Please [landle this volume with care. he University of Connecticut libraries. Storrs BOOK 160.J639 v-1 =■ ' JOHNSON # LOGIC 3 T153 00004104 1 LOGIC PART I CAMBRIDGE UNIVERSITY PRESS C. F. CLAY, Manager LONDON : FETTER LANE, E.G. 4 NEW YORK : THE MACMILLAN CO. BOMBAY \ CALCUTTA I MACMILLAN AND CO., Ltd. MADRAS j TORONTO : THE MACMILLAN CO. OF CANADA, Ltd. TOKYO : MARUZEN-KABUSHIKI-KAISHA ALL RIGHTS RESERVED L O vj I C( :3(;3 PART I •I BY W. E. JOHNaON, M.A. FELLOW OF king's COLLEGE, CAMBRIDGE, SIDGWICK LECTURER IN MORAL SCIENCE IN THE UNIVERSITY OF CAMBRIDGE. TTANTA peT ei a^h TO AAweec CAMBRIDGE AT THE UNIVERSITY PRESS 1921 «MAN IS A RATIONAL ANIMAL' Definition PREFACE THE present work is intended to cover the whole field of Logic as ordinarily understood. It includes an outline of elementary Formal Logic, which should >' be read in close connection with Dr Keynes's classical — - work, in which the last word has been said on most ^ of the fundamental problems of the subject. As regards Material Logic, I have taken Mill's System of Logic as the first basis of discussion, which however is sub- /^v jected to important criticisms mostly on the lines of ^ the so-called conceptualist logicians. I have to express my great obligations to my former pupil, Miss Naomi Bentwich, without whose encourage- ^ ment and valuable assistance in the composition and •fr arrangement of the work, it would not have been pro- duced in its present form. W. E. J. March 30, 192 1. CONTENTS INTRODUCTION PAGE § I. Definition of Logic. Grounds for including the theory of induction . xiii § 2. Thinking includes perceptual judgments ...... xvii § 3. All ulterior motives of the pursuit of truth irrelevant to the analysis and criticism of thought .......... xvii § 4. Relation of Logic to the Art of Thinking . . . . . . xix § 5. Logic, Aesthetics and Ethics as the three normative studies . . . xx § 6. Relation of Logic to Universal Grammar ...... xxi § 7. Logic and Mathematics. Methodology. Pure and Applied Mathematics xxii § 8. Historical sketch of the problems connecting Logic with Philosophy and with Psychology. Realism, Conceptualism and Nominalism. Material- istic and Empirical Logic xxvii § 9. Special features of the author's treatment of Logic , . . xxxiii CHAPTER I THE PROPOSITION § I. Sentence, assertion and proposition ....... i § 2. Assertion as conscious belief ........ 4 § 3. Necessity for recognising the mental attitude in logic .... 6 § 4. Relation between grammatical and logical analysis .... 8 § 5. Connected functioning of substantive and adjective in the proposition . 9 § 6. Criticism of the view that the essential nature of the proposition is a statement of identity . . . . . . . . . .13 § 7. The proposition regarded as characterising the fact, and its analogies with the adjective 14 CHAPTER II THE PRIMITIVE PROPOSITION § I. Account of the psychologically primitive form of assertion — exclamatory or impersonal ......... . . 18 § 2. The need of separate presentment for the most elementary forms of judgment ............ 20 § 3. Criticism of Mr Bradley's dictum ' distinction implies difference ' . .22 § 4. Illustrations of less primitive forms of assertion 23 viii CONTENTS CHAPTER III COMPOUND PROPOSITIONS PAGE § I . Definition of compound and simple as applied to propositions . . 26 § 2. Distinction between the different forms of compound propositions; the logical conjunctions; distinction between the enumerative and conjunctive 'and' ■27 § 3. The first law of thought called 'The Law of Double Negation' . . 29 §4. The laws of conjunctive propositions ....... 29 § 5. The four forms of composite proposition and their immediate implications 30 § 6. Complementary and supplementary propositions and their rules . . 35 § 7. Criticism of the paradoxical forms of the composite, and preliminary explanation of the solution of the paradox ...... 38 § 8. The bearing of the distinction between hypothesis and assertion upon the paradox ........... 44 § 9. Interpretation of compound propositions as expressing possible conjunc- tives and necessary composites. Table of all the possible relations of one proposition to another ......... 47 CHAPTER IV SECONDARY PROPOSITIONS AND MODALITY § I. Definition of primary and secondary propositions and of pre-propositional adjectives ............ 50 § 2. Modal adjectives as a species of pre-propositional. The adjectives 'true' and ' false ' in this connection . . . . . . . • 5 1 § 3. The opposition of modal adjectives ....... 53 §4. Antithesis between 'certified' and 'uncertified' 55 § 5. The rule that holds universally between two antithetical modals . . 56 §6. Threefold meanings of ' necessary ' and ' possible ' .... 59 Antithesis of 'certified' and 'uncertified' as epistemic; antithesis of 'nomic' and 'contingent' as constitutive ...... 61 § 8. Comparison of Mill's distinction between verbal and real propositions, with Kant's distinction between analytic and synthetic propositions . 62 § 9. Summary of the treatment of modal adjectives in connection with secon- dary propositions .......... 65 CHAPTER V NEGATION § I . The nature of pure negation and its different degrees of significance . 66 § 2. The importance of negation determined by its relevance to a specific purpose ............ 69 § 3. Can there be a proposition where there is nothing corresponding to the subject-term? With illustrations of elementary forms of obversion and contradiction ........... 70 §4. An account of the incomplete proposition '5" is' .... 73 § 5. The hypothetical element in the proposition ' 6" is ' . . . . 77 §6. Summary: determinateness of fact contrasted with indeterminateness of knowledge 79 /• CONTENTS ix CHAPTER VI THE PROPER NAME AND THE ARTICLE PACK §1. Distinction between the proper and the descriptive name ... 80 § 2. Proper name explained in terms of the introductory and referential articles ............ 82 §3. Distinction between the connotation and the etymology of a name . 84 § 4. Distinctions between the four kinds of article or applicatives . ' . 85 § 5. The demonstrative applicatives ........ 88 § 6. Ostensive definition 89 CHAPTER Vn GENERAL NAMES, DEFINITION AND ANALYSIS §1. Connection between general names and the applicatives ... 97 § 2. Distinction between connotation and comprehension. Can adjectives be predicated of abstract names ? . . . . . . .100 § 3. The nature of the analysis involved in definition. Definition by means of substitution of phrase. The indefinable ...... 103 § 4. Analytic and synthetic modes of definition ...... 106 § 5. Partition, resolution and analysis ........ 109 CHAPTER VHI ENUMERATIONS AND CLASSES § I . The relation ' comprising ' as defining the nature of an enumeration and an item. The three operators for enumerations . . . . -113 §2. Enumerations of different orders . . . . . . . .116 §3. Distinction between ' comprised ' and * included ' . . . .120 § 4. Connection between enumerations and classes and the question whether ' class ' is a genuine entity . . . . . . . . .121 § 5. The nature and reality of the class as determined by the type of adjective significantly predicable of it . . . . . . . .126 § 6. Attempted proof of the genuineness of 'class' by the so-called principle of Abstraction. Charge of petitio principii and ignoratio elenchi against the alleged proof . . . . . . . . . . .128 CHAPTER IX THE GENERAL PROPOSITION AND ITS IMPLICATIONS § 1. Pure general propositions. Thing . . . . . . . .130 § 2. Analysis of the general proposition, showing how adjectives alone function in the predicate . . . . . . . . . . .130 § 3. Different interpretations of the universal and particular proposition, and the resultant modifications in their relations of implication and opposition 135 J. L. b X CONTENTS PAGE §4. Comparison with traditional scheme 139 § 5. Development of the immediate implications of the universal proposition as ordinarily interpreted ; with tables of immediate inferences . . 140 § 6. Development of the analogies between universals and particulars on the one side and necessary composites and possible conjunctions on the other 143 § 7. Formulation of general propositions in terms of classes. The principle underlying Euler's and Venn's diagrams . . . . . .146 §8. Summary . . . . . . . . . . . -155 CHAPTER X EXISTENTIAL, SUBSISTENTIAL AND NARRATIVE PROPOSITIONS § I. The proper principle for the Classification of Propositions . . . 156 §2. Philosophical distinction between Existential and Subsistential . .158 § 3. The so-called existential formulation of Propositions : Two meanings of the Universe of Discourse ......... 159 §4. The interpretation of mythical propositions as elliptically secondary . 164 § 5. The Narrative Proposition 166 § 6. Distinction between fictitious and historical narratives .... 167 § 7. The relation between the existence of a class and the existence of an individual 170 CHAPTER XI THE DETERMINABLE § I . The fundamentum divisionis, for which the name Determinable is to stand. The analogy and distinction between the relation of an individual to its class and that of an adjectival determinate to its determinable . 173 § 2. Can abstracts be divided into singular and general? The corresponding distinction is properly between the comparatively determinate and the comparatively indeterminate. The notion of a determinable as generating its determinates . . . . . . . . . . .177 § 3. The increase of intension that determines a decrease of extension to be supplemented by the superdeterminateness of adjectival characterisation. Classification as starting from a summum genus constituted by mere determinables and terminating in an ultima species, constituted by abso- lute determinates. Illustration from Botany . . . . .178 § 4. Relations between determinates under a given determinable. Incom- patibility. Order of qualitative betweenness, continuous or discrete. Complex determinables . . . . . . . . .181 § 5. Important consequences of the distinction between the absolutely deter- minate and the comparatively indeterminate . . - . . .183 CONTENTS xi CHAPTER XII THE RELATION OF IDENTITY PAGE § I. Identity goes along with Otherness, the two being co-opponent relations. Verbal and Factual Identification involve the same kind of identity, the nature of the propositions only differing . . . . . .186 § 2. False sense in which identity has been said to imply difference • . 188 §3. Relations like and unlike; similar and dissimilar; agreeing and dis- agreeing ............ 189 § 4. Adjectives comparable and disparate. Distensive magnitude. DifiFerence properly adjectival , . . . . . . . . .190 § 5. Adjectival comparison implies substantival otherness. False view of the notion of 'numerical' difference ........ 192 §6. Criticism of the Leibnizian dictum : the identity of indiscernibles . 193 § 7. Can identity be defined? Identity carries co-implication and substituta- bility. Exception for secondary proposition . . . . . .195 § 8. Identity and the Continuant, psychical or physical. Relation of Occurrent to Continuant. The causal bond that constitutes the unity of Experience 199 CHAPTER XIII RELATIONS OR TRANSITIVE ADJECTIVES § I. Monadic, diadic, triadic, etc. adjectives and propositions. Every relative implies its correlative .......... 203 %i. The substantive-couple characterised by an adjective-couple . . . 205 § 3. Illustrations of 'analogy' extending beyond arithmetical ratio . . 307 § 4. The coupling-tie and the characterising-tie, expressed in the grammatical rules of governance and accordance ....... 209 § 5. The paradox of relation. Distinction between tie and relation. The relation 'characterised' as unit relation entering into all adjectives and relations ... ......... 211 § 6. How adjectives, relations and propositions may function as substantives 214 § 7. Relational propositions of triadic and higher orders. The cognates of a relation : permutations and bracketing. Relations between terms and possibilia . . . . . . . . . . . .216 § 8. The problem of the relation of assertion to the proposition. How ideas are involved in propositions . . . . . . . .219 CHAPTER XIV LAWS OF THOUGHT § I. The logical meaning of consistency as distinguished from truth. The Laws of Thought may possibly be not enumerable as independents . 222 §2. The value and functions of truisms ....... 223 §3. In what sense are the principles of logic imperatives? The distinction between normative and positive sciences dealing with psychological material ............ 224 32 xii CONTENTS PAGE § 4. The fundamental formulae as used implicitly in building up the logical system itself ........... 226 §5. The three laws of identity : Transitive, Symmetrical, Reflexive . . 227 § 6. The four Principles of Propositional Determination ; and their import. Expression of the same for Adjectival Determination. These in generalised form. The distinction between false and not-true, true and not-false. These compared with male and female .... 228 § 7. Defence of the first Principle of Propositional Determination. Truth as temporally unalterable and as capable of only one meaning . . . 233 § 8. The four Principles of Adjectival Determination, as bringing out the relation of an adjective to its determinable. Postulate of the absolutely determinate character of the real. Substantival Categories distinguished by the Adjectival Determinables by which they are characterisable. The possibility of exhaustively comprehending the range of variation of a determinable. Summary ......... 237 § 9. The four Principles of Connectional Determination. These embody the purely formal properties of the causal relation. Its complementary aspects — Agreement and Difference. The fundamental postulate involved in the existence of Laws of Nature. The plurality and uniqueness both of cause and of effect .......... 242 § 10. The bearing of these principles upon the problem of internal and external relations 249 INDEX 253 INTRODUCTION § I . Logic is most comprehensively and least contro- versially defined as the analysis and criticism of thought. This definition involves the least possible departure from the common understanding of the term and is not intended to restrict or extend its scope in any unusual way. The scope of logic has tended to expand in two directions — backwards into the domain of metaphysics, and forwards into that of science. These tendencies show that no rigid distinction need be drawn on the one side between logic and metaphysics, nor on the other between logic and science. The limits imposed by any writer are justified so far as his exposition ex- hibits unity ; it is, in fact, much more important to remove confusions and errors within the subjects dis- cussed under the head of logic, than to assign precise limits to its scope. It is, I hold, of less importance to determine the line of demarcation between loeic and philosophy than that between logic and science ; so that my treatment of logic might be called philosophical in comparison with that of those who implicitly or ex- plicitly separate their criticism and analysis from what in their view should be relegated to epistemology and ontology. This account of the scope of logic does not differ in any essential respects from that given, for example, in Mill's long introductory chapter. The special feature of Mill's logic is the great prominence given to the theory of induction, in contrast to most of his predecessors xiv INTRODUCTION and contemporaries, including Whately. Whately does not omit reference to induction any more than Mill omits syllogism : where they differ is that Whately asserts that in order to be valid any inductive inference must be formulated syllogistically, and that therefore the principle for induction is dependent on the principle of syllogism. Mill opposes this view ; but as regards the scope of logic there is no disagreement between them : they differ simply on the question of the rela- tions of deduction to induction. If any writer deliberately or on principle dismisses from logic the theory of inductive inference, it must be on one of three grounds : either (a) that no inductive inference is valid ; or (d) that different criteria of validity apply to different sciences ; or (c) that the problem of the validity of induction constitutes a topic to be in- cluded in some study other than that named logic. As regards (a), this is the view which seems to be held by Venn in his Empirical Logic where, in the chapter on the subjective foundations of induction, he acknow- ledges that as a matter of fact human beings do make directly inductive inferences, even with a feeling of conviction, but that no warrant for such conviction can be found. Another aspect of his view of induction is expounded in the chapter on the objective foundations of induction, in which he classifies the different kinds of uniformity — such as sequence, co-existence, perma- nence, rhythm — which are used as major premisses, expressive of actual fact, by means of which specific uniformities under each general head are established as valid. When then he is asked what reasonable ground there is for accepting these major premisses as true, he INTRODUCTION xv maintains in effect that they have to be assumed, in order to give security to the conclusions inductively inferred. In using the word assumption, there seems to be some ambiguity, namely whether it is to be under- stood to mean ' assumed to be true although known to be false ' or ' assumed to be true although unprov- able.' I take Venn to mean the latter, and that the attitude towards this assumption is merely one of felt certainty — felt, indeed, by all human beings, but having no root in our rational nature, and only exhibiting a common psychological disposition or character. This view, that there is no inductive principle that is self- evidently or demonstrably true, seems to be held by many other logicians, though none of them, I think, put it as explicitly as Venn. So while he and others include induction in their logical exposition, they neg- lect what I take to be the essential justification for its inclusion, namely as affording a systematic criticism of the question of its validity. As regards (d), many excellent text-books have been written in these days treating of the principles and methods peculiar to different sciences ; it is not denied by their authors that this treatment is logical ; but, if not explicitly stated, yet it seems to be suggested that in comparing the logic of one science with that of another the sole result is to exhibit differences, and that no one set of principles applies to all the different sciences. If this were the fact there would be some excuse for excluding the treatment of induction from the scope of logic, on the ground that the discussion of each of the separate principles should be relegated to its own department of science. But if, as I hold in agreement with most xvi INTRODUCTION other logicians, there must be a community of principle discoverable in all sciences, then the discussion of this must be included in logic. As regards (c) the question raised seems to be : ' Given the topic induction, what name shall be oriven to the science that includes it in its treatment ? ' rather than the converse question ' Given the name logic, shall it be defined so as to in- clude, or so as to exclude, induction ?' If we put the question in the first form, the answer is of course purely arbitrary ; we might give it the name Epagogics. But if the question is put in the second form, the answer is not in the same sense arbitrary, assuming that there is general unanimity as regards the usage of the name logic to denote a science whose central or essential function is to criticise thought as valid or invalid. That induction should be included in logic thus defined follows from the undeniable fact that we do infer in- ductively, and that some persons in reference to some problems do infer invalidly. Even if this were not the fact, it is certainly of scientific importance to render explicit what everyone implicitly recognises in their inferences — as much for the case of induction as for that of syllogism or other formal types of inference. It has even been hinted that nobody makes mistakes in formal inference ; and yet — in despite of this, if true — no one questions the value of systematising the principles under which people may unconsciously reason ; and what holds of formal inference would certainly hold a fortiori of the processes of inductive inference which present many more serious opportunities for fallacy. § 2, As regards the term ' thought ' which enters INTRODUCTION xvii into my definition, its application is intended to include perceptual judgments which are commonly contrasted with rather than subsumed under thought, for the reason that thought is conceived as purely abstract while perception contains an element of concreteness. But properly speaking even in perceptual judgment there is an element of abstraction ; and on the other hand no thought involves mere abstraction. It follows, therefore, that the processes of thinking and of percep- tual judgment have an essential identity of character which justifies their treatment in a single systematic whole. It is the distinction between sense-experience and perceptual judgment, and not that between per- ceptual judgment and thought, that must be emphasised. The essential feature of perceptual judgment in con- trast to mere sense-experience is that it involves activity, and that this activity is controlled by the purpose of attaining truth ; further it is the presence of this purpose which distinguishes thought from other forms of mental activity. Thought may therefore be defined as mental activity controlled by a single purpose, the attainment of truth. § 3. Now it is true, as often urged, that thought is motived not solely by the purpose of attaining truth, but rather by the intention of realising a particular end in some specific form and under certain specific cir- cumstances. But I have to maintain that any other or further purpose which may prompt us to undertake the activity of thinking is irrelevant to the nature of thought as such, this other purpose serving only to determine the direction of activity. When such activity is actually in operation its course is wholly independent of the xviii INTRODUCTION prompting motive and guided by the single purpose of attaining trutli. For instance, our desire for food may prompt us to search for it ; but this resolve, once taken, leads to a thinking process the purpose of which is to come to some conclusion as to where food is likely to be found, and the sole aim of this process is to discover what is true on the matter in hand. This being so, the logical treatment of thought must be disencumbered from all reference to any ulterior purpose. Whether truth is ever pursued without any ulterior purpose is a psychological question which may fairly be asked ; and if introspection is to be trusted must certainly be answered in the affirmative, although the enquiry whether true knowledge has intrinsic value or not belongs to ethics. That the attainment of truth for its own sake constitutes a genuine motive force is further confirmed by recognising the fact that people do actually attach value to true knowledge, as is incontestably proved by their willingness to defy the prospect of social disapprobation, persecution, and even martyrdom incurred by the utterance and promulgation of what they hold to be true. At the same time, it must be pointed out that the aim of the thinking process is not the attainment of truth in general, but always of truth in regard to some determinate question under consideration. This is closely analogous to the psycho- logical fact that what we desire is never pleasure in general, but always — if the doctrine of psychological hedonism is to be accepted — some specific experience which is represented as pleasurable. Any thinking process is normally initiated by a question and terminated by an answer ; what dis- INTRODUCTION xix tinguishes one thinking process from another is the difference of the question proposed. The bond of unity amongst the phases of a single process does not necessarily entail unbroken temporal continuity, but only identity of the question proposed. Indeed any thought process may be temporarily interrupted before the proposed question has been answered. It must be left as a topic for psychology to investigate the causes of such suspension, and how far the advance made serves as a starting point for further advances. Logic, on the other hand, is concerned with the nature of the advance as an advance and criticises the process from the point of view of validity or invalidity. § 4. The above definition of logic as the analysis and criticism of thought should be compared with that of the Scholastics, who laid emphasis on the point that logic is concerned with the art of thinking, where art is nearly equivalent to the modern term technique, and has an understood reference to activity with an end in view. The study of the art of thinking as thus under- stood is of use in instructing us how to proceed when thinking out any problem : for instance, it lays down rules of classification and division for the clearing up of obscurities and inconsistencies in thought ; rules for the recall and selection of knowledge appropriate to any given problem ; etc. Descartes' Discourse on Method is a classical illustration of this species of science. Modern examples of excellent treatises on these lines are to be found in Alfred Sidgwick, and other neo-pragmatists. It is a science of the highest value, and need only be separated from logic on the ground of the difference of purpose; inasmuch as its direct XX INTRODUCTION purpose is the attainment of valid thought, whereas logic is the study of the conditions of valid thought, and as such it does not exclude the study of the art. § 5. Alongside of the use of the term 'art' to mean technique, there is a more modern usage where it implies reference to aesthetic feelings and judgments. Nowadays discussions as to whether an objective standard for these feelings and judgments should be recognised are very prominent. The nature of the feelings and judgments that enter into aesthetic appreciation belongs to psychology ; but if we agree that there is a discoverable objective standard, then the treatment of the subject of aesthetics is to be distinguished from the psychological treatment, precisely as the treatment of thought in logic is distinguished from that in psychology. Aesthetics, in this sense, raises very similar problems to those presented in Ethics ; and it is frequently said that as normative Logic, Aesthetics and Ethics are related in the same way to the three psychological factors, thought, feeling and volition respectively. Each of the normative studies may be said to be based on a standard of value, the precise determination of which it is their function to formulate; in each, imperatives are laid down which are acknowledged by the in- dividual, not on any external authority, but as self- imposed ; and, in each, the ultimate appeal is to the individual's intuitive judgment. There is, however, a closer resemblance between Ethics and Aesthetics in their relations to volitions and feelings respectively, than between either of them and Logic; inasmuch as there are apparently fundamental differences of opinion INTRODUCTION xxi as to the ultimate ethical and aesthetical standards, that give to the studies of Ethics and Aesthetics a con- troversial character absent from Logic about whose standards there is no genuine disagreement. As regards the relation of Ethics to Logic, the question sometimes arises as to which subject is supreme. The answer to this question depends entirely upon the nature of the supremacy intended: the imperatives for thought become imperatives for conduct only on condition that true judgments have intrinsic value and false judgments intrinsic disvalue ; and thus, from the point of view of conduct, Logic is subordinate to Ethics. On the other hand, ethical enquiry — like any other scientific investi- gation — has to avoid violating logical principles, so that from the point of view of true thought Logic is supreme over Ethics. § 6. Our discussion so far has led us to consider the relations of Logic to Philosophy in general, Psychology, Aesthetics and Ethics. Another subject to which it is closely allied and from which it is yet distinct is Grammar, the alliance h^mg piHma facie accounted for by the common concern of the two studies with language. The connection between thought and language presents a problem for the science of Psychology ; but, so far as thinking or the communication of thought in- volves the use of words, the provinces of Logic and Grammar coincide ; that is to say universal Grammar, which excludes what pertains to different languages and includes only what is common to all languages, should be subsumed under Logic. For the modes in which words are combined — which constitute the sub- ject matter for Grammar — cannot be expounded or xxu INTRODUCTION understood except as reflecting the modes in which thoughts are combined; and this combination is effected by means of such logical operations as negation, con- junction, disjunction, alternation, implication and so on, represented by the words not, and, not both, or, if, etc. To justify the subordination of Grammar to Logic we have only to realise that the analysis of the sentence in Grammar corresponds to the analysis of thought in Logic, and that grammatical criticism is confined to securing that the sentence precisely represents the thought, any further criticism of the proposition coming exclusively within the province of Logic. It may be pointed out in this connection as specially significant both for the linguist and for the logician, that languages differ in the degree of their capacity to exhibit through their structure intimacy between words and thoughts. § 7. Amongst all the sciences over which logic must rule, there is one that occupies a unique place. The constituents of thought which are in the most narrow sense logical are those which give form to the construct, connecting alien elements by modes which give specific significance to the whole. The first group of these is expressed by ties, conjunctional words, prepositional words, and modes of verbal inflection. But as the form of thought is further elaborated there enter new kinds of terms, namely specific adjectives which have a con- stant meaning definable in terms of pure thought, or else are to be admitted and understood as indefinables. The most generic form of such adjectives directly expresses the result of such mental acts of comparison as like, unlike, different from, agreeing with. Owing to the purely logical nature of these relations, universal INTRODUCTION xxiii formulae in which they are introduced can be constructed by mere abstract thought. The preHminary condition for this construction is the separating of what is given to constitute a plurality, and thus to introduce a formal factor which can only be verbally expressed by the separations and juxtapositions of the substantial words. The very general relation that separation effects is that most indeterminate relation otherness. When the complementary notions of separateness and together- ness are joined to constitute a unity, there enters the idea of number, and we are in the domain of mathe- matics. The extraordinary capacity for development that marks mathematics is due to the precision with which the relations of comparison are capable of being amplified. Through the substitutions that are thus rendered possible, the range of application of mathe- matical formulae is extended beyond the bounds which would otherwise delimit logic. Any material that might be presented to thought upon which the same precise operations of comparison could be performed, would lead to the same forms as mathematics. For example ideas, not only of difference, but of determinable degrees of difference, bring the material into relations of intrinsic order, and out of these relations emanate relations between relations, so that theoretically the science develops into a highly complicated system. The point then, where we may venture to say that logic actually passes into mathematics is where the specific indefinable adjectives above referred to give new material for further log"ical combinations. Here it is of great importance to point to the xxiv INTRODUCTION relative nature of the distinction between form and matter. Logic begins with a sharp contrast between matter, as what is given as merely shapeless, and form, as that which thought imposes. But as we advance to mathematics, we impose a new element of form in introducing the relation otherness and its developments ; and this being operated on by thought takes the place of new matter: in short, what is introduced as matter is form in the making. All this could be summed up by saying that for elemental logic, mathematical notions would constitute matter ; whereas when the step into mathematics is once taken these same elements are just those in accordance with which thought advances in constructing more and more com- plicated forms. This view of the relation of logic to mathematics will be worked out in Part II of the present work under 'Demonstration,' where the pro- cedure of building up mathematical science is shown to involve the very same principles as are used in the logical structure. All the sciences, including mathematics, over which logic has supreme control, have been properly described as applied logic. But mathematics is applied logic in a certain very unique sense, for mathematics is nothing but an extension of logical formulae introducing none but purely logical factors ; while every other science borrows its material from experiential sources, and can only use logical principles when or after such material is supplied. Within mathematics we have again the same kind of distinction, namely that between pure and applied mathematics, as it has been called. In pure mathematics, the mathematician can give free INTRODUCTION xxv play to his imagination in constructing forms that are restricted only by principles of logical consistency, and he develops the implications that are derivable from what may be indifferently regarded either as definitions of his fictitious constructs or as hypothetically enter- tained first axioms. In order that these axioms and the theorems therefrom derived may be considered as true, recourse must be had to the real world, and if applicable, the axioms come to be assertorically entertained as premisses, and the derived propositions as the developed conclusions. This application of mathematics to reality constitutes applied mathematics. Taking geometry as our first example, while there is no limit to constructing conceived spaces other than Euclidian, their application to reality demands the enquiry whether our space is or. is not Euclidian. This is answered by an appeal to our immediate intuitions directed to our spatial experiences, and it is this appeal that is outside the range of pure mathematics. Again the merely logical conception of betweenness, which develops into that of serial orders of lower or higher forms of complexity, is in the first instance a product of pure logical constructiveness, and would yield implications from which a system of implicates could be developed. But such a hypotheti- cally conceived body of propositions would have no basis in the real but for the applicability of the defined conceptions to what is given in non-mathematical intuition. This applicability holds not only in the domain of spatial order, but also in that of the qualitative relations of difference which impose serial order amongst sense impressions. Regarded in the light of its control over all sciences J.L. C xxvi INTRODUCTION logic has been called by the name 'Methodology'; that is to say while the forms of logic implicitly control the conclusions of science, logic itself includes the study which renders explicit the ways according to which its authority is exercised. The department of logic known as methodology constitutes the third part of the present work, which is entitled 'The logical foundations of Science.' Another illustration of applied mathematics is to quantity. Quantity is not a mere direct development from number, since a new conception, namely that of equality of units, enters as a distinctive factor which is not purely logical. It is true that equality for merely formal developments could be defined as a certain relation having the formal properties of symmetry and transitiveness, and if to this conception is added the fundamental operation plus ( + ), definable as a certain relation having the formal properties commutative and associative, the whole system of quantitative science could be developed without recourse to any but pure mathematical principles. But even in this range of thought quantities of different types would need recognition. For example, given the notion of length as the first spatial quantity, a new quantity is derived by multiplying length by length, which is called area; here 'multiplied' need not be more specifically defined than a certain relation having the formal properties commutative and associative. Again where a quantum of space is divided by a quantum of time, we have velocity, and in this way a totally new type of quantity is constructed and we pass from geometry to kinematics. Another quantity called mass is such that when multi- INTRODUCTION xxvii plied by velocity there is engendered the new quantity called momentum, and when multiplied by velocity squared, energy; and in the introduction of these new species of quantity we pass from kinematics to dynamics. This is the terminus on these lines of applied mathe- matics ; and dynamics may be defined as the science that uses the three independently definable species of quantity time, space and mass. In every extension, then, of mathematics no new idea or mode of thought need accompany the work of the calculus. It is only when the formulae have to be applied to reality, and thus to be entertained categorically, that a process of thought other than merely mathematical enters in, and intuition is directed to what is given in some form of experience. The ideas which enterinto the mathematical sciences thus constructed have a form which renders them amenable to purely logical processes of indefinite degrees of complexity; this distinguishes them from the non-mathematical or 'natural' sciences that intro- duce ideas dependent simply upon brute matter, unamenable to logical analysis, logic entering only in the application to these ideas of classification, and the principles of inductive inference. § 8. Having considered logic in its relation to the different sciences, we may now pass to a discussion of its more philosophical aspects. Logicians have been classified as nominalists, conceptualists, and realists or materialists, according as they think it worth while to discuss words, thoughts or things. Names that are apt to be understood as synonyms for these have been applied to different philosophical opinions; and this fact is indicative of the change which has occurred in xxviii INTRODUCTION the course of the history of philosophy, where the ground has been shifted from ontology to psychology, and later from psychology to logic. To take realism first. It is the name given to the Platonic view which formed the basis of Aristotle's controversy with Plato. Plato in discussing the relation between the universal and the individual, attributes r^^/ existence in the truest or most ultimate sense to the universal, holding that the particular individual has reality only so far as it partakes of the nature of the universal, towards which it strives as the end {ivreXeKy)) of its existence. Aristotle, opposing this view, holds that the universal exists not apart from the particular but in it. A new psychological significance came to be attached to the term Realism, when the question of reality was raised not about the thing, but about the possible idea of the thing, these two concepts being taken to be equivalent. The so-called nominalist school of philo- sophers maintained the psychological view that we had no idea corresponding to a general name, along with the ontological view according to which the particular individual or concrete alone existed, and no existence could be attributed to the universal ; generality, for them, attached only to names in use, and had no objective application. On the psychological point at issue the opponents of this view have been known as conceptualists, and in maintaining their opposition were led to make a psychological distinction of great im- portance between images and ideas. In common with the nominalists, they held that images are necessarily concrete, particular or individual, but they maintained that we can also frame ideas which can properly be INTRODUCTION xxix called abstract or o-eneral. Both schools assumed that images were equivalent to or at least resembled per- ceptions, and further that the latter were obviously concrete and particular. Berkeley represents the nomi- nalist school, and his subtle difference from Locke — who definitely held that we can frame general ideas, though with difficulty — comes out clearly when he dis- putes the possibility of a general idea of a triangle (instanced by Locke) which shall be neither equilateral nor isosceles nor scalene, and from which we can in thought abstract the shape from variations of colour. In my view Locke and Berkeley were both wrong, even where they agreed ; inasmuch as neither images nor perceptions reflect the concreteness and particularity of the individual thing, which should be described as determinate, in contrast to the indeterminateness of the mental processes. In fact there has been a confusion in the description of our thoughts, images and percepts, between the distinction of the universal from the particular, and that of the indeterminate from the de- terminate. The modern term 'o-eneric,' which has been applied to images, should be extended also to percepts, on the ground that they share with images the character of indeterminateness — a character which must be rigidly distinguished from general or uni- versal as properly applied to ideas or concepts. Nominalism has yet another meaning when applied as a special logical theory ; in this sense it denotes the theory according to which the proposition is an indica- tion of the names that have been arbitrarily chosen to denote things or classes of things, and predicates merely what follows from the consistent use of these names. XXX INTRODUCTION Propositions are thus used as mere formulae and re- peated in thought when necessary, without demanding any consideration of their meaning ; so that the only ultimate foundations or premisses of knowledge are definitions, no other propositions of the nature of axioms being required. This view still clings to some modern philosophical expositionsofarithmeticand purelogic,and is rather subtly akin to the view that the first premisses for science are nothing but postulates or hypotheses which, if consistently held, lead to the discovery of truth. As regards Conceptualism, it is doubtful whether, as applied to the work of such writers as Hamilton and Sigwart, it can be properly regarded as a distinctive logical theory. For the prominent use of the word concept and its associate judgment points not neces- sarily to any difference of logical theory between those who use these words, and those who prefer the words 'term' or 'name' and 'proposition,' but merely to the common recognition that thought has form as well as verbal expression. If, however, the conceptualist proceeds to limit the scope of logic to the consideration of the forms of thought alone, then he must maintain that the truth of a judgment is tested by the form that connects the content as conceived ; and conceptualism becomes equivalent to formalism. The criterion for the formalist is indeed mere consistency or coherence in fact ; that for the conceptualist proper, clearness or distinctness in thought. The latter is expressed negatively by Herbert Spencer : what is clearly not conceivable is false ; positively by Descartes : what is clearly conceivable is true. It follows immediately from this view that truth concerns only conceived content ; INTRODUCTION xxxi so that the direct objects of thought are not things, but our ideas about things, and judgment contains no reference to things but only to adjectives. On this understanding, the conceptuaHst's view is that we can only deal with things as conceived, and that it is the mode under which we conceive them that determines the adjectives themselves and their relations as consti- tuting the content of the judgment. In this way they are led to deny all relations as subsisting between things — a denial which is simply equivalent to denying the one supreme relation otherness ; for otherness may be said to be the one determinable relation to which all specific relations stand as determinates. Hence it is enough for this school of philosophy to deny the single relation otherness, and in this denial to adopt the position of monism. The view, if carried out rigidly, goes beyond that of Spinoza, who asserted that thought was other than extension, and even that the one Substance had an infinity of other attributes, though not conceivable by us. It is an odd fact that Lotze, in particular, explicitly rejects relations only, as expressive of the nature of Reality ; but in consistency he ought to have included in his rejection ordinary adjectives. From this point of view, the only kind of singular categorical judgment concerns Reality as a whole and not any one of its several separable parts : it predicates character of the indivisible one, not of this or that unit in the one. Individual units, in fact, are conceived as the result of the imposition of thought to which nothing in the one corresponds. Thus the Monist's first principle is to deny the Pluralist's fundamental assumption that the Real, as given to thought, is given as many and as such involves existential otherness. xxxii INTRODUCTION The conceptualist's account of the character of the singular judgment leads to a similar account of that of the particular and the universal judgment. The view is consistently borne out by his interpretation of