courtesy, for Existence expresses the whole fact ; or we may call it Subjective Identity. A Feeling revived without perceptible change might perhaps be called Treatise of Human Nature : B. I. P. III. 2. t Essay of Human Understanding : . 1L c. 27, I. 48 Theory of Logic. the same as that before experienced; but the dimness and indefiniteness of the Subject makes comparison difficult, and we can seldom be sure of more than Similarity. Feelings, viewed as Qualities of the Object, and thus acquiring Position, are spoken of as persistently the same although absent from consciousness, being regarded as "permanent possibilities" of experience ; and these Qualities when again realized in conscious- ness are, if we believe in their latent Continuity, said to be identical with what we before experienced : this we may call Objective Identity. And as to Sameness, I conceive, that if two Qualities are not distinguish- able in themselves, but only in their relations to other Qualities, we may call them the same; though certainly not identical. Thus the colour of two pieces of silver, though not identical, I should call the same, and not merely similar. The Identity of a Relation, like that of other Simple Terms, depends on persistent Sameness ; but also on the Identity of its Terms severally; for an identical Relation can tie only two Terms. A Rela- tion is Subjective or Objective according to the nature of its Terms ; and its Identity will be estimated accordingly. And if its Terms be one of them Objec- tive and the other Subjective, as in the Likeness between an idea and an object, the nature of the Relation, I conceive, follows the weaker, or Sub- jective, Term. Thus the Likeness of one shilling to Immediate Relationship of Single Terms. 49 another, is not identical with the Likeness of the first shilling to a third, but only the same. And since the nature of Subject-Object Relations follows the weaker part, the Likeness of my present idea of the church-steeple to that object, is not identical with the Likeness to that object which my idea of it may bear to-morrow, but only similar to it; for though I attribute Identity to the object, I cannot to the idea in nearly so perfect a way ; and therefore since there are virtually three Terms, namely, two ideas and an object, two of them cannot be tied to the third with less than two Relations. The Identity of Compound Terms, too, depends upon Continuity and Sameness ; but here, especially, instead of Sameness, Similarity is often accepted, if the differences are according to Nature. The perfect Identity of a Substance involves the per- sistent Coexistence of identical parts and qualities. How far a change of state (involving Unlikeness) may be admitted without loss of Identity, is an un- settled question ; and to pursue it here would lead us too far. It depends to some extent upon the rank of the qualities undergoing change whether they be essential or accidental.* Again, two equal quantities of the same Substance, say two sovereigns, most people would not call the same ; but, perhaps, it would be better to call them the same, since their qualities are * Cf. ch. vii. ii. 50 Theory of Logic. the same each with each ; though not identical, since as compounds they differ (and have always differed) in Position. However, among concrete objects exact Sameness is rare, and at most we can only estimate it within the limits of observation. A Causal Instance can have Identity only by Position, and in the briefest way from moment to moment; for it is the nature of a Causal Instance to be transitory. A similar question regards the Unity of a Causal Instance : the Efficient Relations which are open to observation are always more or less compound. Shall we say that all that has been done upon the earth by sunshine from the beginning is due to one Cause; or shall we limit each Efficient Relation to the transmission of a single ray? We shall see hereafter that the answer to such questions depends, more or less, upon our convenience. Causal Instances may of course be similar to one another to the degree of Sameness. The Identity of Individuals differs characteristically from that of Substances. A man remains identical, although he loses a limb, or although a certain Co- existence of youth gives .place to another Coexistence of age. He only ceases to be identical when not only the Coexistence of qualities has been dissolved, but also the Cyclic Succession of Coexistences has run out or been interrupted. The demand for Sameness in order to Identity seems in such cases to be re- stricted to the vital organs, and even there is expected But in these cases the converse Relation of the Rela- tions is, as we have seen, more than Compatibility, namely, Implication. 5//z. Incompatibility. Relations that cannot coincide may be called In- compatible : a is incompatible with i\ and this Incompatibility is reciprocal. 56 Theory of Logic. Hence no more than two positive Relations can coincide, or tie an identical pair of Terms, namely, a with co a with v And I may add that there is no denial of the Incom- patibility of Likeness and Unlikeness involved in speaking of two Compound Terms as both like and unlike. For we saw that Compound Terms are tied with Compound Relations; and the Incompatibility of any two Relations means that they cannot coincide, and not that they cannot be compounded. Com- pound Terms may very well be alike in some qualities and unlike in others.* Incompatibility might also be called Obverse Imme- diate Implication : since a Relation that is incom- patible with another, if it obtains, implicates the absence of the other Relation ; as Likeness implicates the absence of Unlikeness. 6///. Alteruternity. Every Term is related in some way to every other, and that in each kind of Relation : is either Like or Unlike, Successive or Nonsuccessive, Coexistent or Noncoexistent. But in each of these homogeneous pairs the Positive Relation is incompatible with its Counter. * Cf. chap. iii. 6. Immediate Relationship of Single Terms. 57 Such a position, in which of two Relations one must obtain and both cannot, may be called Alteru- ternity. Thus If a do not obtain 77 must and conversely. There are other cases where one of two Relations must obtain, but both may do so; and this may be called Imperfect Alteruternity. Thus If e do not obtain o must for where e is not v must be, and v implicates o and o co , o> e But we have seen that e and o are compatible. Theory of Logic. Hexagon of Comparison. 5. Qualities and Defects. Suppose that there exists in Nature a certain sum of possible Qualities or modes of Qualities. They do not all coexist in any one Compound Term, but are Immediate Relationship of Single Terms. 59 shared amongst Compound Terms. Every Com- pound Term is a definite conflux of general Qualities :* these it is said to have or possess : the other Qualities, which it has not, may be called its Defects. Every general Quality is either a Quality (Appurtenance) or a Defect of each Compound Term ; but cannot be both. Thus Appurtenance and Defection are Alter- utern forms of any Quality with respect to any Term. Since the Defects of one Term are Qualities of another, any Term with as many others as possess all its Defects, may together be called Complementary as to the sum of possible Qualities. 6. Converse Relationship. Two related Terms both enter into their Relation- ship, but not always both in the same way. In Relations of Likeness, or Coexistence, both Terms are affected alike ; but in a Relation of Succession each Term is differently affected. Any Immediate Relation of two Terms may be viewed from both sides : the side of either Term being taken, the Relation thence regarded may be called Direct ; and from the other side it will then be said to be viewed in its Converse. These are Equivalent Aspects : the Relationship itself is not affected by our point of view, and therefore we may take whatever point of view we please. * Bain : Logic, Introduction, 10, II. 60 Theory of Logic. Perhaps it is not intrinsically more absurd to convert at length the Relations of Single Terms, than the Relations of Classes. However, I gladly avail myself of Mill's ironical statement of this opera- tion,* slightly altering the order of the principles. 1. "When one thing is like (or unlike) another, the other is like or unlike the first." If A a B , B a A. If A t) B , B 77 A. 2. "When one thing is before another, the other is after." When one thing is after another, the other is before. If A v B , B a A For Nonsuccession. f If A e B , 3. "When one thing is along with another, the other is along with the first." If A w B , B o> A For Noncoexistence. If A o B , B o A It will be observed that all Relations allow of Simple Conversion, except Relations of Succession : and these when Positive may be said to be con- verted by Inversion, as the sign indicates ; but when Counter are too indefinite to be converted at all. * Examination of Hamilton, c. xxi. Immediate Relationship of Single Terms. 61 PART II. MEDIATE RELATIONSHIP OF SINGLE TERMS &C. i . Immediate and Mediate Implication. In speaking above of the Implication of one Rela- tion by another we touched the constitutive principle of Logic. Logic might be defined as the Science that investigates the most general conditions of the Implication of Relations. The fundamental assump- tion is, that certain Relations among phenomena involve other Relations ; or, that there exist constant Correlations ; that is, that certain Relations are themselves constant Correlatives (ante. Part i, 2) : and the question of Logic is, what are these Correla- tions ? One of them we have just met with, namely, Correlation by necessary Coincidence, which may be called Biterminal Correlation ; where the Relations compared are conjoined at both ends, or tie an iden- tical pair of Terms. It may be symbolised thus : B If we call any Relation directly known explicit ; any Relation not directly known, but implied in explicit Relations, may be called implicit. In Biter- minal Correlations an explicit and an implicit Rela- 62 Theory of Logic. tion coincide ; and such Implication may be called Immediate. But there are cases in which a Relation between two Terms is implicated in explicit Relations with which it does not coincide in Relations which obtain between its own Terms severally, and some other Term or Terms ; and such Implication may be called Mediate. Where there are more than one Relation that do not coincide, there must be more than two Terms. The Mediate Implication of a Relation is at the same time a Mediate Relationship of Terms. The Rela- tion of two Terms to one another may admit the intervention of one other Term or of many ; but all cases of Mediate Relationship are reducible to two, which may be called the Units of the Mediate Rela- tionship of Terms, or of the Mediate Implication of Relations. 2. Units of Mediate Implication. It was formerly supposed that the Unit of all Mediate Implication (in Logic) was a Correlation of three Terms; such as we have in the Axiom, "Things which are equal to the same thing are equal one to another ; " and this was also supposed to be ex- emplified by the Syllogism. An equally important Unit of Mediate Implication has, however, been discovered in a certain Correlation of four Terms. The Units of Mediate Implication may be thus stated : Mediate Relationship of Single Terms, &c. 63 1 . Where the Relation of two Terms to one another is implied in the Relations which they severally bear to a third : as if A co B, and B o> C ; we know that A to C : 2. Where a Relationship between two Terms is implied in the Relations which they severally bear to two other Terms, and in the Relation which these two other Terms bear to one another; as if A a C, and B a D, and A v B, that is taken as evidence that C v D (Causa- tion). The discovery of these Units of Mediate Relation in their generality is due to Mr. Spencer.* And in- asmuch as where there are three Terms the Relations compared have one Term in common, Mr. Spencer in expounding the theory of Reasoning, calls the cor- responding intuition one of " conjoined relations ; " and since where there are four Terms, the Relations compared have no common Term, he calls the cor- responding intuition one of "disjoined relations." This terminology has well-marked merits ; and it is with much diffidence that I propose, for the purposes of Logic, to speak instead of Triterminal and Quadri- terminal Correlations. These two genera of Correlations, regarded as intuitions, Mr. Spencer symbolizes thus : * Spencer's Psychology, Part. VI., ch. viii. 64 Theory of Logic. B A And these symbols admirably represent the conjunct and disjunct character of the Relations compared, and at the same time the Triterminality and Quadri- terminality of the several Correlations ; but I must venture to alter to some extent for the purposes of this Essay the symbol of Triterminal Correlation. It will be observed that (as Mr. Spencer writes the two symbols) whereas in the case of Quadriterminal Correlation, the Relations compared are one of them explicit (A : B) and the other implicit (C : D) ; in the case of Triterminal Correlation, both the Relations compared are represented as explicit (A:B:D). I see no reasons for this, but many why it should be otherwise ; and I propose to get rid of the discrepancy by writing the symbol of Triterminal Correlation in such a way as to suggest the comparison of an explicit Relation (B : C) with an implicit Relation (A : C), thus : B Only thus is it made apparent that (except in d fortiori Correlations see below) in both orders of Mediate Relationship of Single Terms, &c. 65 Correlations, the Relation of the Relations compared is always a Relation of Equality. Moreover this change in the symbol of Triterminal Correlations is, I think, justified by Psychological considerations ; and in order to show this I may be excused a brief digression. 3. Psychological Digression on the Intuition of Conjunct Relations. Mr. Spencer has shown how the Quadriterminal Intuition arises,* but has hardly, as it seems to me, been successful in doing as much for the Triterminal ; f although the processes are very similar. If, wishing to determine the Relation of A to C, I effect this by means of B, what is the mental process ? Suppose that the Relations are of Coexistence : the result of our enquiry may be formulated thus : A to B co C .:. A D, is psychologically dependent not only upon the Correlation, o> = o> ; but also upon the Relations, A a C , B a D. The inference a> D, is like the imaginary completion of a picture, which we have seen before, and of which a part is now shown us again. 4. Rule of Triterminal Correlation. What now are the most general laws of the Mediate Implication of Relations ? Rule of Triterminal Correlation. Two Terms homogeneously related to a third, and one of them positively, are related to one another as the other is related to the third. 70 Theory of Logic. I call this a Rule rather than an Axiom, for it is too general to be quite self-evident; and moreover (as we shall see) one or two slightly exceptional cases have to be allowed for. The true Axioms are, I conceive, the following special laws of the different orders of fundamental Relations, laws which em- body the above Rule, but can hardly be said to be deductively derived from it : rather is it itself arrived at by generalization from them. i. Likeness and Sameness. J C.'. AT? C C.'. (No positive) (Too indefinite). AaB JC.'. 2. Coexistence or Simultaneity. AcoBo)C.-.AcoC A signifies Simultaneity). AuBuC.'.AuC (a fortiori] \ J AtoB v C .-. Au C AcoB eC .'. AeC AuB eC .-. AuBo C . (Too indefinite) (No positive) (Too indefinite). Let us symbolize one of these Correlations with concrete Terms, Mediate Relationship of Single Terms, &c. 71 Plato Socrates -> Aristotle. In these cases the related Terms should generally be homogeneous ; * events comparing with events, and more stable existences with one another according to their kind, whether Simple Terms or Compound : the Relations especially must be homo- geneous ; since Relations of Time do not compare with Relations of Connature. Hence in compari- son with Relations of Succession, w must mean a Relation of Simultaneity between Terms which are both of them transitory : otherwise, for instance, the Correlation, Ao> Bu C .-. A v C, where A is perdurable, will not obtain. And this suggests the observation that Correlations of Coexistence are only certainly true where Coexist- ence is equivalent to Simultaneity or the Concomitance or Coinherence of Qualities. The Relations, Thebes - Animal qualities. CO For this argument is only concerned with the coexistence of Single Terms in an individual. But should one argue This horse (of which I only see the hind quarters) must have a head, because all horses have heads that is a reference to the nature of Classes ; and such a Correlation cannot be repre- sented as merely Triterminal. 5. Rule of Quadriterminal Correlation. Two Terms that are severally the same as, or like, certain other Terms which are related pairwise to one another, are themselves in the same way related. . Or the Rule may be stated thus : If there be two Terms related to one another, and a third Term the same as (or like) one of them, there shall be a fourth Term the same as (or like) the other; and these third and fourth Terms shall be related to one another as the former two Terms are related. Or thus : If a Term C be the same as (or like) a Term A that is related to another Term B, there shall be a fourth Term D the same as (or like) B, and related to C as B is to A. 74 Theory of Logic. This principle is perhaps less self-evident than the former ; and even its special aspects in the laws of the Correlation of the various fundamental kinds of Relations are not all sufficiently certain to be called Axioms. i. Likeness and Sameness. Qualitative Relations of Likeness need not be compared in this way. For suppose we wish to find a Correlation which implicates the Relation C = D, such a Correlation is indeed given in the expression where A = C and B=D. But the Relation C=D is more clearly implicated in two Triterminal Correla- tions thus : C^ArrrB.'.C^B C = B = D .-. C = D. The Logical application of the Rule of Quadriter- minal Correlation is to Relations of Succession and Coexistence. 2. Coexistence. (Let A = C and B = D). Aco B = C o)D A o B = C o D. 3. Succession (Let A = C and B = D). AU B = CuD A e B = C e D. To take concrete illustrations : Mediate Relationship of Single Terms, &c. 75 Men as a Class x = f Any member (unspe- cified) of the Class e Y = -\ e Risibility ' Or again : Heating metals as a Class every circumstance in common, except that C and E are present in the first and absent in the second : since where E is present C is present, and on the withdrawal of C, E also disappears, it is concluded that C is the Cause or Part-Cause ofE. /3. From the second clause of the Law of Causation we learn That the same Effect always recurs on the recur- rence of the same Cause ; that is to say * Bain : Logic ; Book III. ch. v. 6. Of the Discovery of Classes, &c. 125 1 . Whenever C is present E is present ; wherefore when an antecedent cannot be introduced without the consequent appearing, such ante- cedent must be the Cause, or a part of the Cause. Hence a second aspect of the Method of Difference. (2) If an instance where a phenomenon does not occur and an instance where it does occur have every cir- cumstance in common, except one, that one occurring only in the second ; the circumstance absent in the first and present in the second, is the Cause, or part of the Cause, of the given phenomenon. For the two cases before and after the introduc- tion of the Antecedent differ only in this cir- cumstance and the appearance of the Effect : AB jABC - ana - r^F"- a o are independent Causal Instances ; C a b is the Cause of E : for E must have some Cause ; and since A and B are merged in a and d, by hypothesis there is no other Cause for E, than C. C may be unknown, but we know that it must exist ; for since E = C, nE = nC .'. abE = ABX ; and whatever X is it must be C. ABC If C be discovered we have the case - abE AB and the supposable case j-, and thus apply d the Method of Difference. 