particulars as possible conjunctions, i.e. of adjectives that we can conjoin in conception ; and of universals as necessary conjunctions, i.e. of adjectives that we must conjoin in conception. Symbolically : ' Some things that are / are ^' is to mean '/ and q can be conjoined and can be disjoined' ; 'Everything or nothing that is/ is ^' is to mean '■p and q must be conjoined or must be disjoined.' What is true in this view is that the operations not, and, 7iot-both, if, or,a.re supplied by thought; and that nothing in the merely objective world manifests the mere absence of a character, or the mere indeterminateness of the alternative operation, or dependence as expressed by im- plication. These relations are not manifested to thought, but analytically or synthetically discovered or rather im- posed dy thought. The view is most strikingly expressed by Mr Bradley in his dictum : only if what is possible is necessitated will it be actualised; and again, only if what is necessary is possible will it be actualised. From conceptualism we pass back again to realism in its new sense as applying to logic, and in this appli- cation it is usually denoted by the term materialism or empiricism. We are thus led back again to Venn, and less explicitly to Mill, who contrasts the formalism or conceptualism of Hamilton with his own logical stand- point. Taking empiricism to mean that all knowledge is obtained by experience alone (as Mill only seems to have held) the doctrine amounts to maintaining that all inference is ultimately of the nature of pure induction. But taking it to mean that no knowledge gained by INTRODUCTION xxxiii experience can be validly universalised (as Venn seems to hold) then the doctrine amounts to maintaining that no inference of the nature of pure induction is valid, and that hence only deduction is guaranteed by logic. In default of any explication of which of these two views is meant by empiricism or materialism, we can only conclude that the term stands for that department of logic that is concerned with an analysis of the process of induction. But here we must note that the distinction in character between induction and deduction is not properly expressed by the antithesis of matter and form ; since the relations amongst premisses and conclusion which constitute the form of an inference hold for the validity of induction as for that of deduction; and con- versely, reference to the matter of the propositions is required equally for the truth of a deductive inference as for that of an inductive inference. This obvious fact has been forgotten, owing to the great prominence given by inductive logicians to the treatment of the preliminary processes of observation, search, arrange- ment, comparison of material data, and the formation of formulae that shall hold for the facts collected, and the aid required by experimentation. In consequence, stress is laid on the securing of correctly described premisses in the case of induction ; whereas in the case of deduction stress is laid only on securing validity for the form of inference. § 9. In conclusion I propose to enumerate the most important features in the treatment of logical theory to be developed in the course of this work: (a) The epistemic aspect of thought is included within the province of logic, and contrasted with the constitutive aspect ; the former is a recognition that xxxiv INTRODUCTION knowledge depends upon the variable conditions and capacities for its acquisition ; the latter refers to the content of knowledge which has in itself a logically analysable form. Such fallacies 2l^ petitio principii really require reference to the epistemic aspect of thought, while fallacies of the strictly formal type refer exclu- sively to the constitutive aspect. Again the whole theory of modality which develops into probability is essentially epistemic, indicating as it does the relation of the con- tent of the proposition to the thinker. Thus a distinc- tion is clearly drawn between the proposition and the attitude of assertion or judgment ; and while, on this view, the proposition is identifiable when in variable relations to different thinkers, the necessity is empha- sised of conceiving the proposition in terms of assertion, the act of assertion being thus taken as the complete fact to be analysed and criticised. It is this intimate connection between the assertion and the proposition which gives meaning to the identification of the adjec- tives true and false with the imperatives 'to be accepted' and 'to be rejected.' {b) The proposition itself, which is customarily re- solved into subject and predicate, is more precisely analysed by showing that the substantive alone can function as subject, and the adjective as predicate, and that these stand to one another in the relation of characterisation: the substantive being that which is characterised, the predicate that which characterises. Since an appropriate adjective can be predicated of a subject belonging to any category, including adjective, relation and proposition, the subject as thus functioning becomes a quasi-substantive. The substantive proper seems to coincide with the category 'existent,' while INTRODUCTION xxxv if any category other than substantive stands as subject its logical nature is not thereby altered, but rather the adjectives proper to it fall under correspondingly special sub-categories determined by the category to which the subject belongs. (c) Adjectives are fundamentally distinguishable into determinables and determinates, the relation be- tween which is primarily a matter of degree, a deter- minable being the extreme of indeterminateness under which adjectives of different degrees of determinateness are subsumed. The relation of a determinate to its determinable resembles that of an individual to a class, but differs in some important respects. For instance, taking any given determinate, there is only one deter- minable to which it can belong. Moreover any one determinable is a literal siimmum genus not subsumable under any higher genus ; and the absolute determinate is a literal infima species under which no other deter- minate is subsumable. {d) Relations are treated as a specific kind of ad- jective, and are called transitive adjectives in distinction from ordinary adjectives which are intransitive. The adjectival nature of relations is apt to be obscured by the inclusion under relative terms of what are merely substantives defined by relational characterisation. All that holds universally of adjectives, including the rela- tion of determinates to their determinable, holds of relations as such. {e) Under the head of induction, fundamentally different types are distinguished. First : the very ele- mentary process of intuitive induction, which lies at the basis of the distinction between form and matter, and by which all the formal principles of logic are estab- xxxvi INTRODUCTION lished. Next: summary induction, more usually called perfect induction, which establishes conclusions of limited universality by means of mere enumeration. Such a summary universal stands as premiss for an unlimited universal conclusion, obtained by what is called induc- tio per siinplice7n emtmerationem. What is specially important in my treatment is the function of summary induction in the specifically geometrical form of infer- ence. Thirdly : demonstrative induction, which employs no other principles than those which have been recog- nised in deduction. This species of induction is directly employed in inferring from a single experimental in- stance an unlimited universal ; and it is this species of induction which gives the true form to the methods formulated by Mill and Bacon. Lastly we distinguish induction proper, which is conceived as essentially problematic, and as thus re-introducing the epistemic aspect in the form of probability. (y) The specific notion of cause as applying to events is distinguished from the generic notion of mere determination according to a universal formula. As specific, cause relates exclusively to states or conditions temporally alterable and also referable to place ; and, in this application of the notion of determination, the effect and cause are homogeneous. Not only is the character of the effect regulated by that of the cause, but the date and place of the latter is determined by the date and place of the former. The universal positional relation, as it may be called, of cause to effect is that of con- tiguity, which is to be conceived in the form of the coincidence of the temporal or spatial boundary of that which constitutes the cause with that which constitutes the effect. This absolute contiguity disallows any gap INTRODUCTION xxxvii between the cause process and the effect process; so that contiguity is strictly defined as equivalent to con- tinuity. This further implies that when, as is always permissible, we conceive a phase of the causal process as temporally or spatially separated from a phase of the effect process, we must also conceive of that which goes on in the interval bridging cause and effect to be part of one continuous process. This is possible because time and space are themselves continuous. Thus change and movement are connectionally continuous, in the special sense that the character manifested at one instant of time or at one point of space differs from that manifested at another instant of time or at another point of space, in a degree the smallness of which depends upon that of the temporal or spatial interval. Again superimposed upon the continuity of this process, there is a discontinuity of the second order, ultimately due to the discontinuous occupation of space by different kinds of matter, {£-) The notions of cause and substance reciprocally imply one another, the latter being that which continues to exist and in which alterable states or conditions inhere. These alterable states constitute what may be called the occurrent or, in accordance with scholastic usage, the occasional causal factor. The occurrent is distinguished from and essentially connected with the continuant or the material factor in causation. The occurrent and continuant factors are thus united in our complete conception of substance, neither being con- ceivable apart from the other. This analysis gives meaning to the conception of the properties of the con- tinuant, as potential causes which are actualised in accordance with unchanging rules by the relatively xxxviii INTRODUCTION incidental occurrences that come into beinor either from o within or from without the continuant. In the former case the process is immanent, cause and effect being manifestations of the changeless nature of the continuant, and the temporal relation between cause and effect is here that of succession. In the latter case, the causality is transeunt, the patient being that whose state is determined, the agent being that whose alterable relation to the agent is determinative. In transeunt causality the temporal relation of cause to effect is literal simul- taneity, and the critical instant at which the cause operates is that in which there is also literal geometrical contact of cause agent with effect patient. There are two fundamentally distinct types of transeunt causality. In the one case no change of state in the agent accompanies the change of state in the patient, and we have action without any direct reaction; in the other case change of state in the one directly entails change of state in the other of such a nature that the latter may be formulated as a function of the former, and here action always involves an assignable reaction. The latter case holds invariably of inter-physical causality, and again of inter-psychical causality within the sphere of a single individual's experience. But in physico- psychical causality, as also in psycho-physical causality, action never directly determines reaction, owing to the absolute disparateness between the physical and psychical in regard to the characters of the states which are predicable of the one and of the other. It is here where my treatment of logical questions transgresses into the domain of ontology; but it must be admitted that all logicians who treat these subjects inevitably transgress in the same manner. INTRODUCTION xxxix (h) The position assumed by probability in logical discussion has always been dubious. On the one side the topic has been assumed to be the exclusive property of the mathematician, or rather more precisely, the arithmetician. On this view the quantity called prob- ability is a mere abstract fraction, and the rules of probability are merely those of arithmetic. The fraction is, in short, the ratio of two numbers, the number holding for a species to that holding for its proximate genus, this ratio being necessarily a proper fraction, the limits of which are zero and unity. If this view were correct, there would be no separate topic to be called probability. A precisely reverse account of prob- ability is that it is a measure of a certain psychological attitude of thought to which the most obvious names that could be given are belief or doubt, taken as subject to different degrees. On either of these two extreme views probability would have no particular connection with logic. The psychological account would be sepa- rated from logic, inasmuch as it would concern solely the causal explanation of different degrees of belief, and would thus give rise to no principle of rational criticism. The mere arithmetical account of probability ought in the first instance to be corrected by the recognition that the topic has its mental side. This correction requires that probability should not be expressed by a merely abstract fraction, but rather as a fraction of a certain mental quantity which may be called certainty. The psychological conditions of the variable degrees in which doubt may approximate to certainty are as such outside the province of logic; but when these various degrees are such as reason would dictate, we may speak of reasonable doubt as an assignable fraction of certitude, xl INTRODUCTION thus bringing the subject into the sphere of logic. Further the quantity or degree called probability at- taches exclusively to the proposition; not however to the proposition as such, but to the proposition regarded as based upon rationally certified knowledge acquired by any supposed thinker. The degree of probability is therefore referential to such knowledge, but is wholly independent of the individual thinker, being dependent solely on his rational nature, and the knowledge which he has rationally acquired. The whole development of this aspect of the subject is to be called formal probability, and constitutes the one subject of the fourth Part of this work. The treat- ment of probability there developed must be distin- guished from that of informal probability, that is required in discussing the foundations of science as treated in my third Part- for there, while the log-ic of inductive infer- ence is made to depend upon the principles of probability and not upon any big fact about nature, yet probability is only introduced on broad and indeterminately quan- titative lines. This treatment leads to an attempted enumeration of broadly formulated criteria for the evaluation of the degrees of probability to be attached to the generalisations of inductive inference. These criteria are merely expressions of what is popularly felt, and their rational justification can only be repre- sented as depending upon postulates: that is, specula- tions that are neither intuitively self-evident nor ex- perientially verifiable, but merely demanded by reason in order to supply an incentive to the endeavour to systematise the world of reality and thus give to prac- tical action an adequate prompting motive. CHAPTER I THE PROPOSITION § I. A SYSTEMATIC treatment of logic must begin by regarding the proposition as the unit from which the whole body of logical principles may be developed, A proposition is that of which truth and falsity can be significantly predicated. Some logicians have taken the judgment as their central topic, and it will be necessary to examine the distinction between what I have called a proposition and what appears to be meant by a judgment. It has been very generally held that the proposition is the verbal expression of the judgment; this, however, seems to be an error, because such characterisations as true or false cannot be predicated of a mere verbal expression, for which appropriate adjectives would be 'obscure,' ' ungrammatical,' 'am- biguous,' etc. There appear then to be three notions which, though intimately connected, must be clearly distinguished: namely (i) what may be called the sen- tence; (2) the proposition; and (3) the judgment. The sentence may be summarily defined as the verbal ex- pression of a judgment or of a proposition; it remains, therefore, to distinguish and interrelate the proposition and the judgment. The natural use of the term judgment is to denote an act or attitude or process which may constitute an incident in the mental history of an individual. As so conceived, we should have further to distinguish the J.L. I 2 CHAPTER I changing phases of a process (which might alternately involve interrogation, doubt, tentative affirmation or negation) from the terminus of such process in which a final decision replaces the variations undergone during what is commonly called suspense of judgment. It would thus be more natural to speak of passing judg- ment upon a proposition proposed in thought than to identify judgment as such with the proposition. This more natural usage (which is that which I shall adopt) entails the necessity of recognising the distinction be- tween various attitudes of thought on the one hand, and the object towards which that thought may be directed on the other; and even further, when necessary, of recognising the adoption of any of these alterable attitudes of thought as a datable occurrence within the total experience of some one individual thinker. There will thus be many fundamental attributes that must be predicated of the judgment upon a proposition different from, and often diametrically opposed to, those attributes that are to be predicated of the proposition itself. In this account the judgment is the more compre- hensive or concrete term, since when seriously treated it involves the two terms thinker and proposition and, in addition, the occurrent and alterable relation that may subsist between them. In thus drawing attention to mental process in my exposition of logical doctrine, I am taking what has been unfortunately termed a * sub- jective ' point of view. For the term ' subjective ' should be substituted 'epistemic'; and in discarding the familiar antithesis subjective and objective, it is better for the pur- poses of Logic to substitute the antithesis epistemic and constitutive. The epistemic side of logical doctrine points THE PROPOSITION 3 to the quite universally acknowledged kinship of Logic with Epistemology, and, in using this term in preference to subjective, we can avoid any confusion between what belongs to Psychology as opposed to what belongs to Logic. As to the term constitutive — a term for which philosophers are indebted to Kant — it has the force of ' objective ' inasmuch as it points to the constitution of such an object of thought-construction as the proposition when treated independently of this or that thinker. I may anticipate what will be treated fully in the later part of logical doctrine, by pointing out that the dis- tinction and connection between the epistemic and constitutive sides of logical problems plays an important part in the theory of Probability; and, in my view, it ought to assume the same importance throughout the whole of the study of Logic. Now, as regards the relation of the proposition to any such act as may be called judgment, my special contention is that the proposition cannot be usefully defined in isolation, but only in connection with some such attitude or act of thought ; and I prefer to take the notion o{ asserting as central amongst these variations of attitude — which will therefore be spoken of as variations in the assertive attitude. I shall also maintain that the fundamental adjectives true and false which are (perhaps universally) predicated of mere propositions as such, derive their significance from the fact that the proposition is not so to speak a self-subsistent entity, but only a factor in the concrete act of judgment. Thus, though we may predicate of a certain proposition — say ' matter exists ' — that it is true or that it is false, what this ulti- mately means is, that any and every thinker who might 4 CHAPTER I at any time assert the proposition would be either exempt or not exempt from error. In other words, the criticism which reason may offer is directed — not to the proposition — but to the asserting of the proposition; and hence the customary expression that such and such a proposition is false merely means that anyone's assertion of the proposition would be erroneous. The equivalence of these two forms of criticism follows from the fundamental principle that an attitude of assertion is to be approved or condemned in total independence of the person asserting or of the time of his assertion, and in exclusive dependence upon the content of his assertion. This fundamental principle of Logic will come up for detailed treatment when the so-called Laws of Thought are explicitly discussed. In order to mark the important distinction, and at the same time the close connection, between the proposition and the act of assertion, I propose to take the term ' assertum ' as a synonym for 'proposition' when such terminology may seem convenient. Thus, the assertum will coincide, not exactly with what has been asserted, but with what is in its nature assertible. § 2. Many philosophers have used the term belief in its various phases as a substitute either for judgment or for assertion ; in fact, when the mental aspect of any problem assumes special prominence, the term belief as applied to the proposition is more naturally suggested than any other. While the object of belief is always a proposition, the proposition may be merely entertained in thought for future consideration, either without being believed, or in a more or less specific attitude opposed to belief, such as disbelief or doubt. To doubt THE PROPOSITION 5 a proposition implies that we neither believe nor dis- believe it, while belief and disbelief as opposed to doubt have in common the mental characteristic of assurance. Thus there are three opposed attitudes towards a pro- position, included in the distinction between assurance and doubt; — the former of which may be either (assured) belief or (assured) disbelief, and the latter of which ap- pears further to be susceptible of varying felt degrees. The close association amongst all the terms here intro- duced brings into obvious prominence the mental side, which such terms as judgment or assertion seem hardly to emphasise. It would however, I think, be found that there is in reality no relevant distinction between the implications of the two terms ' judgment ' and ' belief.' Those logicians who have spoken exclusively of judg- ment, conception, reasoning, etc., have had in view more complicated processes, the products of which have been explicitly formulated; while those who have used belief and cognate terms have included more primitive and simple processes, the products of which may not have been explicitly formulated. Since the traditional logic has treated 07ily the more developed processes, the term judgment and its associates is perhaps prefer- able for this somewhat limited view of the scope of Logic, while the use of the term belief — which must certainly be understood to include the higher as well as the lower processes — points to a wider conception of the province of Logic. To put the matter shortly, I hold it to be of fundamental importance to insist that there is some factor common to the lower and higher stages, and that this common factor, to which the name belief has been given, is necessarily directed to what 6 CHAPTER I in Logic is called a proposition \ Assertion, in the sense here adopted, is to be understood to involve belief, and may be defined as equivalent to conscious belief. This de- finition restricts the term in two ways: in that, firstly, to assert does not merely mean to 2/^/^r (without belief) ; and secondly, merely to believe unconsciously is not to assert. § 3. In speaking of variations of attitude towards the proposition, an assumption is involved that there is a single entity called the proposition that is the same whatever may be the attitude adopted towards it. Ordinary language supplies us with names for such different attitudes along with cognate names for the proposition : thus we associate ' to assume ' with ' an assumption ' ; 'to suppose ' with * a supposition ' ; * to propose ' with ' a proposition ' ; ' to postulate ' with ' a postulate'; 'to presume' with 'a presumption'; etc.^ Consider the two verbs 'to assume' and ' to presume.' It will be acknowledged that these denote attitudes between which some subtle distinction may be under- ^ Readers of Psychology should be warned that, when psycho- logists contrast 'imagination' with 'belief,' each term indicates an attitude to a proposition; while, when they contrast 'imagination' with 'perception,' the processes to which they refer do not involve any attitude towards a proposition. There is no common element of meaning in these two applications of the word 'imagination.' ^ In further illustration of this point we may select certain prominent logical terms such as hypothesis, postulate, axiom. Each of these terms indicates the peculiar attitude to be assumed towards the proposition in question by any thinker : thus a hypothesis stands for a proposition which awaits further scientific investigation before being finally accepted or rejected ; a postulate stands for a proposition which cannot be brought to the test of experience, but the truth of which is demanded by the thinker; and an axiom is a proposition the truth of which is self-evident to the thinker. THE PROPOSITION 7 stood; and thus it might appear that in correspond- ence with this distinction there must be a similar subtle distinction between an assumption and a presump- tion. Unfortunately substantival words such as these are apt to suggest a difference in nature between that which in the one case is presumed and in the other assumed; but this suggestion must be rejected, and it must be maintained on the contrary that the content of a proposition preserves its identity unmodified, inde- pendently of all variations of assertive attitude and of personal and temporal reference. This independence holds also in regard to what has been termed ' logical ' in contrast with ' psychological ' assertion. The phrase logically asserted, applied to this or that proposition, is only metaphorically legitimate, and literally equivalent to * asserted on purely rational grounds by any or all rational persons.' In other words, the predicate 'as- serted ' conveys no meaning when taken apart from a person asserting. Adopting as we do the general view that no logical treatment is finally sound which does not take account of the mental attitude in thought, it follows that the fundamental terms * true ' and ' false ' can only derive their meaning from the point of view of criticising a certain possible mental attitude. We are thus bound to distinguish the object of this attitude (the assertum) from the attitude itself which may vary independently of the object; but we can only avoid contradiction or vagueness if, while permitting ourselves to distinguish between the attitude and its object, we at the same time refuse to separate them. We may further explain the adjectives ' true ' and ' false ' so as to bring out what 8 CHAPTER I characterises logic in contrast with — or rather in its relation to — psychology : namely that logic formulates standards or imperatives which as such have no sig- nificance except as imposed upon mental acts. Thus we may say that the application of the adjectives true and false coincides with the application of the imperatives ' to be accepted ' and ' to be rejected ' respectively. We may add that these imperatives are imposed by the thinker — in the exercise of his reason — upon himself. In maintaining this coincidence between the two im- peratives on the one hand and the two adjectives (true and false) on the other, it must not be taken that we are able thus to define the adjectives true and false. On the contrary, we are forced to insist that they are in- definable. We are only indicating that a reference to mental attitude is presupposed when Logic recognises the distinction between true and false in its formulation of standards for testing the correctness of a judgment or assertion. § 4. So far we have taken the proposition as a unit of which the adjectives true and false may be predicated. Before proceeding to analyse the proposition into its component parts, a word must be said in regard to the relation of logic to universal grammar, and in particular the relation between grammatical and logical analysis. Properly speaking, grammatical analysis cannot be re- garded as dealing merely with words and their combi- nations. The understanding of the grammatical structure of a sentence — which includes such relations as those of subject to predicate, and of subordinate to co-ordinate clauses — requires us to penetrate below the mere verbal construction and to consider the formal structure of THE PROPOSITION 9 thought. Hence, on the one hand, grammar cannot be intelligently studied unless it is treated as a department of logic ; and, on the other hand, logic cannot proceed without such a preliminary account of linguistic structure as is commonly relegated to grammar. In short, uni- versal grammar (as it is called) must be subsumed under Logic. On this view, a slight alteration in grammatical nomenclature will be required, whereby, for the usual names of the parts of speech, we substitute substantive- word or substantive-phrase, adjective-word or adjective- phrase, preposition- word or phrase, etc., reserving the terms substantive, adjective, preposition, etc., for the different kinds of entity to which the several parts of speech correspond. § 5. To turn now to the analysis of the proposition. We find that in every proposition we are determining in thought the character of an object presented to thought to be thus determined. In the most fundamental sense, then, we may speak of a determinandum and a deter- minans: the determinandum is defined as what is pre- sented to be determined or characterised by thought or cognition; the determinans as what does characterise or determine in thought that which is given to be de- termined. We shall regard the substantive (used in its widest grammatical sense) as the determinandum, and the adjective as the determinans. Neither of these terms can be defined except in their relation to one another as each functions in a possible proposition. As it has frequently been said, the proposition is par excellence the unit of thought. This dictum means that the logical nature of any components into which we may analyse the proposition can only be defined by the mode in 10 CHAPTER I which they enter into relation within it. For example, when I use determinandum for the substantive and determinans for the adjective, I am only defining the one in terms of the other, inasmuch as the common factor ' determine ' is contained in both. This account goes beyond that which has become commonplace among many philosophers, namely, that the subject of a pro- position is ultimately something which cannot be defined in the way in which a predicate or adjective can be defined ; for to this we have to add that the predicate of a proposition is ultimately something which cannot be defined in the way in which a subject or substantive can be defined. These two statements present the natures of subject and predicate purely negatively, the positive element being supplied by the terms ' determinans ' and * determinandum.' We have now to examine the nature of the connec- tion involved in every case where adjective and sub- stantive are joined; for example 'a cold sensation,' 'a tall man.' In order to understand the verbal juxta- position of substantive and adjective, we must recognise a latent element of form in this construct, which differen- tiates it from other constructs — which also are necessarily expressed by a juxtaposition of words. This element of form constitutes what I shall call the characterising tie. The general term * tie ' is used to denote what is not a component of a construct, but is involved in under- standing the specific form of unity that gives significance to the construct ; and the specific term ' characterising tie ' denotes what is involved in understanding the junction of substantive with adjective. The invariable verbal expression for the characterising tie is the verb THE PROPOSITION n * to be ' in one or other of its different modes. To think of ' a tall man ' or of 'a cold sensation ' is to think of ' a man as being tall,' ' a sensation as being cold.' Here the word ' being ' expresses the characterising tie, and the fact that in some cases the word may be omitted is further evidence that the tie is not an additional com- ponent in the construct, but a mere formal element, indicating the connection of substantive to adjective. This is its peculiar and sole function ; and, as the ex- pression of the unique connection that subsists between substantive and adjective, it is entirely unmodifiable. The distinction and connection between substantive and adjective correspond to — and, in my view, explain — the distinction and connection between particular and universale Ultimately a universal means an adjective that may characterise a particular, and a particular means a substantive that may be characterised by a universal. The terms particular (or substantive) and universal (or adjective) cannot be defined as functioning in isolation, but only as they enter into union with one another. There is some danger of confusing two different uses of the verb 'to characterise,' which may be partly re- sponsible for the historical dispute concerning the relation of particular to universal. Primarily the term ' charac- terise ' should be used to connect substantive with adjective in the form ' such and such a quality or ad- jective characterises such and such an object or sub- stantive.' On the other hand, in the phrase ' the thinker characterises such or such an object, 'characterises means ^ Here the terms particular and universal are used in the sense current in philosophy, and not in their familiar application in elementary logic, where they stand for sub-divisions of the proposition. 12 CHAPTER I ' cognitively determines the character of.' Owing to this elliptical use of the term, the particular has been conceived of as ' an uncharacterised object,' and this would mean literally * an object without any character ' ; but since actually every object must have character, the only proper meaning for the phrase ' uncharacterised object ' is 'an object whose character has not been cognitively determined.' If then the term ' exist' may be predicated equally of a universal as of a particular, then we may agree with the Aristotelian dictum that the universal exists, not apart from, but in the particular ; and by this is meant that the adjective exists, not apart from, but as characterising its substantive; to which must be added that the substantive exists, not apart from, but as characterised by its adjective. Now in thought the substantive and the adjective may be said to be separately and independently represented ; hence thinking effects a severance between the adjective and the substantive, these being reunited in the asserted proposition — not only by the characterising tie, but also by what we may call the assertive tie. The blending of the assertive with the characterising tie is expressed in language by the transition from the participial, sub- ordinate, or relative clause, to the finite or declaratory form of the principal verb. Thus in passing from ' a child fearing a dog ' to ' a child fears a dog,' the charac- terising tie joins the same elements, in the same way, in both cases; but is, in the latter, blended with the assertive tie. That the ties are thus blended is further shown by the modifications ' is-not,' ' may be,' ' must be,' by which the verb ' to be ' is inflected in order to indicate variations in the assertive attitude while the character- THE PROPOSITION 13 ising relation remains unchanged. The copula ' is ' of traditional logic is thus seen to be a blend of the charac- terising with the assertive tie. § 6. We must now criticise a view, explicitly opposed to our own, as to the nature of the copula is. There has been for a long period an assumption that the pro- position in some way or other asserts the relation of identity. This relation of identity, it is admitted, is not one of complete or absolute identity, but involves also a relation of difference : thus the proposition ' Socrates is mortal' is transformed into 'Socrates is -a mortal being ' — where ' Socrates ' and ' a mortal being ' are affirmed to be identical in denotation but different in connotation. Have logicians quite recognised the ex- treme elaborateness of this verbal transformation } The adjective ' mortal ' has first to be turned into a sub- stantive in using the word ' a mortal being ' ; secondly, the indefinite article has to be introduced, since it is clear that Socrates is not identical with every mortal; thirdly, the indefinite article has to be carefully defined as meaning one or other; fourthly, the relation of the adjective ' mortal ' to the substantive ' being ' which it characterises still remains to be elucidated; fifthly, another adjective(a relational adjective) n-dmoly identical is introduced in the compound phrase ' is identical with.' The proposition finally becomes: ' Socrates is identical with one or other being that is mortal.' Here the two adjectives ' mortal ' and ' identical with ' are each intro- duced after is. Now, if ' is identical with ' is to be substituted for is in each case, then we shall arrive at an infinite regress. Thus, in the first place, ' Socrates is identical with X ' (say) must be rendered ' Socrates 14 CHAPTER I is identical with a being that is identical with X ' where the force of is still remains unexplained. And in the second place ' one or other being that is mortal ' must be rendered ' one or other being that is identical with a mortal,' where again is still remains to be explained. In each case, if an infinite regress is to be avoided, the word is that remains must be interpreted as representing the unique mode in which the fundamentally distinct categories substantive and adjective are joined. § 7. Having so far considered the proposition in its mental or subjective aspect, we have next to examine it in what may be called its objective aspect. Whereas a proposition is related subjectively to assertion, we shall find that it is related objectively lofact^. Our conclusion, briefly expressed, is that any proposition characterises some fact, so that the relation of proposition to fact is the same as that of adjective to substantive. Bradley has represented a proposition as ultimately an adjective characterising Reality, and Bosanquet as an adjective characterising that fragment of Reality with which we are in immediate contact. In adopting the principle that a proposition may be said, in general, to characterise a fact, I am including with some modification what is common to these two points of view. One parallel that can be drawn between the relation of an adjective to a substantive and that of a proposition to a fact is that, corresponding to a single given sub- stantive, there are an indefinite number of adjectives which are truly predicable of it, just as there are many different propositions which truly characterise any given ^ Otherwise expressed : The proposition, subjectively regarded, is an assertibile; objectively regarded, d^possibile. THE PROPOSITION 15 fact. Thus we do not say that corresponding to a single fact there is a single proposition, but on the contrary, corresponding to a single fact there is an indefinite number of distinct propositions. Again, just as amongst the adjectives which can be truly predicated as charac- terising a given substantive, some are related to others as relatively more determinate ; so, amongst the several propositions which truly characterise a single fact, some characterise it more determinately and thus imply those which characterise the same fact less determinately. We may therefore regard the process of development in thought as starting from a fact given to be charac- terised, and proceeding from a less to a continually more determinate characterisation. Again there is an exact parallel between the relation of contradiction or contrariety amongst adjectives that could be predicated of a given substantive, and amongst propositions which could be formulated as characterising a given fact. Thus the impossibility of predicating certain pairs of adjectives of the same substantive in- volves the same principle as the impossibility of charac- terising the same fact by certain pairs of propositions: such pairsof adjectivesand propositions ar&incompatible, and this relation of mcom.patibility lies at the root of the notion of contradiction. We may illustrate the re- lation of incompatibility amongst adjectives by red and green regarded as characterising the same patch. It is upon this relation of incompatibility that the idea of the contradictory not-red depends; for not-red means some adjective incompatible with red, and predicates indeterminately what is predicated determinately by green, or by blue, or hy yellow,Q.\.Q„ Amongst propositions i6 CHAPTER I the relation of incompatibility may be illustrated by * Every p \s qu' and ' Every pv is non-^,' which are more determinate forms of the pair of contradictory propositions ' Every p is q' and ' Some p is non-^.' These latter derive their significance as mutually con- tradictory from the principle that the actual fact must be such that it could be characterised either by such a relatively determinate proposition as ' Every" ^ is qu' or by such a relatively determinate proposition as ' Every /z^ is non-^.' This account of the relation of contradiction as ultimately derived from that of incompatibility or con- trariety (whether applied to adjectives regarded as characterising substantives or to propositions regarded as characterising facts) brings out in another aspect the principle that any given substantive or any given fact may be truly characterised by a more or by a less determinate adjective or proposition : a topic which will be further developed in later chapters. The above logical exposition of the nature of a proposition leads to a consideration of the philosophical problem of the relation of thought to reality in one of its aspects. It is at the present day agreed that this relation cannot be taken to be identity, and the notion of correspondence has been put forward in its place. The above account enables us to give a more definite exposition of what more precisely this so-called corre- spondence entails : the truth of a judgment (expressed in a proposition) may be said to mean that the propo- sition is in accordance with a certain fact, while any proposition whose falsity would necessarily follow from the truth of the former is in discordance with that fact. THE PROPOSITION 17 In this way the somewhat vague conception of the correspondence between thought and reaHty is replaced by the relation of accordance with a certain fact at- tributed to the true proposition, and of discordance with the same fact attributed to the associated false propo- sition. J.L. i8 CHAPTER II THE PRIMITIVE PROPOSITION § I. The form of proposition which appears to be psychologically prior even to the most elementary pro- position that can be explicitly analysed is the exclama- tory or impersonal. Propositions of this kind, which are more or less unformulated and which, may be taken to indicate the early stages in a developing process, will here be called primitive. The most formless of such primitive propositions is the exclamatory assertion illustrated by such an utterance as 'Lightning!' This appears to contain only a characterising adjective with- out any assigned subject which is so characterised. Now it is true that any proposition can be regarded as a characterisation of the universe of reality regarded as a sort of unitary whole ; but this way of conceiving the nature of the proposition in general, must be also associated with the possibility of using adjectives as characterising a part rather than merely the whole of reality ; and certainly the case here is one in which we are bound to recognise the lightning as having, so to speak, an assignable place within the universe, and not merely as an adjective attached to the universe as a whole. The lightning as an actual occurrence must occupy a determinate position, in reference both to time and to space ; but it is obvious that no reference to such determinate position is itself contained in the merely exclamatory assertion. i\ny implicit reference to place THE PRIMITIVE PROPOSITION 19 or time can only be rendered explicit when the judg- ment has been further developed ; in the undeveloped judgment the reference is indeterminate, and any judg- ment which might be developed from this primitive form would assert what was unasserted in the original. In logical analysis it is of the utmost importance to avoid putting into an assertion what further development of the percipient's thought might elicit on the basis of the original. We ask then, how such judgment in its most primitive and undeveloped form can be conceived as referring to a subject when its verbal expression includes no such reference .'* Now we may speak of the presented occasions or occurrences that give rise to such in- completely formulated judgments as manifestations of reality. The exclamatory judgment 'Lightning' may thus be rendered formally complete by taking as subject term 'a manifestation of reality. ' Here I do not propose to take simply as the equivalent of the exclamatory judg- ment 'Reality is being manifested in the lightning,' but rather 'A pa7'ticular portion of reality manifests the character (indicated by the adjectival import of the word) lightning.' In short, what is asserted by the percipient is 'a manifestation of lightning\' This phrase for representing the assertum contains of course the characterising tie but not the assertive tie. The asser- tive tie may be introduced by employing the form: ''There is 3. manifestation of lio-htnino',' which raises the interesting problem as to the significance of the word ^ In grammatical phraseology, the expression 'manifestation of reality' illustrates the siibjective genitive, while 'manifestation of lightning' illustrates the objective genitive. 2 — 2 20 CHAPTER II 'there.' Like many other words in current language it is used here in a metaphorical, or perhaps rather in a general or abstract sense. Literally 'there' means 'in that place,' so that in its original significance it involves the demonstrative article, and furthermore — which is the new matter of interest — a reference to position in space. Moreover the tense of the verb is points to the present time. If these references were developed still more precisely, the assertion would become: ' There and now — in that place and at this time — is a mani- festation of lightning.' What remains as the significant element in the word 'there is,' in the absence of any definite reference to position in time or space, must be an indefinite reference to position in time and space. Otherwise the exclamatory assertion can only be ex- pressed by omitting the word 'there' altogether, and the assertion to which we are reduced — when the subject implicit in the exclamation is made explicit — becomes, as above, 'A manifestation of lightning^' § 2. The phrase 'there is' points to an important presupposition underlying the possibility of this most primitive form of perceptual judgment: namely, that things should be presented apart or in separation in order that any characterising judgment may be directed now to one and then again to another. Thus separation of presentment is a presupposition of cognition or judg- ment. Here I use the word 'presentment' not as equivalent to cognition, but as something presupposed in all — even the most primitive — acts of cognition. The ^ As an illustration of how words lose their philological origin and become merely metaphorical, consider the expressions : 'There is a God,' 'There is an integer between 5 and 7.' THE PRIMITIVE PROPOSITION 21 word 'present' — with the accent on the second syllable^ — is in English equivalent to ' give ' ; in this sense a pre- sentation is eqivalent to a datum — where by ' datum ' is meant not a piece of given knowledge, but a piece of given reality that is to be characterised in knowledge. Thus the presentation or the datum is what I have otherwise called the determinandum — that which is given or presented to thought to be thought about. This expresses briefly, the meaning of the primitive ' this.' The ' this ' as thus defined is not rich in predicates and adjectives, but at the same time it cannot be said to be empty of adjectives or predicates, because, in the mean- ing of thisness, abstraction is made from all predicates or adjectives. But the ' this ' cannot be explicated apart from an implicit reference to the ' that,' in the sense that the ' this ' must be for the percipient presented in separation from the ' that ' : one determinandum is one to which its own adjectives may be assigned, just because the other must be presented in separation or apart from the one, before the most primitive form of articulate judgment is possible". Briefly separateness is befoi-e relating; more specifically, it is the presupposition which makes it possible in more highly developed perception to define the relations (temporal or spatial) between those things which are first presented merely as separate. ^ When accented on the first syllable, its meaning combines a reference both to space and to time; so that the word presentation contains in its meaning the three factors in our analysis, viz. the given, the here, and the now. ^ It is here presumed that such mental processes as sense differ- entiation, etc., in which the experient is merely passive or recipient, must have been developed prior to the exercise of judgment, to furnish the material upon which the activity of thought can operate. 22 CHAPTER II It is in this quite ultimate sense that I demur to Mr Bradley's dictum : ' distinction implies difference.' His dictum means, as far as I understand, what in my own terminology I should express by the phrase : ' otherness presupposes comparison ' (the comparison, in particular, in which the relation of difference is asserted). Now in my view this dictum is exactly wrong : the assertion of 'otherness' does not presuppose or require a previous assertion of any relation of agreement or of difference. It does not even presuppose the possibility of asserting in the future any particular relation of agreement or of difference. The first important relation which will be elicited from otherness is, in fact, not any relation of agreement or difference at all, but a temporal or spatial relation ; and thus the primitive assertion of otherness is only occasioned and rendered possible from the fact of separateness in presentation. When presen- tations are separate, then we can count one, two, three; further, we can connect them by temporal relations such as before and after, or by spatial relations such as above or below ; and finally by relations of comparison such as like or unlike. These examples indicate my view of the quite primary nature of separateness of presentment, since it is for me the pre-requisite for all those acts of connecting with which logic or philosophy — and we may add psychology — is throughout concerned. In illustration, I have briefly referred only to relations of number, relations of time and space, and lastly to relations of comparison in a quite general sense. Summarising this attempt to indicate the precise logical character of such primitive judgments as the ex- clamatory or impersonal, and their relation to more THE PRIMITIVE PROPOSITION 23 highly developed judgments : we have found that to assert ' Lightning ! ' is to characterise, not reality as a whole, but a separate part of reality — to use an inade- quate expression — and that the possibility for this primitive assertion to develop into higher, more inter- related forms of judgment, is wholly dependent on the attribution of an adjective to a part of reality presented in separation from other presentables. For the purpose of further elucidation we may bring two or three assertions into connection with one another, which might be briefly formulated thus : * Lightning now ! ' ' Lightning again !' ' Thunder then ! ' The first two judgments when connected, involve two manifesta- tions of the same character denominated lightning, which are two because they have been separately pre- sented. The use of the terms ' now ' and ' then ' does not necessarily presuppose a developed system of temporal relations ; but they indicate at least the possi- bility of defining relations in time between separately presented manifestations. Again the exclamation 'Thunder!' when taken in connection with the exclama- tion ' Lightning ! ' already presupposes — not only that the manifestations are given somehow in separation — but further that the percipient has characterised the separated manifestations by different adjectives. I will not here discuss whether these manifestations (as I have called them) are, in their primitive recognition, merely the individual's sense-experiences of sound and light, or whether from first to last they are something other than sense-experiences. In either case our logical point will be the same, when it is agreed that they are given separately, and that their separate presentment is the 24 CHAPTER II precondition for any further development of thought or of perception. The view put forward here is so far equivalent to Kant's in that I regard space and time as the conditions of the otherness of sense-experiences upon which the possibility of cognising determinate spatial and temporal relations depends, and that this characteristic of space and time is what constitutes Sense-Experience into a manifold, i.e. a plurality of experiences, which we can proceed to count as many only because of their separate presentment. Taking more elaborate examples of these primitive forms of perceptual judgment: 'This is a flash of lightning,' ' This (same) flash of lightning is brighter than that (other),' * This (same) flash appeared before that clap of thunder ' ; we note that in the predesigna- tions 'this' and 'that' the percipient has passed beyond the indefinite article ' a,' and has identified a certain manifestation as that of which more than one characteri- sation can be predicated — e.g. 'lightning' and 'brighter than that.' It is this identification which gives to the articles ' this ' and ' that ' a significance which may be called referential, to be distinguished from their use as demonstratives; and in this alternation between the demonstrative and the referential usage, we can trace, I think, the very primitive way in which thought de- velops: first, in fixing attention upon a phenomenon by pointing to its position ; and, next, in identifying it as the same in character when it is changing its spatial relations. All that is theoretically required for identifica- tion is the retention — or rather the detention — of our cognition or judgment upon a certain manifestation ; but, when attributing different qualities or relations to what THE PRIMITIVE PROPOSITION 25 continues to function as the smne logical subject, we are assisted by the temporal continuance of a phenomenon, either with unaltered quality, or in an unaltered position, or in a continuously changing position, etc. Thus the changes involving variations in space, time, and quality amongst different manifestations of the same pheno- menon constitute the groundwork upon which the several judgments of relation are built. In asserting ' There was a flash of lightning that was very brilliant and that preceded a clap of thunder ' we are grasping the identity of a certain manifestation, thus used in two propositions, of which one predicates a relation in time to a clap of thunder, and the other, a quality characterising the flash itself. Any such con- nected judgment contains implicitly the relation of identity, in that the manifestation is maintained as an object in thought, while we form two judgments with respect to it. It is only, in short, in the act of joining two different characterisations that any meaning for identity can be found. In an elementary judgment which predicates only one adjective, no scope or signifi- cance for the notion of identifying a subject as such can be afforded. Thus the three factors in the thinking process which the ' this ' reveals are : ( i ) the given — which is equivalent to the ' it ' in 'It lightens ! ' (2) the demonstrative — which, by indicating spatial position, helps towards unique identification, (3) the referential — which marks the achievement of this process of identifi- cation. As will be seen from the discussion in a subse- quent chapter, these three elements of significance in the 'this' bring it into line with the proper name. 26 CHAPTER III COMPOUND PROPOSITIONS § I. Having examined the proposition in its more philosophical aspects, and in particular from the point of view of its analysis, the present chapter will be mainly devoted to a strictly formal account of the proposition, and will be entirely concerned with the synthesis of propositions considered apart from their analysis. The chapter is intended to supply a general introduction to the fundamental principles of Formal Logic; and for- mulae will first be laid down without any attempt at criticism or justification — which will be reserved for subsequent discussion. For this purpose we begin by considering the different ways in which a new proposi- tion may be constructed out of one or more given propositions. In the first place, given a single proposition, we may construct its negative — expressed by the prefix not — not -/being taken as equivalent to/-false. Next, we con- sider the construction of a proposition out of two or more given propositions. The proposition thus constructed will be called compound, and the component proposi- tions out of which it is constructed, may be called simple^ relatively to the compound, although they need not be in any absolute sense simple. The prefix not may be attached, not only to any simple proposition, but also to a compound proposition, any of whose components again may be negative. COMPOUND PROPOSITIONS 27 The different forms that may be assumed by com- pound propositions are indicated by different conjunc- tions. A proposition in whose construction the only formal elements involved are negation and the logical conjunctions is called a Conjunctioiial Function of its component propositions. The term conjunctional func- tion must be understood to include functions in which negation ^r any one or more of the logical conjunctions is absent. We have to point out that the compound proposition is to be regarded, not as a mere plurality of propositions, but as a single proposition, of which truth or falsity can be significantly predicated irrespectively of the truth or falsity of any of its several components. Furthermore, the meaning of each of the component propositions must be understood to be assignable irre- spectively of the compound into which it enters, so that the meaning which it is understood to convey when considered in isolation is unaffected by the mode in which it is combined with other propositions. § 2. We will proceed to enumerate the several modes of logical conjunction by which a compound proposition may be constructed out of two component propositions, say/ and q. Of all such modes of conjunction, the most fundamental is that expressed by the word and: this mode will be called par excellence conjunctive, and the components thus joined will be called conjuncts. Thus the compound propositions — (a) ^p and g,' {b) '^ and not-^,' {c) 'not-/ and q,' (d) 'not-/ and not-j^,' are the conjunctive functions of the conjuncts /, g", />, not-^ ; not-/, q ; not-/, not-^ ; respectively. There are thus four distinct conjunctive forms of proposition involving the two propositions/, q, taken positively or negatively. 28 CHAPTER III The significance of the conjunctional and will be best understood in the first instance, by contrasting it with the enumerative and. For example, we use the merely enumerative and when we speak of constructing any compound proposition out of the components/ ^;z^^. Here we are not specifying any mode in which/ and q are to be combined so as to constitute one form of unity rather than another; we are treating the components (so to speak) severally, not combinatorially. In other words, the enumeration — p and q — yields two propo- sitions, the enumeration — p and q and r — yields three propositions, etc.; but the conjunctive '/ and q,' or the conjunctive 'p and q and r etc., yields one proposition. Again, of the enumerated propositions — -/ and q and r and... — some may be true and others false; but the conjunctive proposition 'p and q and r and... ' must be either definitively true or definitively false. Thus in con- joining two or more propositions we are realising, not merely the force of each considered separately, but their joint force. The difference is conclusively proved from the consideration that we may infer from the conjunc- tive proposition '/ and q' a set of propositions none of which could be inferred from / alone or q alone. The same holds, of course, where three or more con- juncts are involved: thus, with p, q, r, as components, seven distinct groups^ of propositions are generated : viz. the three groups consisting of propositions implied by p, by q, by r respectively ; the three groups con- sisting of propositions implied by '/ and q' by 'p and r,' by 'q and r respectively; and lastly, the group con- sisting of propositions implied by p and q and r.' ^ The term group is here used in its precise mathematical significance. COMPOUND PROPOSITIONS 29 § 3. In our first presentation of formal principles we shall introduce certain familiarly understood notions, such as equivalence, inference etc., without any attempt at showing how some of them might be defined in terms of others. The same plan will be adopted in regard to the question of the demonstrability of the formal principles themselves; these will be put forward as familiarly ac- ceptable, without any attempt at showing how some of them might be proved by means of others. Ultimately, certain notions must be taken as intelligible without definition, and certain propositions must be taken as assertible without demonstration. All other notions (intrinsically logical) will have to be defined as de- pendent upon those that have been put forward with- out definition ; and all other propositions (intrinsically logical) will have to be demonstrated -^js, dependent upon those that have been put forward without demonstration. But we shall not, in our first outline, raise the question of the dependence or independence of the notions and propositions laid down. Thus the formal law which holds of Negation is called the Law of Double Negation: viz. not-not-/=/. § 4. We now lay down the formal laws which hold of compound propositions constructed by means of the conjunction and. They are as follows: Laws of Conjunctive Propositions 1. The Reiterative Law: p and/ =p. 2. The Commutative Law: p and q q and/. 3. The Associative Law: (/ and q) and r=p and (^ and r). 30 CHAPTER III Here the notion of equivahnce (expressed by the short- hand symbol =) is taken as ultimate and therefore as not requiring to be defined. These laws and similar formal principles are apt to be condemned as trivial. Their significance will be best appreciated by reverting to the distinction between the mental acts of assertion and progression in thought on the one hand, and the propo- sitions to which thought is directed on the other. Thus the laws above formulated indicate, in general, equiva- lence as regards the propositions asserted, in spite of variations in the modes in which they come before thought. Thus the content of what is asserted is not affected, firstly, by any r^-assertion ; nor, secondly, by any different order amongst assertions ; nor, thirdly, by any different grouping of the assertions. § 5. Having considered the Conjunctive form of proposition, we turn next to the consideration of the remaining fundamental conjunctional forms. These will be classed under the one head Co^nposite for reasons which will be apparent later. So far, compound propo- sitions have been divided into the two species Con- junctive and Co77iposite, and we shall now proceed to subdivide the latter into four sub-species, each of which has its appropriate conjunctional expression, viz. : (i) The Direct-Implicative function of/, q : — U/> then q. (2) The Counter-Implicative function of/, g : — U q then/. (3) The Disjunctive function of/, q : — Not-both/ and q. (4) The Alternative function of/, q : — Either/ or q. In the implicative function Tf/ then q,' p is implicans^ and q implicate ; in the counter-implicative function Tf ^ then/,'/ is implicate and q is implicans^* in the dis- ' The plural of implicans must be written : iniplicants. COMPOUND PROPOSITIONS 31 junctive function 'Not-both / and q^ p and q are dis- juncts; and in the alternative function 'Either/ or q,' p and q are alternants. These four functions of p, q, are distinct and independent of one another. The technical names that have been chosen are obviously in accordance with ordinary linguistic usage. The impli- cative and counter-implicative functions are said to be Co^nplementaiy to one another, as also the disjunctive and alternative functions. Each of the four other pairs, viz. (i) and (3); (i) and (4); (2) and (3); {2) and (4) may be called a pair of Supplementary propositions. These names are conveniently retained in the memory by help of The Square of Independence supplementary ^ Wo ^^.^ (2) Now when we bring into antithesis the four con- junctive functions : (i) p and not-^ ; (2) not-/ and q\ (3) / and q; (4) not-/ and not-^; with the four composite functions: (i) if/ then ^; (2) if-then/; (3) not-both / and ^ ; (4) either/ or ^; we shall find that each of the composite propositions is equivalent to the negation of the corresponding con- junctive. This is directly seen in the case (3) of the conjunctive and the disjunctive functions of/, q. Thus, 32 CHAPTER III 'Not-both/ and q' is the direct negative of 'Both / and q' Again, for case (4) 'Either / or ^' is the obvious negative of 'Neither p nor q' The relations of negation for all cases may be derived by first systematically tabulating the equivalences which hold amongst the composite functions, as below : abbreviating not-/ and not-^ into the forms ^ and q respectively. Table of Equivalences of the Composite Functions Counter- Implicative Implicative Disjunctive Alternative Form Form Form Form 1. li p then q = \i g then ^ = Not both p and ^ = Either p or q 2. \{ p then g = \i q then / = Not both p and ^ = Either / or q 3. \i p then q=\{ q then / = Not both p and ^ = Either ^ or q 4. \i p then q=\i q then /=Not both p and ^ = Either p ox q In the above table it will be observed: {a) That each composite function can be expressed in four equivalent forms : thus, any two propositions in the same row are equivalent, while any two propositions in different rows are distinct and independent. (<5) That the propositions represented along the principal diagonal are expressed in terms of the positive components /, q\ being in fact identical respectively with the implicative, the counter-implicative, the dis- junctive and the alternative functions of/, q. {c) That all the remaining propositions are ex- pressed as functions of/ and not-^^, or of not-/ and q, or of not-/ and not-^. We may translate the equivalences tabulated above in the form of equivalences of functions, thus : I. The implicative function of/, q-, the counter- implicative function of not^, not-^; the disjunctive func- COMPOUND PROPOSITIONS 33 tion of/, not-^ ; and the alternative function of not-/, q, are all equivalent to one another. Again, 2. The counter-implicative function of/, q\ the im- plicative function of not-/, not-^; the disjunctive function of not-/, q\ and the alternative function of/, not-^, are all equivalent to one another. Again, 3. The disjunctive function of/, q\ the alternative function of not-/, xioX.-q\ the implicative function of/, not-^; and the counter-implicative function of not-/, q^ are all equivalent to one another. Again, 4. The alternative function of/, and ^. 3. The disjunctive 'Not both/ and ^' „ „ „ '/ and $^.' 4. The alternative 'Either /> or q' „ „ „ '/ and ^.' Thus inasmuch as no composite function is equiva- lent to any conjunctive function, we have justified our division of compound propositions into the two funda- mentally opposed species Conjunctive and Composite. J.L. 3 34 CHAPTER III The distinction and relation between these compo- site forms of proposition may be further brought out by tabulating the inferences in which a simple conclusion is drawn from the conjunction of a composite with a simple premiss. In traditional logic, the Latin verbs ponere (to lay down or assert) and tollere (to raise up or deny) have been used in describing these different modes of argument. The gerundponendo (by affirming) or/^//^?^^(9 (by denying) indicates the nature of the (simple) premiss that occurs ; while the participle ponens (affirm- ing) or tollens (denying) indicates the nature of the (simple) conclusion that occurs : the validity or invalidity of the argument depending on the nature of the com- posite premiss. There are therefore four modes to be considered corresponding to the four varieties of the composite proposition, thus : Table of Valid Modes Modus 1. Ponendo Ponens 2. Tollendo Tollens If/ then ^; but/; If-then/; but/ : .*. q 3. Ponendo Tollens : Not both/ and q\ but/; .•. q 4. Tollendo Ponens : Either/ or ^;but/;.". ^ Form of Composite Premiss q The Implicative. The Counter-Implic. The Disjunctive. The Alternative. The customary fallacies in inferences of this type may be exhibited as due to the confusion between a composite proposition and its complementary: Table of Invalid Modes Modus 1. Ponendo Ponens : If ^ then/; but/ 2. Tollendo Tollens : If/ then q ; but / 3. Ponendo Tollens : Either/ or ^;but/ 4. Tollendo Ponens : Not both/ and q ; but / Form of Composite Premiss The Counter-Implic. The Implicative. The Alternative. The Disjunctive. COMPOUND PROPOSITIONS 35 The rules for correct inference from the above table of valid modes may be thus stated : 1. From an implicative, combined with the affirma- tion of its implicans, we may infer the affirmation of its implicate. 2. From an implicative, combined with the denial of its implicate, we may infer the denial of its implicans. 3. From a disjunctive, combined with the affirma- tion of one of its disjuncts, we may infer the denial of the other disjunct. 4. From an alternative, combined with the denial of one of its alternants, we may infer the affirmation of the other alternant. § 6. We ought here to refer to an historic con- troversy as regards the interpretation of the conjunction 'or.' It has been held by one party of logicians that what I have called the Alternative form of proposition, viz., that expressed by either-or, should be interpreted so as to include what I have called the Disjunctive, viz., that expressed by not-both. This view has undoubtedly been (perhaps unwittingly) fostered by the almost uni- versal misemployment of the term Disjimctive to stand for what ought to be called Alternative. This prevalent confusion in terminology has led to a real blunder com- mitted by logicians. The blunder consists in the falla- cious use of the Ponendo Tollens as exhibited in the table above given. Consider the argument : 'A will be either first or second'; 'It is found that A is second'; there- fore A is not first.' Here the conclusion is represented as following from the promising qualifications of the candi- date y^, whereas it really follows from the premiss 'A can- not be both first and second.' In fact, the Alternative 36 CHAPTER III proposition which is put as premiss is absolutely irrele- vant to the conclusion, which would be equally correctly inferred whether the alternative predication were false or true. It remains then to consider whether the logician can properly impose the one interpretation of the alter- native form of proposition rather than the other. The reply here, as in other similar cases, is that, in the matter of verbal interpretation, the logician can impose legislation — not upon others — but only upon himself. However, where any form of verbal expression is ad- mittedly ambiguous, it is better to adopt the interpreta- tion which gives the smaller rather than the greater force to a form of proposition, since otherwise there is danger of attaching to the judgment an item of significance beyond that intended by the asserter. This principle of interpretation has the further advantage that it compels the speaker when necessary to state unmistakeably and explicitly what may have been implicitly and perhaps confusedly present in his mind. I have therefore adopted as my interpretation of the form Either-or that smaller import according to which it does not include Not-both. Those logicians who have insisted on what is called the 'exclusive' interpretation of the alternative form of pro- position (i.e. the interpretation according to which ^zV/^^r- or includes Not-both) seem sometimes to have been guilty of a confusion between what a proposition asserts, and what may happen to be known independently of the proposition. Thus it may very well be the case that the alternants in an alternative proposition are almost always 'exclusive' to one another; but this, so far from proving that the alternative proposition affirms COMPOUND PROPOSITIONS 37 this exclusiveness, rather suggests that the exclusive- ness is a fact commonly known independently of the special information supplied by the alternative proposi- tion itself. In this connection, the significance of the term com- plementary which I have applied to the implicative and counter-implicative as well as to the disjunctive and alternative, may be brought out. Propositions are ap- propriately called complementary when a special im- portance attaches to their conjoint assertion \ Thus it may be regarded as an ideal of science to establish a pair of propositions in which the implicans of the one is the implicate of the other; and again to establish a number of propositions which are mutually co-disjunct and collectively co-alternate. The term complementary is especially applicable where propositions are conjoined in either of these ways, because separately the propo- sitions represent the fact partially, and taken together they represent the same fact with relative completeness. We next consider the inferences that can be drawn from the conjunction of two supplementary propositions. These may be tabulated in two forms, the first of which brings out the fundamental notion of the Dilemma] and the second that of the Reductio ad Im,possibile. First Table for the Conjunction of Supplementaries The Dilemma (1) *If^ then q^ and (4) 'If/ then q' : therefore, q. (3) 'If/ then ^' and (2) 'If/ then q' : therefore, q. (2) 'If ^ then/' and (4) 'If j? then/': therefore,/. (3) 'If {? then/' and (i) 'If-then/': therefore,/. ^ Thus complementary propositions might be defined as those which are frequently confused in thought and frequently conjoined in fact. 38 CHAPTER III The above table illustrates the following principle: The conjunction of two implicatives, containing a common implicate but contradictory implicants, yields the affirmation of the simple proposition standing as common implicate. Or otherwise: Any proposed proposition must be true when its truth would be implied both by the supposition of the truth and by the supposition of the falsity of some other proposition. Second Table for the Conjunction of Supplementaries The Rednctio ad Impossibile (i) 'If ^ then/' and (4) 'If ^ then /^': therefore, $r. (3) 'If ^ then/' and (2) 'If-then/': therefore,^. (2) 'If/ then ^' and (4) 'If/ then ^': therefore,/. (3) 'If/ then ^' and (i) 'If/ then $^': therefore,/. This second table illustrates the following principle : The conjunction of two implicatives, containing a common implicans but contradictory implicates, yields the denial of the simple proposition standing as com- mon implicans. Or otherwise: Any proposed proposition must be false when the supposition of its truth would imply (by one line of argument) the truth and (by another line of argument) the falsity of some other proposition. § 7. In tabulating the formulae for Composite pro- positions as above I have merely systematised (with slight extensions and modifications of terminology) what has been long taught in traditional logic ; and it is only in these later days that criticisms have been directed against the traditional formulae, especially on the ground that their uncritical acceptance has been found COMPOUND PROPOSITIONS 39 to lead to certain paradoxical consequences, which may- be called the Paradoxes of Implication. In this con- nection it is (I think) desirable to explain what is meant by a paradox. When a thinker accepts step by step the principles or formulae propounded by the logician until a formula is reached which conflicts with his common-sense, then it is that he is confronted with a paradox. The paradox arises — not from a merely blind submission to the authority of logic, or from any arbi- trary or unusual use of terms on the logician's part — but from the very nature of the case, as apprehended in the exercise of powers of reasoning with which every- one is endowed. In particular, the paradoxes of impli- cation are not due to any unnatural use of the term implication, nor to the positing oi 2.x\.y fundamental for- mula that appears otherwise than acceptable to com- mon sense. It is the formulae that are derived — by apparently unexceptionable means from apparently un- exceptionable first principles — that appear to be excep- tionable. Let us trace the steps by which we reach a typical paradox. Consider the alternative 'Not-^ or q.' If this alternative were conjoined with the assertion '/,' we should infer 'q! Hence, 'Not-/» or q' is equivalent to Tf / then q! Similarly '/ or ^'is equivalent to Tf not-/ then q! Now it is obvious that the less deter- minate statement 'p ox q' could always be inferred from the more determinate statement '/': e.g. from the rela- tively determinate statement '^ is a solicitor' we could infer *^ is a solicitor or a barrister' i.e. 'y^ is a lawyer.' Hence, whatever proposition q may stand for, we can infer '/ or ^' from 'p' \ or again, whatever/ may stand 40 CHAPTER III for we can infer 'not-/ or q' from q. Hence (i) given */' we may infer 'If not-^ then q^ and (ii) given '^' we may infer 'If/ then q,' whatever propositions/ and q may stand for. These two consequences of the uncritical acceptance of traditional formulae have been expressed thus: (i) A y^/^-i? proposition (e.g. not-/ when / has been asserted) implies any proposition (e.g. q) ; (ii) A true proposition (e.g. q, when q has been asserted) is implied by any proposition (e.g. /). Thus '2 + 3 = 7' would imply that 'It will rain to-morrow'; and 'It will rain to-morrow' would imply that '2 + 3 = 5.' That these two implicative statements are technically correct is shown by translating them into their equivalent alterna- tive forms, viz.: (i) 'Either 2 + 3 is unequal to 7 or it will rain to-morrow'; (ii) 'Either it will not rain to-morrow or 2 + 3 = 5.' ^^ ^^y certainly say that one or other of the two alternants in (i) as also in (ii) is true, the other being of course doubtful. Taking 'If/ then q' to stand for the paradoxically reached implicative in both cases, we have shown that (i) from the denial of / (the implicans), and (ii) from the affirmation of q (the implicate) we may pass to the assertion 'If/ then q! This is, of course, only another way of saying that the implicative 'If / then q' is equivalent to the alternative '/ false or q true.' Thus when we know that ' If/ then q' is true, it follows that we know that 'either/ is false or q is true' ; but it does not follow that either 'we know that / is false' or 'we know that q is true.' The paradoxically reached im- plicative merely brings out the fact that this may be so in some cases: i.e. when asserting 'If/ then ^,' there are cases in which we know that '/ is false,' and there COMPOUND PROPOSITIONS 41 are cases in which we know that 'q is true.' But it is proper to enquire whether in actual language — literary or colloquial — the implicative form of proposition is ever introduced in this paradoxical manner. On the one hand, we find such expressions as: 'If that boy comes back, I'll eat my head'; 'If you jump over that hedge, I'll give you a thousand pounds' ; ' If universal peace is to come tomorrow, the nature of mankind must be very different from what philosophers, scientists and historians have taken it to be'; etc., etc. Such phrases are always in- terpreted as expressing the speakers intention to deny the implicans ; the reason being that the keaj^eris assumed to be ready to deny the unplicate. Again, on the other hand, we find such forms as: 'If Shakespeare knew no Greek, he was not incapable of creating great tragedies.' 'If Britain is a tiny island, on the British Empire the sun never sets.' 'If Boswell was a fool, he wrote a work that will live longer than that of many a wiser man. ' 'If Lloyd George has had none of the advantages of a public school education, it cannot be maintained that he is an unintelligent politician.' Such phrases are always interpreted as expressing the speaker s intention to affir7n the implicate, the reason being that the hearer may be assumed to be willing to affirm the implicans. Looking more closely into the matter we find that when a speaker adopts the implicative form to express his denial of the implicans, he tacitly expects his hearer to supplement his statement with a tollendo tollens\ and when he adopts it to express his affirmation of the implicate, he expects the hearer to supplement it with 2iponendo ponens. Furthermore, inasmuch as the alter- native form of proposition requires to be supplemented 42 CHAPTER III by a tollendo {ponens) and the disjunctive by 2iponendo {tollens), we find that an impHcative intended to ex- press the denial of its impHcans is quite naturally ex- pressed otherwise as an alternative: e.g. 'That boy won't come back or I'll eat my head,' to which the hearer is supposed to add 'But you won't eat your head'; therefore (I am to believe that) 'the boy won't come back' [tollendo ponens)', and we find that an implicative intended to express the affirmation of its implicate is quite naturally expressed otherwise as a disjunctive : e.g. 'It cannot be held that Shakespeare both knew no Greek and was incapable of creating great tragedies,' to which the hearer is supposed to add ' But Shakespeare knew no Greek,' and therefore (I am to believe that) 'he was capable of creating great tragedies ' (ponendo tollens). We have yet to explain how the appearance of paradox is to be removed in the general case of a com- posite being inferred from the denial of an implicans (or disjunct) or from the affirmation of an implicate (or alternant). Now the ordinary purpose to which an implicative (or, more generally, a composite) proposition is put is inference : so much so that most persons would hesitate to assert the relation expressed in a composite proposition unless they were prepared to use it for purposes of inference in one or other of the four modes, ponendo ponens, etc. In other words. Implication is naturally regarded as tantamount to Potential Inference. Now when (i) we have inferred 'If/ then q' from the denial of '/»,' can we proceed from 'If/ then q' con- joined with '/' to infer '^'? In this case we join the affirmation of */' with a premiss which has been in- ferred from the denial of / ; and this involves Contra- COMPOUND PROPOSITIONS 43 diction, so that such an inference is impossible. Again, when (ii) we have inferred 'If/ then q' from the affirm- ation of '^,' can we proceed from 'If/ then q' con- joined with '/' to infer '^'? In this case we profess to infer V' t>y means of a premiss which was itself inferred from '^'; and this involves Circularity, so that this inference again must be rejected. The solution of the paradox is therefore found in the consideration that though we may correctly infer an implicative from the denial of its implicans, or from the affirmation of its implicate, or a disjunctive from the denial of one of its disjuncts, or an alternative from the affirmation of one of its alternants, yet the implicative, disjunctive or al- ternative so reached cannot be applied for purposes of further inference without committing the logical fallacy either of contradiction or of circularity. Now it must be observed that the rhetorical or colloquial introduction of a paradoxical composite, which is meant to be inter- preted as the simple affirmation or denial of one of its components, achieves its intention by introducing — as the other component of the composite — a proposition whose falsity or truth (as the case may be) is palpably obvious to the hearer. The hearer is then expected to supplement the composite by joining it with the obvious affirmation or denial of the added component, and thereby, in interpreting the intention of the speaker, to arrive at the proposition as conclusion which the speaker took as his first premiss. Accordingly the process of interpretation consists in taking the same propositions in the same mode and arrangement as would have en- tailed circularity if adopted by the speaker. § 8. The distinction between an implicative pro- 44 CHAPTER III position that can and one that cannot be used for in- ferential purposes may now be further elucidated by reference to the distinction between Hypothesis and Assertion. In order that an implicative may be used for inference, both the implicans and the implicate must be entertained hypothetically. In the case oi ponendo ponens the process of inference consists in passing to the assertion of the implicate by means of the assertion of the implicans, so that the propositions that were entertained hypothetically in the implicative, come to be adopted assertively in the process of inference. The same holds, mutatis 77tutandis, for the other modes. Now when we have inferred an implicative from the affirmation of its implicate or from the denial of its implicans — as in the case of the implicative which ap- pears paradoxical — the two components of the impli- cative thus reached cannot both be regarded as having been entertained hypothetically; and hence the prin- ciple according to which inference is a process of pass- ing from propositions entertained hypothetically to the same propositions taken assertorically, would be vio- lated if we used the composite for inference. This con- sideration constitutes a further explanation of how the paradoxes in question are solved. The above analysis may be symbolically represented by placing under the letter standing for a proposition the sign \- to stand for assertorically adopted and the sign H for hypothetically entertained. Thus the fundamental formula for correct inference may be rendered : From '/ would imply ^' with/; we may infer q, H HI- I- COMPOUND PROPOSITIONS 45 where, in the implicative premiss, both implicans and impHcate are entertained hypothetically. Now the following inferences, which lead to para- doxical consequences, may be considered correct: i.e. {a) From g, we may infer '/ would imply (/.' {d) From p, we may infer '/ would imply g.' h h H But the implicative conclusions here reached cannot be used for further inference: i.e. (f) From '■p would imply g' with/; we cannot infer a. (d) From '/ would imply g' with/; we cannot infer g. h H h I- For in (c) the implicate, and in {d) the implicans enters assertorically, and these inferences therefore con- travene the above fundamental formula which requires that both implicate and implicans should enter hypo- thetically. Thus while admitting [a) that 'a true pro- position would be implied by any proposition,' yet we cannot admit (c) that 'a true proposition can be inferred from any proposition.' Similarly, while admitting {b) that 'a false proposition would imply any proposition,' yet we cannot admit {d) that 'from a false proposition we can infer any proposition.' In fact, the attempted inference (c), where the conclusion has already been as- serted, would entail cii'cularity ; and the attempted infer- ence {d), where the premiss has already been denied, would involve contradiction. Still maintaining the equivalence of the composite propositions expressible in the implicative, the counter- implicative, the alternative or the disjunctive form, each of these four forms will give rise to a like paradox. The 46 CHAPTER III following table gives all the cases in which we reach a Paradoxical Composite; that is, a Composite which cannot be used for inference, either in the modus po- nendo ponens, tollendo to/lens, ponendo tollens or tollendo ponens. The sign of assertion in each composite must be interpreted to mean asserted to be true when the term to which it is attached agrees with the premiss, and asserted to be false when it contradicts the premiss. Table of Paradoxical Composites {a) From g we may properly infer h (i) p or q=\ip then q—\{q then^ = Not both^ and q, Hf-H|-f-H Hh or (2) / or ^=If/ then 0- = ^ ^ then 2J = Not both^ and ^. Hf-Hf-I-H H|- {b) From q we may properly infer h (3) P or q = \ip then q=-\iq then/ = Not both/ and q, HI-HHHH Hh or (4) ^ or ^ = If j> then ^ = If 0" then / = Not both/ and ^. H|-Hf-f-H H|- The above composites can never be used for further inference. Thus : in line ( i ), the attempted inference '/ .•. q'' would be circular and 'not-^ .*. not-/' would be contradictory ; in line (2), the attempted inference 'not-/ .•. ^' would be circular, and 'not-^ .•./' would be contradictory; in line (3), the attempted inference 'not-/ .'. not-^' would be circular, and '^ •'•/' would be contradictory; in line (4), the attempted inference '/ .'