128 Theory of Logic. Lastly since E is equal to C, 3. C + x must be greater than E, and cannot be the cause of E alone; wherefore Consequents whose Causes are fully known, cannot be the Effects of other simultaneous Causes. Hence the Method of Residuary Causes. Subduct from any phenomenon such part as previous probation has shown to be the Cause of certain Con- sequents, and the Residue of the phenomenon is the Cause of the remaining Consequents. Here E may be unknown, but we know that it must exist : and when it is discovered we may apply the Method of Difference as in the last case. 1 8. Concomitant Variations. It will be observed that the general statement of this Method (y i) is too wide to be borne out by the Law of Conservation alone, i . It refers to phenomena of any kind and not merely to quantities of energy : 2, it takes account not only of Efficient, but also of Coeffectional Relations. For of these, too, the Terms vary together ; and a " bond of concomitance " which- is of a Causal nature, but not Efficient, must be Coeffectional. Such now is the persistence of Rela- tions among modes of energy; which is almost another expression for the sameness of the Effects of the same Causes : and this I take to be the surest Of the Discovery of Classes, &c. 129 basis of so much of the Canon of Variations as can- not be derived from Conservation. But there is a treacherous vagueness about the statement, similar to what we noticed in the rule of Quadriterminal Corre- lation ; and Prof. Jevons justly remarks that it needs to be interpreted with caution.* It seems to appeal to experiences not definite enough to be embodied in the Law of Causation, but which have generally accompanied the experiences from which that Law has been formulated, and have generated about it, so to speak, a penumbra of expectation. 19. Supplementary Methods. ,f The above methods of sifting experience are not always sufficient for the purpose. They are liable to "frustration," chiefly in two ways: i, by the Vicari- ousness of Causes; 2, by the Intermixture of Effects. 20. Vicarwusness of Causes. If Vicarious Causes be possible (and we have seen that they must be practically recognized) the Method of Agreement is apt to fail. If, for instance, from the A C C D two cases > - - we conclude that C is the Cause aE Ed Principles of Science : vol. II. ch. xxii. 2. + Cf. Mill : Logic ; Book III. ch. x., xi., &c. Bain : Logic ; Book III. ch, viiL, x., &c. 1 30 Theory of Logic. of E, this is only valid on the assumption that E can have only one Cause ; for if it may have more than one, A may be the Cause of E in the one case, and D in the other. The Method of Difference is not thus affected. If A "D to the case we add an antecedent C, and get E, Cv O it is certain that C is a Cause or Part-Cause of E ; though it may not be the whole or the only Cause. Nor does the Vicariousness of Causes prevent the exclusion of a circumstance from among the supposed A B Causes of a given phenomenon. If to the case 7-, D Cv be added, and E do not follow it, it is certain that D is not the whole Cause of E (unless A or B be a counteracting force). This defect of the Method of Agreement is to some extent remedied by the following means. To enter upon the subject at length, indeed, would require a preliminary account of the Theory of Probabilities. But this is a branch of Mathematics, rather than of Logic, and although often discussed in Logical works * with a practical aim, hardly falls within the scope of this Essay. I will therefore confine myself as much as possible to considerations merely Logical. Let C and E be the phenomena whose connection we are investigating : and suppose that we cannot * Venn : Logic of Chance. Jevons : Principles of Science ; vol. I. ch. x., &c. Bain : Logic ; Book III. ch. ix. Mill : Logic ; ch. xvii., xviil Of the Discovery of Classes, &c. 1 3 r apply to them any of the Methods of Difference, or Variations, cannot bring them directly under the Axiom of Causation ; and are therefore compelled to resort to an enumeration of instances. Positive and negative together, there are four possible kinds of instances of the concomitance of C and E : 1 . Both may occur together. 2. Both maybe absent together. 3. C may occur without E. 4. E may occur without C. 1. Suppose that C and E often occur together. According to the Law of Causation Cause and Effect are a constant sequence ; but in the infinite variety of Nature constant sequences are after all exceptional ; and therefore there is a high degree of probability that a constant sequence is one of Cause and Effect, or of events related by Causation. If then C and E often occur together there is a probability, varying with other circumstances, that there is some Causal Relation between them: which is measured by the improbability that they would have occurred together had there been no Causal Relation between them ; an improbability that increases rapidly as the number of instances of positive concomitance increases. 2. Suppose that C and E are both of them absent together: this is also evidence of their Causal con- nection. For inconstancy of Relation being the commoner case, it is improbable that separable Terms should not occur separately. And this evi- K 2 132 Theory of Logic. dence is strengthened according to the frequency of their occurrence together ; for the frequency of their occurrence is some evidence of their frequency in Nature, and according to their frequency in Nature is the probability that they would occur separately if there were no Causal Relation between them. On these two sets of considerations rests the Joint Method of Agreement and Difference. If two or more instances where the phenomenon occurs have only one circumstance in common ; while two or more instances where it does not occur have nothing in common save the absence of that one circumstance ; the circumstance wherein alone the two sets of instances differ, is the Effect, or the Cause, or a necessary part of the Cause of the given phenomenon. Variety is sought in the circumstances of the positive instances in the hope of " bringing out all the Causes ; " and in the circumstances of the negative instances to multiply opportunities of separate occurrence. 3. C (the supposed Cause) may be present without E. In such a case we know that either C is not the Cause, (by the second clause of the Law of Causa- tion), or that it has been counteracted. We may not know the conditions well enough to be sure that C has not been counteracted, for that would give the Of the Discovery of Classes, &c. 133 Indirect Method of Difference, which we suppose inapplicable. We must therefore try to find C without E in as various circumstances as possible, in order to get rid of the counteracting forces. It is very improbable that C should be in all circum- stances counteracted. 4. E (the Effect) may be present without C ; and this must show that either C is not the Cause, or only one of two or more Vicarious Causes. The former supposition is confirmed if on varying the circum- stances, C does not appear in conjunction with E oftener than would be accounted for by pure chance. From these two sets of considerations then we may deduce an Indirect Joint Method. If two or more instances where the phenomenon does not occur have only one circumstance in common ; while two or more instances where it does occur have nothing in common, save the absence of that one circumstance ; the circumstance wherein alone the two sets of instances differ is not the Cause of the given phenomenon. This Method may be employed where simple Indirect Difference is inapplicable, to sift out specious but unreal Antecedents from among a selection of possible Causes. Other means of dealing with Vicarious Causes are : i. If C and D both pass for Causes of E, by 134 Theory of Logic. particularizing E, and examining it in detail, some difference ought from time to time to be detected in it, according to the difference of its Antecedents. For it seems very improbable that different Causes should produce Effects in quantity and quality exactly equal. 2. If C and D be Vicarious Causes, by the third clause of the Law, on the concurrence of C and D, E ought to be augmented, at least, where E is a Resultant, and C and D not in mutual counteraction. Before quitting this subject I may remark that the particular evidence (if any be needed) for the highest general truths and Axioms is not tested by a Method of Agreement merely, but rather by a Joint Method. For that evidence is not only the most frequent experience, but experience without contradiction. From the absence of contradictory experience arises the impossibility of conceiving the opposite ; and so the employment of the indirect intuitive method may be regarded as a short way of applying the negative side of a Joint Method. 21. Composition of Causes, and Intermixture of Effects. Causes acting side by side sometimes produce Effects that are distinguishable by simple inspection, as when a violin and a piano are played in concert. But sometimes circumstances acting together produce a joint Effect in which their respective operations are Of the Discovery of Classes, &c. 135 not directly cognizable. Such intermixed Effects are of two kinds : 1. Where the Part-Effects of the several co- operating circumstances or conditions are homo- geneous with one another and with their Antece- dents. Such composition is called by Mill, mechanical : * the Effect, Mr. Lewes calls a Resultant, f 2. Where the Effect is not homogeneous with the Antecedents ; as when in chemical com- bination the properties of different bodies dis- appear, and are replaced by those of the com- pound. Such composition is called by Mr. Mill, heteropathic ; Mr. Lewes calls the Effect an Emergent. I would avoid speaking of these cases as exhibit- ing a composition of Causes, or an intermixture of Effects : they present only complex Causes and com- plex Effects ; for the Cause is the sum of the necessary Antecedents, and the Effect is the sum of the neces- sary Consequents : so that the Causes supposed to be combined are really Part-Causes, which may some- times be conveniently styled Conditions ; and the Effects supposed to be intermixed are only Part- Effects. Similarly the counteraction of a possible Cause constitutes a new and more complex Cause, which also has a complex Effect : and, indeed, in every * Logic : Book III. ch. vi. t Lewes : Problems of Life and Mind. 136 Theory of Logic. case of the cooperation of Part-Causes, which are not homogeneous and in the same direction, there is more or less counteraction. Regarding such complex Causal Instances as wholes, and supposing them to be cognizable as such, they are determinable by the Experimental Methods. But a further question arises, namely, to discover what parts, elements, or components of the complex Cause, and what of the complex Effect, cor- respond with one another. In the second of the above cases, this problem has been only very imperfectly solved : it is known that the weight of an Emergent is equal to the combined weights of its Antecedents ; but in what other ways they have contributed to it, is not yet discoverable by any method. It is otherwise with the first case; a Resultant may be analyzed, mentally if not actually, and its components may be assigned to the corresponding components of the Cause, by what is called the Deductive Method. By the same method we must estimate the action of an unknown or hypothetical Cause. The Deductive Method. The Deductive Method has, according to Mill and Prof. Bain, three stages : * i . Induction : the nature and power of the several Part-Causes contributing to the composite * Mill : Lo^ic ; Book III. ch. xi. Bain : Logic ; Book III. ch. x. Of the Discovery of Classes, &c. 137 Effect must be known or discovered (or supposed) ; and this must be ultimately by Induction (or Hypothesis). 2. Deduction : the Effect which these Part-Causes would have in a certain combination must next, be computed. 3. Verification : the computed Effect must be com- pared with the real Effect which was the starting point of the investigation. If these agree, there is so much evidence that we know the Causes and how they are combined ; and this would be conclusive, but for the Vicarious- ness of Causes. If they do not agree, there must be an error somewhere: either (i) we do not know the right Causes, or not all ot them, or have assumed too many, or a wrong combination of them ; or (ii) have not rightly ascertained by Induction their nature and power ; or (iii) have made some mistake in the Deduction. Suppose we see E, an instance of a composite Effect, which we wish to trace to its Cause or Conditions. If it is not even known what the Con- ditions are, we must guess them : this is a hypothetic Subsumption ; * by certain marks the Instance is tentatively classed either exactly or analogically with other known Instances. Next taking the Causes to be A and B, we learn by Induction or otherwise *33- 1 38 Theory of Logic. (ultimately by Induction) the general nature and power of each say a and b , and thence compute by Deduction their combined Effect in this particular case say (a + b) . The Effect (a + b) thus antici- pated is then compared with E : if they agree, we know that A and B so combined are a possible Cause of E ; and if they, account for E in general, and no other Cause is equally probable, they are at last taken to be the Cause. But if (a + b) and E do not agree, we must retrace our steps and look ( for error as above indicated. So much then as to the methods of investigating the coherence of Causal Instances : in an Essay of this kind so much may suffice. For a more complete account the reader, who may not yet be acquainted with the treatises of Mill and Prof. Bain, is referred to those works. 22. Probation of Classes of Substances and Individuals. The probation of Classes of Substances and the probation of Classes of Individuals may be con- veniently dealt with together. In so far as the probation of Classes of Individuals consists in testing the coherence of Coexistent qualities, it proceeds by the same methods, justified by nearly the same considerations as the probation of Classes of Sub- Of the Discovery of Classes, &c. 139 stances. In the case of Individuals there is indeed a Succession as well as a Coexistence of properties to be considered, but sufficient evidence of the Succes- sion of properties will often be found in the process of testing the coherence of the Coexistent properties. If other proof be required, it can only be conducted by the methods of testing Causation. Thus the Probation of Classes of Individuals presents no new problem, but only combines the two problems furnished severally by Causal Instances and Sub- stances. 23. Probation of Coherent Coexistence. We come then to consider the means of testing the constancy of Relations of Coexistence. And here there are no such resources of method available as in the inquiry concerning coherence of Succession. For those resources- were derived from the Law of Causation in general, to which all particular cases of Causation might be affiliated ; and there is no comparable Law of Coexistence in general, to which all particular cases of Coexistence can be referred. Relations of coherent Coexistence have no common marks which permit them to be defined universally and recognized by infallible signs : they agree only in being coherent, and this is the point we have to discover. Thus every case of coherent Coexistence must be received upon some kind of evidence not 140 Theory of Logic. applicable to all cases : it does not follow that each case presents an isolated problem. Constancy of Relationship which cannot be proved by reference to some axiom, must (as before re- marked) depend on an inferior amount of evidence of the same kind as that by which the axioms are themselves established ; that is, upon uncon- tradicted agreement in experience, with the pre- sumption of Nature's uniformity. Agreement in experience sufficient to establish an axiom may be said to give certainty ; less evidence gives only some degree of probability. Hence if each case of Co- existence has to have its coherence tested by simple agreement in experience, or without the sanction of an axiom, none will be more than probably constant, 1 and very few will be so in a high degree. Cases of Coexistence however are not entirely isolated. Although as a class they have no common quality, but the one to be discovered, they may be grouped into subordinate classes of richer attribu- tion. The device of subdivision, already resorted to in order to define more fully the Rule of Quadri- terminal Correlation in general, must now again be adopted with regard to this class of Quadriterminal Correlations namely, Correlations of Coexistences. The chief subordinate classes are: i. Coexistence dependent on Causation ; 2. Coexistences which establish Natural Kinds. Of the Discovery of Classes, drV. 141 . 24. Coexistences due to Causation. Many cases of Coexistence are consequences of Causation : and so far as Coexistence is a result of Causation, it has the coherence and certainty which belong to all Relations implicated in Causal Instances ; contingent only on the existence of the Cause. Of this nature, says Professor Bain,* are the numerous Coexistences of Order in Place, which are always the redistribution of some prior distribution ; and we shall see that there are other cases besides these. Indeed it is possible that all Coexistences are ulti- mately due to Causation ; and it is hardly too much to say that this is a regulative principle of Philosophy : for Philosophy seeks complete generality, and this is not attained so long as there are two distinct kinds of coherence, by Succession and also by Coexistence. We cannot hope however to render the facts of Coexistence perfectly intelligible. Could we explain all present Coexistences by reference to some past state of the Universe and Causation ; still, following the regress of Causes further back, we must come at last to Chaos. For the earliest discoverable distribu- tion of existences was, so far as our understanding reaches, accidental ; and from this taint of the incomprehensible, its consequences can never be wholly free. Logic : Book III. ch. iii. I. 142 Theory of Logic. 25. Natural Kinds. Mr. Mill's doctrine that Kinds have a real existence in Nature, also does something to relieve us of the distraction of regarding all coherence of Coexistence as only to be tested by exhaustive enumeration of particulars. "There are some classes the things contained in which differ from other things only in certain particulars which may be numbered, while others differ in more than can be numbered, more even than we can ever expect to know." This dis- tinction agrees generally with that between Artificial and Natural Classes. The members of a Natural Kind agree among themselves, differ from the members of other Kinds in a multitude of qualities. Attribution conferred by numerous qualities, then, is the . mark of a Kind, and therefore a mark of the coherence of those qualities which confer the attribu- tion ; if we find a few specimens agreeing in qualities so numerous, we may expect to find the same qualities cohering in the same way throughout an indefinitely extensive constituency. Still the prin- ciple is a little vague ; and unfortunately it does not gain much in definiteness, while it loses in generality, if again we subdivide Natural Kinds into the Organic and Inorganic, and consider these separately. i . The Inorganic Natural Kinds are, first, the sixty and odd Chemical Elements. These are the only Inorganic Kinds, and the only Substances, presenting Of the Discovery of Classes, &c. 143 at present the pure problem of Coexistence. All compound Substances are derivative and resolvable, and the Coexistence of their qualities is partly a problem in Causation. Thus as marks of the coherence of attributes in primary Inorganic Kinds we have the mark of Kinds in general, and also Irresolvability. But Irresolvability is relative to the state of Science. 2. The Organic Kinds are the Species of Plants and Animals ; and what constitutes a Species perhaps Naturalists may one day be able to decide; for Mr. Darwin says,* that "every Naturalist knows vaguely what he means when he speaks of a Species." There will then be assignable marks of an Organic Kind, besides those of Kinds in general ; and these will be further marks of coherence. However, in the coherence of the qualities of all Organic Kinds Causation is involved : what is known of generation and heredity forbids the supposition that an Indi- vidual apparently representing such a Kind may be a solitary specimen. And as the nature of the specimen, so the existence of Organic Kinds in general, is deducible from Causation ; for the theory of Natural Selection, which rests upon Causation, shows how, in the Organic World, Kinds of great uniformity must be produced by the destruction of varieties unsuited to the environment. Nor indeed * Origin of Species, ch. ii. 144 Theory of Logic. will it appear incredible to the reader of Mr. Spencer's Chapter on Segregation,* that the existence of the Inorganic Kinds should hereafter be deducible. Thus the coherence of the qualities included in the definition of any one Natural Kind, has the sanction of these wider definitions of Kinds, Organic, or In- organic, and in general : and the wider definitions though not indeed so universal as the definition of Causal Instances, nor by any means so exact, are yet of very high generality, and in some sort affiliated to Causation itself. But the dependence of Kinds on Causation, whilst guaranteeing their sameness so long as the Causes which moulded them remain the same, also ensures their variation or destruction, should those Causes in a sufficient degree themselves vary. 26. Superordinate Kinds. The comparison of Natural Kinds with one another reveals the possession of attributes in common ; and hence arise Laws of Coexistence more general than the definitions of special Kinds, being definitions of higher Kinds, generic or other, that is, of Kinds of Kinds. Such for instance is the Coexistence of Inertia and Gravity, part of the definition of Matter, one of the higher genera of concretes. The qualities whose Coexistence is expressed in the definitions of * First Principles, Part II. ch. xxi. Of the Discovery of Classes, &c. 145 Superordinate Kinds are those which are least liable to incoherence in the members of Subordinate Kinds. For such definitions are generalized from the lower Kinds, and are thus supported both by the Doctrine of Kinds in many applications, and by much experience in detail ; and, besides, the qualities they include are fundamental. 