. not-j'' would be circular, and '^ .". not-/' would be contradictory. The paradox of implication assumes many forms, some of which are not easily recognised as involving mere varieties of the same fundamental principle. But COMPOUND PROPOSITIONS 47 I believe that they can all be resolved by the consider- ation that we cannot witJiout qualification apply a com- posite and (in particular) an implicative proposition to the further process of inference. Such application is possible only when the composite has been reached irrespectively of any assertion of the truth or falsity of its components. In other words, it is a necessary con- dition for further inference that the components of a composite should really have been entertained hypo- thetically when asserting that composite. § 9. The theory of compound propositions leads to a special development when in the conjunctives the components are taken — not, as hitherto, assertorically — but hypothetically as in the composites. The conjunc- tives will now be naturally expressed by such words as possible or compatible, while the composite forms which respectively contradict the conjunctives will be expressed by such words as necessary or impossible. If we select the negative form for these conjunctives, we should write as contradictory pairs : Conjunctives {possible) Composites {necessary) a. p does not imply q b. p is not implied by q c. p is not co-disjunct to q d. p is not co-alternate to q a. p implies q b. p is implied by q c. p is co-disjunct to q d. p is co-alternate to q Or Otherwise, using the term 'possible' throughout, the four conjunctives will assume the form that the several conjunctions — pg,pg, pq ^nd pq — are respectively /^.r- sid/e. Here the word possible is equivalent to being merely hypothetically entertained, so that the several conjunctives are now qualified in the same way as are the simple components themselves. Similarly the four 48 CHAPTER III corresponding composites may be expressed negatively by using the term 'impossible,' and will assume the form that the ^ CHAPTER V is false and that * S is,' may be explained by making explicit the descriptive, adjectival, or connotative factor in the term symbolised by S, which factor we shall symbolise by M. The negative premiss then becomes : 'If anything is M it will not be P' : the categorical premiss becomes: 'A certain thing is J/.' In this formulation the symbol 5 does not appear. Now ^ stands for a certain thing which has not yet been identified and which is only presented in thought by the general description M ; or briefly S is to mean * a certain thing which is M.' In transforming the pro- position 'Anything that is M will not be P' into the form ' S will not be /^ ' we introduce the factor ' a cer- tain thing.' And in transforming the proposition ' A certain given thing is J/' into the form '^ is' we have transferred the whole of the adjectival component in the proposition from the predicate to the subject : or other- wise, the two propositions maybe rendered 'Anything that may be given having the character M will not be P ' and ' A certain thing having the character M is given.' Thus it is not strictly correct to use the same symbol 5" in our two propositions, since the only differentiating element of meaning in the term kS in the negative premiss is the adjectival or descriptive com- ponent, whereas in the categorical premiss the substan- tival component enters along with the adjectival. It follows that the analysis given is not restricted to the negative form of our first premiss, since the same kind of syllogism would apply to an affirmative conclusion : the essential characteristic of the first premiss is its hypothetical character, as opposed to the other premiss which is categorical. Thus the affirmative case would NEGATION 79 be rendered * Anything that may be given having the character M will be P^ ' A certain thing with the character M is given' therefore 'This thing having the character M will be P.' § 6. In the course of this final explanation it will be noted that for the formula '6^ is' in which no determinate adjective is predicated, we might substitute 'S is given' or '^ is real.' Now the words 'given' and ' real' though of course grammatically adjectival, are not in the logical sense adjectival, for their meaning does not contain any indication of character or relation. It may be remarked in passing that the application of the term 'real' includes but goes beyond that of the word 'given.' The postulate that has to be assumed is that, however indeterminately we may have been able to characterise it, the real must have some determinate character. We thus return to our first exposition of the force of the incomplete pre- dication ' 5 is ': namely that S, as being real, must have some determinate character although it may be that this character cannot be completely or exactly known by any finite intelligence. 8o CHAPTER VI THE PROPER NAME AND THE ARTICLES § I. A simple proposition '5 is /*' involves as sub- ject a single uniquely determined substantive, and as predicate a single uncompounded adjective; a singular proposition must satisfy the first of these conditions, but its predicate may be simple or compound. Pro- positions of this nature give rise to the question how the reference in the subject can be uniquely determined. Speaking generally, singular names may be divided into two classes according as they contain or do not contain an explicit adjectival or relational component : to the first class belong such terms as 'the smallest planet,' 'the king of England who signed Magna Charta,' 'the cube root of 8 '; to the second class ' Mercury,' 'John,' and ' 2 ' ; and the former will, for convenience, be referred to as descriptive or significant, the latter as proper or non-significant. Compare now the proper name 'Poincare' with the descriptive name 'the President of France.' In order that this latter term may have unique application its component ' France' must have unique application : and hence here, as in almost every case, the uniqueness of a descriptive name is only secured through its reference to a proper name. On the other hand we shall find that many so-called proper names contain a descriptive factor: thus the term 'England' contains the termination 'land' which would be normally understood as bringing the THE PROPER NAME AND THE ARTICLES 8i term ' England ' within the general class expressed by the word 'land.' This, however, would not explicitly hold of the name ' France ' ; but, in point of fact, just as England might be taken to inean ' the land of the Angles,' so might France be taken to mean ' the land of the Franks.' In the same way the terms ' Mr Gladstone ' and ' Lord Beaconsfield ' have the partial significance expressed by the prefixes 'Mr' and 'Lord' respectively. As another example, the name ' Mont Blanc,' though etymologically equivalent to ' white mountain ' and therefore apparently completely signifi- cant, must yet be called a proper name since its appli- cation is not to any white mountain, but to a specific one. We are here in the reverse position to that reached in discussing the term 'England'; for in 'England' we detected the concealed element of significance indicated by the termination 'land,' while in 'Mont Blanc' we have detected the concealed element of non-significance which prevents us from applying the name to any white mountain indiscriminately. Now, attributing to the term ' Mont Blanc ' the maximum of significance that it can bear, and agreeing that there would be a species of in- correctness in using the term for any object which had not the characteristics 'white ' and ' mountain,' yet this admitted significance is not the sufficient ground for applying the term as it is understood by those who use it with a common agreement as to its unique applica- tion. We may therefore say that any name which is commonly called a proper name has, so far as our analysis has proceeded, a residual element of non-signifi- cance over and above such significance as is naturally recognised in the verbal structure of the name. J. L. 6 i59T4^ 82 CHAPTER VI § 2. But if we allow of any name that it contains an element of non-significance, how is it possible that this name should be understood as applying to the same object when used at different times or by different persons or in different and varying connections ? Where the name denotes a substantive, the possibility that it should mean the same substantive when used in different propositions, involves the possibility of substantival identification. A similar process, i.e. adjectival identifi- cation, is involved when the same adjective is used in different connections. Whether we ask how the sub- stantive-name ' Snowdon,' for instance, can be under- stood to stand for a definite substantive, or how the adjective-name 'orange' can be understood to stand for a definite adjective, we are in fact confronted with pre- cisely the same logical problem ; and hence, if we regard the name 'Snowdon' as a proper substantive-name, we must regard 'orange' as a proper adjective-name. The analogy may be pressed a little further and applied to complex names; for just as an adjective name is ex- hibited as significant when it is expressed in the form of adjectives combined in certain relations, so a sub- tantive name is exhibited as significant when it is expressed in the form of substantives combined in cer- tain relations. Note the analogy, for example, between the complex adjective-name ' the colour between red and yellow ' and the complex substantive-name ' the highest mountain in Wales.' Here the former involves the proper adjective-names ' red ' and ' yellow,' just as the latter involves the proper substantive-name 'Wales.' There is, then, in every explication of significance, a residual element in which we reach either a substantive- THE PROPER NAME AND THE ARTICLES 83 name or an adjective-name which can no longer be de- fined inthis form of analysis. And further, the explication of the significance of this residual proper substantive or proper adjective-name involves the conception of iden- tity: in the one case, substantival identity — which is implied when we understand that the substantive de- noted by the word ' Snowdon ' in one proposition is identical with that denoted by ' Snowdon ' in another proposition ; and, in the other case, adjectival identity — which is implied when we understand that the adjective denoted by 'orange' in one proposition is identical with the adjective denoted by 'orange' in another proposition. Our first approximate account of a proper name is then, that the intended application of the given name is to an object — whether it be substantive or adjective — which is identical with the object to which it may have been previously understood as applying in another pro- position. For example, to explain what I mean by 'orange' I could say: 'You understand the word colour: and I shall mean by "orange" the colour which you can discern as characterising the object to which I am pointing. And when you identify the colour of any ob- ject with the colour of this, its colour is to be called "orange".' The possibility of such appeal presupposes that colour can be perceptually identified in different objects, apart from any other agreements or differences that the objects may manifest. In the same way the explication of a proper substantive-name requires a similar appeal, which assumes the possibility of identi- fying a concrete object when it may be presented or thought about in different contexts. Thus, if it was asked whom I meant when I talked of Mr Smith, I 6-2 84 CHAPTER VI might say : ' I mean the man to whom you were intro- duced yesterday in my study.' The agreement, there- fore, which can be maintained in the application of a proper name amongst those who continue to use it with mutual understanding, is secured by what in a quite general way we may call the method of introduction. An object is introduced, and in the introduction a name is given, and when further reference is intended to the same object, the name is repeated which was given in the act of introduction. In this way the nature of the residual element or undefinable factor in adjectives and relations as well as that in the case of proper names is further explained. § 3. It is worth pausing here to point out a confusion frequently made in discussing the nature of the proper name. The confusion is that between the cause which has led people to choose one name rather than another na7ne for a given application, with the reason for ap- plying the name — once chosen — to one object rather than to another object. This confusion again can be paralleled in ordinary adjectival names as well as in substantival names: thus, it is one thing to assign the etymological causes of the use of the name ' indigo ' rather than some other name to denote a particular colour, and another thing to assign the reason for ap- plying the name, when it has once come into common usage, to one of the colours rather than to some other: the reason for this latter is that the colour presented in a given instance is identical with that to which the name indigo was originally given. This is exactly parallel to the ground on which we should justify our applying the name 'Roger Tichborne' to the man presented in court: THE PROPER NAME AND THE ARTICLES 85 namely, the presumed identity of the man before us with the man to whom his godparents had given the name. Why they chose that name rather than some other name is a matter for historical enquiry. But, to repeat, the etymological or historical account of a name must not for a moment be confused with its significance, or with what, in the case of proper names (substantival or adjectival), takes the place of significance as the condition of mutual understanding. § 4. This discussion is closely bound up with the different ways in which the articles, indefinite and de- finite, are used. The indefinite article in its most general and completely indeterminate meaning is illustrated by such assertions as * A man must have been in this room,' 'We need a sweep,' ' You ought to make a move with your bishop.' Now if we compare this use of the article with its meaning when it occurs at the beginning of a narrative as for instance: 'Once upon a time there was a boy who bought a beanstalk,' we note an important difference in its significance. I n the first set of examples the full significance of the article is made explicit by substituting 'some or other': e.g. 'Some or other man must have been in this room'; in the case of a narrative, where the article prepares the way for future references to period, person or place, it means — not 'some or other' — but 'a certain.' Indeed our story might more logically have begun ' At a certain time a certain boy bought a beanstalk.' When the indefinite article is used in this way to introduce some period, person or place not otherwise indicated, it will henceforward be called the Introductory Indefinite, to distinguish it from the Alternative Indefinite. 86 CHAPTER VI Now suppose the narrative to continue: 'This boy was very lazy'; the phrase 'this boy' means 'the boy just mentioned,' the same boy as was introduced lo us by means of the indefinite article. H ere the article ' this, ' or the analogous article 'the,' is used in what may be called its referential sense. The linguistic condition necessary to render such reference definite is that only one object of the class (whether person, period, or place) should have been immediately before mentioned. Other variations of the Referential Definite are such phrases as 'the former' and 'the latter,' which may be required to se- cure definite reference. The above analysisbrings out the necessarily mutual association of the introductory use of th^indefinite article with t\\Q.7^eferentialus& o{ ih.&defi7tite article. Again, instead of beginning the second sentence with the phrase 'this boy,' language permits us to use a pronoun: thus the word 'he,' in general, is sufficient to denote a specific individual understood by the verbal context; so that here the pronoun serves precisely the same logical function as the referential definite article 'the' or 'this.' A still more important further develop- ment of the referential 'the' comes up for consideration when, instead of depending upon immediacy of con- text — as in the preceding cases of 'this' and 'he' — we refer to an historical personage who has a wide circle of acquaintance as (e.g.) 'The well-known sceptical phi- losopher of the eighteenth century.' Here the phrase 'the well-known' functions as a referential definite, though there may have been no immediately previous mention of Hume, it being assumed that a certain phi- losopher will be unambiguously suggested to readers in general, in spite of the fact that there may have been THE PROPER NAME AND THE ARTICLES 87 more than one person answering to the description 'scep- tical philosopher of the eighteenth century.' This ex- tended use of the referential definite is quite interestingly illustrated in Greek, where a proper name is prefixed by the definite article ' 6 ' ; a usage which appears very happily to bring out the precise function of the proper name, as referring back to an individual who was origin- ally introduced in history or otherwise under that name. The same holds in English of geographical proper names, e.g. the Thames, the Hellespont, the Alps, the Isle of Wight, etc. Lastly, in a narrative, the juxta- position of a proper name with the introductory in- definite supplies a substitute for the referential definite. Thus our story about the beanstalk which begins with the introductory indefinite 'a boy' may be continued either by using the phrase 'this boy' — involving the referential article — or by the pronoun 'he'; or thirdly by the proper name which prepares the way for repeated reference to the same boy: 'Once upon a time there was a boy named Jack who bought a beanstalk.' It will be noted therefore that the way in which the proper name occurs in a narrative where it secures continuity of reference, illustrates the same principle as its use in ordinary intercourse, where it ensures agreement amongst different persons as to its single definite appli- cation : in both cases, the understanding of the applica- tion of the name involves reference back to the act of introduction, when the name was originally imposed. There is an important analogy between the singular descriptive name of the kind illustrated by 'the well- known sceptical philosopher of the eighteenth century' and the proper name, in that frequently it is only within 88 • CHAPTER VI a narrower or wider range of context that the proper name may be said to have a uniquely determined appH- cation. Thus, within a family, the name 'John' maybe understood to denote one brother of that name ; whereas, in a certain period in English history, it will denote the king who signed Magna Charta. That uniqueness of reference is relative to a particular context is similarly seen in such phrases as 'the table,' 'the garden,' 'the river,' which though applicable to different objects in different contexts are understood within a given circle or in a given situation to have a uniquely determined application. The article 'the' used in such cases may be called Indefinite Definite, to distinguish it from the most definite of all uses of the article, namely where the unique application is understood without any limita- tion of context — in cases, for example, like 'the sun,' 'the earth.' We have thus divided articles (and what are logic- ally equivalent to articles) into four classes: (i) the Indefinite Indefinite, otherwise the Alternative Indefi- nite; (2) the Definite Indefinite, otherwise the Instan- tial Indefinite, best expressed by the phrase 'a certain,' which includes the Introductory Indefinite; (3) the Indefinite Definite, otherwise the Contextual Definite, which includes the Referential Definite; and (4) the Definite Definite, for which the understood reference is independent of context. § 5. A special form of the contextual definite which is to be distinguished from the referential, is expressed by the terms 'this' and 'that' when used as demonstra- tives. Literally, the demonstrative method is limited to the act of introducing an object within the scope of THE PROPER NAME AND THE ARTICLES 89 perception. But, when we point with the finger, for instance, to a particular person or mountain or star, our attempt to direct the attention of the hearer to the object intended may or may not succeed : if successful, it will be because there is no other conspicuous object belong- ing to the class indicated by the use of the general significant name (person, mountain, star, as the case may be) within the range of space to which we have directed attention. The condition for securing unam- biguity is not that there should be only one object of the specified class within the range indicated, but that there should be. only one such visible object ; and here observe a parallel between the demonstrative definite, and the case illustrated by the example 'the well-known sceptical philosopher of the eighteenth century.' § 6. At this point in our discussion let us consider the special difficulty which attaches to the notion of a proper name. This problem presents a dilemma. If we maintain that the proper name is non-significant in some sense, then it would follow that any propositional phrase that might contain the proper name would be non-significant in the sanie sense. If, on the other hand, we attempt to assign some definite significance to the proper name, this will entail our substituting a uniquely descriptive name as equivalent in meaning to the proper name, in which case the distinction between the de- scriptive name and the proper name would vanish. This problem raises a question relating to the wider problem of the definition of words or phrases. Taking the two words 'valour' and 'courage,' the brief formula 'valour means courage' is seen on reflection to be im- perfectly expressed. Everybody would agree that what 90 CHAPTER VI is intended here is that the two terms va/our Siud courage have the same meaning; i.e. that the quaHty meant by the one term is the same as the quaHty meant by the other. Hence a more correct expression than 'valour means courage' would be 'the word valour means what is meant by the word courage.' Where a phrase in- stead of a single word is under consideration the same principle is involved. For example, 'p is a factor of q' means- what-is-meant-by 'q is divisible by /'; or again 'some benefactor of A' means-what-is-meant-by 'one or other person who has benefited A' These illustra- tions bring out the distinction between {a) the relation which one word or phrase may bear to another word or phrase, and (<$) the relation which a word or phrase may bear to what is called its 'meaning.' Now the propositions which allow us to substitute one phrase for another may be called bi-verbal defi- nitions^; and the relation that is to be affirmed as hold- ing between two such phrases must be expressed in the complex form 'means what is meant by,' or even — when we distinguish between the phrase which has not been understood and that which has been understood — in the still more complicated form 'is to be understood to mean what has been understood to be meant by.' This last complication brings out the purpose that a definition has always to serve; namely the elucidation of a phrase assumed to require explanation in terms of a phrase presumed to be understood. ^ It has been suggested that a more correct substitute for '■bi- verbal definition'' would be '^translation.'' But whichever terminology is employed, the distinction between the kind of definition called translation and some more ultimate definition remains. THE PROPER NAME AND THE ARTICLES 91 This formulation of the bi-verbal definition leads us to consider what, in contrast, we shall call the uni- verbal definition. When we speak of a phrase as being 'already understood,' it is equivalent to saying that the meaning of the phrase is known. The formula that 'phrase p means what is meant by phrase q^ in short, raises the question, What is it that phrase q means? Let us first consider the kind of entity that a phrase could mean. Phrases which would be grammatically or logically distinguished in regard to type or category would mean entities belonging to correspondingly dif- ferent types or categories: thus one phrase would mean a certain proposition; another would mean a certain adjective, another a certain substantive, and so on. Thus 'courage' means a certain adjective or quality-of- conduct, 'horse' means a certain substantive or kind-of- animal. A phrase prefixed by an article such as a, the, some, every, any, requires special consideration. Thus, if we were to substitute for such phrase any phrase that means what is meant by the given phrase, the article or some equivalent would still remain. Thus 'the first novel of which Scott was the author' means what is meant by 'the romance that was written by Scott before any other of his romances.' In this bi-verbal substitu- tion the word 'the' is retained. Now consider, in con- trast to the proposition stating the equivalence in mean- ing of the above phrases, the proposition ' The first novel written by Scott was called Waverley' ', or, inas- much as there is only one novel that is known bearing this name, we may put the statement in the form: 'The first novel written by Scott was the novel called Waver ley! Such a proposition is of nearly the same 92 CHAPTER VI type as 'The author of Waver ley was the author of Marmion' In both of these propositions the relation of identity is asserted in regard to two uniquely de- scriptive terms. But neither of these propositions is verbal ; in neither case could we substitute for the rela- tion of identity the expression 'means what is meant by.' Hence we are not identifying the meaning of the two phrases: i.e. we are not identifying what is meant by one phrase with what is meant by the other. What then is it that we are identifying? In the language of Mill we should say we are identifying what is denoted — and in the language of Frege what is indicated or (as we prefer to say) factually indicated — by the one phrase with what is denoted or factually indicated by the other. Now, as our term suggests, an appeal X.Qfact\s required in order to understand what it is that is factually indi- cated in distinction from what is meant by a certain phrase. Hence, though a knowledge of the usage of language alone is sufficient to know what a phrase means, a knowledge of something more than mere linguistic usage is required to know what a phrase de- notes or factually indicates, whenever we are dealing with a phrase that indicates something different from what it means. The word 'courage' or the phrase 'not flinching from danger' is of such a nature that there is no distinction between what it means and what it indi- cates or denotes; it is only phrases prefixed by an article or similar term for which the distinction between mean- ing and indication arises. Turn now to the peculiarities of the illustration given above: 'The first novel written by Scott was the novel called Wave7'ley' The inter- pretation of this statement is that the object indicated THE PROPER NAME AND THE ARTICLES 93 by the phrase that stands first is the same as that indicated by the phrase 'the novel called Waverlcy' although the meanings of the two phrases differ. Take a parallel case: 'The colour of the object at which I am pointing is identical with the colour that is called red ' ; here again the identity of what is indicated by the two phrases does not carry with it identity in what is meant by the two phrases. In short, where we have an identification of what is indicated in spite of non- identity in what is meant, we recognise that the state- ment of identity is not merely verbal but factual. In the above illustrations we have taken such names as Scott and Waverley to exemplify names universally recognised as proper; while the phrase 'the first novel written by Scott' — or any phrase having the same mean- ing — would be called descriptive in a sense primarily intended as antithetical to proper. Now one step was taken to bridge this antithesis when we used the proper name in the extended phrase 'the novel called Waver- ley '. i.e. the single name Waverley is a proper name and the compound phrase 'the novel called Waverley' is constructed in the form of a descriptive name. We are thus leading up to the view that what is indicated by the descriptive phrase — 'the novel called Waverley' — is identical with what is meant by the proper name 'Waverley.' Thus, in interpreting the simple propo- sition ' Waverley was the first novel written by Scott,' which is recognised at once to be factual not verbal, we are identifying what is factually indicated by the subject and predicate terms respectively ; and in the case of the proper name 'Waverley,' what it factually indicates is indistinguishable from what it means. Hence it seems 94 CHAPTER VI legitimate or possible to define a proper name as a name which ?7teans the same as what \i factually indicates. § 7. We may now introduce the technical term ' osten- sive' which will suggest as its opposite the familiar term 'intensive.' A proper name may be said to be osten- sively definable in contrast to those more ordinary terms which are said to be intensively definable. This osten- sive definition will be only a special instance of a form of definition involving the complex relation 'means what is indicated by' — a relation which is involved in any attempt to define a proper name by means of a de- scriptive name. The particular force of the notion of ostensive definition will now be explained, and it will be found to apply both to an adjectival and to a sub- stantival name. The ordinary proper name applies to an object whose existence extends over some period of time and generally throughout some region of space. The appearance of such an object in perception (or rather of some spatially or temporally limited part of that object) provides the necessary condition for im- posing a name in the act of indicating, presenting or introducing the object to which the name is to apply, and this it is that constitutes ostensive definition. In extending the notion of a proper name to certain adjectives our justification is that ultimately a simple adjective-name — such as red — cannot be defined ana- lytically but only ostensively. Theoretically, we must suppose that any name, singular or general, proper or descriptive, substantival or adjectival, has originally been imposed on a particular occasion by a particular person or group of persons. In the case of ostensively defined names, the occasion on which definition is pos- THE PROPER NAME AND THE ARTICLES 95 sible must be one on which the object is actually pre- sented. When, however, the meaning or application of such a name has afterwards to be explained, or so-to- speak redefined, the only direct method is to secure for the enquirer another presentation of the object in question. Thus John Smith, having been presented to his family at birth — which we may take to be the occa- sion on which the name was imposed — must be pre- sented again to the person ignorant of its application. Hence, in introducing a man under the name John Smith, we are using the same ostensive method as was required in the original definition, but such mention of the name does not, properly speaking, constitute defi- nition. We are stating, in effect, the proposition — which is not merely verbal — that 'the person introduced is identical with the person upon whom the name was originally imposed.' This case of an ordinary substan- tival proper name is analogous to that of an adjectival name — say cochineal — which originally could only have been ostensively defined, and which must therefore be ostensively redefined for the person ignorant of its application, in the form 'the colour of this presented object is identical with that upon which the name cochi- neal was originally imposed' — a statement which again is not merely verbal. When a proper name is called arbitrary, this arbitrariness attaches only to the original act of imposition ; but, when the application of the name is afterwards explained, such explanation is no longer arbitrary, since to be correct the real proposition that the substantive or adjective presented is identical with that upon which the name was originally imposed, must hold good, and this statement may be either true or 96 CHAPTER VI false, apart from linguistic convention. Furthermore, when ostensive definition is employed, it must be ob- served that we do not say that the proper name means what is meant by such a phrase as 'the object to which I am pointing' (which after all is only an instance of a descriptive phrase), but we say that the proper name means what is indicated hy the descriptive phrase 'the object to which I am pointing.' For it is obvious in this case, as in the more general account of a descrip- tive phrase, that however we may further explicate the meaning of the phrase 'the object to which I am pointing,' the substituted phrase would not have the nature of a proper name but necessarily of a descrip- tive name. When, then, finally we agree with the general posi- tion of the best logicians that the proper name (as Mill says) is non-connotative, this does not amount to say- ing that the proper name is non-significant or has no meaning; rather we find, negatively, that the proper name does not mean the same as anything that could be meant by a descriptive or connotative phrase ; and positively, that it does precisely jnean what could be indicatedhy some appropriate descriptive phrase. This exposition holds both for the names of objects which can be presented and thus ostensively defined ; and also for the names of objects removed in time or place, for the definition of which a descriptive phrase (which is other than ostensive) must be employed. 97 CHAPTER VII GENERAL NAMES; DEFINITION AND ANALYSIS § I . Having, in the preceding chapter, distinguished the different kinds of articles, we now turn to a common characteristic in the use of an article, namely, its attach- ment to a general name. The general name has usually been differentiated by reference to number, and roughly defined as a name predicable of more than one object. In fact, however, there are general names such as ' in- teger between 3 and 4 ' or ' snake in Ireland' that are predicable of no object, while ' integer between 3 and 5 ' and ' pole-star ' are general names predicable of only one object. There is therefore nothing in the meaning of a general name which could determine the number of objects to which it is applicable. Rejecting this re- ference to number, we may point out that a universal characteristic of the general name is its connection with the article — the use of the grammatical term 'article' being extended to include this, that, some, every, any, etc. All terms of this kind serve to determine the in- tended application of reference in a proposition, and hence might more properly be called applicatives or seledives. Now a general name is distinguished as that to which any applicative can be significantly prefixed: thus the applicative, on the one hand, requires a general name, while, on the other hand, it follows from the essence of the general name that to it any applicative is J.L. 7 98 CHAPTER VII significantly attachable. And a further and connected characteristic of the general name is that it can always be used in the plural, or, in fact, with any numerical prefix. If now we further consider the typical applicatives ' every ' and ' some,' we find that in the last analysis they entail the conception of unlimited generality, and language reflects this unlimited reference by uniting these applicatives with the substantival term ' thing ' to form the single words ' everything ' and ' something,' where the word ' thing ' stands only for abstract generality without any such limitation as could possibly be defined by adjectival characterisation. If any account of the name ' thing ' here could be given in philosophical ter- minology, we should have to say that ' thing ' stands for the category of all categories; or more precisely, that the generality implied in this use of the word ' thing ' is so absolute that it must include all logical categories, such as proposition, adjective, substantive, etc. On the other hand, the general term with which ordinary logic is almost exclusively concerned, is re- moved by two steps from this absolute generality: for example, the general term ' man ' comes under the logical category 'substantive,' which itself is a limitation of the absolute generality peculiar to the word ' thing.' Thus in philosophical logic we ought to be prepared to define the category ' substantive ' by some limiting characterisation within the absolute general 'thing'; and similarly of the category 'adjective' or 'proposition,' etc. And again, in ordinary logic, any general name is understood to be defined first, as coming under an understood category — in particular the category sub- ^ GENERAL NAMES; DEFINITION AND ANALYSIS 99 stantive — and secondly as delimited by a certain adjec- tival characterisation. As regards any specified general substantive-name, its nature as general is brought out by showing its meaning to be resolvable into some specific adjective or conjunction of adjectives, juxtaposed to the category 'substantive' itself. The consideration that to the general name any applicative can be prefixed, distinguishes it from the singular name, whether descriptive or proper. We have now to bring out a characteristic shared in common by the general name and the singular descriptive name, which distinguishes them both from the proper name : namely that the two former always (except in the case of the absolute general name ' thing') contain in their meaning an adjectival factor, whereas the strictly proper name contains in its meaning no adjectival factor. This account may seem a somewhat arbitrary way of settling the prolonged controversy initiated by Mill as to whether proper names are non-connotative. In default of a definition of a proper name, however, it is impossible to decide whether any given name, such as London, is to be called proper. I propose, therefore, to define the word proper as equivalent to non-connotative, non-de- scriptive or non-significant (since these three terms are themselves synonymous), and the only debatable point which remains is as to whether any names can properly be called 'proper'.' A similar remark applies to the question whether all general names are connotative, since it would seem necessary to define a general name ■* This brief account of the Proper Name has been discussed and qualified in the preceding chapter; but for the present purpose it is adequate. 100 CHAPTER VII (except the absolutely general name * thing') as equi- valent to one which is connotative. § 2. At this point we must enquire into the precise meaning of the word connotation, and this enquiry necessitates the introduction of the more general notions of intension and extension. While extension stands for a set of substantives, intension stands for a set of adjec- tives ; and moreover the two terms are used in correlation with one another, the substantives comprised in the extension being characterised by the adjectives com- prised in the intension. Now of all the adjectives that may characterise each of a set of substantives, a certain sub-set will have been used for determining the appli- cation of the general name, and the range of extension thereby determined constitutes the denotation of that name. Thus the specific function of connotation is that it is used to determine denotation; and hence any other adjectives that may characterise all the substantives comprised in the denotation, do not determine the denotation, but rather are determined by it. The entire and often innumerable conjunction of adjectives deter- mined by the denotation has been called by Dr Keynes, the Comprehension. Thus Dr Keynes's exposition may be summed up in the statement that connotation in the first instance determines denotation, which in its turn determines comprehension. The controversy on the question of what adjectives should be included in the connotation of the general term, has arisen from the false supposition that the logician starts with a known range of denotation, and that with this datum he has to dis- cover, amongst the known common characterspeculiar to the members of the class, those which shall constitute the GENERAL NAMES; DEFINITION AND ANALYSIS lor connotation. But this is reversing the order of priority, since there is no means of deHmiting the range of de- notation of a term otherwise than by laying down a finite enumeration of adjectives which, taken together, con- stitute the test for applying the name. This test, which is provided by the connotation, may have remained unchanged, or have varied in the course of time, or again have been technically determined by science in such a way as to conflict more or less with common usage ; but in every case what constitutes the connotation is invariably a conjunction of adjectives which, within these limits of time or context, is used to determine the application of the name. We are indebted to Mill for an elaborate and com- plete treatment of connotative names, including the connection between connotation and definition ; but the formula that the definition of a general name is the un- folding of its connotation, must be corrected by the added reference to the substantival element, since con- notation by itself is purely adjectival. It is obviously impermissible, for instance, to substitute for the sub- stantival name * a man ' a mere adjective ' human ' or the abstract term ' humanity ' ; rather ' a man ' should be defined as 'a human being' or ' a being characterised by the attribute humanity.' In other words, a substan- tive-name must be so defined as to show that it is sub- stantival. The word ' being ' here has the force of the word 'substantive,' so that pushed to its logical con- clusion, ' a man ' should be defined as ' a human sub- stantive'; and it is the adjective 'human' which alone may require further analysis in our definition, not the word 'being' or 'substantive.' 102 CHAPTER VII The same principle will apply to the definition of a general adjectival name, if it is admitted that certain adjectives and relations (which may be called secondary) can be properly predicated of other adjectives (which in this connection might be called primary). For example, in the proposition : ' the unpunctuality of his arrival was annoying,' we appear to be predicating of the primary quality or adjective represented by the abstract name 'unpunctuality' the further or secondary adjective ' annoying.' By a secondary adjective we mean an adjective of an adjective, and if the primary adjective is expressed grammatically as a substantive, the secondary adjective would be expressed grammatically as an ad- jective, but if the primary adjective retains its gram- matical form as an adjective, then its secondaryadjective is expressed by an adverb, which is logically equivalent to an adjective of an adjective. Thus the following series of propositions: 'A is moving,' 'the movement of A is rapid,' ' the rapidity of the movement of A is surprising' — involve the primary adjective 'moving,' the secondary adjective 'rapid,' and what we must here call the tertiary adjective 'surprising.' When the primary adjective 'moving' retains its adjectival form, the secondary adjective predicated of it would be expressed as an adverb; thus 'A is moving rapidly'; or when the secondary adjective ' rapid ' retains its adjectival form, the tertiary adjective predicated of it is again expressed as an adverb, as in 'A's movement was sur- prisingly rapid.' These examples seem to confirm the view that adjectives can properly be predicated of ad- jectives as such. If this analysis is correct, we should expect to find certain general adjectival names (ana- GENERAL NAMES; DEFINITION AND ANALYSIS 103 logous to general substantival names) whose meaning could be elicited in terms of the secondary adjectives implied in their use. Thus to take Mill's example, the name 'fault' which ispredicable of such primary qualities or adjectives as 'laziness,' 'unpunctuality,' 'untidiness,' is predicable of such qualities of conduct on the ground that these are characterised by the further or secondary quality ' faultiness.' In this way the definition of a general adjectival name would be formulated in terms of the secondary adjective or conjunction of adjectives, constituting its connotation, juxtaposed to the category adjective itself. § 3. Turning now to another aspect of our topic, we shall consider the nature of definition only in the form of bi-verbal substitution, where the reference is restricted to words or phrases; in contradistinction to ideas or things, which some philosophers have under- taken to define. The question of the definition of words requires a wider treatment than that generally accorded to it in logic; for under the influence, I pre- sume, of scholastic doctrine, definition has been tacitly restricted to the case of substantive terms, to which the traditional formula ' per differentiam et genus ' is alone applicable. It seems, indeed, as if logicians had shrunk in terror from the task of defining other than substantive- terms, setting aside preposition words, conjunction words, pronominal words and even adjectives, as if these, as such, were outside the scope of logical definition. The problem of definition, it is clear, must extend to any^or^, however it may be classified by grammar. We must certainly come to some mutual understanding of the meanings of such words as ' and,' ' or,' ' if,' inasmuch as 104 CHAPTER VII these and many other such words have a meaning, the understanding of which is essential in the pursuit of logic. It is frequently impossible, however, to define a word taken in isolation, and in such cases, we must construct, as that which is to be directly defined, a verbal phrase containing the word. We should not perhaps go far wrong if in every such case we took the completed propositional phrase as that which is to be defined, although it is often possible of course to take only some part of a proposition and succeed in meeting the re- quirements. We repeat then that our problem is how to define a given verbal phrase ; and the answer is to substitute for it another verbal phrase. This is the complete and quite universal account of the procedure of definition, which justifies our restriction of the topic to bi-verbal definition ; its obvious purpose is fulfilled if the substituted phrase is understood. (Cp. preceding chapter.) In this connection it is worth noting that, when what has to be defined is a verbal phrase rather than a single word, we may italicise — so to speak — that part of the phrase for which an explanation is asked. In such cases the remaining components of the phrase may be, and generally ought to be, repeated in the phrase constituting the definition. In symbols, let us say that the phrase ' abed ' requires definition, where the components 'be are combined in the whole 'abed' in such way that the combination 'abed' is not understood. Suppose the symbol ' apqd ' to represent our definition ; then we shall have defined 'abed' (where the component 'be' required explanation), by the phrase 'apqd' in which 'pq' explicitly replaces 'be^ and is offered in expla- GENERAL NAMES; DEFINITION AND ANALYSIS 105 nation. A definition such as that symboHsed above is rejected in current logical text-books on the ground that it commits the sin of tautology; for it repeats verbatim the symbol 'ad' in the definition given as explanation of 'abed! But this mode of definition, so far from being a ground of condemnation, exactly answers in the most adequate sense the requirements. The more exactly we repeat in our definition the actual words and their form of combination, used in the phrase to be explained, the more precisely do we meet the demands for an explanation. Typical instances of this principle will occur to anyone who reflects on the subject. It follows that no general or merely formal criticism of a definition can be made by any logical rule ; the question whether any proposed definition is good or not being entirely relative to the enquirer's knowledge and ignorance of meanings. This by no means precludes the possibility of definitions which would be generally useful, because any obscurity or ambiguity which one person might feel is likely to be felt by others ; a little common sense is in general all that is necessary. This account leads at once to one conclusion, which is perhaps tacitly understood by all logicians and philo- sophers; i.e., that, inasmuch as the only way to explain one verbal phrase is to substitute another, therefore no successive chain of explanatory phrases can serve the purpose of ultimate explanation, if that chain is endless. Hence perhaps the important point in the theory of explanatory definition is that it must stop. In other words, by a shorter or longer process, every definition must end with the indefinable. A certain misunder- standing as to what in logic is meant by the indefinable io6 CHAPTER VII must here be removed ; for it has been frequently sup- posed that the indefinable means that which is admittedly not understood. But so far from meaning the ' not- understood,' the indefinable means that which is under- stood ; and philosophy or logic may ultimately adopt a term as indefinable only where, because it is understood, it does not require a further process of definition. Phi- losophy must never stop with the indefinable, in the sense of reaching a component of thought expressed obscurely or with an admitted margin of doubt as to meaning. The indefinable does not therefore mean that which is presented as having no understood meaning, but that whose meaning is so directly and universally understood, that it would be mere intellectual dishonesty to ask for further definition. § 4. There has been practical unanimity in re- garding definition as a process which essentially in- volves analysis. We have above reached the idea of an indefinable; and it has been almost universal, I believe, to regard the indefinable as equivalent to what is incapable of analysis — a view which is obvious if definition and analysis coincide. But, if definition is not to be merely equated to analysis, then we may pause before we regard anything as being indefinable on the ground of its being, in some proper sense, unanalysable. In my own view, definition assumes so many varied forms that its equivalence to analysis seems to be highly dubious. Before entering into further detail on this point it will be well to consider what is meant by the word analysis in philosophy and logic. Associated with it we often find a reference to parts and wholes: thus, analysis is GENERAL NAMES; DEFINITION AND ANALYSIS 107 often said to mean the separation of a whole into its parts. But ' separation ' does not adequately repre- sent the process. For instance, grammatical analysis does not mean taking the several words of a sentence and determining the part of speech and inflection, etc., of each word, but its object is to show how the signifi- cance of the sentence is determined by the mode in which the several words are combined. Similarly the analysis of a psychosis is not merely the cataloguing of such elements as knowing, feeling, acting, but rather the representation of the essential nature of a given psychosis as determined by the mode in which these factors combine. What holds of grammatical and psy- chological analysis holds of every kind of analysis ; and, since the important process is — not the mere revelation of the parts contained — but rather the indication of their mode of combination within the whole, analysis is better defined as the exhibition of a given object in the form of a synthesis of parts into a whole. In this way we can say that any process of analysis can also be described as a process of synthesis; but this does not amount to saying that analysis means the same as synthesis, any more than that the relation 'grandfather' is the same as the relation ' grandson,' although the fact that A is the grandfather of B is the same fact as that B is the grandson oi A. In short, analysis is the inverse of syn- thesis ; i.e. when the whole X is analysed into its several components a, b, c, d; then a, b, c, d have to X the in- verse relation which X has to a, b, c and d. In this way it is clear that to analyse X simply means the same as to exhibit X as a synthesis. Now instead of taking ^ as a term to be defined. io8 CHAPTER VII and exhibiting it as a synthesis of a, b, c and d, let us take the term a and define it by showing how it functions in a whole X where it is combined with b, c and d. This points to two modes of definition, viz. analytic and synthetic. In analytic definition we pass from an unanalysed — i.e. an apparently simple — to an analysed equivalent; in synthetic definition we exhibit the nature of what is simple, not by representing it as a complex, but by bringing it into synthetic connec- tion in a complex of which it is a component. Or more shortly: analytic definition is explaining a complex in terms of its components, and synthetic definition is explaining components in terms of a complex. A few illustrations will help to make clear exactly what we mean by synthetic definition. Take, in arithmetic, the definition of 'factor' or 'multiple.' We first construct a certain complex, involving integers illustratively sym- bolised by a, b, c\ — namely, the proposition axb = c; this complex, being understood, is used to define the terms that require definition, and the definition assumes the following form : a is said to be a factor of c, or c is said to be a multiple of a, when the relation ex- pressed in the proposition aY.b = c holds. Here we do not resolve the meanings of the terms 'factor 'or 'multiple' into their simple components of meaning, but define them by showing in what way they enter as components into the understood construct — viz. the particular equa- tional proposition. Again taking the algebraic definition of ' logarithm ' : we begin by constructing a certain whole — expressed for convenience in terms of general illustrative symbols b, I, p, — viz. the proposition that ' b to the power / equals /. ' This constructed complex — GENERAL NAMES; DEFINITION AND ANALYSIS 109 the nature of which is presumed to be understood — is the medium in terms of which the notion of a logarithm will be defined as follows: '/ is the logarithm of/ to the base b ' means what is meant by the proposition : ' <^ to the power / equals p! Again take the definition of 'sine': here, starting with an angle A bounded at the point O by the lines OX, OR, we make the following construction ; taking some point P in OR, and dropping the perpendicular PM upon OX, then the sine of A means the ratio of MP to OP. The constructed com- plex here, namely of a certain right-angled triangle, has first to be indicated and understood, and by its means we are enabled to define what may be called a component of this complex. Another example of this form of de- finition is afforded in any attempt to define the words 'substantive' and 'adjective.' Here we may presuppose that the notion of ' proposition ' is understood — e.g. as that of which 'true' or 'false' may be significantly pre- dicated — and further presupposing that the notion of characterisation is understood, the first account of sub- stantive and adjective will be that they are combined in a proposition in a mode expressible either in the form that ' a certain substantive is characterised by a certain adjective, 'or that 'a certain adjective characterises a certain substantive.' Here we give at the same time the definitions of substantive and of adjective by showing how, as components in the whole, i.e. the proposition, they have to be combined \ § 5. The general notions of analysis and synthesis ^ All the definitions occurring in a symbolic system, whether Logical or Mathematical, should in my view be synthetic (in the above sense) and never analytic. no CHAPTER VII are often explained in terms of parts and whole ; but these latter terms should be used for a process essen- tially different from analysis. At least three processes which are commonly confused are here to be carefully distinguished, viz., partition, resolution, and analysis proper; probably other more subtle variations must finally be recognised. By partition is meant transforming what is first presented as a mere unit by exhibiting it in the form of a whole consisting of parts; it is perhaps more generally defined as the process of dividing a whole into its part ; but it is of the first importance to point out that until a thing is presented as having parts, it cannot be said to be a whole. This conception of part and whole should be strictly limited to three types of cases : ( i ) to an ag- gregate, and to a number as the adjective of an aggregate; {2) to what occupies space and to the space which it occupies ; {3) to what fills time and to the time filled. These three cases bring out the essential nature of the conception, viz., that the parts must always be conceived as homogeneous with one another and with the whole which they constitute; and further that a certain character called magnitude is predicable of any whole, the measure of which is equal to the sum of the magnitudes predicable of the parts\ Next, consider the term resolution. This is very shortly explained as the process of exhibiting a com- posite in terms of its components; but such a definition is open to the same kind of criticism as we have levelled ^ No intensive or qualitative characteristic of an object can be regarded as a 'whole' of which a magnitude can be predicated by addition of the magnitudes of its 'parts.' GENERAL NAMES; DEFINITION AND ANALYSIS in against the popular definition of partition, and by a parallel emendation we shall say that resolution means the exhibition of what is presented as simple in the form of a composite of which the components are assigned. An example of psychological interest is the resolution of a chord heard into its component notes, or again of a note into its component tones, where in either case the combination describes or accounts for the sound as heard. While, on Helmholtz's theory of auditory sensa- tions, the physiological process here involved would be represented as a whole capable of partition, it remains none the less true that psychological apprehension pre- sents the sound as a composite to be resolved. Then thirdly, I should restrict the word analysis to a process which is distinctively logical, and which as- sumes its simplest form when we combine various adjectives as predicable of one and the same substantive, by means of the mere conjunction 'and.' A simple ex- ample will bring out the distinction between resolution and analysis. We have shown what is meant by resolving a note into its component tones. Now the character of the note is described under certain aspects, such as pitch, intensity and timbre, and this constitutes an analysis of its character. These three characteristics are predicated of the sound, not in the sense of re- solving the sound into various component sounds, but in the sense of characterising the sound itself, whether it be composite or simple. Thus taking timbre for instance as one of the constituent characters, if a note contains three partial tones this would count as 3 in its resolution but i only in its analysis ; if on the other hand the note were simple — i.e. contained only one 112 CHAPTER VII component tone — this would count as i in the resolution, while the analysis of this single note yields the same 3 characters (pitch, timbre and intensity) as that of the composite note. The value of this illustration is that it conclusively disposes of the assumption that a plurality of predications characterising an object depends at all upon its partition or resolution; that is, upon regarding the object either as a whole consisting of parts or as a composite resolvable into components. So far we have considered that form of analysis which exhibits its object as a synthesis of constituents conjoined by the conjunction 'and,' and yielding what will be termed a compound. In contrast to a compound synthesis we must consider also what must be called a complex synthesis — namely one in which the material constituents are heterogeneous, including substantives of different kinds and adjectives of different order — monadic, diadic, triadic' — which, qua heterogeneous, are united in different modes from that of simple con- junction. Thus the word 'courageous' yields the complex synthesis 'not flinching from danger' ; where the material constituents are 'danger' and 'flinching from,' of which the former is expressed substantivally and the latter as a diadic adjective. From this fairly simple example it will be seen that the possible forms of complexity that analysis may yield are inexhaustible. The discussion of this topic will be continued from a somewhat different aspect in a subsequent chapter on relations. ^ See Chapter IX. "3 CHAPTER VIII ENUMERATIONS AND CLASSES § I. An enumeration is an assignment of certain items which may be said to be comprised in the enumera- tion. We attach therefore to an enumeration the con- ception of unity as appHed to the whole along with plurality of the items comprised in this whole. In naming the items to be comprised in an enumeration as a, b, c, d, e, for instance, it will generally be implied that we shall not repeat any item previously named; also, that the order of assignment is indifferent. For the purposes of elementary illustration, we shall consider that all the items ultimately to be enumerated have a finitely as- signed number (say) 1 2 : which may be named respec- tively a, b, c, d, e,f,g, k, k, I, m, n. Such an enumeration might be called our enumerative universe. Thus taking any assigned enumeration included in this universe, we may speak of the remainder to this enumeration — by which will be meant the items comprised in the universe, but not comprised in the first assigned enumeration. The notion of remainder is therefore associated with the notion of not\ although the two must be strictly distinguished. The remainder to an assigned enumera- tion is the simplest function of a single enumeration with which we shall be concerned. We next consider the typical functions of two enumerations — E, F ^2.y: J. L. 8 114 CHAPTER VIII namely ' E into /%' and ' E with F.' By the former is meant the largest enumeration which is included both in E and F\ by the latter, the smallest enumeration that includes both E and F. Thus, let E be \a, b, c, d, e,/] and let F be [d, e,f, g, k~\', then 'E into F' will be [d, e,f']\ and ' E with F' will be [a, b, c, d, e,f,g, ^]. Anticipating elementary arithmetical notions, we may at once assert the generalisations: first, that the number for ' E into F' cannot be greater than the number for E or the number for F; and secondly, that the number for ' E with F' cannot be less than the number for E or the number for F. HE and F are identical, then in every sense of the word equals, E into F=E with F = E = F. This represents one limiting case. If, on the other hand, some items comprised in E are com- prised in F, the number for ' E with F' is less than the sum of the numbers for E and F respectively, and if there are no items comprised both in E and in F, then the number for ' E with F' will equal the sum of the numbers for E and F respectively, while the number for ' E into F' is zero. In the former case E and F would be said to be not exclusive, and, in the latter, ex- clusive of one another. For the general case: The number for {^E mX.oF) + the number for {E with F) = the number for ^-f-the number for F. For the purpose of further development, we will abbreviate the term remainder into R\ The symbol R , i.e., remainder to, and the prepositions with, into may be called '■ operators,' because each indicates a certain operation to be performed upon one or upon two enumerations by means of which another related single enumeration of the same order is to be constructed. ENUMERATIONS AND GLASSES 115 The relation of the given operation to the enumeration which is to be constructed will be called the relation of yielding. Now, noting that R [«, b, c, d, e, f] yields \_g, k, k, I, m, 7i] and that R' \d, e, /, g, /{] yields [a, b, c, k, I, m, Tt], we may illustrate the import of these operators in the eight following formulations where the symbol = stands iov yields: ( 1 ) \a, b, c, d, e, /] into \d, e, f,g,h'\= \d, e, /] (2) ^' {a, b, c, d, e,/] into 7?' [d, e,/, g, h\ = \k, /, w, «] (3) R [rt, b, c, d, e,f] into [d, ej, g, /i] = [g, /t] (4) [a, b, c, d, e,/] into /?' [d, e,/,g, /i] = [a, b, c] where (i) with (2) with (3) with (4) = the enumerative universe. (5) J?' [a, b, c, d, e,/] with R' [d, e,/,g, k] = [a, b, c, /i, k, /, m, n] (6) [a, b, c, d, e,/] with [d, e, /,g, /i] = [a, b, c, d, e,/, g, h] (7) [a, b, c, d, e,/] with R[d, e,/,g, /i] = [a, b, c, d, e,/, k, /, m, n] (8) R [a, b, c, d, e,/] with [d, e,/,g, /i] = [d, e,j, g, A, k, /, ;«, «] where (5) into (6) into (7) into (8) = the enumerative zero. As regards these eight formulae we observe that each of the pairs (i) and (5), (2) and (6), (3) and (7), and (4) and (8) give two enumerations related the one to the other as remainder. What holds in one illustration can be formulated in general terms. Let E and F be any two enumerations, then : the operation 'E into F' yields-the-enumeration-yielded-by the opera- tion R' [RE with R'F). Since the relation 'remainder to' as also the relation ' yields- what-is-yielded-by' are re- versible or symmetrical, this single formula includes all the eight formulae which have been illustrated above. But, we may for the sake of emphasis, express the principle again in eight formulations where = will now stand for 'yields-what-is-yielded-by.' Ii6 CHAPTER VIII (i) E into F = the remainder to /?'£■ with ^'/^ (2) I?'E into R'F= „ „ „ E with F (3) R'E into F = „ „ „ E with jR'F (4) E into ^'F = „ „ „ R'E with F where (i) with (2) with (3) with (4) = the enumerative universe. (5) R'EmthR'F= „ „ „ ^intoF (6) E with F = „ „ „ jR'E into R'F (7) E with /?'F = „ „ „ J?'E into F (8) TP^-with/^ = „ „ „ EmtoR'F where (5) into (6) into (7) into (8) = the enumerative zero. From any one of the above eight formulae we can read off any other. Where any one mode of constructing an enumeration is equivalent in the above sense of = to a certain other mode of constructing an enumeration, it is obvious that the equivalence will imply equality of number, although the reverse does not hold : that is, we may have equality of number for two enumerations while the items comprised in them are not necessarily the same. § 2. We have now to consider how a single enumera- tion may be taken as an item to be enumerated along with other enumerations so as to constitute an enumera- tion of enumerations, that is an enumeration comprising, as its items, units which are themselves enumerations. This conception of an enumeration comprising enumera- tions must not be confused with an enumeration zW/^^^m^ enumerations. Thus: if *i^ includes E' then the items comprised in E are the same as some of the items com- prised in F, and here E and F comprise the same types or kinds of items. But if ' F comprises E^ then the items comprised in F will be of a higher order or type than the items comprised in E. Using the term item (in the first instance) to stand for an entity of order ENUMERATIONS AND CLASSES 117 zero, i.e., one which is not itself an enumeration, an enumeration comprising such items will be of the first order, and an enumeration comprising enumerations of the first order will be of the second order ; and so on. In passing from enumerations of the first order, viz., those which comprise mere items, to enumerations of the second order which comprise enumerations of items, we may symbolise the distinction by using square brackets. Thus an enumeration of the first order may be illustrated thus : \_a, b, c, d, e, f, g, h, k, /, m, n\ in- volving one size of bracket. Now, with these same twelve items we may illustrate several enumerations of the second order which will involve two sizes of brackets as follows: [[^, b\ [c, d, e,f,g], \_h, k, I, m\ [«]J where the item-enumerations are exclusive and four in number. Again: [a, b, c], [c, d, e,f], [a,f, k, k, /], [_d, m, n\ where the item-enumerations are not exclusive of one another and again are four in number. Or again : {a, b\ {c, d, e], [b,f, h\ \d, k\ [e, m, n]j where some pairs of the item-enumerations are exclusive and others not, the total number being five. In all these illustrations, a comma is used to separate the items to be enumerated in constituting an enumera- tion, and the square brackets are used where required to indicate what is to be taken as an item. Similarly, ii8 CHAPTER VIII using the same twelve items we may illustrate an enumeration of the third order; which will involve three sizes of brackets thus : [a, d], [c, ci, ^]J, [[<^,/ h\ [d, k'jj, [[^, m, n\ where there are three items which are enumerations of the second order, of which the first two comprises as items two first order enumerations, while the third com- prises only one first order enumeration. Comparing our illustration of a first-order enumeration with the first illustration of a second-order enumeration, we must note that the item n which is of zero order is to be dis- tinguished from the item \n\ which is of the first order, and comparing this last with our third order enumera- tion, we must distinguish the item \e, m, n] which is of the first order from \e, m, n\ which is of the second order. The distinction between n and \n] is, therefore, that the former is to count as one along with other items in constituting an enumeration of the. Jirst order, while \n] is to count as one along with other items in consti- tuting an enumeration of the second order. Similarly, the distinction between [e, in, n] and [e, m, n] , is that the former is to count as one along with other items in constituting an enumeration of the second order, while the latter is to count as one along with other items in constituting an enumeration of the ^/iird order. On precisely similar grounds we must distinguish between. say, \_a, ^], \_c, d, e^ and [a, d'], [c, d, ^] for the former represents an enumeration of the second order ENUMERATIONS AND CLASSES 119 comprising two items which are enumerations of the first order, while the latter represents an enumeration of the third order comprising one item of the second order. In the above treatment we have in effect defined the notions 'item' and 'enumeration,' not as having absolute significance, but as having relative significance, in the sense that the two notions are indicated by the re- lative term 'comprising' and its correlative 'comprised in.' In other words our proper topic has been the development of the kind of relation expressed by the verb comprise. Further to illustrate the principle that the operators into and with yield an enumeration of the same order as the enumerations operated upon, we will apply these operators to enumerations of the second order. When abbreviating the expression for an enumeration by sub- stituting a single letter E or F, we shall use as indices I, 2, 3, ... to indicate the different orders to which any enumeration may belong. Thus: Let^' stand for \\a, b, c\, [c, d], [a, e,f,g\ [a, /i, k, /]J and F' stand for [[a, b, c\ {c, d, e], [a, e, /,£■'], [b, k, /, nt] then the operation'^' into/"'' yields \\^a, b, c], [_a, e,f,g]^ and the operation 'E^ with F-' yields [[«, b, c], [c, d\ [c, d, e\ \a, e,f,g\, [a,k, K /]. \b, k, I, m'] Thus the operation E' into F^ yields C and the opera- tion E- with F" yields H^, where it may be seen that G^ and H"" stand for enumerations of the second order. We shall also require a symbol for the result of using the operator into, where the enumerations are exclusive Thus: 120 CHAPTER VIII of one another. According as the enumeration yielded in this case is of the first, second, or third order, it will be symbolised by [o], or [o] or [o] The operation \_a, d, ^] into [d, e] yields [o]; and the operation [a, b, c], [c, cf\ into [a, d, e], [c,f~\ yields [o] . Thus amongst all possible enumerations of the first order we must include [o] ; and amongst all those of the second order we must include [o] . An enumera- tion characterised as being o may be called an empty enumeration, the symbol o having in every case the same meaning: the single, double, etc., bracket adds a further character to the character zero. The symbol o is obviously selected to indicate that the number of items in an empty enumeration is zero. As regards items of zero order none can be called empty; and there- fore none can be symbolised as o. § 3. At this point we must explain more precisely the distinction between being comprised in and being included in. Thus, each of the three items a, d, c is comprised in the enumeration \a, b, c], and no others. The relation of comprising thus always correlates an item or an enumeration of a certain order with an enu- meration of the next higher order. On the other hand \_d] is included in [a, b, c], thus showing that the relation of inclusion is a relation between enumerations of the same order. The above account suggests the elementaryproblem, how many enumerations are included in the enumeration [a, b, c] (taken to be an enumeration of the first order). ENUMERATIONS AND CLASSES 121 Since [o] being of the first order is included in every first order enumeration, the following first order enu- merations will exhaust all those that are included in \_a, b, c'], namely: First, that which comprises no item : [o] : Secondly, those which comprise one item: [_a\ and \_b'] and [c] : Thirdly, those which comprise two items : \_a, d] and [a, c] and [d, c~\: Fourthly, that which comprises three items : [a, b, ^]. Thus, within the enumeration [a, b, c~\ the number of distinct enumerations included is 8; for, in selecting any enumeration which shall be included in \_a, b, c~\, we have, with respect to each of these three items, the ^zuo alternatives of admitting or omitting it. Hence the number of our choices is 2x2x2. Similarly in any enumeration comprising (say) n items, 2" enumera- tions will be included. Thus, we may write down an enumeration of the second order which shall comprise all the enumerations of the first order included in [^a, b, c]. Thus: [[o], [a'], [b], [c], [a, b], [a, c], [b, c], [a, b, ^]] . In general terms then: The enumeration of the second order, that shall comprise as its items all the enu- merations of the first order included in a given enu- meration of the first order comprising n items, will comprise 2" items of the first order. § 4. Having treated of enumerations we may now consider the relation between an enumeration and a class. Whether a class may or may not be considered as an enumeration of a special kind, it will be agreed that there is involved in the notion of a class an element 122 CHAPTER VIII entirely absent from that of a mere enumeration. In the language of Mill, the denotation of a class may be said to be determined by connotation ; i.e. by a certain conjunction of adjectives. But here it is of the utmost importance to note that, on the one hand, the substan- tival items constituting the denotation are united merely by the enumerative ' and ' ; but, on the other hand, the adjectival items constituting the connotation are united by the conjunctional ' and\' In fact, what is common to every logician's employment of the term class is that its limits are determined — not merely, if at all, by a mere enumeration of items — but essentially by the character or conjunction of characters that can be truly predicated of this and of that item that is to be comprised. The distinction between an enumeration and a class is closely connected with that between an extensional and an intensional point of view, so that logicians have con- trasted the extension of a class with its intension or the intensional conception of a class with its extensional conception. Phrases of this kind have in fact been in- troduced in various parts of this work; but a more direct way of attacking our problem would be to speak — not of the extension and the intension of a class — but of the extension of an intension and of the intension of an extension, where the preposition ^requires to be logically defined. More explicitly I propose to speak of an intension as determining a certain extension, or conversely of an extension as being determined by a certain intension. Thus the relation determining and its correlative determined-by will indicate the required connection and distinction. We shall not generally ^ See Chapter III. ENUMERATIONS AND CLASSES 123 speak of an extension determining an intension or of an intension being determined-by an extension, but the relation of determining will be always from the intension to the extension. In short, it is this direction of deter- mination which justifies Mill's use of the term con- notation, and when we are conceiving the converse case of an extension determining an intension, then we may adopt for the intension in this case the convenient term comprehensioii as introduced by Dr Keynes. The word determining as used above is of course elliptical. In speaking of a given intension or conjunction of adjectives as determining an extension, what of course is always understood is that this or that item is or is not to be comprised in the extension according as it is or is not characterised by the given conjunction of adjectives. Now it will be found that the larger and more familiar part of logical theory is actually concerned — not with the notion of extension — but solely with that of intension, and that it is only when arithmetical predicates come into consideration that the notion of extension seems to be required. Thus, taking the pro- position: ' Everything having the character m has the character/,' we may, for any English letter standing illustrativelyforan adjective, introduce the corresponding Greek letter in a purely symbolic sense to stand for the class determined by that adjective. Thus the above distributively expressed proposition may be rendered: * the class /z is included in the class tt.' Again, if we conjoin with the above proposition : ' Everything having the character/ has the character m,' we reach the form of proposition : ' The class /a coincides with the class TT.' Now the relation coincides is analogous to the relation 124 CHAPTER VIII of co-implication, in that both are transitive, symmetrical and reflexive; i.e., they have the properties of equi- valence or identity. In this way we may speak of the identity of a class determined by one adjective with that determined by another (merely as expressing a symbolic or abbreviated formula) without implying that there is any real entity to be called an extension or a class to which the strict relation of identity could be applied. All this is assumed in the next chapter, where we shall represent the force of propositions by means of closed figures. In spite then of the prominent em- ployment of the word class both in the treatment of propositions and still more in that of the principles of syllogism, it may be maintained that there is no real reference in thought to the class as an extension, but only a figurative or metaphorical application of the word which serves tobring out certain analogies between such notions as inclusion, exclusion, and exhaustion which apply primarily to parts and wholes and are transferred as relations between propositions and their constituent elements. Some logicians have even gone so far as to say that the spatial relations amongst plane closed figures represent the actual mode of thought by means of which we are able to comprehend logical relations. I, however, reject this extreme point of view, but suggest that the mere fact that we are able to represent logical relations by analogy with relations amongst spatial figures almost justifies our maintaining that the idea of an extension determined by an intension is a logically valid concept. The full significance of such a scheme as Euler's diagrams for representing class-relationships has, in my view, been inadequately ENUMERATIONS AND CLASSES 125 recognised. It should be pointed out that the boundary- line of a closed figure may be taken as the proper ana- logue of the intension, while the area within that boundary is the proper analogue of the extension. This suggestion brings out the following analogies: firstly, that it is intension which determines extension in the same way as a boundary line determines the enclosed area and separates this area from the remaining area outside;, secondly, that we can apprehend in thought the full determining intension in the same way as we can op- tically grasp the single boundary in its entirety; and thirdly, that in general we cannot in thought enumerate all the items which are to be comprised in the extension, just as we cannot exhaustively present to the eye the several and innumerable points within the givenenclosed area. On the other hand, though the several points cannot be exhaustively presented to the eye and yet the area presents itself ocularly as a unitary whole, similarly it would seem that though we cannot ex- haustively enumerate in thought the members of a class yet we can conceive the class or rather the extension as a unitary whole. Again we may make within the area actual dots of a finite number which thus constitute a literal (though of course not exhaustive) enumeration, and thus the force of the diagram, as providing analogues to logical relations, is still further brought out, in that we may think one by one of the objects which we have selected — not arbitrarily — but on the ground that each of them is actually characterised by the adjectives which determine the class. Whether this analogy between a psychical image or perception of an area and the logical conception of a class, justifies our regarding the latter as 126 CHAPTER VIII a genuine concept, is a debatable psychological problem. Dismissing this problem we must return to the strictly logical question whether a class is a genuine entity. § 5. If we provisionally allow a class comprised of individual existents to be an existent, then as an existent it is of a different order from any individual comprised in it. Similarly a class comprising adjectives — if it is to be called an adjective — must be an adjective of a different order from any adjective comprised in it. Similarly for a class of propositions. Therefore the question whether a class is a genuine entity admits at any rate that it is not of the same order of being as any item which it comprises. Whether this or that is a genuine entity can only be answered when we have provided a test of genuineness. The only general test which I can conceive of is as to whether the entity intended to be meant (in using such a word as class) can serve as subject of which some predicate can be truly asserted^ Thus, as an illustration of \!\^^ general question, we may ask whether a proposition is a genuine entity, and taking the proposition matter exists, the reply would be in the affirmative, inasmuch as we can make the assertion 'that matter exists was rejected by Berkeley.' Similarly with regard to the genuineness of a 'class' which is the topic under consideration. Taking for example the class apostles, we may assert ' that this class numbers twelve.' Inasmuch as this statement will be admitted to be true, the only relevant question that could arise would be as to whether the number twelve is predicated — not of the kind of entity called a class — but rather of its determining adjective. To this it may ^ Cf. the treatment of ' .S is ' in Chapter V. ENUMERATIONS AND CLASSES 127 be replied that since different intensions may determine the same class, the number predicated cannot be said to be predicated of one of these determining intensions rather than of any other. In other words, we may say that the adjective twelve is of such a kind that, taking any two co-impHcative determining intensions, if it can be truly predicated of one it can be also truly predicated of the other. Since then it is indifferent of which of these several co-implicatives the adjective twelve is to be predicated, it seems to follow that it is not predicated of any of them whatever. Let us take/ and q to stand for two co-implicative intensions. This means more precisely that, if anything is characterised as being/, it will be characterised also as being q ; and conversely, if anything is characterised as being q, it will be also characterised as being/. It is then clear that the relation called co-implication is both symmetrical and transitive. Let us then assume the question at issue, namely that there is such an entity as a class. Then our conception of a class involves the universal statement that any given intension determines one and only one class. In this way the relation of co-implication subsisting between / and q, may be resolved into the statement ^2X p de- termines a certain unique entity which is the same as that which is determined by q\ that entity being what we have taken to be a class. Here the relation deter- mining, which relates an intension to its extension, is what is called a many-one relation, because there may be many different intensions which determine a single extension. Similarly, the relation determined-dy, which relates an extension to its intension, is what is called a one-many relation, because there is only one extension which is determined by many different intensions. 128 CHAPTER VIII § 6. We may generalise the above by taking, as a typical illustration, the relation of a man to the country which he inhabits. Since there is only one country which a man inhabits the relation inhabits is many-one, and its converse inhabited-by is one -many. Now let us combine these two correlatives in the following form : 'A inhabits the country that is inhabited by B' The relation thus constructed oi Ax.o B (which may be called the rela- tion compatriot -of) is obviously symmetrical and tran- sitive. It is symmetrical because thereis onlyone country which A can inhabit and one country only which B can inhabit, so that to say that A and B inhabit the same country exhibits a symmetrical relation, since the terms A and B may be interchanged. Again, the relation is transitive since if A and B inhabit the same country, and B and C also inhabit the same country, it follows, since no one inhabits more than one country, that A and C inhabit the same country. The identity of the country inhabited by this, that and another man is a symmetrical and transitive relation, and it is upon these properties of identity that the symmetry and transitive- ness of the relation compatriot-of do.'^Q.nds. In fact, to speak of the country inhabited by A would not be a legitimate expression unless there was one and only one country which a man could be said to inhabit. Thus it will be seen that, when we combine in this sort of way any many-one relation with its correlative (which is necessarily one-many), then we have constructed a relation which has the two properties transitive and symmetrical. But the reverse of this does not obviously hold — that is, given a transitive and symmetrical relation it does not obviously follow that this relation can be resolved into a combination of a certain many-one ENUMERATIONS AND CLASSES 129 relation with its correlative. If, however, this converse proposition may be taken as axiomatic, it would follow, from the two properties — symmetrical and transitive — which hold of co-implication between two intensions/ and q, that there is a certain thing which/ determines and which is determined by q. To this kind of entity we apply the name class. It is therefore only by assuming the theorem that any given relation that is symmetrical and transitive can be resolved as above in terms of a many-one relation and its converse that we can prove that the notion of a class represents a genuine entity. But, in making use of the analogy between (a) 'the country inhabited by any of the men that are compatriots of one another' and [d) 'the class determined by any of the intensions that are co-implicants of one another,' it must be further pointed out that, just as the 'country' under consideration is not the same as the compatriots taken in their totality, so the 'class' under consideration is not the same as the co-implicants taken in their totality. Since, then, the 'compatriots' or the 'co-implicants' taken in their totality would constitute a c/ass, the attempt to prove the above theorem would entail a petitio principii, when applied to the question of the genuineness of the notion 'class.' Merely from the symmetry and transitiveness of the relation {s) compatriot, we cannot prove that there is a certain many-one relation {r) inhabit, or (an entity x) viz. a certain country distinct from the class of compatriots all of whom inhabit (r) that country {pc). Our certainty that ' when a; is 5 to b, then a \sr \.o x and x'lsrtob for some x' is due to the fact that, in order to construct any s, we must first have been given r. J.L. 9 130 CHAPTER IX THE GENERAL PROPOSITION AND ITS IMMEDIATE IMPLICATIONS ^ § I. We may define a general proposition as one in which the subject is constructed by prefixing an appH- cative to a general name. According to this definition, the only kind of proposition which is not general would be that in which the subject is expressed by a proper name; and the general proposition would include two forms of singular proposition : namely, where the general subject-term is prefixed by 'a certain' or by 'the.' We must first point out that there are what may be called pure general propositions, where the general term is represented by the word 'thing' in its absolute univer- sality: for example 'Everything is finite,' 'some things are extended,' In these propositions the subject-term has merely substantival without any adjectival significance. But ordinary propositions in every-day use apply to subjects adjectivally restricted; in other words, there is an adjectival significance in the subject-term as well as in the predicate term. § 2. Using the capital letters P and Q for general or class terms and the corresponding small letters/ and q to represent their adjectival significance, then the correct ^ This Chapter should be read in close connection with Chapter III. GENERAL PROPOSITION AND ITS IMPLICATIONS 131 expression for the affirmatives, universal and particular, would be: Every P is q, Some P is q^ which brings out the substantival significance in the subject-term and the adjectival significance of the predi- cate term. Now these forms are at once seen to be equivalent respectively to A. Everything that \s p is q, I. Something that is/ is ^\ where the adjective / occurs as predicate in the sub- ordinate clause, and the introduction of the word 'thing' indicates an ultimate reference to the absolutely general substantive. The negative general categoricals may first be expressed in the forms : Everything that \s p is-not q, Something that is p is-not q^ where, in attaching the negative to the copula, it is to be understood that the adjective q is denied, in the first case, of everything that is p\ and, in the second case, of something that is /. But these negatives are ex- pressed less ambiguously as the contradictories of / and A respectively, thus: E. Nothing that is/ is q^ O. Not-everything that is/ is q, ^ The subject-terms in these two readings are often contrasted as being the one in denotation and the other in connotation. This disregards the fact that the meaning of the term P contains a con- notative as well as a denotative factor, while the phrase 'thing that is p' contains a denotative as well as a connotative factor. Hence the two readings are not properly to be contrasted. 