2 7 . A ccidental Conjunctions. i. Contrasted with those qualities of a Substance or Individual which confer attributes on Superordinate Kinds, are those of its qualities which do not even enter into the definition of its own Kind. Of such qualities some (called Propria*) are derivable from the Kind-attributes and partake of their coherence ; but others are not known to be in any way constantly coexistent with the other qualities ; and these are called Accidents,* and in their Relations to the other qualities of the members of a Kind, or to one another, they may be said to be accidentally conjoined. To the constancy of such Conjunctions, the Doctrine of Natural Kinds extends no sanction ; so that in their case we are reduced to probation by simple enume- ration. By this means we may find an accident to be constantly concomitant with the attributes of a Kind within certain limits of observation, as the blackness of crows ; or we may reach only an * Ch. viL, ii. 146 Theory of Logic. approximate generalization, as ' most metals are whitish.' 2. We may extend the name Accidental Conjunc- tions to certain other cases. Two or more qualities may happen to coexist frequently, or always, to an extent not conterminous with any Natural Kind or Kinds : such is the Non-coexistence of scarlet colour with scent in flowers. And here again since there is no general mark of the constancy of such Relations, we can only test it by simple enumeration of examples. 28. Classification of Laws of Coexistence. It is only, I conceive, in these last two cases that simple enumeration alone is relied on for testing the coherence of Coexistences ; but here no doubt it is desirable that every specimen should be examined. It is however, questionable, whether merely Acci- dental Conjunctions should be dignified with the name of Laws. Would it not be better to confine that name as regards Coexistence to the following cases : 1 . Consequences of Causation. 2. Definitions of Summa Genera, where these are of Plural Attribution. 3. Coherence of Generic Attributes and part of the Difference of a Species with its remaining Difference ; or of its whole Difference with the Generic Attributes. 4. Coherence of Generic or Specific Attributes with Of the Discovery of Classes, &c. 147 Propria ; which however may often be viewed as a consequence of Causation. Laws of Coexistence, thus understood, are supported by an amount of evidence somewhere between axiomatic certainty and simple enumeration of examples. 29. Causation disguised as Coexistence. Besides that cases of Coexistence are often due to Causation, some cases which seem to be of simple Coexistence, may really be of direct Causation. In such cases, according to Prof. Bain,* the means of distinguishing Causation from Coexistence are chiefly two : i, to try to detect sequence in the apparently simultaneous ; 2, to trace expenditure of energy. 30. Definition and Probation. The process of Probation by simple enumeration is a continuation of that process of collecting examples which is preparatory to Definition. The same process is likely to bring to light whatever cases exist suitable for the employment of the other Probative methods. Definition is thus a preparation for, and an aid to, Probation; and in return Probation aids Definition. For a first Definition is not likely to be perfect. To say nothing of the connotation of common names, the * Logic : Book III. ch. vi. 2. 148 Theory of Logic. early Definitions of Science are nearly always subject to much dispute and modification. After the first tentative Definition of a Class by finding Terms with common qualities, the work of Probation sets in ; the coherence of these common properties has to be tested. The result may be that some Relations sup- posed to be constant break down, whilst other Rela- tions suggest themselves as more constant. The work of Definition then takes a fresh start : and so on. Thus by a continuous and alternative process of Definition and Probation, Classes are discovered and established. 31. Laws. The result of definition is a Definition itself. Cer- tain attributes are fixed upon as marking a Class, because the corresponding qualities are common to certain Terms. When disregarding the Class and its Constituents, we fix our attention upon the Quali- ties themselves and their Coherence, we are said to contemplate a Law. Coherence may be of Succession or Coexistence ; and as there are Terms and Classes* so there are Laws based upon these Relations, Laws of Succession, Laws of Coexistence, and of course Laws of the combination of these Laws. Classes may be more or less extensive, and so Laws may be more or less general. The most general Laws or Axioms are called Ultimate ; the less general Secondary: and Secondary Laws are either Of the Discovery of Classes, &c. 149 Derivative or Empirical. Laws are said to be Deriv- ative when they can be shown to be special cases of Ultimate Laws. Empirical Laws are generally believed to be special cases of Ultimate Laws, but cannot yet be shown to be so. And it is usual to consider all Laws of Coexistence, except those which can be derived from Laws of Causation, as Empirical ; though perhaps Imperfectly Derivative would be a better name for the Laws of Natural Kinds. In short, Laws, being only Definitions, or parts of Definitions, differently viewed, are discovered, tested, and valued by the same rules. 32. Explanation.* To establish a Class or a Law is to generalize, to find similar Relations obtaining amongst similar Terms in an indefinite number of cases. The same process which is Classification as to the generality of the Terms concerned, and the discovery of a Law as to the Relations of their properties, is called with regard to any particular Term or Terms referred to the Class or Law, Explanation. Explanation is thus implicated in the Definitive-Probative process. When Classes are classified, and thereby higher Laws dis- covered, there is a further step of Explanation. Mr. Mill found three modes of Explanation : * Mill : Logic ; Book III. ch. xii. Bain : Logic ; Book III. ch. xiL 150 Theory of Logic. 1. "Explaining a Joint-Effect by assigning the Laws of the separate Causes ; " as when the course of a pro- jectile is shown to be due partly to the energy of its discharge, partly to gravitation, partly to the resistance of the air. 2. Explaining "by discovering an intermediate link or links between an Antecedent and Conse- quent ; " as when the scientific supplements the popular view of a Causal Instance by finding in it a series of Causal Instances. 3. Explaining several Terms by merging them in one Class or Law, or several Classes or Laws by merging them in one more general. All these modes of Explanation involve generali- zation : i . A Joint-Effect is a special case of the concurrence of Causes which may exist apart, or in other combinations. 2. The more a Causal Instance is narrowed, the less liable it is to interruption, and the more frequently it will occur in its completeness. 3. The third mode is, as Prof. Bain remarks, " gene- ralization pure and simple." When a phenomenon is explained by being likened to many others, it may often happen that amongst the many some are commoner or better known than the first phenomenon. In such cases the phenomenon in question is not only explained but familiarised. This, however, is by accident. The phenomenon is familiarized not by generalizing its properties, but by specializing them ; not by showing the extent of Of the Discovery of Classes, &c. 1 5 1 their prevalence, but by finding resemblances in this or that particular case. To confound Explanation with familiarization, generalization with specializa- tion, is perhaps the fallacy of fallacies, of which anthropomorphism or heautomorphism (if I may be allowed the expression) is the Hydra-headed example. There is an egotism of intelligence, as well as of desire ; and it is equally at enmity with Philosophy. 33. Subsumption. When Classes and Laws have been established new instances may be discovered and recognized as coming under them. To this process I propose to restrict the name, Subsumption. Subsumption is a kind of Explanation ; but whereas Explanation of some kind is involved in the Definitive-Probative process ; Subsumption supposes that that process has been to a certain extent completed, that Classes have been already established ; and is itself the process by which it is determined whether a given Term is, or is not, included in a given Class, or whether a given Class is, or is not, included in a higher Class. This involves at some stage the par- ticular examination of the Term or Class to be sub- sumed ; in order to find, in the case of Terms, whether they have the qualities common to the con- stituents of the Class in which it is proposed to include them ; in the case of a Class, whether its 152 Theory of Logic. attribution includes the attribution of the Class under which it is to be subsumed. In other words (and this may throw light upon future passages) the Sub- sumption of one Class under another, involves a recognition of the Concomitance in the lower Class of those attributes in which the two Classes agree with those in which they differ (generic and diffe- rential attributes) : thus if Cat is subsumed under Animal, we have Animality . Similarly, the conclusion that of two Classes neither can be subsumed under the other, involves a recog- nition of the Nonconcomitance of their reciprocally differential attributes: thus if Cat is not subsumed under Dog, nor Dog under Cat, we have Felinity^ Caninity. We must note, however, that a positive Relation, o> or aw, between the Attributions of different Classes can seldom be simply converted, like the same Relation between Single Terms, or between Qualities in the Members of one Class. Animality M Felinity is only true within the Class of Cats. Animality in general is two Terms in relation to Felinity; one of which does, and the other does not coexist with it. Definition and Probation then are the processes by which Classes are discovered and established, whereby Of the Discovery of Classes, &c. 153 at the same time Laws are formulated and proved, and particular phenomena explained. By Subsump- tion fresh members, or components, or instances of known Classes and Laws are recognized and referred to their own. And Subsumption may be Immediate or Mediate : it is Immediate when the Relation of a new Term or Class to the Class under which it is to be subsumed is directly investigated ; Mediate when its Relation to the subsuming Class is proved by its Relation to another Class, whose Rela- tion to the subsuming Class is known. Plainly Sub- sumption is a mode of Probation, and Subsumption under an Axiom is the most perfect Probation. I have used the words Induction and Deduction as little as possible, and would gladly see Logic freed of both. They are names, I conceive, not of modes of proof, but of modes of inference of modes of infer- ence which differ in the comparative extent of their data and conclusions and in this sense Mr. Spencer uses and defines them.* Of course in order to proof there must generally be an inference to be proved ; but whether an inference be inductive or deductive, it must be proved in the same ways : a deductive inference may be susceptible only of empirical proof, and an inductive inference may be demonstrable. In Logic, too, the departments usually called In- ductive and Deductive have had their boundaries * Psychology : Part VI., ch. viii. 154 Theory of Logic. much blurred, and Induction has come near to be con- founded with empiricism. Thus Prof. Bain says that the Experimental Methods, which used to be called Inductive, are Deductive. And this, I think, must be manifest to everyone; or is there any reason why Subsumption under the Axiom of the Syllogism should be called Deduction, rather than Subsumption under the Axiom of Causation ? But if Induction is deprived of the Experimental Methods, little else than simple enumeration remains to it. We now come to the subject of the Relations of Classes, the substance of nearly all Scholastic Logic. There are two leading questions : i . Given any Immediate Relation of Classes, to find all the Equiva- lent and Implicated Relations between them ; 2. Given any Relations between more than two Classes, to find under what conditions other definite Relations are implicated. CHAPTER VII. OF THE IMMEDIATE RELATIONSHIP OF CLASSES. i . Inclusion and Exclusion.* THE Relationship of Classes considered in Logic is with respect to the identity or nonidentity of the Terms which constitute them. Two or more Classes may have many Terms in common, or they may have none. Perhaps then it would be more correct to speak of Classes so related as Coincident or Nonco- incident ; but we shall obtain greater facility and flexibility of expression by calling them Inclusive or Exclusive. Inclusion and Exclusion may be regarded as the forms of Class-Relationship. A Class is said to include another Class, if it includes in its Constituency the Constituency of the other Class, and to exclude another Class, if it does not include in its own Constituency any Constituent of the other Class. Sometimes if we could count the constituency of a certain Class, we should all the while be counting the constituency of another Class ; then the first Class is included in the second. Some- * Cf. Leibnitz : Definitiones Logicae. $5$ Theory of Logic. times if we could count the constituency of a certain Class we should not all the time count a single constituent of a certain other Class: such Classes exclude one another. Thus, if we could count all the cats in the world, we should all the while be counting animals, but never any dog: cats are in- cluded by the Class Animal, and excluded by the Class Dog. Again, Inclusion and Exclusion may be either Total or Partial. The Terms of one Class may be identical with a part of the constituency of another Class, but not with the whole : the first Class then partly includes and partly excludes the second. The Class White-animals includes some cats and excludes others. It is supposable that we should know exactly how many of one Class were included or excluded by another: 9999 cats might be white animals. Such information, however, is not to be had in the case of Natural Classes, but only sometimes in the case of Artificial Classes; we might ascertain, for instance, that out of 12 town-councillors 9 were publicans. Or we might know that half, or more, or less than half of one Class were members of another. And De Morgan and Hamilton have proposed to take account of these more definite modes of Partial Relationship ; but they have not been generally recognized by Logicians. It is usual only to take account of Total, and indefinitely Partial Relationship, and to express the former by the signs All and No (All cats are Of the Immediate Relationship of Classes. 157 animals, No cats are dogs) ; the latter by Some. That is, whenever it is not known that the whole of one Class is included or excluded by another, though a part of it certainly is ; this is signified by saying Some are (Some cats are white, Some animals are not cats). 2 . Knowledge and Reality. We must distinguish three conditions of a pheno~ menon : 1 . As it really is ; 2 . As we know it ; 3. As our knowledge of it is expressed. With the third condition Scholastic Logic is largely occupied, but it properly belongs to Rhetoric, and we avoid the consideration of it here as much as possible. We endeavour to deal with the Relations of Classes themselves : but of course this is only possible in so far as those Relations are known to us. Between our knowledge of a Relation and its reality there may be a great hiatus. A Class may totally include another whilst we are only informed of a Partial Inclusion, or even of a Partial Exclusion. We must judge accord- ing to our knowledge and make allowance for its possible shortcomings. Hence what may be called the Rule of Continence : Assume no Relation to be j stronger (more Inclusive, Exclusive, or Constant) than ] there is evidence for. 158 Theory of Logic. On the other hand, since we can never know too much, I may add a Rule of Husbandry : Assume no Relation to be weaker (less Inclusive, Exclusive, or Constant) than there is evidence for. 3. Designation. Any Relationship of Classes as known to us may be Indesignate or Designate. If we are told that the class Animal includes cats, but not whether the In- clusion be Total or Partial, the Relationship is Indesignate. If we know that All cats are animals, the Relationship is designated as Total by the word AIL Thus All, Some, &c., may be called the Desig- nations of Class-relationship. Again, a Relationship may be designated in one or both Terms, may be Unidesignate or Bidesig- nate. f All cats are animals' is a Unidesignate; 'All cats are some animals,' is a Bidesignate Relation ship. By the Rule of Continence any Relationship in so far as Indesignate should be treated as Partial. It is said that cats are jealous ; but unless- it is affirmed that all are, we must assume that only some are. Where, however, a Class or part of a Class is given as excluded from another, it is excluded from the whole, although this be not expressed by designation. Thus Some animals are not fish, means that some animals are totally excluded from the Class Fish ; or to express Of the Immediate Relationship of Classes. 159 the Relationship by Bidesignation, Some animals are not any fish. And this seeming breach of the Rule of Continence will presently be justified ( 6). 4. Qualitative and Quantitative Aspects of the Relationship of Classes. Plainly if the Inclusion and Exclusion of Classes be viewed altogether as a Relation between their Constituencies with respect to number, it is a Quanti- tative, and not a Qualitative intuition. Even if we only speak of All and some, and do not use numerical designations, the Relations in question are no less quantitative for being indefinite. However, although Logicians have usually talked of Classes as quantities of Terms, it was not this aspect of the Class which they really had in view. If they said, the Class A is included in the Class V, they meant not merely or principally that the Constituents of A are Constituents of V, but that the Constituents of A have the Qualities which confer Attributes on V : it is for this reason that they are (or are identical with) Constituents of V; and it is this Qualitative Relationship of A and V with which Logicians are ultimately concerned : the Quantitative Relationship they use as the implicated coincident and mark of the Qualitative. The constant concomitance of certain quantitative and qualitative Relations amongst Classes will presently be proved ( 6), and thenceforth taken for granted. 160 Theory of Logic. 5. Conditions of Subsumption. ist. Sulsumption of Terms. We have seen that Classes consist of Terms in so far as these have qualities in common. When Classes have been formed, then, on the discovery of any new Term, the question arises, ' To what Class, or Classes, does it belong r ' It might happen that the discovery of new Terms would lead to an alteration in existing classifications ; but to examine this case would only be to return to the considerations of the previous Chapter : we here suppose that the new Term is a member of known Classes. - A Term is a Member of every Class whose Attribu- tion is realized in its Qualities. Hence a Compound Term may be a member of many Classes : it is a member of as many Classes as it has qualities, for every quality is by its nature the basis of a Class ; and may be a member of as many more Classes as there are possible combinations amongst its qualities. To ask to what Class a Term belongs is, then, to ask what its qualities are ; and to find out this the Term must be examined, at least so far as to discover its fundamental characteristics, which are marks of the others. After examination it is subsumed under all the Classes whose attributions it realizes. Or the question may be, ' Is the newly-discovered Term a member of this or that particular Class ? ' If on examination it is found to have the qualities which Of the Immediate Relationship of Classes. 