132 CHAPTER IX where the negative is prefixed to the propositions as a whole, and must not be falsely supposed to qualify the subject-term. By expressing the propositions A, /, E, O, in the above forms the true logical nature both of obversion and of conversion can be explained. Thus the negative that is introduced or omitted by the process of obversion, is to be attached to the adjectival factor alone in the predicate; and hence the four propositions: A. Every \ ^- No ^- ^ • • } -thmsf that \s p \s a I. Some I O. Not-every) become (respectively) by obversion E,. No A I,. Every Oi,. Not-every I I,. Some -thing that is p is non-^ where E/,, A;,, O^, Ij„ are of the forms E, A, O, I, but having in each case non-^ for predicate in place of q. Here the suffix b indicates that the proposition is obtained by obversion. Applying again the process of obversion to E^, Aj,, Of,, /^, we obtain \ -thing that is/ is non-non-^. Atb- Every E,,. No I^t. Some I Oil,. Not-every) Now, by the principle of double negation, non-non-^ GENERAL PROPOSITION AND ITS IMPLICATIONS 133 is equivalent to q. Hence Ai,^,, E/,/,, Ii,^, Of,/,, are re- spectively equivalent to A, B, I, O — showing that the propositions obtained by obversion are equipollent, i.e. formally coimplicant. The process of simple conversion may be exhibited by interpolating intermediate steps, where first the ad- jectival factor in the subject is removed to the predi- cate; then, by the commutative law for conjunctives, the adjectives are transposed; and finally, the first ad- jectival factor in the predicate is removed back to the subject: E. Nothing that is/ is ^ = Nothing \sp and q = Nothing is q and/ = Nothing that is q is/. /. Something that is/ is ^ = Something is/ and q = Something is q and/ = Something that is q is/. The forms A and O, not being directly convertible, must first be obverted and then converted, giving the contrapositive, i.e. the converted obverse, thus: A. Everything that is/ is q = Nothing that is/ is non-^ = Nothing that is non-^ is/. O. Not everything that is/ is ^ = Something that is/ is non-^ = Something that is non-^ is/. We have so far taken E, 'No P is q' to be equiva- lent to 'Nothing is/^'; and /, 'Some P is q' to be equivalent to 'Something is pq.' In other words we 134 CHAPTER IX have reformulated the E and / propositions, in which the subject-term is restricted by the adjective/, by using as subject-term the unrestricted reference expressed by the word 'thing.' In the same way, A, 'Every P is q and O, 'Not-every P is q' may be reformulated thus: A, 'Everything is / or ^'; and O, 'Not-everything is p or q' \ since '/ or q' is equivalent to 'q if/.' In these reformulations, the form of the proposition as universal or particular and as negative or affirmative is unaltered : the ^-proposition is expressed as an ^-proposition, the / as an /, the ^ as an y^, the O as an O. We have, in short, merely reduced all the propositions to the same common denominator (so to speak) by using the nar- rowest reference for our subject-term which will be suf- ficiently wide to include all the subject-terms that may occur in any given connected system of statements. This kind of transformation has been called 'existential,* but since the term 'existential' has been so persistently misunderstood, it will be preferable to speak of 'instan- tial' instead of 'existential' formulation. In this mode of formulation, a further question arises, namely that of interpretation', in particular, as to whether the propo- sition — given to be reformulated — is to be understood to include (implicitly or explicitly) the statement that there are instances characterised by /, where / is the adjective connoted by the subject-term: i.e. to include the affirmative statement 'Something is/.' Here it is to be observed that the instantial statement 'Some- thing is/' is implicitly contained in 'Something is pq^ but not in 'Nothing \spq.' §3. Wewilldistinguish the proposition which ^^w/^^W the instantial affirmation of its subject-adjective from GENERAL PROPOSITION AND ITS IMPLICATIONS 135 the (otherwise) same proposition which does not contain this affirmation by using the suffix/" for the former, and n for the latter. In order to change (say) T,, into 7}, where 7"stands for any proposition having an adjectival factor (say p) in its subject-term, we must add to T,, the statement 'Something is />,' i.e. we must conjunc- tively combine with T„ the instantial affirmation. And, in order to change (say) 77 into T,^, we must subtract from Ty the statement ' Something is/,' i.e. we must al- ternatively combine with 7]. the instantial denial. Thus : given T,„ 77will be expressed ' T„ «/2^ Something is/'; given 77, T,^ will be expressed ' Tyor Nothing is/.' Applying these expressions to the forms ^„, If,A,^, O^ we have: Eu means 'Nothing is pq.^ 1/ „ 'Something is /^.' An „ 'Nothing is pq.^ Oj ,, 'Something is /^.' Ef means 'Nothing \s pq ««c/ Something is/.' /„ „ 'Something is /^ (7r Nothing is /.' A/ „ 'Nothing '\^ pq and Something is/.' 0,1 „ 'Something is/^ or Nothing is /.' From the meanings of the symbols, as thus shown, the following rules will be evident: {a) for obtaining the co-opponent : interchange f with n and A with 0\ or y with n and E with / (with- out any other change); [b] for obtaining a sub-implicant; change either y into n, or A into /, or E into O (without any other change). Since super-implication is the reverse of sub-impli- cation, rule {b) reversed shows how to obtain a super- 136 CHAPTER IX impHcant. Moreover, since a sub-opponent is a sub-im- plicant of the co-opponent and a super-opponent is a super-implicant of the co-opponent, rules (a) and [d) combined show how to obtain a sub-opponent or super- opponent. Lastly, since a sub-implicant of a sub-im- plicant is a sub-implicant, and a super-implicant of a super-implicant is a super-implicant, all the several sub- implicants, super-implicants, sub-opponents or super- opponents can be found. In the following mnemonic diagram, an arrow is used to point from a super-impli- cant to a sub-implicant, and co-opponents are placed diagonally opposite. The diagram is required in place of the ordinary square of implication and opposition, because of the distinction introduced between two pos- sible interpretations of ^, B, /, or O. Here, £/, E„, 0} are super-implicants of 0,„ and are .". super-opponents of A/, A}, A„, If are super-implicants of /„, and are .•. super-opponents of £>, /„, If, An are sub-implicants of Af, and are .•. sub-opponents of 0„, q /'. Not everything is/ or ^ = / is not co-alternate to ^ The four relations and their contradictories here exhibited lead by combination to seven possible rela- tions corresponding exacdy to those shown in Part I Chapter III. Thus: ^ and /i'. Every jZ> is ^ and Every ^ is/ =p is co-implicant to q A and O'. Every / is ^ and Not every q \s p = p\s super-implicant to q A' and O. Every q \sp and Not every p is q = p is sub-implicant to q O and O' and / and /'. = / is independent of q E'andl. Everything isp or q and Some/ is ^ = / is sub-opponent to q E and /'. No/ is q and Not everything is/ or ^ = / is super-opponent to q EandE'. Nopis q and Everything isporq = / is co-opponent to q Thus the same seven relations in which propositions may stand to one another hold of the relations in which GENERAL PROPOSITION AND ITS IMPLICATIONS 145 adjectives may stand when entering into the subject and predicate of universal and particular propositions. If / is formally independent of ^, then all the rela- tions tabulated above are material, but if p is formally related to q, then it must be related in one or other of the six ways, which remain when the relation of independ- ence is excluded. For example, taking the five adjec- tives useful, har^nful, useless, harmless, and expedient, which are formally related, and adding pleasant which is formally unrelated, we find that: (i) expedient is equipollent to useful, (ii) useful is super-altern to harmless, (iii) useless is sub-altern to harmful, (iv) [useful is independent of pleasant], (v) useless is sub-contrary to harmless, (vi) useful is contrary to harmful, (vii) useful is contradictory to useless. The same relations hold for the five numerical adjec- tives — greater than 7, less than 4, less than 8, less than 3, not less than 8, to which we may add 'even' in place of 'pleasant.' Thus: ■^ sub- contrary A" pleasant J.L. 146 CHAPTER IX § 7 . Yet another way of representing the categorical proposition is in terms — not of the adjectives/ and q — but of the substantive classes P and Q which these adjectives determine. Any class must be conceived in extension as a part of the universe of substantives, where the universe is sufficiently widely extended to include all the classes which occur in any set of interconnected propositions. The absolutely widest substantive uni- verse is that which we have represented by the word 'thing'; and corresponding to any more restricted uni- verse, the same word 'thing' can be used with a corre- spondingly understood restriction. The following three technical terms may now be introduced : {a) the part of the universe which remains when any given class is subtracted will be denominated the remainder (or co-remainder) to the class; {b) the smallest class that includes both of two given classes P and Q will be denominated ' P with Q' \ and (c) the largest class that is included in both of two given classes P and Q will be denominated 'P into Q^' With the help of these three class functions, the following funda- mental relations between the class functions and the adjectival functions which determine them may be expressed : (i) The class determined by the negative not-/ = the remainder class to P. (2) The class determined by the adjectival alternation '/ or ^' = the class 'P with Q.' (3) The class determined by the adjectival conjunction '/ and q =the class 'P into Q.' ^ These two functions of P and Q have many of the properties of the arithmetical l.c.m, and h.c.f. See also Chapter VIII. GENERAL PROPOSITION AND ITS IMPLICATIONS 147 Thus the notions 'not,' 'or,' 'and' which must be ap- plied to predications (and here to adjectives) correspond respectively to the notions 'remainder,' 'with,' 'into,' which apply to substantive classes. Traditional logic has encouraged confusion between these two types of notion by employing the terms which are only proper for adjectival functions, for class functions also\ This usage, while it has the advantage of brevity and facili- tates the logical transformations that the learner has to carry out, suffers from the serious objection of leading to confusion between the two types of function. Thus the notion of the remainder as a relation between classes is founded upon that of non-identity as a relation be- tween substantive individuals. For when the class X is said to be the remainder to the class Y, part of what is meant is that no individual comprised in the one class is identical with any individual comprised in the other. Now this relation of non-identity has been repeatedly confounded with that of negation ; so much so, that an important school of philosophy seems to hold that diversity or non-identity mwo\v^s> prima facie a contra- diction ; in other words, that the togetherness of non- identical substantives in the universe of reality involves the joint affirmation and negation of one and the same predicate. The correspondences between adjectival relation- ships and class relationships will now be shown by first taking each of the four universals A, A', E, E' . Thus: ^ The reversed confusion is committed when adjectival predications are spoken of as 'co-exclusive' or 'co-exhaustive' instead of 'co- disjunct' or 'co-alternate.' 148 CHAPTER IX Adjectival Relations Distributive Relations Class Relations Direct Implicative / implies q A. 'Every p is q P is-included-in Q Counter Implicative p is implied by q A'. Every q \s p P includes Q Disjunctive p and q are co- disjunct E. Nop is q P and Q are co- exclusive Alternative p and q are co- alternate £'. Everything is ^ or ^ P and Q are co- exhaustive Since (9, O' , /, /' respectively contradict A, A\ E, E', we derive the following combinatory results: (i) p is co-implicant to q (ii) p is super-impli- cant to q (iii) p is sub-impli- cant to q (iv) p is independent oi q (v) p is sub-oppo- nent to q (vi) p is super-oppo- nent to q (vii) p is co-opponent to q P is co-incident to Q P is sub-incident to Q P is super-incident to Q {A and^'.) Every p \s q and Every q\sp {A and 0'.) Every p is q and Not every q '\s p {A' and O.) Every q '\s p and Not every p is q (Oand 0'^ Some but not every p is\ P is inter-sectant to Q and I" q and Some but not /and /'.)) every norv-p is non-^ i^E' and /. ) Everything is either/ or /*is super-remainder to Q q and Something is both/ and q {E a.T\d I'.) Nothing is both/ and ^ and Not everything is / or ^ (E and E'.) Nothing is both p and q and Everything is either/ or q P is sub-remainder to Q P is co-remainder to Q Here the class-relationships must be compared and contrasted with the adjective-relationships. In parti- cular a 'super '-relation for the adjectives always yields a 'sub '-relation for the classes — illustrating the general principle that a more determinate connotation yields a narrower denotation, and a less determinate connotation GENERAL PROPOSITION AND ITS IMPLICATIONS 149 yields a wider denotation. The above seven relations may be at once expressed on Euler's scheme. Thus: 0) p co-implicant to g (ii) (iii) p super-implicant Toq p sub-implicant to q (iv) p independent of q P co-incident to Q co-opponent to q P sub-inddent to Q P super-incident to Q p super-opponent loq p sub-opponent to q P Q c© P inter-sectant to Q P co-remainder to Q (vii) P sub-remainder to Q (vi) P sujjer-remainder to O (V) These diagrams, due to Euler, illustrate the adapta- tion of diagrammatic representation to propositions ex- pressed, as above, in terms of classes. The employment of diagrams in Logic requires some special explanation. A class is represented by a closed figure, while any- thing that is comprised in the class may be represented by a point within this figure; and anything not com- prised in the class, by a point outside the figure. It is further requisite that the all-inclusive class (otherwise called the universe) whether restricted or unrestricted should be represented also by a closed figure within which all the specific classes adjectivally delimited should fall. Thus the boundary line may be taken to represent the adjective by which the scope of the class is determined, 150 CHAPTER IX while the area within this boundary-Hne represents the class itself. In the figure below the class /* is represented as determined by the adjective/; the universe is repre- sented by the square, and what is outside the circle represents the class-remainder to P, which will be sym- bolized as P' , the boundary of which is indicated by/. For two adjectives p and q, we must use two areas having a part in common with a remainder to both. The thickened outline in (3) separates the class '/*with Q' from 'P' into Q'' \ and the thickened line in (4) separates the class ' P into Q' from ' P' with Q' ' The diagram shows to the eye the correspondences between the adjectival and the class functions ; viz., that the class determined by the adjectival alternation 'p or q' is the class ^ P with Q^ and that determined by the adjectival conjunction 'p and q' is the class ' P into Q.' These diagrams, first employed by Dr Venn, do not represent any proposition, but the framework into which proposi- tions may be fitted. Thus it is shown, for instance, that, using two determining adjectives — p and q — the uni- verse is divided into 2x2 classes, namely: P into Q, P into Q\ P' into Q, and P' into Q', determined respec- tively by the adjectival conjunctions '/ and^,' 'p and q, 'p and q,' 'p and q.' Again: taking three determining adjectives p, q, r, we must draw three closed figures GENERAL PROPOSITION AND ITS IMPLICATIONS 151 in such a way that every resulting sub-class shall be represented ; namely the 2x2x2 classes P into Q into R, P into Q into R\ P' into Q into R, P' into Q into R', P into Q into R, P into Q' into R\ P' into ^' into R, P' into ^' into R\ as determined by the corresponding adjectival conjunctions. Into these frameworks propositions are fitted in the following manner. Beginning with a single determining adjective/, consider the four propositions: A. Every- thing is/; E. Nothing is/; /. Something is/; O. Not everything is /. These four propositions can be ex- pressed in terms of the classes P and P' thus: A. P exhausts the universe or P' is empty ; E. P \s empty or P' exhausts the universe ; /. P is occupied or P' does not exhaust the universe; O. P does not exhaust the. universe or /" is occupied. The import of each of these propositions may therefore be expressed by means of the opposite conceptions oi occupied and empty : crowded horizontal shading will be used to indicate empty, and a single straight line to indicate occupied: Everything iSj)> |g;Nothing is/> Something is/ Not everything is/ This shows that the universals may be expressed as denying and the particulars as affirming occupation. Taking now two adjectives / and ^, we have eight dis- tinct propositions A, A', E, E\ O, 0\ /, /', where 152 CHAPTER IX again the universals deny and the particulars affirm the occupation of certain sub-classes. P IS included in Q P includes Q P and Q are co-exclusive P and Q are co-ezhausiive 9- An Every p \s q A'n Every g is p E„ No p is 9 £^ Ever^ahing is p or 9 These are respectively contradicted by (1 ED D Of Not all p is ^ Of Not all 9 is p If Something is p and g I{ Not everything is p or 17 In this system the strict relation of contradiction is indicated as of O to ^, O' Mo A', I to E, I' to E', by the single straight line and the shading occurring in the same sub-class, namely PQ', P'Q. PQ, P'Q'. It must be observed that in Venn's system (i) the circles are drawn in every case as overlapping one another without exhausting the universe, and (2) that the specific proposition is represented by marking some sub-class as occupied or as empty. Next let us combine the diagrams representing the several universals, which contain no instantial affirma- tion, with the diagram representing 'Something is /' for A and E, and 'Something is not-/' for A' and E'\ or again with the diagram representing 'Something is q' for A and E' , and 'Something \?> q' iox A' and E. GENERAL PROPOSITION AND ITS IMPLICATIONS 153 A Nothing is pq and something is p E Nothing is pq and something is p K Nothmg IS pq and something is p E' Nothmg IS pq and something is p A Nothing is pq and something is q F. Nothing is pq and something is q (S) A' Nothing is pq and something is q E Nothing is J>q and something is q Now since a single line means that something is to be found in one or other of the two sub-classes which it crosses, and since the shading denies it for one of the two, it follows that something is to be found in the other. Thus the first four diagrams prove to the eye respectively that: From 'Every/ is ^i" rt«fl^ Something is /' we can infer /. 'Some/ is ^' From'No p'xsqand Something is /' we can infer (9. 'Some/ is non-^' From 'Every^ is/ a«^ Something is non-/' wecan infer/'. 'Some non-/isnon-^ From ' Every- thing is/ or ^ rtwrt^ Something is non-/' we can infer O'. 'Some non-/ is q' And similarly for the second four, where the affirmative instantial involves non-^ or q. These inferences illustrate the general principle that in order to infer a proposition giving instantial affirma- tion, we must have a premiss giving instantial affirma- tion. From A,^ alone we cannot infer /^, but from A,^ jointly with 'Something is/' we can infer /^. 154 CHAPTER IX Finally let us use Venn's diagrams to represent the seven relations which result from the possible com- binations of the eight elementary propositions A„, AJ, F F ' T T' O n' A and A' and / and /' (vi.) A and Cf and / and /' /4'and O and / and /' (vi) (V) / and /' and O and (y E and /' and O and O' E' and / and O and O' Band^'andOandO' In comparing this scheme with that of Euler (pre- viously given) two points arise. Eulerdraws these figures from the outset so as to represent the class-relationships both as regards instantial affirmations and instantial denials, so that the figures directly express propositiona information. But in Venn both these factors in the proposition have to be specifically marked and in order to represent a completely determined class-relationship all the four sub-classes must be marked. In spite of this apparent difference, an optical comparison of this last scheme with the Eulerian scheme on p. 149 will disclose their essential equivalence. The practical distinction, however, remains that in Ruler's scheme each uncom- GENERAL PROPOSITION AND ITS IMPLICATIONS 155 pounded categorical must be represented by an alter- native of figures, viz. : A by (i) or (ii) ; and O by (iii) or (iv) or (v) or (vi) or (vii) A' by (i)or (iii) ; and O' by (ii) or (iv) or (v) or (vi) or (vii) E by ( vi) or (vii) ; and / by (i) or (ii) or (iii) or (iv) or (v) E' by (v) or (vii) ; and I' by (i) or (ii) or (iii) or (iv) or (vi) and conversely, each diagram represents a conjunction of propositions (i) =A and A\ (ii) = ^ and O' , (iii)=^' and O. (iv) =/ and /'and 6> and O' , (v) =E' and /, (vi) =^ and I', {nX\) = E and E' . On the other hand Venn's diagrams represent each of the uncompounded propositions by its appropriate 'mark- ing' of the proper sub-class, and are thus immediately adapted to the conjunction of two or more affirmatively or negatively instantial pieces of information. § 8. All the above inferential operations are per- formed upon adjectival factors, these occurring always as predicates in a principal or subordinate clause; and, as is impressively brought out in the so-called ' exist- ential ' formulation of the proposition, a residual substantival factor always remains in the subject, though for linguistic convenience it may appear also in the predicate. The importance of this feature may have been obscured owing to the complicated detail with which the inferences have been treated ; and, in con- clusion, it is therefore to the point to emphasize the connection between the account of inference in this chapter and that of the functioning of substantive and adjective given in Chapter I. 156 CHAPTER X EXISTENTIAL, SUBSISTENTIAL AND NARRATIVE PROPOSITIONS § I. Before directly approaching the topic to be dealt with in this chapter, it will be necessary to con- sider the general question of the classification of pro- positions. In previous chapters, several classifications of propositions under different fu7tdanienta divisionis have been given : for example they have been divided into simple and compound, the latter being subdivided into conjunctive and composite; and again into uncer- tified and certified, the latter being subdivided into formally and experientially certified; even the distinc- tion between true and false yields an exclusive and exhaustive division of propositions. On the other hand, many well-known so-called classifications of proposi- tions break the purely formal rules of logical division, in that the sub-classes are not mutually exclusive, and often can hardly be regarded as collectively exhaustive. The most notorious example of this is the classification of propositions upon which Kant based his enumeration of the categories, and which comprised such sub-classes as singular, particular, universal, affirmative, negative, hypothetical, categorical, assertoric and problematic. Regarded as a classification of propositions this involves a flagrant violation of the formal rules of division; for a categorical proposition may be singular or universal, negative or affirmative, problematic or assertoric, etc. What is of real logical value, and was indeed intended EXISTENTIAL, SUBSISTENTIAL AND NARRATIVE 157 by Kant, is a classification not of propositions, but of the several formal relations which may enter within the structure of a proposition more or less simple or com- plex. For example, 'If the American harvest is bad, the European prices of corn are high' is properly enough denominated hypothetical because the central or principal relation asserted is that of implication; but further analysis discloses the categorical nature of the two implicationally related clauses, and the universality of the statement as understood to refer to any or every year; and furthermore such relations as contemporaneity and causality may be taken as implicitly asserted in such a proposition. Of distinctions amongst the different forms of relation that may enter into a proposition we may select as one of the most important that between the relation of characterisation and the relation of im- plication, which, properly speaking, should take the place of the distinction amongst propositions between categorical and hypothetical. Of these two relations — characterisation and implication — the former holds only of an adjective to a substantive, the latter only of one proposition to another proposition. Again, the distinc- tions of modality are not, properly speaking, distinctions between propositions, but distinctions between the dif- ferent adjectives that can be significantly predicated of propositions. In short the sole logical purport of a so-called classification of propositions is, by means of an analysis of propositions of various forms of com- plexity, to disclose the different modes in which their components are bound into a unity. With special reference to the topic of this chapter we may pass to such logicians as Lotze, Bosanquet 158 CHAPTER X and others who have attempted to classify propositions on philosophical rather than purely formal principles. In particular Sigwart has distinguished propositions under three — not necessarily exclusive or exhaustive — heads corresponding to what, in the title, we have called existential, subsistential and narrative. § 2. We proceed then to examine in the first place what is meant by an existential proposition. The most general and appropriate sense in which the word 'exist- ential' is predicated of a proposition is where the pro- position refers to existence in that narrower sense in which existence is distinguished from subsistence, as two sub-divisions of reality. Thus the proposition '3 plus 4 equals 7' must be regarded not as existential, but as subsistential, if that terminology be permitted. From an examination of the illustrations given by philosophers of the existent (as distinct from the subsistent) it may be gathered that the term is equivalent to that which is manifested in time or space. This interpretation may be justified by considering the etymology of the word 'existent' which is closely connected with 'external,' and is further confirmed by the fact that the typical so-called external relations are temporal or spatial. On the other hand it has been maintained, for example, that the number 3, or the relation of equality between 3 plus 4 and 7, or the relation of causality subsists rather than exists. If this conception is to be general- ised what subsists is primarily an adjective, whether ordinary or relational; whereas what in the more exact sense may be said to exist is a substantive proper. We may therefore regard the terms 'existent' or 'substan- tive proper' as meaning 'what is manifested in time or EXISTENTIAL, SUBSISTENTIAL AND NARRATIVE 159 space.' Thus an existential proposition is distinguished from a subsistential proposition in that the latter makes predications about adjectives (including propositions) as such. It may, however, be maintained that such a proposition as '3 plus 4 equals 7' should properly be interpreted as existential on the ground that it applies to all possible existent groupings of classes numbered 3, 4, and 7; or again that the proposition 'Unpunctu- ality is irritating' is existential on the ground that it means nothing more nor less than that 'All unpunctual arrivals are irritating,' where the term arrival, with its implicit temporal and spatial reference, obviously stands for an existent. Or yet again that the proposition ' Heat causes wax to melt' is existential on the ground that it merely expresses the universal proposition that *A11 cases in which heat enters wax, are cases in which the wax is melted,' where merely temporal and spatial rela- tions of possible occurrences are involved. If then a subsistential proposition is to be distinguished from an existential, it must be on the ground that propositions in which the explicit predications concern adjectives or relations, have a special significance beyond what they undoubtedly imply existentially\ § 3. But turning from the more philosophical to the strictly formal usage of the term existential, we find that by such logicians as Venn, Keynes and Russell, existential and subsistential propositions are indiffer- ently denominated existential, and that the term exist- ential is used without any reference to the substance of the proposition, but rather to a certain mode in which ' This question will be treated in more detail in a subsequent chapter. i6o CHAPTER X 2i\\y general proposition (particular or universal) may be formulated. This entirely distinct and peculiar use of the term 'existential' has given rise to endless confu- sion ; and, on this account, the term should be entirely discarded and replaced by some such term as instantial, or, more accurately, indeterminately instantial. At this point we must explain the distinction between determin- ately instantial and indeterminately instantial. While the former corresponds roughly to narrative propositions, of which we shall treat later, the latter are most natur- ally prefixed by the phrase 'there is' or 'there are'; e.g. there is a God; there are horses; there are no sea-serpents ; there is an integer between 3 and 5 ; there are prime integers between 4 and 15; there is no in- teger between 3 and 4. Of these, the first three would be called existential, in the philosophical sense, the last three subsistential. A minor distinction amongst such indeterminately instantial propositions, the disregard of which has not infrequently led to confusion, is that between the affirmatively instantial and the negatively instantial. In short, the essential nature of a particular or of a universal proposition is expressed by formulating the former as affirmatively instantial, and the latter as negatively instantial^ The further development of the topic of such pro- positions, or rather of such propositional formulations, requires us to introduce the phrase 'universe of dis- course,' to which frequent reference is made in formal expositions of the so-called existential import of pro- positions. There are two applications of this phrase, which demand different criticisms. One quite harmless 1 See Chapter VIII. EXISTENTIAL. SUBSISTENTIAL AND NARRATIVE i6i application of the expression 'universe of discourse,' points merely to the familiarly elliptical nature of con- versation. Thus the reference of such a proposition as 'All voters are males' is understood to be limited, say, to the present time (19 14), the English nation, and election to Parliament. In spite of the fact that some nations now and all nations will confer the franchise upon women, in spite of the fact that for the Board of Guardians and other offices women take part in voting, the proposition 'all voters are males' is perfectly intel- ligible in its context. The phrase, though elliptical — like all phrases in discussion or conversation — does not require the explicit introduction of every well-under- stood qualification. I n our view, therefore, logicians have unnecessarily paraded this application of the notion of a universe of discourse, where it means merely that the ordinary reader is expected to supply the restrictions intended by the writer. A limited universe understood as indicating the subject-matter of a single work, such as geometry, which refers exclusively to spatial figures, illustrates the same simple relation to the universe as a whole. Understood in this general sense, the universe of discourse has to the universe the relation of part to whole, and the notion is certainly harmless if trivial. But the other application of the phrase requires more serious criticism. Here 'the universe of discourse' is presented to the reader, not as inside, but as out- side what is commonly called the universe. In this usage the phrase seems to imply that there are several uni- verses related to one another as Europe is to Africa rather than as France is to Europe, taking Europe in both cases to stand for the universe. The distinction J. L. II i62 CHAPTER X between the two uses of the phrase is evident when we pass from such an example as 'All voters are males' to 'Some fairies are malevolent' which well illustrates the use now under consideration. The former is understood to refer to a limited part of the universe of persons, whereas the latter refers to no part whatever of this universe, and on this account is said to be concerned with the 'universe of imagination' conceived as outside and separate from the universe of reality. Now if, when speaking of specific universes, such as the universe of imagination, the universe of ideation, and the universe of physical reality, we meant merely universes comprising images, ideas, or physical realities, then all these three are in the strictest sense included within the one single universe of existents, to which they are related merely as parts to whole. Anything that is comprised in the universe of images must be an image; anything that is comprised in the universe of ideas must be an idea (both of these being psychical) ; and similarly, anything comprised in the universe of physical realities must be physical. What logicians seem to have confused, and requires only common sense to distinguish, is between a horse and either the idea of a horse or the image of a horse ; and accordingly a proposition about horses is concerned with different material from any proposition about ideas or images of horses. When then a proposi- tion is spoken of as being false in the universe of reality and yet true in the universe of imagination or ideation, this involves the tacit assertion that the same proposi- tion can be both true and false; whereas in fact the contents of the two propositions, one of which is said to be true and the other false, are different. The affirmation EXISTENTIAL, SUBSISTENTIAL AND NARRATIVE 163 or denial that there are sea-serpents is different from the affirmation or denial that there are imaofes of sea- serpents ; which again is different from the affirmation or denial that there are ideas of sea-serpents. It is absurd to say that the same things exist in one universe and do not exist in another: wherever this appears to be the case the things asserted or denied to exist are different. What is here said of sea-serpents holds equally of horses or of dragons; as regards the latter, it is supposed that because dragons are acknowledged not to exist in the universe of physical reality, there must be some uni- verse in which they do exist in order that we may intelligently use the term 'dragon.' Now it is a purely psychological question whether at this or that moment of time an image of a horse or equally of a dragon is in course of mental construction; it may be that we may intelligently read or think about dragons or horses without mentally constructing any images of such crea- tures. Properly speaking there is no such thing as the image of a horse or the image of a dragon, because the constructing of images by one person at one time is, as an occurrence, distinct from such a construction by another person at another time, however closely these images may agree with one another in character. Hence, if existence is predicated of any image of this or that kind, it must be remembered that by existence is here meant manifestation in time, and that therefore there exist as many images of any kind of thing as there are occurrences of the constructive act. What holds of images holds, strictly speaking, also of ideas, though not so obviously ; the existence or non-existence of the idea of any object, if idea stands II— 2 i64 CHAPTER X for mental process, must mean the occurrence or non- occurrence of an act of thinking about the object during this or that period of time. The term 'idea,' however, may be understood in a less literally psychological sense : thus intelligently to entertain a proposition in thought would seem to entail our entertaining ideas correspond- ing to the several terms in the proposition. But in this connection we may refer to Mill's pronouncement in regard to the import of propositions in relation to ideas. His dictum is that propositions are not about ideas, but about things; and by this he intended to assert that a proposition is concerned with the things which it ex- pressly talks of, and not with any mental process that may be involved in the assent to or understanding of the proposition. In short, although any genuine act of assertion requires as a preliminary process the under- standino- of the terms and combination of terms that constitute a proposition, yet it is not this process to which the proposition refers. This, of course, holds, whether the matter of the proposition is physical reality or mental reality: we must understand what is meant by the association of ideas or by an emotion of anger in a psychological proposition, just as we must under- stand what is meant by dragons or horses in proposi- tions describing such creatures ; while, on the other hand, the propositions in neither case are concerned with these processes of understanding. § 4. We next proceed to consider in what sense truth and falsity can be predicated of propositions such as 'Some fairies are malevolent' or 'No Greek gods are without human frailties.' These may be otherwise rendered: 'There are malevolent fairies,' 'There are no EXISTENTIAL, SUBSISTENTIAL AND NARRATIVE 165 Greek gods without human frailties.' If these be taken literally, as merely primary propositions, nothing can be said but that the first is necessarily false and the second necessarily true, because there are actually no fairies and no Greek gods in the real universe. But within the real universe there are to be found descriptions of fairies and of Greek gods in stories or legends, and hence it may be true that some fairies have been described as malevolent, and it may he false that no Greek gods have been described as without human frailties. If then we distinguish these secondary propositions — to be recog- nised as such by the introduction of the word 'describe' — from the original primary propositions, the establish- ment of their truth or falsity is seen to depend upon special evidence. The universe of descriptions is simply part of the universe of reality; indeed it seems strangely to have escaped logicians that books and the persons who wrote them belong to one real world, and that therefore the universe to which we refer for verification of propositions concerning the descriptions of fairies or of Greek gods is simply and precisely the same universe as that to which we refer for verification of propositions concerning Frenchmen and geologists. It is therefore evident that, only when we have transformed such pri- mary propositions into their secondary correspondents, any question of interest arises as to their truth or falsity. This contention finally forbids us to speak of various universes of discourse which are outside the one uni- verse of reality. The briefest mode of indicating the peculiarity of propositions of the type illustrated is to say quite simply that they are elliptical ; not elliptical in the sense of limiting the subject-term to a narrower sphere i66 CHAPTER X included in the universe, but elliptical in the sense of being expressed as primary propositions and understood as secondary. Thus our first example should properly be expressed 'Story-books describe some fairies as being malevolent,' and our second 'Homer describes all the Greek gods as subject to human frailties'; and in these transformed shapes the propositions are seen at once to be verifiable in exactly the same way as any other propositions; namely by reference to the one real uni- verse of books and persons. § 5. We pass now to the logical significance of the term narrative in its application to propositions. The notion of a narrative proposition is not restricted to the type of proposition characteristic of a work of fiction or history, since it includes statements made in ordinary conversation etc., where there may be no intention to develop the account of an incident into a connected story. Moreover histories and novels are composed of others besides narrative propositions — the non-narrative propositions being generally what we may call comments on the incidents, characters, situations or emotions de- scribed. Novels (or even histories) might indeed be classified according as their narrative or commentary elements predominate; compare for example Scott with Thackeray, or S. R. Gardiner with Macaulay. A nar- rative proposition may be more precisely defined as one whose subject-term is prefixed by introductory or refer- ential applicatives ; whereas non-narrative propositions are prefixed by such distributives as 'every,' 'some' or similar phrases. Now distributives serve as predesigna- tions of adjectivally significant subjects, while com- mentary propositions may be distinguished from such EXISTENTIAL, SUBSISTENTIAL AND NARRATIVE 167 narrativepropositions as may happen to use subjects con- taining an adjectival element because in the latter case the adjective has no general reference. For example, in the course of a narrative the proposition may occur 'A shabbily-dressed gentleman entered the room,' and this may be followed later on by 'the shabby gentleman withdrew,' where the adjective 'shabby' enters in the context merely first as introductory and later as refer- ential. In fact it is not by the consideration merely of the grammatical structure of a sentence, but rather by the logical nexus of the propositions that the distinction can be established; the narrative and non-narrative elements in any literary work being not necessarily ex- pressed in separable sentences. Hence the reader may easily pick out the commentary elements, these being recognisable by their reference to persons and things in general as distinct from the persons and things entering directly into the plot. My account of narrative propositions covers a wider range than is apparently intended by Sigwart; but for both of us the distinction between narrative and non- narrative rests upon that between the substantive and the adjective. In Sigwart's definition the subject in the narrative proposition is merely substantival, while the subject as well as the predicate in the non-narrative proposition contains adjectival elements. My applica- tion of the term 'narrative' on the other hand, includes cases in which the subject term maycontain an adjectival element the significance of which is purely introductory or referential. § 6. A new problem, bearing upon the existential import of propositions is raised when we contrast i68 CHAPTER X fictitious with historical narratives. We may take for illustration the proposition 'Mr Pecksniff is a hypocrite' and first ask what is meant by Mr Pecksniff. Now a provisional answer to this would be 'A certain archi- tect, living near Salisbury, in the beginning of the nine- teenth century.' The term by which we have replaced Mr Pecksniff seems to have an obvious reference to the universe of reality, and more particularly to the universe of things happening and existing in time and place. But the question as to whether there was any architect in Salisbury at that time would be irrelevant, and therefore the proposition would appear not to be about any architect then living near Salisbury. The difficulty here points to a peculiarity in the use of the predesignation 'a certain.' If for Pecksniff we had substituted 'some architect' instead of 'a certain archi- tect,' the proposition 'Some architect living then near Salisbury was a hypocrite' would have been amenable to the ordinary modes of verification. But the form of statement 'A certain architect was a hypocrite' appears not to represent a proposition, inasmuch as it cannot be either affirmed or denied, since the architect to whom the writer refers is not indicated. What holds then of the reader or hearer of such a proposition does not hold of the writer or speaker \ Though the hearer is unable to give the direct contradictory of the proposi- tion, yet the speaker may propound the two alternatives ^ According as logicians exclusively interpret propositions from the point of view of the speaker (writer) or hearer, they are to be classed respectively as conceptualists or nominalists. The difference between these two points of view lies at the root of many logical controversies. EXISTENTIAL, SUBSISTENTIAL AND NARRATIVE 169 that a certain 5" is or is not P, provided that he has his own individual means of identifying the 5" to whom he is referring in thought. In fact the most common usage of the phrase 'a certain' involves de- liberate concealment for various harmless purposes on the part of the speaker. Thus, when I say ' A certain boy now in this room has stolen my purse,' I deliber- ately preclude any hearer from strictly contradicting or agreeing with me, though of course he could deny the proposition by asserting not the contradictory but a contrary, namely: 'No boy in this room has stolen the purse.' As regards a narrative, fictitious or historical, how- ever, where any substantival reference must always be interpreted in accordance with its nexus with the intro- ductory 'a certain' (coupled or not with a proper name), the writer and reader are so far in the same position that neither the one nor the other is concerned with the question of ultimate identification. The referential 'the' is prefixed to an object identical with that to which the introductory 'a' was first prefixed, but outside and beyond this nexus there is no further possibility of identi- fication. Hence the whole body of propositions in a fic- titious narrative is not entertained with a view to the consideration of their truth or falsity, and might be called pure suppositions. The scholastic logicians introduced the phrase 'suppositio materialis' which would illustrate the sense in which 'supposition' has just been used; but modern logicians have interpreted this phrase as equivalent to what they call the universe of discourse.' It is obvious, however, that the two conceptions are totally distinct, inasmuch as the former consists of 170 CHAPTER X classes of propositions included in the universe of all pos- sible propositions, whereas the latter consists of classes of substantives included in the universe of all possible substantives. In contrasting a work of fiction with an historical work, the propositions laid down in the latter are put forward as to be accepted as true on the authority of the writer. But in both cases, whether history or fiction, it still holds that there is no means for ulti- mately identifying the characters introduced either on the part of the reader or the writer; we can only say that in history it is believed that these characters are identifiable with persons who have actually existed, whereas in fiction no such belief is involved, § 7. Within propositions which are fictitious, the distinctions between those which introduce beings to which there are, and to which there are not, similar beings in the world of reality, gives rise to a further problem \ Thus we may contrast the various statements about the architect Mr Pecksniff in Ma7^tin Chuzzlewit with the various statements that might be made about the fairy Puck in a fairy-tale. It will be noted that in the former case the general class (architect) to which reference is made, actually exists, whereas in the latter the class (fairy) to which reference is made, does not exist; while neither the individual Pecksniff nor the individual Puck does or ever did exist. This imme- diately gives rise to the question of the distinction or ^ Of course there is a further distinction between fictions (novels, dramas) which describe characters and incidents such as might occur in the real world, and fairy-tales (myths, legends) which give descrip- tions such as never can occur, in that judgments as to their natural- ness or 'realism' in the former case are more often relevant than in the latter. EXISTENTIAL, SUBSISTENTIAL AND NARRATIVE 171 relation in the significance of the word 'exist' as applied, firstly to a class, and secondly to an individual. We have above pointed out the peculiar and comparatively modern application of the word 'exist' on the part of formal logi- cians, who express the proposition: 'There are archi- tects' in the form 'The class architect exists'; and the proposition 'There are no fairies' in the form 'The class fairy does not exist.' A more precise formulation of these propositions is obtained by taking C to stand illustratively for any class, the affirmation of whose existence is thus rendered: 'There is at least one indi- vidual, say /*, which is comprised in C; or rather, since a class is determined by connotation: 'There is at least one individual, say P, which is characterised by the conjunction of adjectives constituting the connota- tion of the name C! Now here, I maintain, that the symbol P stands for a proper or uniquely descriptive name, and hence that the conception of the existence of a class — indicated by a connotative name — requires the conception of the existence of an individual — indi- cated by a proper or uniquely descriptive name. Now we may agree that there is no such individual as Peck- sniff, and that there is no such individual as Puck; although in the first case the class 'architect' — which might be used in the description of Pecksniff — would be said to exist, while the class 'fairy' — which might be used in the description of Puck — would be said not to exist. If then we brought forward Sir Christo- pher Wren and Mr Pecksniff as instances of architects, or Oliver Cromwell and Mr Pecksniff as instances of hypocrites, would this substantiate the affirmation that there are at least two individuals comprised in the class 172 CHAPTER X architect, or in the class hypocrite? If this question is answered in the negative it must be on the ground that, in some sense of the term 'exist' which is not ap- propriate to classes, Mr Pecksniff does not and never did exist, and hence he cannot count as one when we are enumerating the members comprised in any given class. Furthermore, since the numerical predication 'at least one' is highly indeterminate and could be in this or that case replaced by the relatively determinate 'at least n where n stands for this or that number, the affirmation that 'the class C exists' is only a special and less determinate case of the affirmation that 'the class C comprises at least n items,' and the number n cannot be counted as such unless all the n items exist. The conclusion therefore follows that the sense of the word 'exist' when predicated of a class is dependent upon that of the word 'exist' when predicated of an item or individual indicated by a proper or uniquely descriptive name\ ^ This contention is directed against the position held in the Principia Mathematica, where E ! is ultimately defined in terms of 3, whereas in my view 3 is to be ultimately defined in terms of E ! T-7Z CHAPTER XI THE DETERMINABLE § I. In this chapter we propose to discuss a certain characteristic of the adjective as such, which perhaps throws the strongest light upon the antithesis between it and the substantive. Here it will be apposite to con- sider the traditional account of the principles of logical division where a class (of substantives) is represented as consisting of sub-classes. This process is governed by the following rules: (i) the sub-classes must be mutually exclusive ; (2) they must be collectively ex- haustive of the class to be divided; (3) division of the class into its co-ordinate sub-classes must be based upon some one 'fundamentum divisionis.' The first two of these rules may be said to be purely formal, and do not raise any problem of immediate interest; but the technical term fundamentum divisionis — though per- haps readily understood by the learner — is actually in- troduced without explicit account of its connection with, or its bearing upon, ideas which have entered into the previous logical exposition. To illustrate the notion we are told, for instance, that, when a class of things is to be divided according to colour, or to size, or to some other aspect in which they can be compared, then the colour, size, or other aspect constitutes the fundamen- tum divisionis. Now although, grammatically speaking, words like colour and size are substantival, they are in 174 CHAPTER XI fact abstract names which stand for adjectives ; so that the fundamentum divisionis is, in the first place, an adjective, and in the second, an adjective of the par- ticular kind illustrated by 'colour' when considered in its relation to red, blue, green, etc. Superficially this relation appears to be the same as that of a single object to some class of which it is a member: thus two such propositions as 'Red is a colour' and 'Plato is a man' appear to be identical in form; in both, the subject ap- pears as definite and singular, and, in both, the notion of a class to which these singular subjects are referred appears to be involved. Our immediate purpose is to admit the analogy, but to emphasise the differences between these two kinds of propositions, in which com- mon logic would have said we refer a certain object to a class. I propose to call such terms as colour and shape determinables in relation to such terms as red and cir- cular which will be called determinates] and, in intro- ducing this new terminology, to examine the distinction between the relation of red to colour and the relation of Plato to man. To predicate colour or shape of an object obviously characterises it less determinately than to predicate of it red or circular-, hence the former adjectives may be said negatively to be indeterminate compared with the latter. But, to supplement this nega- tive account of the determinable, we may point out that any one determinable such as colour is distinctly other than such a determinable as shape or tone; i.e. colour is not adequately described as indeterminate, since it is, metaphorically speaking, that from which the specific determinates, red, yellow, green, etc., emanate ; while THE DETERMINABLE 175 from shape emanate another completely different series of determinates such as triangular, square, octagonal, etc. Thus our idea of this or that determinable has a distinctly- positive content which would be quite inadequately re- presented by the word 'indeterminate.' Further, what have been assumed to be determinables — e.g. colour, pitch, etc. — are ultimately different, in the important sense that they cannot be subsumed under some one higher determinable, with the result that they are in- comparable with one another; while it is the essential nature of determinates under any one determinable to be comparable with one another. The familiar phrase 'incomparable' is thus synonymous with 'belonging to different determinables,' and 'comparable' with 'belong- ing to the same determinable'; not that this is the actual meaning of the terms, but that enquiry into the reason for the comparability or incomparability of two qualities will elicit the fact that they belong to the same or to dif- ferent determinables respectively. This phrase 'belong- ing to' is also more usually used of a member of a class in relation to its class: we have, then, to contrast the significance of the relation 'belonging to' when applied in one case to a determinate and its determinable, and in the other to an individual and its class. If it is asked why a number of different individuals are said to belong to the same class, the answer is that all these different individuals are characterised by some the same adjec- tive or combination of adjectives. But can the same reason be given for grouping red, yellow and green (say) in one class under the name colour.'* What is most prominently notable about red, green and yellow is that they are different, and even, as we may say, opponent 176 CHAPTER XI to one another ; is there any (secondary) adjective which analysis would reveal as characterising all these differ- ent (primary) adjectives? In my view there is no such (secondary) adjective ; in fact, the several colours are put into the same group and given the same name colour, not on the ground of any partial agreement, but on the ground of the special kind of difference which distinguishes one colour from another; whereas no such difference exists between a colour and a shape. Thus red and circular are adjectives between which there is no relation except that of non-identity or otherness; whereas red and blue, besides being related as non-identical, have a relation which can be properly called a relation of difference, where difference means more than mere otherness. What is here true of colour is true of shape, pitch, feeling-tone, pressure, and so on: the ground for grouping determinates under one and the same deter- minable is not any partial agreement between them that could be revealed by analysis, but the unique and peculiar kind of difference that subsists between the several determinates under the same determinable, and which does not subsist between any one of them and an adjective under some other determinable. If this is granted, the relations asserted in the two propositions 'Red is a colour' and 'Plato is a man,' ^on^ for anally equivalent, must yet be contrasted on the ground that the latter but not the former is based upon an adjectival predication. For the latter is equivalent to predicating the adjective 'human' of 'Plato,' v/hile, without denying that some adjectives may properly be predicated of (the adjective) red, yet the proposition ' Red is a colour' is not equivalent to predicating any adjective of red. THE DETERMINABLE 177 § 2. Bearing in mind this distinction, the question arises whether what are called abstract names can be divided in the same way as concrete names into singular and general ; in other words, whether adjectives can be divided into these two classes. The answer seems to be that adjectives can be divided into two classes more or less analogous to the singular and general which dis- tinguish substantives, but that the two different kinds of adjectives are preferably distinguished as determinate and indeterminate. When, in considering different de- grees of determinateness, the predication of oneadjective is found to imply another, but not conversely, then the former we shall call a super-determinate of the latter and the latter a sub-determinate of the former. Thus the relation of super-determinate to sub-determinate means not only that the former is more determinate than the latter, but also that the predication of the former would imply that of the latter. A simple example can be taken from the determinable 'number': thus 7 is super-determinate to 'greater than 3'; the adjective 'greater than 3,' though not itself a summum determin- able, may be called determinable, inasmuch as it is not merely indeterminate but capable of being further de- termined in the sense that it generates a definite series of determinates. To illustrate more precisely what is meant by 'generates'; let us take the determinable 'less than 4 ' ; then ' less than 4 ' generates ' 3 ' and ' 2 ' and ' i ' in the sense that the understanding of the meaning of the former carries with it the notion of the latter. Now no substantive class-name generates its members in this way; take, for instance, 'the apostles of Jesus,' the under- standing of this class-name carries with it the notion J. L. 12 178 CHAPTER XI 'men summoned by Jesus to follow him,' but it does not generate 'Peter and John and James and Matthew etc.,' and this fact constitutes one important difference between the relation of sub-determinate to super-deter- minate adjectives and that of general to singular sub- stantives. § 3. Another equally significant difference is brought out by considering that aspect of substantive-classes in Avhich — to use the terminology of formal logic — increase of intension is accompanied by decrease of extension. The phrase 'increase of intension' conjures up the notion of adding on one attribute after another, by the logical process called conjunction ; so that, taking/, q, r, to be three adjectives, increase in intension would be illustrated by regarding p, q, r conjoined as giving a greater intension than p, q ; and />, q as giving greater intension than p. We have now to point out that the increased determination of adjectival predication which leads to a narrowing of extension may consist — not in a process of conjunction of separate adjectives — but in the process of passing from a comparatively indeter- minate adjective to a comparatively more determinate adjective under the same determinable. Thus there is a genuine difference between that process of increased determination which conjunctivally introduces foreign adjectives, and that other process by which without in- creasing, so to speak, the number of adjectives, we define them more determinately. In fact, the foreign adjective which appears to be added on in the conjunctive process, is really not intro- duced from outside, but is itself a determinate under another determinable, present from the start, though THE DETERMINABLE 179 suppressed in the explicit connotation of the genus. We propose to use a capital letter to stand for a de- terminable, and the corresponding small letter with various dashes to stand for its determinates. Thus, in passing from the genus p to the species pq, we are really passing from/^ ^o pq; or again the apparent in- crease of intension from p to pq \.o pqr is more correctly symbolised as a passing ir om pQR lo pqR to pqr. In the successive process of dividing a summum genus into the next subordinate sub-genera, and this again into sub-sub-genera, the summum genus ought to be repre- sented by a conjunction of determinables, say PQRS T\ the genera next subordinate X.o\.]\\s,hYpQRST,p'QRST, p"QRST, etc., and the genera next subordinate to the first of these hy pqRST, pq'RST, pq"RST, and so on down to the infima species represented by determinates. Thus: PQRST pQRST p'QRST p"QRST. pqRST pq'RST pq" RST In this way we represent from the outset the nature of the ultimate individuals under the summum genus, as being characterisable jointly by the determinables PQRST, while any genus or species is represented by these same determinables, one or more of which are replaced by determinates. This meets a criticism which has often been directed against the formal account of the inverse variation of extension and intension, since we see now that the same number of adjectives should be used in giving the connotation of the wider as of the narrower class. To illustrate these symbols from a botanical classification of plants : let the determinable i8o CHAPTER XI P stand for the number of cotyledons, Q for the dis- position of the stamens, R for the form of the corolla, S for the attachment of the petals and sepals, and T for the divisibility of the calyx. Then PQRST represents the summum genus 'plants' as describable under these five heads, but otherwise undetermined in character. Then p, p', p", might stand respectively for having no cotyledons, having i, and having 2, thus representing the defining characteristic of each of the three classes — acotyledon, monocotyledon, and dicotyledon — by the symhoXs PQRST, p'QRST, fQRST. Again q, q\ q" , might stand respectively for the stamens being under, around or upon the carpels, thus representing the three sub-divisions — hypogynous, perigynous, epigynous — of dicotyledons, 2.s fqRST, p"q'RST,p"q"RST. Taking regular and irregular to be the two possible forms of corolla, then the next sub-division under /'y/?^' 7^ will be p"q'rST and p"q'i^ST. Again s and s' may stand respectively for separability and inseparability of the calyx and corolla, and yield the further sub-divisions, sdiy p"q'rsT, p"q'rs'T. The calyx may contain only one part or 3 or 4 or 5 or 6, and if these are represented respectively by t, t', I", t'", t"", a relatively determinate characterisation is finally symbolised hy p"q'rs't"' say. There are cases for which a modification of this general scheme is required. The cases are those in which one particular sub-division is definable by the absence of an element upon which the predication of other determinables depend, while in the sub-divisions co-ordinate with this the element in question is present. For example, the class of plants called acotyledons might be defined by the absence of any corolla, etc., and THE DETERMINABLE i8i hence such variations as that of the form of the corolla or the disposition of the stamens, etc., are inapplicable to this particular sub-division. § 4. Now adjectives under the same determinable are related to one another in various ways. One rela- tional characteristic holds in all cases; namely that, if any determinate adjective characterises a given sub- stantive, then it is impossible that any other determi- nate under the same determinable should characterise the same substantive: e.g. the proposition that 'this surface is red' is incompatible with the proposition 'this (same) surface is blue.' It has been usual to modify the above statement by adding the qualification — at the same time and at the same place; this qualification applies where the substantive extends through some period of time and over some region of space, in which case the existent substantive, having temporal or spatial parts, may be said to be extended. For this reason the qualification would perhaps better be attached to the substantive itself, and we should say that, where opponent adjectives are predicated, reference is made to different substantives, since any one part of an extended substantive is existentially other than any other part. A second characteristic of many determinates under the same determinable is that the differences between different pairs of determinates can be compared with one another; so that if a, b, c, are three determinates, there are cases in which we may say that the differ- ence between a and c is greater than that between a and b; e.g. the difference between red and yellow is greater than that between red and orange. In this case the several determinates are to be conceived as i82 CHAPTER XI necessarily assuming a certain serial order, which de- velops from the idea of what may be called 'adjectival betweenness.' The term 'between' is used here in a fa- miliar metaphorical sense derived from spatial relations, and is figuratively imaged most naturally in spatial form. Thus if ^ is qualitatively between a and c, and c qualita- tively between b and d, and so on, the whole series has its order directly determined by the nature of the adjectives themselves. The further distinctions amongst series as interminable or as cyclic, and again of series of more than one order of dimensions, lead to logical complexi- ties which need not be entered into here. Suffice it to say that this characteristic, which holds of so many determinates, gives significance to another well-known rule for logical division: divisio non facial saltum: one meaning of which appears to be that we contemplate not merely enumerating a set of coordinate sub-classes, but enumerating them in a certain order. The rule pre- scribes that the order in which the sub-classes are enu- merated should correspond to the order of 'betweenness' predicable of their differentiating characteristics. The order of betweenness which characterises the determinates just considered may be either discrete or continuous. I n the case of discrete series there is one de- terminate that can be assigned as next after any given determinate; but, in the case of a continuous series, a determinate can always be conceived as between any two given determinates, so that there are no two deter- minates which can be said to be next to one another in the serial order. 1 1 follows from this account of continuity that, between any two determinates which may be said to have a finite adjectival difference, may be interpolated THE DETERMINABLE 183 an indefinite number of determinates having a finite dif- ference, and this number becomes infinite as the differ- ences become infinitesimal. Amongst continuous series further differences between the interminable and the cyclic, and again between those of one or more order of dimensions, hold as in discrete series. The reference here to determinables of higher or lower dimension requires explanation. Our familiar ex- ample of colour will explain the point: a colour may vary according to its hue, brightness and saturation ; so that the precise determination of a colour requires us to define three variables which are more or less inde- pendent of one another in their capacity of co-variation ; but in one important sense they are not independent of one another, since they could not be manifested in se- paration. The determinable colour is therefore single, though complex, in the sense that the several consti- tuent characters upon whose variations its variability depends are inseparable. § 5. Returning to the conception of the absolutely determinate adjective, we have to note an important distinction between absolutely determinate and com- paratively indeterminate predications. The distinction may thus be formulated: If, of two substantives the same determinate adjective can be predicated, then all the adjectives and relations definable in terms of the determinable, that can be predicated of the one, could be predicated of the other. But if, of two substantives the same indeter7ninate adjective can be predicated, then only certain of the adjectives and relations definable in terms of the determinable, that can be predicated of the one, can be predicated of the other. To illustrate i84 CHAPTER XI first the case of an indeterminate predication; let us take the numerical adjective 'greater than 7'; then of any collection of which this numerical adjective could be predicated, other adjectives such as 'greater than 5' and 'greater than 3' could also be predicated; but some collections that are ' greater than 7' such as the apostles, are greater than 1 1 and divisible by 4 for instance, whereas other collections that are 'greater than 7,' such as the muses, are less than 1 1 and are not divisible by 4: hence it is only some of the numerical adjectives that are predicable of the muses that are also predicable of the apostles, although the adjective 'greater than 7' is predicable of them both. Turning now to the case of determinate predication; if, instead of defining a col- lection by the indeterminate adjective 'greater than 7,' we had defined it by the determinate adjective 'twelve,' then any numerical adjective that is predicable of one collection of twelve, say the apostles, would be predic- able of any other collection of twelve, say the months of the year or the sons of Israel; for example, 'greater than II,' 'divisible by 4,' 'a factor of 96.' What we have here seen to hold of determinate and indetermi- nate number holds of any other determinable. The case of colour lends itself easily for illustration on account of the specific names which have been assigned to its determinates : thus, if the colours of two different objects are the same shade of yellow, then though these two objects may differ in any number of other respects such as shape and size, yet we may say that any colour- property of the one object will agree with the colour- property of the other; if the colour of one is more brilliant or less saturated than the colour of an orange, THE DETERMINABLE 185 then the same will hold for the colour of the other. In fact, whatever sensational determinable we take, whether it be colour, or sound, or smell, the determinate characterisations under any such determinable would lead to the same forms of generalisation that have been developed by science only in the sphere of quantity. It is agreed that in the sphere of sense perception, dif- ferences of quality strictly speaking hold only of the mental or sensational, and that the physical can only be defined in quantitative terms. Thus in the Weber- Fechner experiments the experient judges of equiva- lence or difference in the intensity or quality of his sensations, with which are correlated quantitative dif- ferences in the stimuli. The attempts that psycholo- gists have made to discover formulae of correlation between the stimuli on the one hand and the sensations on the other hand show that determinateness in a quali- tative or intensive scale carries with it the same logical consequences as does determinateness of magnitude for physically measurable quantities. Furthermore deter- minateness in either case is only approximately attain- able, whether we rely upon the immediate judgments of perception or are able to utilize instruments of measure- ment. The practical impossibility of literally determi- nate characterisation must be contrasted with the uni- versally adopted postulate that the characters of things which we can only characterise more or less indetermi- nately, are, in actual fact, absolutely determinated ' The notion of the Determinable will be shown in later chapters to have importance in a large number of applications. i86 CHAPTER XII THE RELATION OF IDENTITY § I. The occasion for using the relation of identity is where a common term appears in different connec- tions ; thus we use the idea of identity always along with the idea of difference. The logical relation between difference — or more properly otherness — and identity, is that of co-opponency : that is, taking A and B as any two terms, it cannot be that A is both identical with and other than B, and it must be that A is either identical with or other than B. Thus the relation between identity and otherness is reciprocal. It must therefore be explained that we cannot define otherness as meaning non-identity any more than we can define identity as meaning non-otherness. The conceptions of identity and of otherness must be separately and independently understood before we can assert the above axioms. The most trivial and apparently insignificant use of the relation of identity is expressed in the formula ' x is-identical-with x^ where what is primarily meant is that in repeated occurrences of the word x, either in a special context or irrespective of context, the word shall mean in any later occurrence what it meant in an earlier occurrence. Thus, even in this very elementary usage, the idea of identity goes along with the idea of otherness ; for identity applies to what is meant by THE RELATION OF IDENTITY 187 the word, and otherness to its several occurrences. Underlying this characteristic of language there is the corresponding characteristic of thought ; thus, in using 'x is-identical-with x' in reference to entities and not mere words, we return in thought to the object pre- viously thought of, so that identity applies to what constitutes the odj'ec^ of thinking, and otherness to the several recurrent ac^s of thought. A less elementary usage of the relation of identity occurs in the definition of words or phrases ; thus, if x and y stand for two different phrases, we may speak of X being identical with y, although the phrases are palpably different. Here otherness applies to the phrases and identity to what is meant by the phrases. Verbal identification of x with y may be contrasted with factual identification ; here the relation of identity applies (as before) to the objects denoted by the words ; but the propositio7t asserting identity in the one case is of a different nature from the proposition asserting identity in the other case : for in the first case it is verbal, in the second factual. The relation of identity asserted in the two propositions : ' Courage is the mean between timidity and foolhardiness ' and ' Courage is the one virtue required of a soldier ' is the same : but the natures of the propositions differ, since the first — being put forward as a definition — is verbal, and the second is factual. More generally, we may distinguish the different grounds — such as rational, ex- periential or linguistic — upon which the assertion of any specific logical relation is based ; but these differences in the grounds of assertion, do not affect the nature of the relation asserted. Thus, abbreviating ' is identical with ' iS8 CHAPTER XII into ' 2,' from the verbal statement xiy together with the factual statement jj/^.s' we may correctly infer the factual statement xiz. This inference uses the transitive^ pro- perty of identity, and would therefore be impossible unless the relation of identity asserted in a verbal statement was the same as that asserted in a factual statement. § 2. We have shown the proper sense in which difference can be said to be involved in identity, but many philosophers have laid down the dictum that identity implies difference, in the sense, apparently, that when we assert that A is identical with B we are also involved in the assertion that A is different from B. The plausibility of this dictum depends upon a certain looseness in the application of the word * im- plies'; thus the statement that identity implies difference is correct in the sense that the asserting of identity between one pair of terms implies our having implicitly or tacitly asserted difference between another pair of terms. This follows from what has been said above ; e.g. when identifying the colour of this with the colour of that, we are implicitly differentiating ' this ' from ' that ' ; and thus the identification and the dif- ferentiation may properly be said to be component parts of a single mental act. But, to give a more pre- cise statement of this implication, it would be necessary to say that, when A is identical with B in a certain respect, then A is different from B in some other respect. In common language it is of course perfectly legitimate to say of two things that they are identical in respect ^ For the term 'transitive' as a property of the relation of identity see Part 1, Chapter XIV. THE RELATION OF IDENTITY 189 of colour and different in respect of shape. But here the term ' identity ' should be used more precisely : we ought properly to say that, while the colour of this is identical with the colour of that, the shape of this is different from the shape of that. When then identity and difference go together in this way we ought to say, not that the thi7igs are both identical and different, but thgt one of their qualities is identical and another different. To complete the account of the modes in which identity and difference mutually involve one another without confusion, we need only take four such typical elementary propositions as : (i) this is/, (ii) this is q, (iii) that is/, (iv) that is q, where / and q stand for different qualities, and ' this ' and ' that ' for different things. Then, comparing (i) with (ii) or (iii) with (iv), we have identity of thing and otherness of quality ; and, comparing (i) with (iii) or (ii) with (iv), we have otherness of thing and identity of quality ; finally, comparing (i) with (iv) or (ii) with (iii), we have both otherness of thing and otherness of quality. § 3. The relation and distinction between thing and quality may be generalised in terms of the correla- tive notions of substantive and adjective, the latter admitting of further resolution into determinates and their determinates. Thus, when predicating the same adjective/ of this and of that substantive, we shall say that this and that agree as regards the determinable igo CHAPTER XII P ; and when predicating / of this and p' of that substantive, we shall say that this and that disagree as regards the determinable P. Here the term ' disagree ' is used in place of 'differ,' for, strictly speaking, 'dif- ference ' is a relation between adjectives under the same determinable ; and in measuring different degrees of difference amongst such adjectives, we may speak of the substantives as being similar when the degree of difference between the adjectives characterising them is small, and as being dissimilar when the degree of difference is great. Further we may say that two substantives partially disagree when they are charac- terised by the same determinates under certain deter- minates and by different determinates under certain other determinables. But partial agreement must be distinguished from approximate agreement, otherwise called similarity ; and partial disagreement must be distinguished from remote disagreement, otherwise called dissimilarity. The distinction between similarity and dissimilarity involves reference to adjectives under the same determinable, and is obviously a matter of degree ; while partial agreement or disagreement in- volves reference to adjectives under different deter- minables. \ 4. Adjectives under the same determinable are usually said to be comparable ; whereas those under different determinables are said to be incomparable or disparate. This point raises a question of considerable psychological interest as to the possibility of comparing two such disparate characters (say) as red and a trumpet- blast. The possibility of such comparison may perhaps be accounted for by association ; or it may be that some THE RELATION OF IDENTITY 191 real deep-rooted form 'of connection underlies the two characters, which, if it could be explicitly rendered, would have its logical place amongst relations such as those which we are now discussing. But apart from this possible topic of psychological interest, it is not usual to speak of ' red ' and ' a trumpet-blast ' as being either like or unlike ; and it is more usual to restrict the use of the terms like and unlike to qualities belonging to the same determinable, such as colour or sound. If this is admitted, the conclusion at once follows that like and unlike are not proper logical con- tradictories, but are relations of degree ; so that to say of two comparable qualities that they are more or less like is equivalent to saying that they are less or more unlike ; we cannot define the point at which the rela- tion of difference changes from likeness to unlikeness, their opposition being only one of degree. When we compare different degrees of difference between deter- minates under a determinable whose variations are continuous, and judge, for instance, that the difference between A and C is greater than that between A and B, such differences between the determinates may be said to have distensive magnitude. When this distensive magnitude is too small, we fail perceptually to dis- criminate (say) between A and B ; and some psycho- logists have virtually taken 'identity' to be equivalent, in such cases, to ininimum discernible difference. This, however, entails logical contradiction ; for the concep- tion of a ' minimum discernible difference ' implies that we fail to discriminate between qualities, which really are different and not identical ; and that strict identity can only be predicated when difference has reached the 192 CHAPTER XII absolute limit zero. On this ground, it might be urged that identity when appHed to qualities susceptible of continuous variation means zero difference ; and has, therefore, a different significance from identity when applied to things or to qualities which vary discretely. This contention would, in my view, be fallacious ; for it would appear to involve a confusion between the objective conception of identity itself and the subjec- tive limitations in our power of judging identity. On the other hand, difference when applied to adjectives under the same determinable has a certain meaning which is distinct from any meaning of difference ap- plicable to substantives or to adjectives under one and another determinable. As regards the latter, difference can only mean mere otherness ; but as regards the former, difference may mean more than mere otherness ; viz. something that can be measured as greater or smaller. Thus Socrates is merely other than Plato, red is merely other than hard ; but round and square, red and yellow, five and nine are not merely non-identical, but are also such that the difference between them can be apprehended as greater or smaller (say) than that between oblong and square, orange and yellow, seven and nine. § 5. There is yet another aspect of the dictum: identity (of adjective) implies difference (of substan- tive) according to which it could equally well be ren- dered: difference (of adjective) implies difference (of substantive). For, where identity applies to the adjec- tive and difference to the substantive, identity may properly be said to imply difference, in the sense that the identity predicated of an adjective is used along THE RELATION OF IDENTITY 193 with otherness as predicated of compared substantives ; but, in this sense, we may also say that difference im- plies difference; i.e. that difference predicated between two adjectives is used along with otherness as predi- cated of the compared substantives. The dictum should therefore be expressed in more general terms to include identity and difference in respect of the adjectives char- acterising one and another substantive. Thus: Com- parison with respect to any determinable character, whether it yields identity or difference, presupposes otherness of the substantives characterised by the de- terminable in question. In this connection we may examine the contrast commonly drawn between quali- tative difference and numerical difference. This termin- ology is incorrect, for 'numerical difference' simply means otherness— the very notion 'numerical' owing its origin to the conception of mere otherness, which is the basis of number. Again in contrasting qualitative with numerical difference there is the suggestion that other- ness does not apply to qualities or adjectives, whereas in its developments into number otherness is clearly seen to apply precisely in the same way to adjectives as to substantives. I n our view the required distinction is that which was drawn above between the word difference as meaning merely otherness, and the word difference in its exclusive application to adjectives under the same determinable. § 6. Under the head of difference and otherness a special problem to be discussed is that involved in the famous Leibnizian principle 'the identity of indiscern- ibles.' This phrase signifies that 'indiscernibility im- plies identity,' which is an awkward way of saying that J- L. IS 194 CHAPTER XII 'otherness implies discernibility.' Here the term dis- cernibility has not a psychological but a purely onto- log-ical significance. More explicitly, the phrase sig- nifies that a plurality of existent objects is only possible so far as there is some difference in the qualities or relations which can be predicated of them. If, in this phrase, the term relation is interpreted to include such external relations as space or time, then no reasonable criticism of the Leibnizian formula could be maintained; for, as has been contended from another point of view- in Chapter II, 'existential otherness' implies difference in spatial or temporal relations. But this interpretation can hardly be taken to represent Leibniz's meaning, since he denied external relations, and held that this denial demonstrates the non-reality of space. But whereas he pretends to base the denial of space upon his dictum, in reality his dictum would have no plausibility unless it had been previously agreed that space was unreal. What Leibniz certainly meant was that two existent objects could not agree in all their internal characters and rela- tions. The difficulty that here arises in regard to the number of respects and the remoteness of difference that are abstractly necessary for the possibility of two- ness of existence exhibits more emphatically the purely dogmatic nature of the Leibnizian principle, which seems to me in any case to have no logical justification what- ever\ ' Much the same considerations were brought forward in the criticism of Bradley's dictum that 'distinction impHes difference' (see Chapter II). If the Leibnizian and the Bradleyan principles can be in any way distinguished, it is that the former is ontological and the latter epistemological. THE RELATION OF IDENTITY 195 § 7. Having discussed the notion of identity in its contrasts and connections with difference and otherness, we must finally examine the nature of the relation of identity itself, apart from its connections with other re- lations. The problem may be indicated by discussing whether identity is or is not definable. For this purpose it will be desirable to begin by considering the formula xtx, rather than xiy. For xiy can only be interpreted by explicitly distinguishing the word, phrase or symbol which denotes an entity from the entity itself that is denoted ; and moreover the proposition xiy (until defi- nite equivalents for x and y are substituted) can only be adopted hypothetically or illustratively. On the other hand, the straightforward proposition xix is to be as- serted on rational grounds for any value of x. Elimi- nating the symbol x, what is to be universally asserted is that 'Any entity is identical with itself.' Unfortunately the term itself Q.2Si only be defined as 'what is identical with zV; and hence any explication of the formula seems to lead to an infinite regress. This difficulty can be removed by expressing the formula in the negative form: 'No entity is identical with any entity other than itself This is to be understood as a brief way of asserting: 'No entity {pc) is identical with any entity othei'- tha7i what is identical with x! This axiom ex- presses a universal and rationally grounded truth, ex- pressed in terms of the two relations, identity and otherness. I shall attempt to show that the conceptions of identity and of otherness are two independent inde- finables, the understanding of which is required in order intelligently to accept the truth of the above axiom or of any other proposition which explicitly or implicitly 13—2 10 CHAPTER XII involves the notions of identity or of otherness. Now the above axiom of identity (xzx) is never expHcitly used anywhere but in an abstract logical or philosophi- cal context; on the other hand xzy is explicitly used in concrete logical and mathematical formulae; and its usage in such cases always involves the process of substituting y for x. The connection between identity and substitution is roughly expressed in the rule : Given xiy, we may always substitute y for x. More exactly, taking/ to be any predication, 'If Jtr is identical with y, then ;i; is / would imply y is /.' The converse of this is also generally admitted: viz. that 'If for eveiy value of/, :r is / would imply jj/ is/, then x must be identical with jj/.' From these two assertions conjoined it follows that the proposition 'x is identical withjj/' is equipollent or co-implicant with the proposition that ' For every predication/, x\s p would imply jj/ is/.' The problem thus arises whether this equipollence or co-implication can serve as a definition of identity. My ground for rejecting such a view is that the equipollence asserted could not be understood or utilised unless we under- stood what is meant by the assertions 'x is identical with x^ ' y is identical with jy' and '/ is identical with/,' and had accepted these assertions as true, because of our independent understanding of what is meant by the relation of identity. In the converse form of the identity formula we have admitted that if all the adjectives that characterise a substantive x also characterise a substantive y, then x and y are identical. But the hypothesis here is really impossible, for the adjective 'other than y' cannot characterise y. Hence there is one adjective at least THE RELATION OF IDENTITY 197 that must characterise x but not y. What then is the one relational adjective that must characterise any one existent and not any other? It must be existence itself. For to exist means to stand out fro^n amongst other things. Otherness is thus presupposed by existence. In short, if the existent is what is manifested in time and space, and if time and space are wholes divisible into parts, then the only necessarily differentiating mark of one existent must be its temporal and spatial position. This brings us back to the Leibnizian formula, but at the expense of admitting the reality of time and space as the condition of otherness — a condition which is both necessary and sufficient. Although in the sense explained identity always implies the legitimacy of substitution, we cannot say conversely that the legitimacy of substitution always implies identity. For whenever two predications are co-implicative, the one may always be substituted for the other in the same way a§ for substantives which are identical. Thus, for substantives x and jy, we have the formula : If .a: is identical with^, then 'x is/' is co-implica- tive with 'y is/,' where/ is any predication appli- cable to X and y. Corresponding to this, for predications q and r, we have: If ^ is co-implicative with r, then 'q is /^' is co- implicative with 'r is n^ where n is any predication applicable to q and r. Thus, given the co-implication of the two predications {(j) human being, and (r) featherless biped, we can infer that the proposition 'the number of human beings is n 198 CHAPTER XII is co-implicative with the proposition 'the number of featherless bipeds is nj And again given the co-impli- cation of the two propositions (q) 'There is a righteous God' and (r) 'The wicked will be punished,' we can infer that the proposition 'That there is a righteous God is problematic' is co-implicative with the proposition 'That the wicked will be punished is problematic' These examples show that the relation of co-implication be- tween predications has some of the same properties as that of identification between substantives, and therefore co-implication is apt to be conceived (and even by some symbolic logicians has actually been symbolised) as equivalent to identification. It appears to me that it is theoretically possible that the conception of co-implica- tion could be shown to correspond to factual identifica- tion; but this indeed is doubtful, because the relation of co-implication is compound, i.e. it denotes the conjunc- tion of the two correlatives implying and implied by, whereas it seems impossible to reduce the notion of identification to the conjunction of two correlatives. Before dismissing this subject, it must be admitted that both as regards substitution for identified substan- tive terms, and substitution for co-implicative predica- tions, certain limitations seem to be required. For example Mr Russell has familiarised us with illustra- tions for the necessity of this limitation by such ex- amples as that, from the identification of 'Scott' with 'the author of Waver ley,' we cannot, by substituting the one term for the other in such a proposition as 'George IV believed Scott to have written Marmion infer that 'George IV believed the author of Waver ley to have written Marmion.' Or a^ain that the number of the THE RELATION OF IDENTITY 199 apostles is identical with the number of months in the year, does not necessarily imply that anyone who doubts that the number of apostles is 12, would necessarily doubt that the number of months was 12. It appears that the only statements for which such substitutions are invalid are secondary propositions predicating of a primary proposition some or other relation to a thinker. \ 8. To complete the account of identity, it is neces- sary to anticipate what will be elaborated in a later part of the work; and for this purpose the discussion will be restricted to identity of substantives proper, excluding any further reference to the identity of adjectives or predications. A substantive proper I have defined as what is manifested in space and time, or otherwise an existent; and the category existent has been divided into the two sub-categories called respectively occurrent and continuant. Now identity as applied to an occur- rent could be illustrated thus: "The flash of lightning to which I am pointing is identical with the flash of lightning to which you are pointing.' A continuant, on the other hand, means that which continues to exist while its states or relations may be changing ; identity of continuant may therefore be illustrated by some such examples as ' The body which illuminates the earth is identical with the body that warms the earth'; 'The person who was experiencing the tooth-ache is identical with the person who intends to go to the dentist.' This last conception we find discussed at some length under the heading 'Identity' by the earlier English writers Locke, Hume and Reid, who used the term to signify personal identity instead of giving to it the merely rela- tional significance of the more modern conception. In 200 CHAPTER XII effect, for them the assertion or denial of identity is equivalent to the assertion or denial that there is a person who continues to exist throughout a period of time in which his various experiences may be altering in character. The destructive view represented by Hume regarded experiences as what I call occurrents, any one of which is merely replaced by another in the course of time. This destructive view is equivalent to the denial of any psychical continuant. Necessarily an occurrent experience is as such identical with itself; what Hume denied was that kind of connection between one such occurrent and another which constitutes them into alter- able states of one individual self. Hence it was not the general conception of identity, but the reality of the psychical continuant that Hume denied and that Reid maintained. When this old problem is revived at the present day, it is usually formulated as the question whether there is any ultimate philosophical justification for regarding one and another experience as belonging to the same self This phraseology, however, is mis- leading ; for it appears to assume the existence of a self, and to raise only the question as to whether we can refer different experiences to the same self; whereas the real problem is whether there is a self at all. In discussions, by Hume and others, connected with that of 'personal identity,' we find the same problem raised as to the validity of the more general notion 'substance' — i.e. in my terminology, a continuant (whether psychical or physical). Now we might interpret Kant as admit- ting the validity of the conception of a physical con- tinuant while denying that of a psychical continuant. On the other hand, Berkeley more obviously supported THE RELATION OF IDENTITY 201 the notion of a psychical continuant and rejected that of a physical continuant. At the present day, most philosophers who reject these notions regard them as reached by a constructive process, and therefore as being merely convenient fictions. To assume that a notion is necessarily fictitious because it owes its origin to a constructive process is fallacious, especially for those who accept the central Kantian position according to which the objective validity of any conception is estab- lished by showing it to be the result of a synthetic, i.e. constructive act of pure thought. The same relation between identity and the notion of a continuant applies to the physical as to the psychical continuant. In the physical sphere, those who reject the physical continu- ant maintain that any physical event is a mere occur- rent which is replaced by another occurrent; and that a so-called change is merely the temporal succession of one and another occurrent. Those, on the other hand, who accept the notion of a physical continuant maintain the validity of the notion of change as distinct from mere alternation, and as therefore presupposing the con- ception of a physical continuant. We may indicate more positively the distinction between the two views which respectively reject and accept the notion of a continuant, while agreeing in the application of the relation of identity. Those who deny continuance of existence as well as those who affirm it can legitimately collect all the experiences, occurring during some part or the whole of what we call an indi- vidual's total experience, to constitute a class and assert that this collection is identical with itself. On the other hand, for those who affirm the continuant, the collection is 202 CHAPTER XII not a mere plurality but a specific kind of unity ; in other words, they hold that an intimate bond of causality subsists between the experiences attributable to one individual of a kind which does not subsist between experiences arbitrarily selected from the histories of different individuals. The notion of this unique kind of bond is, on this view, the product of a constructive act, but not to be dismissed on this ground as merely fictitious. 