161 confer attribution upon the given Class, or (in other words) to realize that attribution, it is subsumed accordingly. If however it has not the requisite qualities, it is excluded from the given Class, or (in other words) it is subsumed under the Counter Class. 2nd. Subsumption of Classes. Since a Class is an.assemblage of Terms, the process of subsuming Classes does not essentially differ from the process of subsuming Terms. In subsuming a Class indeed we have not to examine its constituents, for this has already been done whilst forming it. But as in subsuming a Term we discover its qualities, and observe what Classes have their attributions realized therein ; so in subsuming a Class we take its attribu- tion as defined, and observe what other Classes have their attributions contained in it. And a Class is subsumed under all other Classes, whose attri- butions are contained in its own attribution. Thus a Class of Plural Attribution may be subsumed under many other Classes: it is subsumed under as many Classes as there are distinct attributes in its attribution, for each of these is the attribution of a Class ; and it may be subsumed under as many more Classes as there are possible combinations amongst its attributes. On the other hand, a Class which is not subsumed under another Class (nor subsumes it) is excluded from 1 62 Theory of Logic. that other Class, inasmuch as it has not that Class's attribution ; is excluded from the other Class, or subsumed under the other's Counter Class. These remarks apply to the subsumption of a Class as a whole ; but a Class may also be partially sub- sumed by the subsumption of some of its Terms. 6. Propositions concerning the Necessary Concomitance of certain Relations between the Constituents and Attributes of Classes. a. Inclusive Relationship. i. If there be two Classes of unequal attribution, and one of them possess all the attributes of the other, the Class of lesser attribution includes the constituents of the other Class, and has other constituents besides. Suppose two Classes A and V of unequal attribution ; let A have the lesser attribution, and let V have all the attributes of A and some besides : the Class A includes the constituents of V and other members besides. For the constituents of V, having the qualities which confer attributes on A, are constituents of A. But the constituents of V cannot be the whole constituency of A, for then they must confer on A the whole attribution of V, which is contrary to the hypothesis. Of the Immediate Relationship of Classes. 163 2. If there be two Classes of unequal constituency, and one of them include all the members of the other, the Class of lesser constituency possesses the attribution of the other Class and other attributes besides. Suppose two Classes, A and V of unequal con- stituency ; let V have the lesser constituency, and let A include all the members of V and some besides ; the Class V possesses all the attributes of A and other attributes besides. For the constituents of V, being constituents of A, have in common the qualities which confer attributes on A, which therefore must be also attributes of V. But the attribution of A cannot be the whole attribution of V, for then the constituents of A would all be constituents of V, which is con- trary to the hypothesis. Corollary i. There cannot be two distinct Natural Classes which wholly coincide either in attri- bution or in constituency ; for such Classes coincide both in attribution and constituency, and are the same Class. ii. If there be two Classes, and some constituents of the one have the qualities which confer attribution on the other, these Classes partially at least include one another. iii. If there be two Classes which partially in- clude one another, the constituents common to M 2 164 Theory of Logic. both have the qualities which confer attributes on both. /3. Exclusive Relationship. 1. If there be two Classes, and no constituents of the one have all the qualities that confer attri- butes on the other, these Classes totally exclude one another. For if any constituent of either Class were a member of the other, it must have the quali- ties which confer attribution on the other. 2. If there be two Classes that totally exclude one another, no constituent of either can have all the qualities that confer attribution on the other. Corollary i. If there be two Classes, and some constituents of the one have not all the qualities that confer attribution on the other, the one Class is partially (at least) excluded by the other. ii. If there be two Classes, and one partially ex- cludes the other, some constituents of the latter Class cannot have all the qualities which confer attribution on the former. These propositions and corollaries explain why a Class, or part of a Class, given as excluded by another, is understood to be totally excluded by it. For if a Class, or part of a Class, be excluded by another, it is Of the Immediate Relationship of Classes. 165 because the constituents of the former Class, or some of them, have not the qualities which confer attribu- tion on the second Class ; and Terms which do not realize the attribution of a Class cannot be any part of it. We see, then, that certain quantitative Relations between the Constituencies of Classes, are constantly concomitant with certain qualitative Relations between their Attributions. Subsumed under always means contained in ; subsumed under the Counter Class always means excluded from the Positive Class : and so on. Hence it is not material which Relation be made explicit ; the concomitant Relation is always implicit ; one is a mark of the other : but the quantitative Relation is more convenient to deal with. 7. Unidesignate Relationship. Of the Unidesignate Relations of Classes, Logicians have usually recognised these four : i. Total Inclusion Total Exclusion AfoAisV ...... E Partial Inclusion Some A is V . * - . .1 Partial Exclusion Some A is not V . O 1 66 Theory of Logic. The letters on the right hand are the symbols commonly used to denote the Relations which they respectively stand over against. We have next to discover all that is involved in the Relations thus given. And it may be observed that in Unidesignate Relations the designation Some is here taken to mean not Some only, but Some, it may be all, or Some, it may be none, according as the Relation is Inclusive or Exclusive ; or, briefly and generally, Some at least. 8. Comparison of Unidesignate Relations of Classes. \st. Implication. Understanding Relations to Coincide when they tie the same pair of Classes : a Relation may be denned to Implicate another when that other must coincide with it. Such Implication springs directly from the nature of a Class as a Whole or Sum of Parts. Many other branches of the Science grow more or less directly from the same root: indeed the Relation of Whole and Part is, if I may so express it, the principal schema of all this latter part of the subject. And perhaps it will be as well to state explicitly some principle similar to Euclid's so-called Axiom, " The Whole is greater than its Part : " as, for instance, The Whole includes every Part ; or, The Whole is identical with the Sum of its Parts. As a whole in relation to parts, a Class may be Of the Immediate Relationship of Classes. 167 regarded in relation to its Constituents or to its Attributes. a. Class and Constituent. (a) Direct Implication. 1 . A Class which includes (or excludes) the whole of another Class, includes (or excludes) every constituent or part of it. Hence i. A implicates I. For if all the members of A be members of V, some must be. ii. E implicates O. For if all the members of A be excluded from V, some must be. [b] Inverse Implication. 2. A Class which does not include (or exclude) part of another Class, cannot include (or exclude) the whole. Hence i. If I do not obtain, A cannot, ii. If O do not obtain, E cannot. One Constituent or Part of a Class, as such, implies other Constituents or Parts, or another Part ; which therefore may be called the Counterpart. /3. Class and Attribute. (a] Direct Implication. 3. A Class which has all the attributes of another Class, has each, or any, of them. (b} Inverse Implication. 4. A Class that has not some attributes of another Class, cannot have all. 1 68 Theory of Logic. 2nd. Compatibility. Relations of Classes that may coincide are Com- patible. The Compatibility of certain Relations of Classes depends partly on the vagueness of the partial desig- nation, partly on the nature of the case, i. I is compatible with A. For if some members of A be members of V, we do not know but that all are so. ii. O is compatible with E. For if some members of A be excluded from V, we do not know but that all are excluded. In these two cases the Compatibility of the Rela- tions depends upon the circumstance that Some may mean All : if it should prove to mean Some only, the Compatibility would be destroyed. Again : iii. I is compatible with O. For though some members of A be members of V, others may not be so. iv. O is compatible with I. For though some members of A be excluded by V, others may be included. In these two cases the Compatibility of the Rela- tions depends upon the circumstance that Some may mean Some only : should it prove to mean A II, the Compatibility would be destroyed. I and O, then are compatible, if the whole truth concerning the Rela- Of the Immediate Relationship of Classes. 169 tions of the two Classes be already known ; I and A, O and E are compatible only on the supposition that the whole truth is not known. $rd. Incompatibility. Relations of Classes that cannot coincide are In- compatible. Thus : i. A is incompatible with E. For if all the members of A be included by V, none can be excluded, ii. E is incompatible with A. For if all the members of A be excluded by V, none can be included. If either A or E obtain between two Classes, then the other cannot ; but Relations may obtain between two Classes which are neither A nor E, nor yet imply them : namely, I and O. Again : iii. A is incompatible with O iv. E w I v I F ** >> >> J ~' vi. O A All these Incompatibilities are implicated in the Incompatibility of A and E, but do not, like those Relations, admit a third case. Incompatibility may also be viewed as Obverse Implication; since either of two Incompatibles, where- ever it obtains, implicates the absence of the other. 1 70 Theory of Logic. Ofth. Alteruternity. It of two conceivable Relations between Classes one must obtain, but both cannot : this is Alteruternity. Thus A and O are incompatible, but If A do not obtain, O must : for if E obtain, it implicates O. And similarly If E do not obtain, I must : for if A obtain, it implicates I. And so conversely. If of two Relations between Classes one must obtain, but both may do so: this is Imperfect Alteruternity. Thus I and O are compatible, but If I do not obtain, O must : for if some members of A be not included by V, they must be excluded. If O do not obtain, I must : for if some members of A be not excluded by V, they must be included. $th. Square of Comparison. The famous Square of Opposition may be a little modified and called the Square of Comparison ; since " opposition " is too strong a word, and very mislead- ing. Relationships of Implication, or Compatibility, cannot be regarded as Opposition, unless in the sense that the symbols of the Relations compared are placed opposite one another ; and to base a technicality on Of the Immediate Relationship of Classes. 171 such a paltry circumstance, is to throw opportunity out of window, and open the door to misunder- standing. Compatibility ; i 72 Theory of Logic. 9. Equivalent Aspects of Unidesignate Relations. \st. Obverse Relationship* We have seen that any Class and Counter Class together include the sum of possible Terms. Any other Class, then, being constituted of Terms, must be included either in the Positive Class, or in the Counter Class, or partly in one, partly in the other. Hence any direct Relation of one Class to another Positive Class, implies an obverse Relation to it, that is, a Relation to its Counter Class. These direct and obverse Relations are equivalent, and we may use whichever suits our purpose. The letter which stands as the symbol of a Class, may with a stroke before it represent the Counter Class. Thus if A be a Positive Class, the Counter Class will be /A. Propositions of Obverse Relationship. 1. In so far as a Class is included in a Positive Class, whether wholly or partially, it is ex- cluded from the correlative Counter Class, or Classes. A. If all A is V, No A is /V. I. If some A is V, Some A is not /V. 2. In so far as a Class is excluded from a Positive Class, whether wholly or partially, it is in- Bain's Logic, Book I., ch. iii., 27. Of the Immediate Relationship of Classes. 1 73 eluded in the correlative Counter-Class, or Classes, or some, or one of them. E. If no A is V, All A is /V. O. If some A is not V, Some A is /V. As any Relation to a Positive Class implies a Relation to the correlative Counter Class ; so any Relation to a Counter Class, implies a Relation to the correlative Positive Class. And as the direct Relations are symbolized by A, E, I, O ; the cor- responding obverse Relations may be represented (like the Counter Class) by the same letters with a stroke before each, thus : /A, /E, /I, /O. 3. In so far as a Class is included in a Counter Class, whether wholly or partially, it is ex- cluded from the correlative Positive Class. /E. If all A is /V, No A is V. /O. If ,s002 A is /V, Some A is V. 4. In so far as a Class is excluded from a Counter Class, whether wholly or partially, it is, or is included in, the correlative Positive Class. /A. If^Ais/V, ^//AisV. /I. If some A is not /V, Some A is V. 2nd. Converse Relationship. A Relationship between two Classes does not always affect both in the same way ; and it is im- portant to note the different ways in which the two Classes are respectively affected in different Rela- tions ; since some of them are liable to be misread 1 74 Theory of Logic. by negligent observers. Any Immediate Relation between two Classes may be viewed from both sides : either side being taken, the Relation thence regarded may be called Direct; and from the point of view of the other Class, the Relation will then be seen in its Converse. The Relationship itself is not altered by our point of view, and therefore we may take whichever suits our convenience. A. A Class totally included in another Class, includes at least a part of that other. Hence If all A is V, Some V is A. ( i .) This is the usual mode of viewing the Converse of A : it is called Conversion by Limitation, because the correlative V, being indesignate, is taken partially, according to the Rule of Continence. Unfortunately, however, there is a custom by which a Relationship once unidesignate, must be always unidesignate ; and so V having now been designated, the Class A loses its designation. The result is that the Class A, now indesignate, is also limited ; and if we attempt to reconvert the Converse of A (the Relation), we get not A itself, but only I : If some V is A, Some A is V (infra Prop. I.) Thus by viewing the Relationship on both sides we seem to lose a part of our information concerning it ; although the Relationship itself is certainly not thereby altered. The usual mode of converting A by Limitation merely, is therefore contrary to the Rule of Husbandry ; and since this Rule seems more Of the Immediate Relationship of Classes. 175 profitable than the custom of preserving the uni- designate character of Relations, I propose a second mode of converting A, to be used whenever con- venient; which may be called Conversion with Bidesignation thus : If all A is V, Some V is all A. (2.) E. Total Exclusion between two Classes is reci- procal. Hence If no A is V, No V is A. I. Partial Inclusion between two Classes is reci- procal. Hence If some A is V, Some V is A. This treatment of E and I is called Simple Con- version, because the designation of the Relationship remains the same. Nothing is lost by leaving the Class A indesignate in the Converse; for we have seen that in E the correlative Class is taken totally ; and in I, where it is taken partially, that is all we know. O. This Relationship is so indefinite that it does not admit of direct Conversion, if we insist on pre- serving its unidesignate character. For if Some A is not V, it may be that No V is A, or that All V is A, together with the implications of these possi- bilities. Accordingly, the usual practice is to con- vert the Obverse of O, and to this process we shall come presently. Here we propose to resort, as in the second Conversion of A, to Conversion with Bide- signation thus : 176 Theory of Logic. A Class which excludes a part of another Class, is itself by that Part-Class wholly excluded. If some A is not V, No V is some A. L Converse of Obverse Relations. Obverse as well as Direct Relations may be viewed from the side of either Class, and are converted on the same principles as the formally-equivalent Direct Relations, thus : /A, like E. If no A is /V, No /V is A. /E, like A. If all A is /V, Some /V is A ( i ). . . . Some /V is all A (2). /I, like O. If someKis not /V, No /V is some A. /O, like I. If some A is /V, Some /V is A. This last Relationship, Some /V is A, is usually taken as the Converse of O, and together with all the above Converse-Obverses, is said to be obtained by Contraposition. It is an obvious extension of this discussion to consider the Obverse of Converse Relations ; but we should meet with no novelty, except in the bidesignate Converses of A and O ; and these will be examined when we come to the Obversion of bide- signate Relations in general. 10. Genus and Species. If of two Classes of unequal Constituency, one includes the other, they are called in relation to one Of the Immediate Relationship of Classes. 177 another Genus and Species ; that is to say the in- cluding Class is called the Genus ; and the Class included, the Species. By Prop, o, 2, 6, this is equivalent to saying that the attribution of the Genus is less than and included in the attribution of the Species. In Logic these names are not necessarily confined to Natural Classes, but may denote any Classes standing to one another in the defined rela- tion : we shall however gain in definiteness by keeping an eye on Natural Classes. Genus and Species are said to be respectively higher and lower Classes. A Genus not included in any higher Genus, is called a Summum Genus ; and the desire for the utmost possible generality of con- ception plainly aims at discovering one all-embracing and absolute Summum Genus. But it seems that there is none : there is none to those who follow Kant* and Prof. Baint in not regarding simple existence as an attribute. By the definition of a Class, "all things" cannot be a Class, since there is nothing to distinguish it from. Or if it be con- tended that the attribute of existence is sufficiently contrasted with nonexistence, then nonexistence must also be an attribute ; but the impossibility of this is shown by the absurdity of supposing such an attri- bute realized in any constituent. Instead of one / * Critique of Pure Reason : pp. 165-367 (Meiklejohn). Beweisgrund /H einer Demonstration des Daseins Gottes. Betracht. I. I. t Bain : Logic } Book I. ch. lii. 23. N 178 Theory of Logic. Summum Genus, there are two coordinate Summa Genera, namely, Terms and Relations. To Prof. Bain's view, that Object and Subject are the true Summa Genera,* I cannot altogether subscribe; but no doubt they are the Summa Genera of Concretes. A Species that includes no lower Species is called an Infima Species. Between a Summum Genus and an Infima Species many Classes may stand in gradation. Each Class is a Species of any Class above it, and a Genus of any Class below it (and within it). Thus a Summum Genus is the Genus of a Species, which in turn is the Genus of a lower Species ; and so on until we come to the Genus of the Infima Species. The Genus next above any Species is called its proximate Genus. 1 1 . Of the Qualities which appertain to a Term with regard to its Subsumption. No Term is subsumed immediately and only under a Summum Genus, but is also a member of some lower Class ; therefore of both a Species and a Genus. It realizes, then, in its qualities the attributions both of a Species and of a Genus. The qualities of a Term which confer attributes on its Species, are called Specific qualities. * Bain : Logic ; Appendix C. Of the Immediate Relationship of Classes. 1 79 Of a Term's Specific qualities those which confer attributes on its Genus, are called Generic qualities. A Specific quality (or qualities) not also Generic, may be called the Specific Difference (subordinately privative determinant of the Genus). To these names of the qualities of a Term, there correspond names of the attributes of its Classes Specific attributes, Generic attributes and Specific Differential attributes. And Specific Differential attributes should be carefully distinguished from the Differential attributes of which we spoke in the fourth Chapter, in as much as the latter were so called with- out reference to any particular Genus. These three kinds of qualities and attributes are sometimes said to be Essential. The * Essence ' is ' a convenient name for those qualities of a Term on account of which if is subsumed under a Class : the Essence of a Class is its defining Attribution. In order that any quality may rank as part of the Essence, i, it is requisite that it be ultimate or independent ; or, rather, that it be not known to be dependent on, or derivable from, any other quality : 2, it is desirable that it be fundamental, or one on which other qualities depend. The qualities appertaining to a Term besides its Essence in relation to any Class are either Propria or Accidents : and Propria are distinguished by this, that they belong to all the members of a Class by derivation from the Essence; whereas it is doubted N 2 180 Theory of Logic. whether an Accident belongs to all the members, or it may be known that to some it does not belong. A Proprium, then, like the Essence of a Term, belongs to it in common with all the constituents of the Class ; but a Proprium differs from the Essence in that it is known to be derivable from, or depen- dent on, the Essence or some part of it. Thus Propria form no part of the Essence or Attribution of a Class, and are not included in a Definition ; and so it is questionable whether they should be called Attributes ; although it would seem that they ought to be, since they are qualities common to all the members. Perhaps it will be sufficient whenever there is danger of a misunderstanding, to signalize them as secondary or derivative attributes. Again, a Proprium may be dependent on the Generic at- tributes, and appertain to the Genus as a whole ; and may then be called a Generic Proprium : or it may be dependent on the Specific Attributes only, and may then be called a Special Proprium. Accident is the name given to a quality of any Term which neither ranks among its Essential qualities, nor is known to be dependent on them : such a quality may be peculiar to a few Terms of a Class or common to many ; but it is not considered to appertain to a Natural Kind. Accidents are indeed said to be Separable or Inseparable from the members of a Kind ; if members have been Of the Immediate Relationship of Classes. 181 known without the Accident, it is said to be Sepa- rable ; if not, Inseparable. And with regard to Inseparable Accidents the questions arise : Why, since they are common to all known members, are they not appurtenances of the Class ? Since they are not derivable from other qualities, why are they not considered Essential to the Class? The reason why Inseparable Accidents are not referred to a Class as a whole, appears to be some suspicion grounded in analogy, that if Nature were exhaus- tively known they would be found to be separable : this suspicion would naturally attach to a quality which had been known to be separable in relation to the Members of other Classes. But perhaps what generally draws this doubt upon a seemingly In- separable Accident is want of fundamental character, where no other qua'lity depends upon it. This defect would especially exclude a quality from Essential rank. Accidents are, of course, never really accidental, but potentially derivable, if not from the essential attributes alone, from these in connection with cir- cumstances ; as is believed to be the case, for instance, with the colour of many animals. And on the other hand, it is not improbable, that wider knowledge will show many qualities now deemed essential, to be themselves derivative. So that the difference' between Essential Attributes, Propria, and Accidents, is in some degree relative to the state of Science ; \ lS2 Theory of Logic. and what qualities are to be classed under each head, is in every case a question of special science. Although Accidents do not appertain to Natural Classes, it is common to find Artificial Classes deter- mined by qualities which would be Accidents in relation to a Natural Class. Inhabitants of London constitute a Class, whose attribution is the circum- stance of living in that city; although to live in London is accidental to a man, and even to an Englishman. Table of the Qualities appertaining to a Term. r Underived and fundamental Attributes, or Essence Specific Attributes (-Generic. Specific Differential. L Derived from the Essence Propria (-Generic. L-Special* -Neither fundamental nor derivable Accidents r-Inseparable. 1-Separable. Of the Immediate Relationship of Classes. 183 1 2 . Propositions concerning Genus and Species. i. A Genus includes a Species and more than a Species ; or, a Species is only a part of its Genus. (Def. of G. and S.^ 10.) It is understood of course that this discussion is of an abstract nature, and does not proceed on the supposition that its propositions will always represent the relations of concrete phenomena. In relation to concrete phenomena the principles of Logic are merely regulative regulative, that is, not of the phenomena themselves but of our conduct in dealing with them. They define, I may say, the conditions of the intelligibility of phenomena, and in their imperative aspect direct us to seek in phenomena their own realization. Thus we are to seek in Nature, for every Species, a Genus including it and more ; but it is not certain that we shall always find one. For instance, the fish, Amphwxzis, is, I believe, a Species which is the only known representative of its Genus. It might indeed be said that this is due to the fixity of the names, Genus and Species, in a Zoological classification ; but the abstractness of Logical principles is a better ground of recon- ciliation. Those Terms by which the constituency of a Genus exceeds the constituency of its Species, maybe called the Counter Species or the Special Counter Class. The Special Counter Class is of course to be distin- 184 Theory of Logic. guished from the Counter Class in general, which has no reference to a particular Genus. Corollary : The general Counter Class of a Species includes the Counter Class of its Genus and other Terms besides (namely, the special Counter Class). 2. A Species includes the attributes of its Genus, and others besides. 3. The Differential Attributes of a Species, are Differential Privations of the Counter Species. For if the constituents of the Counter Species possessed the qualities which confer the Specific Difference, they wonld be con- stituents of the Species. 4. If there be two Classes, and Part of one is in- cluded in, whilst the Counter-Part is excluded from the other Class, the Part of the former Class is a distinguishable Species of it. Let A and V be two Classes such that a Part of A is included in, whilst the Counter- Part is excluded from V : the Part of A in- cluded in V is a distinguishable Species of A. For the Part of A included in V must be so included on account of possessing attri- butes, which are privations of the Counter- Part ; and these attributes, being additional to the attributes common to the Part, and Counter-Part, are Specific.* * In Natural Classes these attributes- to be Specific, must be of Essential rank : the Prop, is only true on this condition. Of the Immediate Relationship of Classes. 185 The Counter-Part is, then, the Counter Species. 5. The known Counter Species is either another Species, or several others. For we have seen that a Term has more than its Generic qualities. First, then, if all the constituents of the known Counter Species agree among themselves in a certain essential quality (or qualities) which is not Generic, the same is a Specific Difference ; and the Counter Species, as a whole, is another Species. Secondly, if not all constituents of the Counter Species, but only some of them, agree in a quality which confers a Specific Difference, these Terms constitute a second Species : and if any of the remaining con- stituents similarly agree, they constitute a third Species. Thus the Counter Species may consist of several Species. Lastly, if in the Counter Species there be no two known Terms that agree in any Essential quality, that is not Generic ; each Term may rank as a Species, and be called a Specific Instance. Corollary : A Genus has more than one Species. It appears to be an assumption of Logic, which may as well be explicitly stated, that Nature is 1 86 Theory of Logic. inexhaustible, or that the natural limits to the production of -instances of any kind are unknown. Hence 6. The number of Species in any Genus is indeter- minate. In any Genus, any Species, or number of Species being taken, the Terms (if any) by which the con- stituency of the Genus exceeds the constituencies of these Species, may be called the Remainder. Species of the same proximate Genus may be called coordinate Species. 7. The sum of coordinate Species is identical with the Genus ; or the constituency of the Genus is distributed among the Species without remainder. This follows from Prop. 4 ; for if there were a Remainder not groupable into Species, its constituent Terms must be Specific Instances. The idea of Specific Instances is supported by the assumption of Nature's inexhaustibility ; for though similar Terms should not be known, it does not follow that they do not exist ; and in some cases what we know of the conditions of the existence of such Terms, is a guarantee that others of the same Species do exist, or have existed. The Specific Difference of a Specific Instance can only be distinguished by analogy, or as the comple- ment of its Differential Privations. Of the Immediate Relationship of Classes. 187 8. In any Genus the Species are mutually ex- clusive. For they are reciprocally Species and Counter Species. Corollary : i . A Species excludes part of its Genus ; that is, the Remainder : or, is excluded by it. 2. Part of a Genus includes the Counter Species. 9. If two Classes be mutually exclusive they may, or may not, agree in some attributes, but cannot agree in all. They do not agree in all by Prop. /3, 2, 6. If Species of one Genus, they agree in their Generic Attributes. If Summa Genera, or exclusively included in different Summa Genera, they have nothing in common. And here it may be remarked once more that although all Relations are Terms, Terminality is an Accident, and not an Attribute of Relations, for it is related to Relationality neither as fundamental nor as derived : else all Classes must have something in common. 10. A Class only partly included in (or excluded from) another Class, may or may not have some of its attribution, but cannot have all. Let A and V be two classes such that part of A is included, and part excluded by V. i. A may be the Genus of V and possess its Generic Attributes. Theory of Logic. 2. A and V may be coordinate, but imperfectly differentiated Species (exceptions to Prop. 8), having the same Generic attributes, and some members of A being moreover marked with the Difference of V. 3. Supposably, A and V, as Classes, may have nothing in common. 4. But A cannot have all the Attributes of V, by Cor. ii., Prop. /3, 2, 6. 13. Dims ton. We saw in the preceding Chapter that the problems of Logic had to do with the discovery and arrange- ment of Classes. One of these problems may be stated thus : Given a Genus to find its Species. The process by which this is accomplished is called Division. Three Canons of Division are usually given, which may be derived from certain propositions in the section concerning Genus and Species. Thus : 1. A Genus includes a Species and more. (Prop, i 12.) Whence what may be called the Canon of Limitation. Each of the parts must contain less than that which is divided. 2. The sum of Coordinate Species is identical with the Genus. (Prop. 7, 12.) Whence the Of the Immediate Relationship of Classes. 189 i Canon of Consummation. All the parts together must be exactly equal to that which is divided. 3. In any Genus the Coordinate Species are mutu- ally exclusive (Prop. 8, 12.) Whence the Canon of Disjunction. The parts must be opposed, that is, mutually exclusive. These Canons help to test a Division already made, but do not tell us how to make it. To learn this we must fall back upon the considerations of the pre- ceding Chapter. Division is the discovery, definition and probation of all the Species of a given Genus. The first step will be to assemble the constituents of the Genus. We then select a quality, or modifi- cation of a quality, appertaining to some of the constituents, and propose it as a Specific Difference. The selected quality should be essential, that is, fundamental and underived ; and, of course, not one conferring an attribute on the Genus. Those con- stituents of the Genus which agree in this quality may form a Species. The Species thus formed may be treated accord- ing to the Canons of Definition: i. Assemble the constituents of the Species : 2. Assemble the constituents of the Counter Species ; that is, those constituents of the Genus which lack the selected Difference. This rearrangement of the constituents of the Genus may serve two purposes : i , it may 190 Theory of Logic. disclose a quality more fit to be made a Specific Difference, and thus lead to the formation of a more natural Species : 2, supposing the quality chosen to be the best, the segregation of Species and Counter Species, enables us to observe what other qualities or modifications of qualities are correlated with the Difference, and what with the absence of it. Then, when enough specimens have been examined, the Species is proved and defined ; and the Counter Species also as to its privations. This method is the celebrated Division by Dicho- tomy ; which, as Prof. Jevons remarks, is the only method by which we can be sure of making a Division exhaustive. Indeed it insures a sound Division in every respect, so far as a sound Division exists in Nature, as we may see by comparing the results already reached with the Canons : i . The Species and the Counter Species are each less than the Genus ; 2, the Species and the Counter Species are together identical with the Genus ; 3, the Species and Counter Species are mutually exclusive. So far then, the Division is sound ; and it is intuitively clear that a Division thus conducted must always be sound ; but we have not carried the present one far enough. The Counter Species has been left in a very vague state, defined only by its privations. But by Prop. 5, 12, the Counter Species is either itself a Species or several others ; we must therefore look for the positive determinants. Of the Immediate Relationship of Classes. 191 The procedure is as before : a quality is selected, which we will suppose to be the best a quality underived, fundamental, and conferring 1 attribution neither on the whole Genus, nor on the first Species. If the Counter Species be but one Species (save an unknown Remainder) this quality will be found to mark all its constituents. The correlated modifica- tions are then noted, and the Species is defined. Or if the Counter Species contain more than one distinct Species, these have to be discovered severally in a similar way. But we have not yet done. Returning to the case in which the Counter Species, or known Counter Species, was one Species, we observe that it had itself a Counter Species. The Counter Species of the second Species is twofold, comprising, i, the first Species and, 2, an unknown Remainder. Now in either of these groups there may possibly be Terms which have not, and Terms which have the Difference of the second Species. As to the first Species it cannot be that all its constituents have the Difference of the second ; but some of them may. As to the unknown Remainder, if any of its Terms have the Difference of the second Species, they are members of it ; if not they constitute a further Counter Species to be treated as before. The result so far may be exhibited in a diagram, borrowed from Prof. Jevons, and adapted to the notation of the present Essay. The groups of letters 1 92 Theory of Logic. stand for Classes ; each letter for an attribute ; and a letter with a stroke before it (/B) for the attribute regarded as a privation. G GB G/B GBC GB/C G/BC G/B/C Here we have G the Genus to be divided : B is taken as the first Specific Difference ; and so G B is the first Species, with G/B as Counter Species. As the second Difference C is taken, and G/B C becomes the second Species. The first Species is then logi- cally represented by GB/C ; and of the other two possible Classes, G/B/C is the assumed Remainder. A question arises as to the existence of G B C, which can only be decided (where C and B are compatible qualities) by examining GB Term by Term. It is a logical desideratum that GBC do not exist ; for then, neglecting the unknown Remainder, we have GB/C and G/BC, coordinate and mutually exclusive Species of G. But if G B C do exist, the Divison will not run clear : for we have in fact GBC and G B/C, coordinate Species of G B ; and GBC and G/BC, coordinate Species of GC. Such cases may occur : it is not the Logician's fault, but Nature's. And it is the double merit of Dichotomy to exhibit a perfect classification where it exists, and to expose Of the Immediate Relationship of Classes. 193 the shortcomings of Nature where a perfect classifi- cation does not exist. For further discussion of this and allied subjects the reader is referred to Prof. Jevons' Principles of Science, Chapter XXX, 14. Bidesignate Relationship. Bidesignate Relations are double the number of the Unidesignate. The following is a list of them with their respective symbols placed opposite, 1. Toto-total Inclusion AllKisallV '. A 2 2. Toto-partial Inclusion All A is some V . . . . A 3. Parti-total Inclusion Some A is all V ... I 2 4. Parti-partial Inclusion Some A is some V . . . . I 5. Toto-total Exclusion No A is any V . . . . E 6. Toto-partial Exclusion No A is some V , . . E t 7. Parti-total Exclusion Some A is not any V . . . O 8. Parti-partial Exclusion Some A is not some V . . . O 2 The names of these Relations are taken from Sir W. Hamilton ; the symbols, A 2 and I 3 , from Mr. 194 Theory of Logic. Spalding : and in each of these symbols the figure 2V placed above the character indicates that the Rela- tion thus denoted is a " better Relation " than the Unidesignates equivalent to the Relations still repre- sented by A and I: and E- a and O, are obvious imitations; where the figure 2, placed below the character in each case, indicates that the Relation. so denoted is a " worse Relation " than the Uni- designates equivalent to the Relations still repre- sented by E and O. There have been doubts as to the Logical legitimacy of Bidesignation : it has been urged that as a rule we neither think nor speak in this form. But whether these objections be sound or not, they can- not excuse us for not treating the subject here. Bidesignation certainly most adequately represents the Relations of Classes as they exist in Nature ; as we often seek, and often discover them. Indeed the Bidesignate Relations of Classes are involved in the doctrine of Genus and Species, and may be deduced from the Props, of 12. 15. Deduction of Bidesignate Relations. 'A 2 . The sum of coordinate Species is identical with the Genus. (Prop. 7, 12.) All G is #//nS Let A be a Genus, and B, C, D coordinate Species, Of the Immediate Relationship of Classes. 195 with Remainder X : if for (B, C, D, X) we substitute V, we may generalize the Relationship thus : All A. is all V. *A. A Species is part only of its Genus (P. i, 12). All S is some G; or, All A. is some V. 'I*. A Genus includes a Species, and more (P. i, 12). Some G is all S ; or, Some A is all V. X I. This.Relationship is exceptional : it occurs where P. 8, 1 2 is not true ; that is where coordinate Species happen not to be mutually exclusive. Some S is some 28 (a second Species) ; or, Some A is some V. *E. Coordinate Species are mutually exclusive (P. 8, 12) No S is any 2$ ; or, No A is any V. 'E 4 . A Species is excluded by part of its Genus (P. 8, Cor. i, 12). No S is some G, or, No A is some V. XD. A Species excludes part of its Genus (P. 8, Cor. i, 12). 196 Theory of Logic. Some Gf is not any S ; or, Some A is not any V. X3 a . A Genus has more than one Species; that is, is divisible (P. 5, Cor. i, 12). Some G is not some G, or, Some A is not some V. . Or we may regard this Relationship as given in the Relationship complementary to V I, for If some S is some 28, Some S is not some 28 : or else S and 28 would not be different Species. It will be observed that in so far as Bidesignate Relationships are based on the doctrine of Genus and Species, the sign Some must be understood to mean Some only (semi-definite Hamilton). But Bidesigna- tion, although involved in the doctrine of Genus and Species, is not entirely dependent upon it. A Rela- tionship of Classes may be given us with Bidesigna- tion, in which Some signifies Some at least ; and such cases may be called, Bidesignates detached from considerations of Genus and Species. I will add something about the Comparison and Equivalent Aspects of Bidesignates of both kinds ; but briefly, since this Chapter threatens to run to dispro- portionate length. To avoid confusion Relations based on the doctrine of Genus and Species may be denoted by symbols marked on the left side thus : Of the Immediate Relationship of Classes. 197 'A 2 , 'A, &c. ; and their Terras (or terminal Classes) may be represented by G, S, 28, &c. : whilst the symbols of detached Bidesignates may go unmarked, and their terms may be represented by the usual A and V. 1 6. Obverse Aspect of the Relationship of Genus and Species. In connection with the doctrine of Genus and i Species, the notion of a Counter Class acquires greater definiteness. The general Counter Class of any Positive Class may present a mere chaos of Terms without division or boundary. But the Counter Class of ~a Species, that is the coordinate Species, or the Remainder of the Genus, is a far more intelligible realm. We know some of the qualities of everything that can be found there. And De Morgan pointed out that it was not the general, but the special Counter Class, or the Re- mainder of some assumed Genus, which we always have in view, when referring explicitly or implicitly to the obverse correlative of any subject of thought or discourse. It is true he does not use this language ; but instead of the Species and Counter Species of a Genus, speaks of contraries within an " Universe "- a new expression which seems scarcely needed. The Genus with its Species and Counter Species do not always correspond to Natural Kinds ; but it is enough if the use of these words be Logically valid. Thus if 198 Theory of Logic. we speak of males with reference to the Counter Class females, we may rightly regard these Classes as Species and Counter-Species; for though male and female are not coordinate Species in Zoology, they certainly are in Logic. Pro-positions Ampliative. 1. A Class which includes a Positive Class, not coinciding with it, includes a part at least of its Counter Class. TT // c r f Some /S (general) ) . If all S is some &,<.' \ is some G. ( All /S (special) ) 2. A Class which includes (or excludes) a Part only of another Class, excludes (or includes) the Counter Part. If some G is all S, I Some G is not any S. 3. A Class which includes a Counter Class, not coinciding with it, includes a part (at least) of the Positive Class. If all /S is some G, All S is some G. If all /V is some (only] A, Some (at least] V is some A. 4. A Class of which a part only is included (or excluded) by a Counter Class, is partly included by the Positive Class. If some G is all /S, Some G is all S. If some (only] A is all /V, Some A is some V. Of the Immediate Relationship of Classes. 199 And so on. Besides the special Counter Class we might take account of the generic Counter Class; which again might be either general or limited by a higher Genus, 17. Bidesignate Relationships detached from the ' Doctrine of Genus and Species-, Not all the eight Bidesignate Relations have been accepted by all Logicians who have accepted the principle of Bidesignation. Sir W. Hamilton and Prof. Baynes accept all eight, but De Morgan, Mr, Spalding, and the Archbishop of York agree in rejecting E 2 and O 2 . It is not, however, easy to see why E 2 and O 2 should be rejected, since our know- ledge concerning the Relations of Classes may con- ceivably exist in those forms ; and besides Toto-partial Exclusion, the form of E 2 , is also the form of the Converse of O ; and Parti-partial Exclusion, the form of O 2 , is also the form of the Obverse of L But a question arises with regard to A 2 : for we know by Cor. i, P. i and 2, 6 that- there cannot be two Classes which wholly coincide. But i. Artificial Classes may coincide in their constituents without coinciding in their explicit attribution ; for their attribution may be a matter of convention ; so that the same Terms may on account of some qualities constitute one Class, and on account of other qualities another Class. 2oo Theory of Logic. 2. A Natural Class possessing a peculiar propriurn or attribute may be regarded as coinciding with any Artificial Class based upon that proprium or attribute : or an accident peculiar to the members of a Natural Class may also be the basis of a coincident Artificial Class. 3. A Natural Class based on two attributes both of which are peculiar to the Class may be viewed as a coincidence of the Classes based on either attribute. Thus gravitating bodies and inert coincide in the Class material bodies ; but this is really only one Class. We conclude then to accept all eight Bidesignates, and proceed to consider their relations to one another. Instead of comparing them in detail, however, it may suffice to give the results of comparison in a diagram: in which the abbreviations may be interpreted Impl. Implication, Comp. Compatibility, Incomp. Incompatibility, Imp. Alter. Imperfect Alteruternity. Of the Immediate Relationship of Classes. 201 ist. Octagon of Comparison. (Relations given in the Square are omitted.} 2nd. Obverse Relationship. i. If two Positive Classes coincide their Counter Classes coincide. A? If all A is ,all V, A II /A is all /V. ( i ) /A 2 . 2O2 Theory of Logic. 2. In so far as a Class is included (or excluded) by a Positive Class, or Part of a Positive Class, it is excluded (or included) by the Correlative Counter Class. A\ All A is all V No A is any /V . . . (2} /A 2 A. All A is .sw;z V A 7 "*? A is <2?zjy I some V ... /A I 2 . Some A is #// V Some A is not #?2jy /V . . . /I 2 I. Some A is .sw;/ V Some A is not #?ZJK /some V . /I E. Afr A is #??jy V AM A is some /V . . . . /E E 2 . No A is s0;;2 V All A. is some /some V . . . /E, O. *Si5W2 A is not any V S0w A is some /V . . . . /O O a . <5"(9w^ A is not .5w;2 V Some A is some /some V . . /O s 3. If two Counter Classes coincide, the correlative Positive Classes coincide : and so on. It will be observed that E does not, like A 2 , admit of double Obversion. We cannot say, that the Counter Classes of mutually exclusive Positives are mutually exclusive : this would be possible only if we knew the Positives to be equivalent to Class and Counter Class; as for instance, if they were two Species Of the Immediate Relationship of Classes. 203 which together coincided with a Genus : we might then, speaking with reference to the Genus, say No S is 28 .-. No /S is / 2 S But generally the Relations of the Counter Classes are too vague to render such an intuition possible. Simple Conversion is applicable to all these Rela- tions, and needs no illustration. 1 8. Of the Addition and Subtraction of Attributes as affecting the Relations of Classes, &c. ist. Abstraction and Generalization. Whether abstraction involves generalization, in other words, whether a decrease of Attribution is always accompanied by an increase of Constituency, is a well-worn, but, I suppose, unanswerable question. We can only say that potentially it is so, but whether actually may lie beyond our knowledge. There may, for instance, be a Class with attribution A B ; and if there exist any Term marked with A and not with B, to subtract B from the attribution of the Class is certainly to increase its constituency: but we may not know of any such Term, and perhaps there really is none ; though if we accept the inexhaustibility of Nature as a Logical assumption, such Terms must always be regarded as potentially existing. As the subtraction of an attribute extends a Class 204 Theory of Logic. potentially, but perhaps not actually ; so the addition of an attribute, potentially, but perhaps not actually, narrows it. 2nd. Class and Class. What is the effect of a change in the attribution of any Class upon its Relations to other Classes ? i. Where two Classes are related as Genus and Species. a. Let the Genus increase in attribution. A Genus may increase in attribution by the discovery of a new essential quality prevailing throughout its constituency. Such an attri- bute must be common to Genus and Species, and cannot affect their Relationship. A Genus may increase in attribution at the expense of a Species by the discovery that a supposed specific difference really extends throughout the Genus ; and should this extend to the whole difference, the Species would be submerged. b. Let the Genus decrease in attribution. A Genus may decrease in attribution by the discovery that a quality supposed essential is really only an inseparable accident of its constituents'. A loss common to Genus and Species can only alter their Relationship by destroying the whole geneiic attribution. A Genus may also decrease in attribution by Of the Immediate Relationship of Classes. 205 the discovery that an attribute supposed generic is really the difference of one or more Species. This might destroy the Genus, or occasion the intermediation of a proximate Genus. And so on. c. Let the Species increase, or decrease, in attri- bution. The increase of a Species' difference by new attributes can only alter its Relation to a Genus by admitting the intermediation of a proximate Genus. And similarly the decrease of specific differ- ence, if not destructive, needs not affect the Relationship with the Genus. 2. Where two Classes are related by reciprocally Partial Inclusion ; that is, where there are two Classes of Terms, and some Terms realize the attributions of both Classes. a. If it be discovered that all the Terms of one Class realize the attribution of the other, that Class is totally included in the other. b. The addition to both of attributes foreign to both, or the subtraction from both of attri- butes common to both, does not alter their Relationship. 3. Where two Classes are mutually exclusive. a. If the difference of one be added to the other, the one includes the other. 206 Theory of Logic. b. The addition to one, or both, of attributes foreign to both ; or the subtraction from one, or both, of attributes foreign to both, leaves the exclusive Relationship unaltered. Since the addition of a constituent to a Class (should it seem desirable) may occasion a decrease of attribution ; the subtraction of a constituent, an increase of attribution ; there is an opening here for a parallel series of propositions concerning the addition and subtraction of constituents. $rd. Term and Class. If any Term, or any number of Terms, be included in a Class, the addition to them of any quality or qualities not incompatible with (or destructive of) the "attribution of the Class, does not exclude them from it : nor does the subtraction of qualities which do not confer the Class-attributes. Thus, to take Archbishop Thompson's illustration, the addition of * suffering ' to a negro does not exclude him from the Class of fellow-creatures ; since it is sufficiently notorious that suffering is compatible with the essential qualities of a fellow-creature : it is indeed a proprium of the Class. But, as Prof. Bain remarks, we cannot argue " Beauty is pleasure ; hence, beauty in excess is pleasure in excess : " for excess is incompatible with Of the Immediate Relationship of Classes. 207 beauty, and must always exclude its subject from the Class of beautiful things ; so that whether it be a pleasant thing we do not learn. Or if we subtract from a negro his freedom, we do not exclude him from the Class of fellow-creatures ; for hitherto freedom has been by no means an attribute of fellow-creatures : though perhaps it, too, becomes a proprium at a certain stage of develop- ment. Similarly if any Term, or number of Terms, be excluded from a Class, the addition of qualities not conferring attributes on the Class, does not bring it within the Class. CHAPTER VIII. OF HYPOTHETICALS. i . Of Hypotheses in General. THE word Hypothesis signifies in general, some- thing laid down to be tested or argued upon; but within this there are at least four shades of meaning which blend into one another. The most marked difference is perhaps between a Hypothesis viewed as an inference, and a Hypothesis viewed as a datum ; but since an inference may become the datum for new inferences, the division even here is not quite distinct. Hypotheses viewed as inferences are common pro- perty of the Theory of Reasoning and Logic regarded as a Science of Proof; for all inferences need proof; and that which we try to prove is nearly always an inference. To infer something is an act of reasoning ; to test the inference belongs to Logic. In this sense an Hypothesis is i . A guess ; or a kind of inference which Mr. Spencer* distinguishes from other kinds according * Spencer : Psychology ; Part VI. ch. viii. Of Hypothetical. 209 to the " numerical ratio between the premised and ] inferred relations." If the inference be from a few known Relations to all that are similar in certain respects, it is Hypothetical ; if from many to all, Inductive properly so called ; if from all to some, Deductive ; and so on. For the purposes of Logic we may call any inference offered for probation a Hypothesis. 2. An inference is especially called a Scientific Hypothesis, when it is elaborated and offered for verification as a Law of Nature. This is the condi- tion ' of new Theories, while their truth is still very doubtful. In order to verify or disprove a Hypo-' thesis we must compare it with the known facts and laws of Nature ; and since we may not be able to do this directly, the Hypothesis should be such that we are able to make further inferences or deductions from it : * and thus it becomes a datum. Regarded as data, Hypotheses may be designed to be true, or approximations to the truth ; or they may be designedly false, and only used as a means of proving something else. 3. Thus a new theory put forward for verification is intended by its inventor to be true. And the ab- stractions of Geometry and other sciences, such as the definition of a line or a point, if considered to be Hypotheses, may be ranged under this head. But * Jevons : Principles of Science ; ch. xxiii. 2 t o Theory of Logic. Prof. Bain * hesitates to call these abstractions Hypo- theses ; and they differ from other Hypotheses in not being at all doubtful ; for in reality they are certainly false, and in ideality certainly true. However they agree with Hypotheses in this that they are something laid down to be argued upon. 4. A false Hypothesis may be assumed as a means to the indirect proof of a true one, when direct proof is not attainable. Thus if we cannot directly prove a line to be equal to another, we may assume it to be either greater or less, and by disproving both of these false Hypotheses, show the necessity of the true one. It is in the interpretation of false Hypotheses that rules of procedure are especially needed, since there may be nothing else to guide us, and we often have to conduct an argument repugnant at every step to our plainest intuitions. 2. Hypothetical Relationship. A Hypothetical Relationship is interpreted as if it were the truth and the whole truth ; as if the explicit data were exact and exhaustive. Here the Rules of Husbandry and Continence are particularly to be borne in mind : it is required to find how to deduce from the Hypothesis all that it contains, and to assume nothing that it does not contain. * Logic : Book III. ch. xiii. 6. Of Hypothetical. 211 And Hypothetical Relationships, corresponding-, or professing to correspond, with matter of fact, must admit of the same analysis. They have accordingly been reduced to two forms in one or other of which any matter-of-fact may likewise be expressed namely, the Conditional and the Disjunctive; for everything is conditioned, and to everything there is an alter- native. 3. Of Conditionals. The word Condition is most strongly associated with Causation, but may be applied to any Term which is a mark of another ; and a Term may be the mark of another by constant Relationship either of Causation or of Coexistence. Including Causal Rela- tion and Coexistence under the single name Concomi- tance, we may define the Condition of a Term to be any constant Concomitant. Constant Concomitance is either Perfect or Imper- fect. Two Terms are perfectly concomitant if they occur together, and neither ever occurs without the other : imperfectly concomitant, if one never occurs without the other, but the other sometimes occurs without the first. Perfect Concomitance is represented by the Relation of Cause and Effect where there is no vicarious Cause, Thus in deducing the Experimental Methods, we were able to write (on the supposition that there were no vicarious Causes) : 212 Theory of Logic. 1 . If E is present C is present. 2. If C is absent E is absent. 3. If C is present E is present. 4. If E is absent C is absent. Similarly if two Terms are perfectly concomitant by Coexistence, as appears to be the case with Gravity and Inertia, we may write 1 . If G is present I is present. 2. If I G 3. If G is absent I is absent. 4. If I G And this thorough going Concomitance is similar to the coincidence of Classes symbolised by A 2 ; or rather it is that coincidence, since the coincidence of Classes depends upon the coextension of qualities. Thus we write All gravitating bodies are all inert. Imperfect Concomitance on the other hand is repre- sented by the Relation of Cause and Effect, if there are vicarious Causes. For then we may write 1 . If C is present, E is present. 2. If E is absent, C is absent. But we cannot write with certainty 3. If E is present, C is present. 4. If C is absent, E is absent. For in either of these cases E may be present in concomitance with the vicarious Cause. Of Hypothetical. 213 And again if two Terms be imperfectly concomitant by Cbexistence, as appears to be the case with Inertia and Extension, we may write 1 . If I is present, E is present. 2. If E is absent, I is absent. But we cannot write 3. If E is present, I is present. 4. If I is absent, E is absent. For Space is regarded as Extension without Inertia, that being its difference from Matter. And this im- perfect Concomitance is similar to the Relationship of Classes symbolised by A :* if the constituency of one Class include the members of a second and other Terms besides ; it is because the attribution of the former Class always accompanies the attribution of the latter, and is sometimes found without it. Thus we may write All inert entities are (some) extended. 4. Of Disjunctives. Hypothetic Alteruternity is called Disjunction. A Disjunction, then, may be Perfect or Imperfect, according as the Alteruternity is perfect or imper- fect ; that is, according as the Terms are or are not mutually exclusive. * Bain : Book I. ch. iii. 31 (Logic). 214 Theory of Logic. A perfect Disjunction is given in every case by Class and Counter Class, and so in every clear Division by Dichotomy. If we knew of any Genus that it contained only two Species, and that these were mutually exclusive, we should know that any mem- ber of the Genus was included in one or the other, and that no member was included in both. Any member of G is either S or CS (Counter Species). We may then write 1. If G is S, it is not CS. 2. If G CS, S. 3. If G is not S, it is CS. 4. IfG CS, S. This, it will be observed, is equivalent to simple Obversion ; and if, instead of two alternatives, we have three or more, it makes no real difference. Any G is either S, or 28, or R (Remainder). i. If G is either S or 28, it is not R ; G is either S or 28 : If G is S, it is not 28 : and so on, by a sort of inverse Dichotomy. Similarly we may have a Hypothesis concerning Cause and Effect : The Effect of C is either E or F, and not both : and this too has four forms. An imperfect Disjunction, on the other hand, is Of Hypothetical*. 2 1 5 given in every Division that does not run clear. Such a Disjunction has only two forms. If the Hypothesis be that animals live either on land or in the water, we may write 1 . If A is not L, it is \V 2. If A W, L: but not 3. If A is L, it is not W 4. IfA,,W, L; for some animals are amphibious. I may add an Obverse equivalent of a sound Dis- junction whether perfect or imperfect : Any G is either S or 28 .-. No G is /(S and 28). If this Obverse is not true, on account of a Re- mainder, the original Disjunction is inadequate. 5. Probation of Hypothetical*. These hypothetical forms may convey either known truths (or untruths) or Hypotheses properly so-called, that is, cases for probation. In either contingency they have to be interpreted. And if a known fact is stated in hypothetic form if in saying If A is, B is, we mean the only doubtful point to be the existence of A at any time ; the interpretation of the Relation 2 1 6 Theory of Logic. is all that concerns us. But if there be any doubt as to the constancy of the Relation A : B, we interpret the Relation chiefly for the sake of discovering the possible modes of testing it. If now a Conditional be given in which the only doubtful element is the occurrence of the conditions ; then, if we know the Concomitance of the Terms to be perfect, no antiquity of Logical custom to the contrary, ought to prevent us from availing ourselves of that knowledge, and interpreting the Relationship, as above, in all four ways ; for this is according to the Rule of Husbandry. But if we know the Con- comitance to be imperfect, or only do not know it to be perfect; then, according to the Rule of Conti- nence, we must only regard it as having two forms. Similarly if we know a Disjunction to be perfect, the four forms may be accepted ; but if we know it to be imperfect, or do not know it to be perfect, we can only accept two. On the other hand, I conceive, if we are interpreting a Hypothetical of any kind for the sake of probation, the Rule of Husbandry directs that, whether given as perfect, or not, we should interpret and try it in all four aspects ; for the Hypothesis may be better than its promise. Thus a Hypothetical Relation of any kind, supposed perfect, may prove imperfect ; or, supposed imperfect, may prove perfect ; or, supposed true, may prove false ; or, supposed false, may prove true. Of Hypotheticals. 1 1 7 And with regard to Disjunctives in particular, a kind of falsehood to be carefully guarded against is inadequacy ; where a Division is not exhaustive ; as when two Species are given as together coinciding with a Genus, though in fact there is a Remainder. After interpretation, the actual probation of Hypo- theticals is of course conducted according to the means appropriate in each case, whether the Relation involved be of Succession or Coexistence. All Hypotheses may, I think, be reduced to one or other of the forms here discussed ; and we have seen Disjunctives readily take the form of Conditionals, without however changing their real nature. For a Conditional is essentially a Hypothesis concerning Concomitance or Nonconcomitance as simple Rela- tions ; a Disjunctive, a Hypothesis concerning Con- comitance or Nonconcomitance as alternative Rela- tions: of which facts, as of interlacing fibres, the whole tree of Logic is compacted, of one substance in root and leaf. The Dilemma, compounded of a Conditional and a Disjunctive, involves no principle peculiar to itself, and needs not be discussed here ; though requiring like all hypothetical forms careful treatment in a work on Rhetoric. CHAPTER IX. OF THE MEDIATE RELATIONSHIP OF CLASSES. i . The Question stated. ALTHOUGH every known Relation of Classes with respect to Inclusion and Exclusion may be regarded as Immediate; it may happen that the Relation of certain Classes to one another is to be most readily discovered not by direct comparison of these Classes among themselves, but by the interference of some other Class to which their Relations are already known. These are cases in which, the Relations of two or more Classes to a third being known, we have proof of their Relations to one another ; and Classes thus mediately compared may be regarded as medi- ately related. Any number of Classes may stand to one another in Mediate Relationship; a Relation between two Classes may be proved to obtain by the intervention of one other Class or of many ; but it is usual to discuss the subject chiefly with regard to the Rela- tionship of three Classes ; this is, the doctrine of the Syllogism, or, as we may call it, Mediate Subsump- Of the Mediate Relationship of Classes. 219 tion. The Relations of two Classes to a third being given, it is required to find their Relations to one another ; or if there are cases where the Relations of two Classes to a third do not show the Relation of those two Classes to one another, we have to deter- mine what those cases are. This problem of the three Classes is the only problem as to the Mediate Relationship of Classes which it is necessary to treat of at length : since the consideration of more than three Classes presents no novelty of principle. 2. Definitions. Of the three Classes 1. That to which the Relations of the other two are already known is called the Middle Class. 2. Those between which a Relation is to be dis- covered by the intervention of the Middle, may be called the Outer Classes. 3. Possible Modes of combining the Unidesignate Relations of Two Classes to a Third* There are six different modes of Unidesignate Relationship between the Middle and one Outer Class. * Cf. De Morgan : First Notions of Logic. 220 Theory of Logic. 1. All K is M. 2 . Some A is M. 3. No A is M. 4. Some A is not M. 5. All^li is A. 6. $0#2 M is not A. Two other Relations verbally different Some M is A No M is A are the same as the second and third cases. And similarly to the other Outer the Middle may be related also in six ways. And since in comparing together the Outers by means of the Middle, any mode in which the Middle can be related to one Outer, may be combined with any mode in which it can be related to the other ; there are in all thirty-six possible ways in which the Relations of a Middle to two Outers may be formulated. But of these thirty- six modes, fifteen are merely superfluous repetitions of some of the others ; so that there are only twenty- one really different ways of stating the Relations of two Classes to a third. It will be found that only ten of the twenty-one com- binations prove direct Relations to subsist between the Outer Classes. Eleven combinations remain : of which three yield evidence of indirect Relations between the Outers, that is, of Relations between their Counter Classes; and eight are altogether inconclusive. Of the Mediate Relationship of Classes. 221 We have to examine the nature both of those com- binations which are forms of Proof, and of those which prove nothing. 4 . Conditions of Mediation. The Relations of Classes whether Mediate or Immediate may be viewed either as to attribution or as to constituency; and it does not matter in which way, since we have seen that one aspect of a Relation is a constant mark of the other. If we view the Relation of two Classes as a Relation of their constituencies, to say that one includes the other is to say that their constituents are (part or all of them) the same Terms ; to say that one excludes the other, is to say that their constituents are (part or all of them) not the same Terms. And to prove such Relations not by a direct examination of the Terms, but by comparison with a third Class, is only possible if the known Relations of the two Classes to the third are such as to show, i, where the Relation to be established is Inclusive, that certain Terms of the Middle are Terms of both the Outer Classes ; or, 2, where the Relation to be established is Exclusive, that certain constituents of the Middle are con- stituents of one of the Outer Classes, and not of the other. Thus in every case in which an Inclusive Relationship is mediately proved, one of the Outer Classes is given as totally including the Middle, and 222 Theory of Logic. the other Outer as also wholly or partially including it, or included in it; that is, all the constituents of the Middle are constituents of one Outer, and some at least are constituents of the other. And so on. 5. Axioms of Mediate Subsumption. Are there any Axioms that generalize the con- ditions of Mediate Subsumption ; and, if so, what are they ? This question as to the presiding Axiom of the Syllogism, has lately been much debated. It had been the usual practice of Logicians to affirm that the Axiom of all Syllogistic reasoning was the famous Dictum : some, however, of whom Kant * was the greatest name, held that the true Axiom was the Nota notae. According to Hamilton,t the Dichim was the peculiar canon of Extensive Syllogisms (Mediate Relation of Classes viewed in their consti- tuencies) ; the Nota notae was the peculiar canon of Intensive Syllogisms (Mediate Relation of Classes viewed in their attributions). Mill rejected altogether the Dictum, on the ground that it begged the question ; and proposed instead Axioms closely resembling the Nota notae, namely : i. "Things which coexist with the same thing coexist with one another: or (still more pre- cisely)" as he observes in his latest Editions * Logik : Allg. Elementarlehre, 93. t Logic : Lecture XVI. Of the Mediate Relationship of Classes. 223 "a thing which coexists with another thing, which other coexists with a third thing, also coexists with that third thing. 2. "A thing which coexists with another thing, with which other thing a third thing does not coexist, is not coexistent with that third thing."* Similar Axioms to these we have already recog- nized as formulating certain modes of Triterminal Correlation ; we noticed, too, the limitations with which they were to be understood ; and we observed that the Relations of Classes were not governed by these laws.f Prof. Bain, again, departing from Mill at this point, apparently prefers to fall back upon the Dictum, only amending it so as to fence it against the imputation of begging the question. His amended statement of it reads : "Whatever is true of a whole class (class inde- finate, fixed by connotation), is true of whatever thing can be affirmed to come under, or belong to, the class (as ascertained by connotation)." J As long as we regard the Syllogism as a Relation- ship of three Classes, the chief objection, from the point of view of this Essay, to the Dictum as worded by Prof. Bain, is that it contains allusions to the theory of Names and Predication, which we regard * System of Logic : Book II. ch. ii. 3. t Ante, ch. iv. part ii., 4. Bain : Logic ; Book II. ch. i. II. 224 Theory of Logic. as belonging to Rhetoric. We are thus again driven to find new statements for old principles; and accord- ingly propose the following Axioms, under which all conclusive cases of the Mediate Relationship of Classes regarded as matter-of-fact may readily be brought. And first, with regard to Classes viewed as to their constituencies Axioms of Constituent Mediate Relationship. 1 . Inclusion : A. Class that includes a second Class, that includes a third, itself includes the third in so far as the third is included in the second.* 2. Exclusion: A Class (or Part-Class) that excludes a Class, that includes a third Class, itself ex- cludes the third Class, in so far as the third is included in the second. A moment's consideration will show the resemblance between these Axioms and the Dictum in its old form. We may write the Dictum thus : Whatever is aifirmed (or denied) of a Class, is affirmed (or denied) of every part of it. But that which is affirmed of a Class is always an attribute ; and every attribute is the basis of a Class. To say ' whatever is affirmed of a Class/ then, amounts to saying * whatever Class includes another Class ; ' and the whole Dictum amounts to this : A Class that includes a Class, includes every part of it. And * Cf. Leibnitz : Definitiones Logicae ; 12, c. Of the Mediate Relationship of Classes. 225 either this ' part ' is specified (marked with specific attributes), and therefore itself a third Class, or repre- sentative of a third Class ; or else, if it is not specified there is no third Class, and no real mediation. But, again, as every attribute is the basis of a Class, so every Class is based upon attributes; a Term or Class can be included in two or more dif- ferent Classes only by realizing their respective attri- butions ; and if it is excluded from any Class, it is for not possessing the requisite qualities. The Inclusion and Exclusion of Classes is, as we often remarked, equivalent to the Concomitance and Nonconcomitance of qualities. We may therefore rewrite the above Axioms in forms better agreeing with Prof. Bain's amended statement of the Dictum. \ Axiom of Attributional Mediate Relationship. 1. Inclusion : A Class whose attribution is in- cluded in the attribution of a second Class, whose attribution is realized in the constituents of a third Class, or in some of them, includes those constituents of the third Class. 2. Exclusion : If the constituents of a Class (or some of them) do not realize the attribution of a second Class, whose .attribution is realized by the constituents of a third Class (or by some of them) ; the constituents of the first and third Classes (or some of them) are not identical. 226 Theory of Logic. 6. Cautions as to Mediate Subsumption. In their quantitative form it was remarked, the Axioms given above resemble the Dictum; and we may deduce from them Cautions of valid Mediation similar to those usually deduced from the Dictum : or the Cautions may be viewed as flowing, like the Axioms, from the nature of the Middle as a mediating Class. 1 . If we are comparing two Classes with a third, we must preserve the identity of the three Classes severally throughout the comparison, or there is no real comparison at all. 2. The Middle must be given as totally related to at least one of the Outer Classes (or to part of one). For else with the vague designation of Logical Relationships, we have no assurance that we are comparing the Outers with the same part of the Middle. 3. No Class not totally related in the premises can be shown to be totally related in the conclusion. For the Middle cannot transfer to one Outer more of the other than itself contains. 4. Where both premises are Exclusive Relations, no direct Relation between the Outers can be proved : for there is no direct mediation. But there may be evidence of some Relation between the Counter Classes. 5. If one premise be an Exclusive Relation the Of the Mediate Relationship of Classes. 227 conclusion must be an Exclusive Relation : for we cannot know that the Middle contains any part of one Outer to transfer to the other. Corollaries: i. Two premises of Partial Relation prove no (Unidesignate) Relation. 2. If one premise be a Partial Relation the conclusion must be Partial. 7. Ten Modes in which the Relations of Two Classes to a Third may prove something as to their Direct Relation to one another. a. Eight with Unidesignate Conclusions ; namely, (a). Three Inclusive.. i. Where the Middle totally includes one Outer, and is totally included in the other. is V: 2. Where the Middle is totally included in both the Outer Classes. AH Mis A; A II Mis V: .'. Some A is V. 3. Where the Middle is totally included in one Outer, and partially in the other. All M is A ; Some M is V : . . Some A is V. (b). Five Exclusive. Q 2 228 Theory of Logic. 4. Where the Middle totally excludes one Outer, and totally includes the other. No AisM; 5. Where the Middle totally excludes one Outer, and is totally included in the other. No AisM; All Mis V: .*. Some V is not A. 6. Where the Middle totally excludes one Outer, and partially includes the other. No A is M ; Some V is M : .;. Some V is not A. 7. Where the Middle totally includes one Outer, and partially excludes the other. All A is M ; Some V is not M : .. Some V is not A. 8. Where the Middle is totally included in one Outer, and partially excluded by the other. All M is A ; Some M is not V : .*. Some A is not V. ft. Two with Bidesignate Conclusions. 9. Where the Middle is totally included in one Outer, and partially excludes the other. All M is A ; Some V is not M : .;. Some V is not some A. Of the Mediate Relationship of Classes. 229 10. Where the Middle partially includes one Outer, and partially excludes the other. Some A is M ; Some V is not M : ..!. Some A is not some V. 8. Reduction of Irregular Cases. Four of these ten cases, or Moods, come readily enough under the Axioms : namely, No. i and No. 3 manifestly realize the Axiom of Inclusion ; and No. 4 and No. 6, the Axiom of Exclusion. These may be called Regular, the others Irregular. The agree- ment of the Irregular cases with the Axioms may be shown in two ways. First, we may deduce from the Axioms secondary principles for immediate application to the Irregular cases. For instance, from the Axiom of Inclusion we may deduce the principle : Classes including the same Class include part of one another : and this applies directly to No. 2. And so from the Axiom of Exclusion we may deduce the principle : A Class that excludes a Class included in a third, partially at least excludes the third : and this applies directly to No. 5. And similarly the other cases may be treated. Or, secondly, we may adopt the inverse process, and reduce the Irregular cases to forms, in which they better suit the Axioms ; as in the Scholastic 230 Theory of Logic. Logic the Moods of all other Figures are reduced to the Moods of the First. I need not reduce No. 2 and No. 5, but will merely treat the remainder. No. 7 may be written Some V is not M = No M is some V ; that is, a part of a Class (Some V) excludes a Class (M), which includes a third class (A) ; therefore, according to the Axiom of Exclusion, Some V is not A. No. & will be recognized as Baroko, and may be reduced in two ways, besides the old reductio ad impossibile, which was not, properly speaking, a reduction at all. We may reduce it under the Axiom of Exclusion by regarding Some M as a whole Class, thus : V excludes Some M ' ; and that Some M (all M] includes Some A ; &c. Or we may bring it under the Axiom of Inclusion by obverting one premise : AllM. is A; Some M is not V = Some M is / V : .'. Some A is /V = Some A is not V. No. 9 is reducible to No. 10 by converting one premise : All M is A Some A is M. No. 10 is reducible by converting one premise: Some V is not M = No M is some V ; Some A. is M : Of the Mediate Relationship of Classes, 231 that is, a Part-Class (Some V] excludes a class that includes part of another class (Some A}; there- fore, according to the Axiom of Exclusion, Some V is not some A = Some A is not some V. These Bidesignate conclusions from Unidesignate premises, have not been usually recognized ; and the information they afford is, to be sure, very meagre : but I remember the Rule of Husbandry, and am unwilling to let the smallest grain of knowledge slip through a crack in the threshing floor. The Relation Some A is not some V (O a ) is compatible even with A 2 ; since if A 3 obtain, O 2 will mean Some A is not some A, or Some V is not some V. Still we may learn this from it : either A and V do not wholly coincide ; or, if they do, the Class (A or V) is divisible. If A and V be one Class, we know that some of it is M, and some is not : and this must result from some observable difference in the qualities of its members, and may be a hint toward developing a classification. The reason why two partial Relations, one of which is exclusive, may yield a conclusion ; but not if both be inclusive ; is that in the former case the Middle may be given as totally related (Caution 2} : Some V is not M = No M is some V. Even so, however, a conclusion is only obtained by treating Part-Classes as wholes. 232 Theory of Logic. 9. Three modes in which the Relations of Two Classes to a Third prove something as to the Relations of their Counter Classes. 1. Where the Middle wholly excludes both Outer Classes. No A is M ; No V is M : . . Some /A is /V. 2. Where the Middle is totally excluded from one Outer and partially from the other. No M is A ; Some M is not V : .-. Some /V is /A. 3. Where the Middle wholly includes both the Outer Classes. .-. Some /A is /V. 10. Reduction of Obverse Cases. The Axioms of the Mediate Subsumption of Classes apply equally to the Mediate Subsumption of Counter Classes, or of mixed Classes and Counter Classes. No. i is reducible to No. 2 Direct : No AisM = Some I A. is /V. No. 2 is reducible to No. 3 Direct : A^Mis A = ^4//Mis/A; Some M is not V = Some M is /V : .: . Some /V is /A. Of the Mediate Relationship of Classes. 233 Here the least quantity of Counter Class common to A and V must be, in No. i, All M, in No. 2, Some M. In the third case the common Counter Class must be at least /M. No. 3 is reducible to No. 5 Direct : M = No Ais/M; .'. Some /V is not A Some /V is /A. It will be observed that by making free use of Obversion, the Axiom of Exclusion may be reduced to that of Inclusion, or the Axiom of Inclusion to that of Exclusion. But this would be no real simpli- fication ; and would in fact increase the trouble of reduction, by rendering necessary more complicated manoeuvres with the machinery of equivalence. 1 1 . Eight Cases in which the Relations of Two Classes to a Third prove nothing as to their Relations to one another. i . Where the Middle includes one Outer totally and the other partially. All A is M ; Some V is M. 2. Where the Middle includes both Outers partially. Some A is M ; Some V is M. 3. Where the Middle totally includes one Outer Class, and is partially excluded by the other. All A is M ; Some M is not V. 234 Theory of Logic. 4. Where the Middle partially includes one Outer, and is partially excluded by the other. Some A is M ; Some M is not V. 5. Where the Middle excludes one Outer totally, and the other partially. No A is M ; Some V is not M. 6. Where the Middle partially excludes both Outers. Some A is not M ; Some V is not M. 7. Where the Middle partially excludes one Outer, and is partially excluded by the other. Some A is not M ; Some M is not V. 8. Where the Middle is partially excluded from both Outers. Some M is not A ; Some M is not V. These eight cases yielding no conclusion are dis- tinguished from the others by this, that the premises admit of all possible modes of Relationship obtaining between the Outer Classes. A may totally include V ; or V, A ; or they may totally exclude one another : and their Counter Classes too may be similarly related in every possible way. But this is not true of those combinations of premises that give conclusions : in them some particular Relation must obtain between the Outers (or parts of them regarded as wholes), or between their Counter Classes ; excluding of course in each case the incompatible Relation. Of the Mediate Relationship of Classes, 235 12. Mediation of Bidesignates. The possible combinations among the Bidesignate Relations of two Classes to a third are sixty-four. Subtracting twenty-eight, which are repetitions of others, there remain thirty-six. These I have cur- sorily examined with a view to sorting them, but need not give the results at length. There appear to be twenty combinations that yield direct con- clusions : seven Inclusive, and thirteen Exclusive. Three give indirect conclusions as to the Relations of the Counter Classes ; and thirteen prove nothing at all. In this computation Bidesignates are regarded as detached from the restrictions of Genus and Species, so that in their designation Some means Some at least. We might further consider the possible combina- tions of Unidesignate with Bidesignate premises. 13. Mood and Figure. The Moods and Figures of Scholastic Logic may, if it appear desirable, be replaced in this system by some such classification as the following : 236 pUnidesig- nate Theory of Logic. Inclusive 2. Exclusive 3. -Direct . plnclusive i. -Regular . -Irregular -Bidesignate c. pUnidesignate 3. l-Bidesignate 2. -Indirect 3. 14. Mediation of Hypothetical* : Cases of Mediate Subsumption may occur in which one or more of the Classes compared is affected by an Hypothesis. If A is B, CisM; ^//MisY: .-. If A is B, Cis V. AisM, ifMisN; All Mis V: .-. IfMisN, A is V. And so on. Such cases come at once under the Axioms, but for the hypothetic element; and this should, I conceive, be regarded as something quite Of the Mediate Relationship of Classes. 237 extraneous ; since it reappears in the conclusion in the same form as in the premises, having been altogether unaltered in the process. Similarly we may have cases involving Disjunctives. Either A, or B, or C, is M ; All M is V : .*. Either A, or B, or C, is V. No A is either M or N ; All V is either M or N : No A is V. In the former of these cases the hypothesis is extraneous : in the latter it is a means of mediation. And to bring this second case under the Axiom, we must regard the Disjunctives, either M or N, as together forming a whole ; just as we previously regarded a Part- Class as forming a whole, when drawing a bidesignate conclusion from unidesignate premises. Suppose, for instance, that S and 28 are the only Species of a given Genus, without Re- mainder : No A is either S or 28 = No A is G. No A is either M or N = No A is (M and N). 15. Sorites. An unknown Relation between two Classes may also be discovered and proved by the intervention of more than one Middle. is V .-. AHAisV. 238 Theory of Logic. Such cases are called Sorites. In the above instance there are two Middles ; and a new Middle would be added with every further step. How many steps a Sorites extends does not matter as long as the sanction of the Axioms of Mediate Relation is retained. We might indeed frame special Axioms of Sorites, such as these : 1. A Term or Class subsumed under a second Class, is subsumed under as many Classes as the second Class is subsumed under. 2. A term or Class subsumed under a second Class, is not subsumed under any Class which is ex- cluded either by the second Class or by any Class under which the second Class is subsumed. There would be corresponding Axioms of the Pro- gressive Sorites. And these Axioms might sometimes be useful : but for safety it is better to break up a Sorites to which any suspicion attaches into links of three Classes, to which the Axioms of the Syllogism may be directly applied. Thus to set aside all doubt whether in the above instance A is V, we may proceed in this way : All A is M ; AMU is 2 M .-. All A is 2 M. 'is 2M; AllzUis V .-. AM A is V. A Sorites, in fact, contains as many cases of Mediate Relation as Middle Classes. Accordingly it is subject throughout to the Rules and Cautions of valid Mediation. If it contains more than one ex- Of the Mediate Relationship of Classes. 239 elusive or partial Relation, however far apart they may be, the evidence is vitiated as if there were only three Classes to be considered, All A is B ; No B is C ; All C is D ; No D is E. Breaking up this chain we get All A is B ; No B is C .-. No A is C No A is C ; All C is D .-. Some D is not A Some D is not A ; No D is E .-. Some /A is /E. And similarly with other occasions of error; the chain may only attenuate, or may quite break in pieces : we must look to the unity and total Relation of each Middle ; and so on. 1 6. How many Terms has a Syllogism? Perhaps in the seventh Chapter and the present one I have sometimes seemed to be forgetting whilst dealing with Classes the speculations of the earlier parts of the book: but I hope that the unity of the whole inquiry will become apparent in the course of the investigation upon which I now enter. It has always been regarded as an unquestionable maxim of Logic that a Syllogism must have three Terms. Both the Dictum and Mill's Axioms assume this: the Terms intended in the former case bejng Classes ; and in the latter case, Attributes. And it lies on the face of the Axioms of the Mediate Relationship of 240 Theory of Logic. Classes brought forward in 5 of this Chapter, that, if by a Term be meant an explicit Class, a Syllogism is supposed to have three Terms. In the fourth Chapter, however, I seemed to adopt certain views of Mr. Spencer's which are set by him in opposition both to the Scholastic account of the Syllogism and to Mill's doctrine ; and of which perhaps the most startling is, that a Syllogism has four Terms : so that I may now appear to be landed in a contradiction. But the truth is that Classes are seldom Terms of the same kind as those of which we treated in the fourth Chapter. * In dealing with Classes as we have lately been doing, we resort to an artifice, an abbreviated mode of expression, which we are liable to pay for by sublation of thought. If throwing away the clogs of language we get our own feet upon the facts, and explore once more the actual Correlations of pheno- mena, we shall probably perceive that a Syllogism comprises more than three Terms, and even more than four. To take an example : how many Terms has this Syllogism ? Men are mortal ; Greeks are men : Greeks are mortal. According to the old view, there are three Terms, Greeks, Men, Mortals, * Cf. ch. vi. 32. Of the Mediate Relationship of Classes. 241 or in comprehension, Mortality, Humanity, Hellenicity : and either way the three Terms slide one into the other, as one shuts up a telescope. According to Mill's Axiom, the Correlation might be symbolized thus : Humanity co Hellenicity , Mortality This, however, does not represent a Relationship of Classes at all ; but only the Concomitance of certain three qualities in the members of one Class, namely Greeks. For Hellenicity is not concomitant with all Humanity, nor Humanity with all Mortality. The evidence thus adduced for the mortality of Greeks is, the mortality of Greeks and no more: but much more is intended when it is argued that Greeks are mortal, because all men are. To rely on Mill's Axiom is to lose all that evidence of the mortality of Greeks which is derived from the mortality of the rest of mankind. So far then I agree with Mr. Spencer that Mill's view is insufficient : but I cannot assent to the view which he appears to take, that the symbol of Quad- 242 Theory of Logic. riterminal Correlation adequately represents the Cor- relation formulated in a Syllogism. Mankind > f Certain men unspecified. Mortality ) \ Mortality. This, it seems to me, is all than can fairly be got into a Symbol of Quadriterminal Correlation, and this represents a Relation of qualities in the members of only two Classes (Humanity and Mortality), not of three a single Subsumption, not a double and Mediate Subsumption. The differential nature of Greeks is here omitted ; wherein perhaps there may lurk something incompatible with Mortality. The Correlation formulated in a Syllogism, therefore, must be represented as Quinqueterminal FIRST SUBSUMPTION Humanity N C/) in general M O V , 2! f '', D W e ^ ^ { G 8 G \ r g r 3 O 2 f Humanity cfl M o 2 D C/3 C ( e ~ a \ G 1 i s > 55 "l\/T/-v*-4-o 1 1<- / Hellenicity Of the Mediate Relationship of Classes. 243 And Quinqueterminal Correlation, it will be noticed, is a union of Quadriterminal and Triterminal Correla- tions. The above Syllogism then really comprises five Terms : 1. Hellenicity. 2. Hellenic Humanity. 3. Mortality of Hellenic Humanity. 4. Non-Hellenic Humanity. 5. Mortality of Non-Hellenic Humanity. Thus we see that in the Axioms of the Syllogism as above stated the three Classes spoken of are, two of them (Humanity and Mortality) divisible each into two portions (Hellenic and Non-Hellenic) ; and one of the two (Mortality) contains a third portion, namely, Non-human Mortality, which is not a Term of the Syllogism. And it may contribute to the right understanding of Logic, as well as to the uni- formity of its formulae (which is a test of truth), if we write the Axiom of the Syllogism thus : Rule of Quinqueterminal Correlation, A Term that coexists with a second Term, that second Term and a third being severally the same as a fourth and a fifth Term, which are related to one another by Co-existence or Succession, is related to the third Term, as the fourth to the fifth, and as the second to the third. R 2 244 Theory of Logic. For that the Rule and its equivalent Axioms apply to Classes of Causal Instances as well as to Kinds will be apparent to anyone who contemplates this symbol : Metal ^ heated. FIRST SUBSUMPTION 1 Metal to heated Expansion/ a Differentia of Iron \ Expansion and observes that this is as much as to say, Ex- panded bodies include heated metals, which include heated iron. 1 7 . Table of the Modes of Implication. We see, then, that there are four principal kinds of the Implication of Relations four modes of Correla- tion in which Relations that are explicit, imply and prove Relations that are not explicit. Let us exhibit this in a Table : Of the Mediate Relationship of Classes. 245 [-Immediate ^-Mediate Biterminal Correlations (Doubly Conjunct). Triterminal Correlations (Singly Conjunct). Quadriterminal Correlations (Disjunct). Quinqueterminal Correlations (Disjunct and Conjunct). * 2. 3 ~ 3 S. 1 o 5- g B 3 P *p r* S & The first three modes appear to be elementary and irreducible: The fourth mode is compounded of the second and third ; but cannot, I think be reduced to them without loss ; all other compound modes as far as I have examined them, are easily reducible and do not need separate discussion. 1 8. Classification. In discussing questions of Mediate Relationship we have now compared Relations of all kinds except those of Genus and Species. The extension of the doctrine of Genus and Species to the subordinate 246 Theory of Logic. Relations of more than two Classes, leads to the con- sideration of grades of Classification higher than the lowest Genus, in which the lowest Genus may be itself contained. It has been impossible to keep such considerations hitherto entirely out of sight ; but this seems to be the best place to bring them explicitly forward. Between a Summum Genus and an Infima Species there may exist a gradation of Classes of unknown extent. So many attributes as a Class has, so many grades of Classification may stand above it : each attribute being in turn the difference of a grade. From the most to the least general grade, at each step downward in the scale, the attributions of Classes increase while their constituencies diminish ; and the subdivision may be continued as long as any tubfK discernible difference remains. The Classifications investigated in Botany, &c., aim at exhibiting this order as it exists in Nature : and certain expressions Of the Mediate Relationship of Classes. 247 have been appropriated to denote the successive stages of decreasing generality, as, for instance Kingdom, Order, Family, Tribe : reserving the words Genus and Species for the last two steps in the descent. The above diagram represents an irregular Classi- fication in four grades. The capital letters stand for Classes, and each for the difference of its Class ; the small letters are for the other attributes. A might be called an Order ; B and C Tribes ; D, E, F, G, H, Genera; K, L, &c., Species. But for the purposes of Logic it is usual to speak of Genus and Species only ; to make these names moveable up and down the scale, and relative only to one another. As we cannot know beforehand how many grades of Classification may exist in Nature, nor can we devise beforehand a suitable terminology, this is a matter for special Science. In Logic, which with regard to special matter is a Science a priori, it is usual to say, with reference to the diagram, for instance, that K and L are Species and as far as appears Species only ; that F (to follow this line) is the Genus of K, and a Species of B, coordinate with D and F; that B is the Genus of F, &c., and a Species of A coordinate with C ; and that A is the Genus of B and C, and so far as appears a Genus only. And perhaps this custom is, on the whole the best ; or at most it might be an allowable innovation to add to the technical terms of general Logic such a 248 Theory of Logic. word as Tribe, in order to designate by appropriate names three Classes (not necessarily Natural Classes) in successive inclusion Tribe, Genus, Species. The Relations of Tribe and Genus, being the same as those of Genus and Species, do not need particular investigation. Plainly now a Natural Classification is a vast Logic- machine ; exhibiting in the most definite way, at a glance, the Inclusions and Exclusions of all Classes both Immediate and Mediate. A complete Classifi- cation would have a place for every thing and every event in the world, according to its closest affinities, displaying the whole hierarchies of Natural Kinds and Causes. And so far as this Classification extended, the labour of proof as to the Relations of Classes, having been accomplished once for all, would ever after be superseded by a glance at the Tables. There would be seen the inclusion of a Species in the Tribe which included its Genus ; the exclusion of a Species from another Species having a different Genus : Sorites would be given along all lines from the Species upward above the Tribes. The notion of the Counter Class, too, attains its greatest clear- ness in this connection by the facilities afforded for defining it. It would naturally in every case not expressly excepted, be circumscribed by the Class next above its Positive : taking any Species, the special Counter Class would be the sum of the other Species of the same Genus; the generic Of the Mediate Relationship of Classes. 249 Counter Class, the sum of the other Genera of the same Tribe. Thus a Natural Classification is a sort of solidified Logic ; and perhaps the best way to begin the study of the Science, is to take some good Classification, and analyse it into the simplest Relations. CHAPTER X. OF SECONDARY RELATIONS. i. Symbols. RELATIONS are Terms ; and in as far as they are Terms only, they are related in the same way as others : and the laws of the Relationship of Terms in general, which we have discussed in the preceding Chapters, are in no way modified when the particular Terms concerned happen to be themselves Relations. There are, however, certain Relations of Relations which do not obtain in the same way between mere Terms that are not Relations. Let us call these Secondary Relations, and assign them symbols as follows : Coincidence i2 Noncoincidence These are the ordinary principles of indirect demon- stration * (instead of the first of them, the second prin- ciple given above under Implication may serve) : as when it is shown that the supposed equality of two lines implicates the equality of two angles, which is in perfect Alteruternity with the known inequality of those angles : or else that the known equality of two lines is incompatible with the supposed inequality of two angles, and therefore implicates the alteru- tern fact of their equality. Again, let the Counter of any Relation a be /0. a ^ (or ) b x c . * . ja A /c. And again, a ^f (or <) b A c . . la A c. * Cf. ante, Ch. IV. Part II. 6. 256 Theory of Logic. Sometimes a Correlation has more than one implication. All these formulae supply means of interpreting hypotheses. Suppose, for instance, we are given the following : A is either B or C, and not both ; and if it is C, it is neither E nor F ; but if it is not C, it is either E or F. This may be written out [A : B] x/r [A : C] ; [A : C] ^ [A : E and A : F] : .-. ist. [A : B] A [A : E or A : F] 2nd. /[A : B] A /[A : E and A : F] 3 rd. /[A : C] A [A : B L A : E or A : F] 4 th. [A : C] A [/(A : B) a /(A : E and A : F)]. And these implications may be interpreted : If A is B, it is either E or F. If A is not B, it is neither E nor F. If A is not C, it is B, and either E or F. If A is C, it is neither B, nor E, nor F. The question arises whether Correlations can be formed by the combination of Primary with Secondary Relations. And the answer is, that such combina- tions are possible as far as the two orders of Rela- Secondary Relations. 257 tions are homogeneous ; that is, in as far as Secondary Relations may be viewed as modes of Concomitance or Nonconcomitance. But we saw in 2 of this Chapter, that whilst Coincidence, Imme- diate Implication, and Compatibility, were modes of Concomitance, actual or possible, the rest were not necessarily modes of Nonconcomitance, having been defined with reference to Coincidence, and not to Concomitance in general. Hence the possible com- binations of Primary Relations with Secondary Incompatibility and Alteruternity, have no necessary implications. Should we, however, construct Primary Relations of Incompatibility and Alteruternity, defined with reference to Concomitance in general ; there would then arise (taking the above symbols in this altered sense) a number of implicative Correlations such as these : au>b\c.'.aoc a o b c .' . a oj c The above then seem to be the most important principles of Secondary Correlation. Many others might be suggested, some of them having implica- tions, and some not ; and other principles yet more remote and more complex may remain to be dis- covered : and there are perhaps still other directions in which the Science may be elaborated. For the 258 Theory of Logic. time one of its two larger branches has noticeably outgrown the other; the theory of Quadriterminal Correlation bears a great disproportion to the theory of Triterminal Correlation. This is because the former theory has assimilated the doctrine of Classes ; and at present it is certainly not easy to guess where the latter theory will find an equal store of prepared pabulum. But some conception no less rich may one day disclose itself; and the life of Science is as long as the pursuit of Science is difficult. THE END. LONDON I BKAUHUKV, AGSEW, & CO., 1'KINTERS, WHITEFRIARS. I, Paternoster Square, London. A LIST OF C. KEGAN PAUL AND CO.'S PUBLICATIONS. ABBEY (Henry). Ballads of Good Deeds, and Other Verses. 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