203 CHAPTER XIII RELATIONS OR TRANSITIVE ADJECTIVES § I. So far we have treated the adjective solely in its reference to the substantive which it characterises. We have now to consider a type of adjective whose meaning when analysed exhibits a reference to some substantive other than that which it characterises. Thus we may characterise a certain child by the adjective ' liking a certain book,' or a certain book by the adjec- tive ' pleasing a certain child.' These adjectives predi- cated respectively of the child and of the book, are complex; and when we take the substantival reference out of this complex, there remains the term 'liking' or 'pleasing.' Such terms do not function as completed adjectives, and will be called relational adjectives. Propositions involving adjectives of this type may be ranged in a series according to the number of substan- tives to which they refer. Thus, in the following ex- amples : 'A is wise,' 'A likes B,' 'A gives X to B,' 'A accuses B at time 7" of C,' the number of substan- tival references are respectively one, two, three and four, and the corresponding adjectives or propositions may be called monadic, diadic, triadic and tetradic. Taking first the two-termed relation, let us con- sider the proposition 'X likes K' or 'X is greater than Y.' The notion of '^ as liking K' or of ' X as being greater than Y' is to be distinguished from the notion of ' -K as liking X ' or ' JK as being greater than 204 CHAPTER XIII X! At the same time the thought of any assigned relation of X to Y involves the thought of a definitely- assignable relation of K to A" ; for example, the thought of X as liking Y involves the thought of Y as pleas- ing X\ and X as greater than Y involves Y as being less than X. Two relational adjectives such as liking and pleasing, or greater than and less than — each of which in this way involves the other — are called cor- relatives, and either one is said to be the converse of the other. When the relation is expressed by a transi- tive verb, the opposition between active and passive expresses the mutual implication of correlatives : thus, 'X likes K' means ' Y is liked by X,' or ' Y pleases X' means 'X is pleased by K.'^ Except in the case of the active and passive voice, there is no general rule of language according to which the converse of a given relative can be expressed, and therefore a special know- ledge of words in current use is required in order to be able to express a relation in its converse form ; as when we pass from 'X is greater than K' to ' K is less than A",' or from ' X likes Y' to ' Y pleases X.' How- ever, the fact expressed in terms of a relative is the same as the fact expressed in terms of its converse, whether the terms employed are philologically cognate or not. It must be pointed out that 'liking Y' or 'liking someone,' etc., is a completed adjective ; and, in general, out of a relational adjective we may construct a com- ^ Comparing ':r sleeps' or ^x is sleeping' with ^x hits }>' or '.r is hitting jv,' and noting that '■sleeps'' is an intransitive and hits a transitive verb^ we ought properly to call sleeping an intransitive and hitting a transitive adjective. Thus a relation is properly defined as a '■ transitive adjective,' the ordinary adjective being distinguished as intransitive. RELATIONS OR TRANSITIVE ADJECTIVES 205 pleted adjective by supplementing the substantival reference. And conversely most ordinary adjectives in use can be analysed so as to elicit a relational element as a component. For instance ' amiable ' contains the relational element 'liked by,' and may be roughly defined 'liked by most people.' Again, substantive words are constructed out of relational adjectives, e.g. 'a shep- herd' which means 'a person who takes care of sheep.' It is noteworthy, however, that to take the substantives ' shepherd ' and ' sheep ' as examples of correlatives involves a double error, since the true correlatives involved in the meaning of shepherd are ' taking care of and ' taken care of by,' which are adjectival and not substantival ; while the meaning of the word ' sheep ' contains no relational element at all. § 2. Our immediate concern will be with diadic adjectives, otherwise called coupling. Given any two substantives — say x and y — we may construct what will be termed a substantive-couple expressed by the phrase ' X to_y,' which is to be distinguished from the substan- tive couple ' y \.o x! Similarly, given any two correlative coupling adjectives — say greater-than and less-than — we construct what will be termed an adjective-couple, expressed by the phrase 'greater-than to. less-than.' The significance of a substantive-couple is to be ex- plained by defining it as that which may be characterised by an adjective-couple ; and the significance of an adjective-couple, by defining it as that which may characterise a substantive-couple. Thus the relation of substantive-couple to adjective-couple is the same as that of an ordinary adjective to an ordinary substan- tive ; and just as the latter are united through the 2o6 CHAPTER XIII characterising tie, so are the former. Again, just as the extension determined by an ordinary adjective comprises the substantives of which the adjective may be truly predicated, so we may say of an adjective- couple that the extension which it determines comprises the substantive-couples of which the adjective-couple may be truly predicated. This relation between the substantive-couple and the adjective-couple is brought out by expressing the proposition 'x is greater thanjv' in the form *:r toj)/ is-as greater than to less than,' and the proposition 'j/ is less than x' in the form 'y to X is-as less than to greater than.' This mode of formulation helps perhaps to explain the process of relational conversion, which may be illus- trated as follows : ( I ) jr is greater than y, .'. (2) X to y is-as greater than to less than, •'• (3) y ^^ ^ ^^"^^ ^^^^ '■^^^ ^^ greater than, • ' • (4) y is less than x. In passing from (1) to (2), the introduction of the term ' less than ' depends merely upon knowledge of the arbitrary usage of language ; but the logical validity of the step rests upon the fundamental principle of thought that every relation has its converse. Each step also requires that the order in which the adjective terms are mentioned is to be understood to correspond to that in which the substantive terms are mentioned. Similar reformulations could be applied to triadic and RELATIONS OR TRANSITIVE ADJECTIVES 207 higher orders of adjectives : thus 'x receives b fromjj/ will be rendered X \ b \y I is-as | receiving : given by : giving to ; and ' y gives b to x ' becomes y -.b -.x I is-as [ giving : given to : receiving from ; where ' : ' stands for ' to,' and the ordering of the words is to be interpreted cyclically. § 3. The general notion of an adjective-couple that can be predicated of a substantive-couple is familiarly illustrated in what is called analogy. Take the proposi- tion ' England to Australia is-as parent to child'; here the predicate is what I call an adjective-couple, consti- tuted by taking the relation ' parent-of ' and its converse *child-of'; while the subject is a substantive-couple composed of the two substantives England and Aus- tralia. The copula ' is-as ' marks a statement of analogy. Another example with the same adjective-couple is ' France to Algiers is-as parent to child.' From these two predications of the same adjective-couple we should infer that ' France is to Algiers as England is to Australia.' This form of proposition, however, differs importantly from that which predicates an adjective- couple of a substantive-couple. For in affirming the equivalence of the relation in which France stands to Algiers with that in which England stands to Australia, there is no indication of the kind of relation in respect of which the two substantive-couples agree. We might, in fact, compare this inference with the inference that 'X is like K' which could be drawn from the two propositions that 'X is red ' and ' Y is red '; or equally from the two propositions that 'X is square ' and ' Y is 2o8 CHAPTER XIII square.' We then see that such terms as similar or analogous when used to connect two substantives or two substantive-couples are quite indeterminate with respect to the ground of similarity or analogy ; so that from the assertions * A is similar to B ' and ' B is similar to C it could never be inferred that ' A is similar to C My formulation of the relational proposi- tion ' A to B is-as / to ^ ' is of course suggested by the arithmetical expression for a ratio and the equality of ratios, but here there is a danger of misunderstanding ; for in my phraseology we could assert that ' 8 to 6 is-as greater by 2 to less by 2,' and again that ' 5 to 3 is-as greater by 2 to less by 2 ' and hence that ' 8 to 6 is-as 5 to 3 ' : but in arithmetic the phraseology '/to ^ ' is understood to denote exclusively a relation under the genus called ra^io, and could not be applied to a rela- tion under any other genus such as difference or dis- tance, etc., etc. Instead of calling such a term as ratio, difference or distance, etc., by the familiar na.mQ £-enus, it ought, properly speaking, to be termed ' relational determinable ' in contradistinction to three-fifths, or minus two, or three yards distant, which are ' relational determinates ' under their respective relational deter- minables. Whereas Ae7^e the phraseology ' a to if is-as / to ^ ' is used for relational determinates under any relational determinable, in arithmetic this phraseology is limited to relational determinates under the one re- lational determinable called ra^io. It must be observed that, when the term analogy is explicitly referred to a given relational determinable, such as 7-a^zo, then, from the assertion that two substantive couples are analogous to the same substantive couple, we may infer that the RELATIONS OR TRANSITIVE ADJECTIVES 209 two are analogous to one another. This is precisely parallel to the case of agreement or similarity when explicitly referred to any given adjectival determinable, such as colour. § 4. This account of relational adjectives leads to a consideration of a species of tie distinct from the characterising tie, which we shall call the coupling tie. In the phrase ' x to y' the word to has been chosen to indicate this tie, and hence the effect of the coupling tie is to construct a substantive-couple. Any prepo- sition or prepositional phrase such as of, by, for, at, with, in, in reference to, indicates the presence of the coupling tie. We must not, however, in general say that the preposition denotes merely a tie ; for a dif- ference of preposition often stands for a difference in the relation predicated. For example ' x is influenced to move towards y' has a different meaning from 'x is influenced to move away from y' ; where the difference of preposition is seen to entail a difference of relation — namely the difference between attraction and repulsion. In fact, prepositions used along with adjectives or verbs express determinate modifications of relation. The essential feature of a tie, on the other hand, is that it is incapable of modification, and hence we frequently find that it does not enter as a separate verbal com- ponent in a sentence. Whenever a tie (whether it be the characterising tie, or the coupling tie, or any other) does not appear as an actual word, there are conventions of language which indicate its presence. In languages in which inflexion is largely used, such as Latin and German, there are two main kinds of grammatical rule ; namely, the rules J. L. 14 210 CHAPTER XIII of concordance and the rules of governance. We shall find that the rules of concordance point to the presence of the characterising tie ; and those of governance to the coupling tie. The rules of concordance are, briefly, that adjectives and verbs must agree in gender, number and case, with the substantives that they characterise ; so that the characterising tie is not necessarily expressed by use of the verb 'to be ' but merely by inflexion. On the other hand, the rules of governance always deter- mine the case — genitive, dative, accusative, or ablative — of the substantive that is introduced along with any transitive verb, relational adjective, or preposition. We find, especially in Latin, that considerable changes in the order of words (which may vary for purposes of rhetorical significance) are permissible because of the inflexions which are understood to indicate (i) by gram- matical agreement, how the words are to be attached in thought by the characterising tie, and (ii) by gram- matical governance, how they are to be attached in thought by the coupling tie. Furthermore, where modification of case occurs (with or without a prepo- sition), not only the coupling tie, but also the special modification of relation that is to be understood is .grammatically indicated. The characteristic of English, in contrast to highly inflexional languages, is that no inflexions are required by rules either of concordance or of governance, except in the two instances : (i) for differences of person and number in many verbs (which illustrate the characterising tie), and (ii) the accusatives — him, her, me, us, them, whom (which illustrate the coupling tie). All the other instances of inflexion in English — for example the possessive pronouns and the RELATIONS OR TRANSITIVE ADJECTIVES 2ir tenses of verbs — are used, not according to any rules of concordance or governance, but to express distinc- tions of meaning. The difference between the two kinds of inflexion — the one being significant and the other syntactic — is brought out by comparing the English 'her father' or 'his mother,' where the differ- ence of gender is significant and not syntactic, with the French ' son pere ' or ' sa mere,' where the difference of gender is syntactic and not significant. In English, the conventional rules of concordance or governance are replaced — except in the two cases mentioned above — by the equally conventional ordering of the words. §5. The coupling tie — which might have been called the prepositional tie, in consideration of the grammati- cal rules of governance, or again the relational tie, in consideration of the philosophical problems that have been raised in regard to the nature of relation — is of fundamental importance in discussing one of the para- doxes that Mr Bradley and others have found in the general notion of relation. The paradox is briefly brought out in the following contention: when we think o{ x as being r to y, we have first to relate x \.o yhy the rela- tion r, and then relate the relation r x.o x by — say f^ — • and r X.O y by — say r", another relation. This again will require that x should be related to r' by the further relation r"' , which will lead to an infinite regress on the side of X, and a similar regress on the side of y. This paradoxical contention is met by pointing out that in constructing an object out of the constituents x, r, andjv, we do not introduce another constituent by the mere act of constituting these constituents into a unity. The pretence of paradox is due to the assumption that to 14 — 2 212 CHAPTER XIII the act of relating or constructing there corresponds a special mode of relation ; so that a tie is confused with a relation. That a tie and a relation are distinct is brought out by considering the fact that if, for a given adjective — whether ordinary or relational — we substi- tute another adjective, we shall have constructed a different unity; but, if we drop the characterising tie with a view to replacing it by some adjective or rela- tion, then either the unity itself is destroyed, or it will be found that the characterising tie remains along with the adjective or relation so introduced. Similarly, the coupling of terms is not a mode of relating them for which another mode of relation could be substituted ; for, if they were uncoupled, again the unity would be destroyed. The distinction between a tie and a relation maybe brought out from another point of view by the con- sideration that the specific difference between one kind of tie and another is determined by the logical nature of the constituents tied. Thus the use of an adjective in general involves the characterising tie, by which it is attached to a substantive ; and the use of a relational adjective in particular further involves the coupling tie by which the two substantive-terms are attached to one another. On the other hand, where terms are related by a genuine relation, their logical nature allows any specific relation to be replaced by any other, this other being in general under the same relational determinable. Now just as we must distinguish between any rela- tion and the relational (or coupling) tie, so I have throughout assumed, and it is of the first importance to emphasise, the distinction between characterisation and RELATIONS OR TRANSITIVE ADJECTIVES 213 the characterising tie. I have, in fact, spoken repeatedly of the relation of adjective to substantive; and this is the relation called characterisation — a specific kind of relation to be distinguished, for instance, from 'liking' or 'exceeding,' etc. While characterisation, then, is a relation, the characterising tie (like any other tie) is a mode of connection, represented usually by the par- ticiple 'being.' This participle may be expanded into 'being-characterised-as-being,' and it will still represent merely the characterising tie. Thus the following series of examples are seen to be but different modes of ex- pressing precisely the same fact: I am thinking (i) of this 'as I tall; (2) of this jas-beingl tall; (3) of this | as- being- characterised -as -being j tall; where the phrase enclosed in vertical lines represents nothing more nor less than the characterising tie. The equivalence be- tween '5 as-being P' and ' S as-being characterised as- being /*,' etc., is precisely analogous to the arithmetical equalities: '5x/*' = '^x i x/*' = '.S'x i x i x /^,' where S stands for any magnitude and P for any numerical multiplier, while the number one takes the place of the relation 'characterised,' and the operator ' x ' takes the place of the tie 'as-being.' We will therefore call 'characterisation' the unit relation, because it may be conceived as a factor in every adjective and in every re- lation. The conception of 'characterised' as a relational factor, analogous to one in the expression ^ x ( i x P), will be shown more precisely by adopting a different mode of bracketing (3) whereby it becomes: (4) I am thinking of 'this |as-being| characterised-as-being-tall.' Now this last expression is formally analogous to(5) I am thinking of this | as-being j taller-than-that. The phrase 214 CHAPTER XIII I as-being |, occurring both in (4) and (5), represents the characterising tie ; but in (5) the relation (with its tie) is expressed by 'taller-than,' and supplemented by the substantive-term 'that,' while in (4) the relation (with its tie) is expressed by 'characterised-as-being' and sup- plemented by the adjective-term 'tall.' Since, moreover, we may expand the tie in (5) into 'as-being-character- ised-as-being,' we see that 'characterised' is a latent factor in every relation, as well as in every ordinary adjective. Thus 'characterised' is a latent factor even in the relation 'characterising': for 'I am thinking of P as characterising S' may be expanded into ' I am think- ing of P I as-being | characterised as characterising S.' This reformulation incidentally shows that, though char- acterised has the properties of a unit relation, yet its converse characterising has not these properties. § 6. Now the form of proposition in which 'character- ised' is introduced explicitly as a relation, derives its significance and its legitimacy from our having taken an adjective — namely 'tall' — as a term. We are therefore extending the application of the notion of a relation, when in this way we take an adjective as term, instead of (as hitherto) a substantive. Indeed no limit can be imposed upon the kind or category of entity which may constitute a 'term' of which adjectives or relations to other entities may be predicated. In particular, we must recognise that certain adjectives may be significantly predicated of adjectives and of propositions, and even of relations ; and that certain relations may be significantly predicated as subsisting between a substantive and an adjective, or between one adjective and another, or between one proposition and another, or even between RELATIONS OR TRANSITIVE ADJECTIVES 215 one relation and another. The adjectives, relations, or propositions of which other adjectives or relations may be predicated must when so connected be called terms, in contrast with the adjectives or relations pre- dicated of them. The logical mode in which adjectives, relations or propositions enter as terms into a construct, is reflected in language by the substantival form assumed by them ; e.g. intolerance, hatred, the enthusiasm of the people, that matter exists, to be or not to be, etc. etc. Underlying the merely nominal question whether adjectives, relations or propositions when functioning as substantives in a construct should be called terms, two philosophical questions arise, which I shall here deal with rather summarily. First, is it literally the same entity which can be treated indifferently either as an adjective in its primary or natural functioning, or as a quasi-substantive of which certain other adjectives or relations may be predicated? To this I give an affirmative answer; and the objections to my view are (I think) met by insisting that the adjectives or rela- tions which may be significantly predicated oi primary adjectives or relations (as they may here be called) belong to a different logical sub-category from these latter, and may be called secondary. Thus, it is not that the primary adjective changes its category when functioning asquasi-substantive, but it is that the second- ary adjective must be said to belong to a special sub- category, determined by the category of the primary adjective of which it may be predicated. But a second question arises: whether the relation or adjective that is apparently predicated of a relation, adjective or pro- position, is really so predicated; or whether it is pre- 2i6 CHAPTER XIII dicated rather of certain substantives to which there is implicit or explicit reference. As regards this question, it must be pointed out that while relations, subsisting primarily between certain entities entail relations be- tween other entities involved in or connected with the former, yet the relations thus entailed are not identical with the primary relation ; so that whether a relation is predicated of this or of that kind of entity needs separate discussion in each type of case. A typical example will be discussed in a succeeding paragraph. § 7. We now propose to analyse relational proposi- tions of more complicated forms. The principles (pre- viously illustrated) in accordance with which a diadic proposition may be reduced to a monadic by hyphening (or bracketing) the predicate, and may be converted by substituting for the given relation its correlative, may be extended to relational propositions of higher orders. In such propositions any one of the substantive- terms may be taken as subject, and a complex containing the remaining substantive-terms as monadic predicate; or any couple of substantive-terms may be taken as subject and a complex containing the remaining sub- stantive-terms as diadic predicate; and so on. In such transformations, any/^r;;2?//^/z^?2J that are made amongst the substantive-terms, will require the substitution of 'cognate' modes of expressing the given relation; and, as in diadic relations, the order of relationality is reduced by constructing a complex predication indicated by a hyphen or bracket. For example, the triadic proposition 'A gives XtoB'is reduced to a diadic by transforming it into 'A gives-.^to B' or 'B is receiver-from-^ of X' or 'X is given-to-^ by A,' which again may be converted, RELATIONS OR TRANSITIVE ADJECTIVES 217 respectively, into ' B receives-X from A,' 'X is given- hy-A to B,' 'A is giver-to-j9 of X' Thus the single radical relation represented by such a verb as 'to give,' from which a triadic proposition is constructed, gives rise to three pairs of converse forms (making six corre- latives), namely: giver-to and receiver-from ; receiver- of and given-to; given-by and giver-of. And, further- more, each of the above six diadics becomes monadic by completing the hyphening of the predicate without any further verbal alteration. In the above illustrations a single radical relation was involved; but another kind of complication arises when, besides a number of substantival components, the proposition contains more than one adjectival or r^/«a;- //^;W component, these being in general indicated by different verbs. For example, the proposition 'A pre- vented B from hurting C contains the three substantive- terms A, B, C, together with two verbs, of which 'pre- vent' is the principal and 'hurt' is sub-ordinate. Such a proposition, treated merely as triadic, may be trans- formed by permutation into (e.g.) 'i5 was prevented by A from hurting C,' and by bracketing into ' C was saved- from-being-hurt-by-^ by ^.' But another form of ana- lysis may be used where — as in the case before us — we are dealing, not only with a plurality of substantive terms, but also with a plurality of radically different relational components. In this analysis, we take as a bracketed constituent a complex containing all the com- ponents of a complete proposition ; viz. the proposition that contains the subordinate relation or adjective. A proposition in this aspect may be called a possibile. The possibile in our illustration is '^'s-hurtingC,' and 2i8 CHAPTER XIII the proposition predicates of the person A the relation 'preventing.' The distinction between the two modes of bracketing — in both of which a triadic proposition is reduced to a diadic — will be shown by comparing: 'A is I preventer-from-hurting-C| by B' with 'A |prevents| the-hurting-of-C-by-^.' In the former the terms of the relation are the two persons 'A' and 'B^ and the rela- tion may be expressed : ^A' to ' B' is-as 'preventer-of-hurt-to-C-from' to 'prevented-from-hurting-C by'; in the latter the terms are the person 'A' and the pos- sibile 'Cs being hurt by B' the relation being: 'A' to 'C's being hurt by B' is-as 'preventer-of to 'prevented-by.' The immediate purpose of this illustration is to show that a proposition that predicates a relation of a term to a possibile, entails also the predication of certain complex relations between this principal term and the various substantive constituents of the possibile. The example under consideration may be more fully ex- pressed by explicit reference to the action of A which was causally preventive: thus, 'y^'s-instructing-Z? pre- vented ^'s hurting- C This more explicit form of state- ment is typical of the causal proposition as such which, properly speaking, always relates on^ possibile to another either by way of production or of prevention. For ex- ample, such a statement as 'The earth causes the fall of the stone' is an elliptical expression for the more fully analysed proposition 'The-proximity-of-the-earth causes the-fall-of-the-stone.' Since, however (as follows from the above remarks), a diadic relation of one pos- RELATIONS OR TRANSITIVE ADJECTIVES 219 sibile to another entails certain more complex relations amongst the substantive constituents of the possibilia, it is formally legitimate to assert that the earth stands in some relation to the fall of the stone, or again that the earth stands in some relation to the stone, though each of these relations is more complex than the diadic relation of causation, which holds between the two possibilia. § 8. The above discussion leads to the special pro- blem of the nature of the relation of assertion. Consider the simplest case : 'A asserts 5 to be characterised by P.' This contains the three terms A, S and P, together with the two verbs or relations 'assert' and 'character- ise/ of which 'assert' is the principal, and 'characterise' the subordinate. Taking the complex as expressing a triadic relation of ^4 to ^ to P; it may be reformulated in several ways: such as, 'A asserts-/^-to-characterise5"; 'P is asserted-by-^-to-characterise 5,' which predicate diadic relations of A to S, and oi P to S respectively. But the more natural mode of expressing the relation is as one of the thinker A to the possibile 'S being (characterised by) P.' My account of the relation of causation as holding primarily between possibilia but also as entailing relations (of a higher order than diadic) amongst the component terms of the possibilia, must be applied to any such relation as that of interrogating, doubting, considering, affirming, denying, etc., in which a thinker may stand to a possibile or assertibile. The relation of the thinker to the proposition as a whole does not preclude — it rather entails — relations to the constituents of the proposition. Thus, the relation of the thinker to the subject S is expressible in terms of/*, 220 CHAPTER XIII viz., * asserts-/'- to-characterise' ; and that of the thinker to the predicate P is expressible in terms of S, viz., *asserts-5'-to-be-characterised-by.' This principle — that assertion or judgment involves a relation of the thinker to each of the constituents of the proposition as well as to the proposition as a unitary- whole — is familiarly (but misleadingly) expressed in some such form as that 'in judgment the thinker asserts a relation of one idea to another.' If by idea was really meant 'the object of a thought' rather than 'the thought of an object,' this statement would, in my view, be essentially correct. Or more briefly : I agree that judg- ment relates the object of one thought to the object of another thought; but I deny that judgment relates the thought of one object to the thought of another. That is to say, though judgment requires us to think about objects, the judgment \sabout\\\^'&^ objects, and not about our thinking about them. The account of judgment that is to be rejected involves a confusion between a primary proposition which is about objects, and a secondary proposition (as it may here be termed) about our ideas of objects ; and, if this identification of a primary with a secondary proposition is consistently carried out, we should have to interpret a secondary proposition as a tertiary ; namely, as being about our ideas about our ideas of objects, and so ad infinituTn. But admitting, as I have done, that in judgment we assert a relation of one object of thought to another, say of S to P, it is neces- sary further to consider what sort of idea we have of 5 or of /• when we judge that 'S is P' \ and here I pro- pose to distinguish between the specific and the generic idea of S or of P. The specific idea of S that is involved RELATIONS OR TRANSITIVE ADJECTIVES 221 in doubting or asserting the proposition, isthe idea of ^as 'characterised by /^'; and this idea includes the ^^;2^r/<; idea of 6" as 'characterisable,' i.e. as of tha nature of a substantive. Similarly the specific idea of P is the idea o{ P as 'characterising 5" and this includes \h^ generic idea of P as 'characterising' ; i.e. as of the nature of an adjective. But here it is to be observed that the relation that may be said to be predicated, viz. that of character- isation, does not subsist between the idea of S and the idea of P^ since each of these ideas is specifically com- pleted in the single complex idea of '^S-as-characterised- by-/^' or of '/^-as-characterising-.S" or again of 'the- characterisation-of-vS-by-/^'; and I hold that these three phrases express different modes of constructing ox\^ and the same construct or complex object of thought. No difference of principle is involved when dealing with an explicitly material relational proposition, such as '^ is R X.O T.' Here the specific idea of 6" is the completed idea of S as 'being R to T'\ that of 7" is the completed idea of 7" as 'being R to 6"; that of 7?, as 'relating 5" to 7"'; and that of R as 'relating Tlo S.' These specific ideas include such generic ideas as that of S and of T as being substantives and of R and R as being a pair of relations correlative to one another. As in our simpler illustration, the completed idea is of a complex object of thought constructed out of three constituents {S, R, T) bound together in a certain form of unity. 222 CHAPTER XIV LAWS OF THOUGHT § I. It has been customary to apply the phrase Laws of Thought to three specific formulae ; but the application of the phrase should be extended to cover all first principles of Logic. By first principles we mean certain propositions whose truth is guaranteed by pure reason. It is often too hastily said that logic as such is not concerned with truth but only with consistency; as if a conclusion were guaranteed by formal logic merely because it is consistent with any arbitrarily assumed premisses. But this entirely misrepresents the function of formal logic, which is not permissive, but rather prohibitive. It guarantees the truth — not of any pro- position that is consistent with the premisses — but only of the proposition whose contradictory is inconsistent with the premisses. And even this statement goes too far; for logic does not allow any arbitrarily chosen pre- misses to be taken as true ; and thus the only conclusions that it can be said in any sense to guarantee are those which have been correctly inferred from premisses that are themselves true. When consistency is placed in a kind of antithesis to truth, it seems often to be assumed that logic is indifferent to truth. That the reverse is the case is shown by the consideration that to say that a conclusion is validly drawn from given premisses is tantamount to asserting the truth of a certain composite proposition, viz. that the premisses imply the conclusion. LAWS OF THOUGHT 223 In enunciating and formulating the fundamental principles of Logic, we shall not enter into the question whether they are all independent of one another, nor into the problem as to how a selection containing the smallest possible number could be made amongst them from which the remainder could be formally derived. This problem is perhaps of purely technical interest, and the attempt at its solution presents a fundamental, if not insuperable, difficulty: namely, that the procedure of deriving new formulae from those which have been put forward as to be accepted without demonstration, is governed implicitly by just those fundamental logical principles which it is our aim to formulate explicitly. We can, therefore, have no assurance that, in explicitly deriving formulae from an enumerated set of first principles, we are not surreptitiously using the very same formulae that we profess to derive. If this objec- tion cannot be removed, then the supposition that the whole logical system is based on a few enumerable first principles falls to the ground. § 2. The charge has been brought against all the fundamental principles of Formal Logic that they are trivial; or otherwise that they are nothing but truisms. Now a truism may be defined as a proposition which is (i) true, and (2) accepted by everybody on mere inspec- tion as true; and these are just the characteristics required of a fundamental principle of logic. Hence to charge the fundamental formulae with being mere truisms is not to condemn them, but to admit that they are fitted to fulfil the function for which they are intended. This function is to enable us to demonstrate further formulae, some of which, though true, are not accepted by every- 224 CHAPTER XIV body on mere inspection as true. It is an actual fact that by means of truisms and truisms alone we can demonstrate truths which are not truisms. The above and similar criticisms directed against the fundamental formulae of Logic will be best met by directly examining this or that formula so as to bring out its precise signifi- cance in view of the different points of view from which it has been criticised ; and we shall adopt this plan as occasion offers. § 3. Before enunciating the fundamental principles in detail, we will enquire into what is implied in speaking of them as 'Laws.' The word lazv is closely connected with the notion of an imperative ; and many logicians of the present day hold that the so-called laws of thought are no more imperatives than are the axioms of arith- metic or geometry. With this view I agree, inasmuch as the axioms of mathematics can themselves be re- garded as having an imperative aspect ; but this is because all truth may be so regarded. The idea of truth and falsity, in my view, carries with it the notion of an imperative, namely of acceptance and rejec- tion — a corollary from the theory which insists on the reference of judgment and assertion to the thinker. For it is only so far as assertion is recognised to be a mental act, that the notion of an imperative becomes relevant. An imperative of reason implies a restraint upon the voluntary act of assertion — a restraint which does not, however, infringe the freedom that charac- terises every volition, since the obligation to think in accordance with truth is self-imposed. Any study of which imperatives constitute the subject-matter has been called a normative science, and normative sciences have LAWS OF THOUGHT 225 been contrasted with positive sciences. But from a certain point of view every science may be said to exercise an imperative function in so far as any mistake or confusion in the judgments of the ordinary man is corrected or criticised by the scientist as such. Every science therefore can without any confusion of thought be regarded as normative ; which is only another way of saying that the notions of truth and falsity as pre- dicable of propositions carry with them the notions 'to be accepted ' or ' to be rejected ' understood as impera- tives. But an explanation can be given for the restricted use of the term normative to logic, aesthetics and ethics: viz., that, while each deals with a certain kind of mental fact, it does not deal with it merely as fact. Every science which deals with man, either in his individual or social capacity, takes as its topic the description of mental facts — including an analysis of how men think, feel and act ; but such a descriptive study of our thoughts, feelings and actions (including their causal relations) treated generally, historically or speculatively, is to be distinguished from the study of precisely the same facts in relation to certain norms or standards, and from the critical examination of these norms or standards them- selves. The division oi sciences in general vato norma- tive and positive is, therefore, unsound, inasmuch as all sciences may be regarded as normative in the sense that they are potentially corrective of mistaken, false or obscure views. This division (into normative and posi- tive) is therefore properly restricted to sciences dealing with psychological material ; thus the positive or descrip- tive treatment of mind — in its thinking, feeling or actino- aspect — is (like all sciences) normative in the sense of J. L. 15 226 CHAPTER XIV being potentially corrective of false judgments on the topics directly dealt with ; while the treatment in Logic, Aesthetics and Ethics of these same processes is norma- tive in the more special sense that these sciences examine and criticise the norms of thought, feeling or action themselves. Within the range for which the antithesis between normative and positive holds, the distinction between a descriptive or causal account of psychological or sociological matters, and an examination of standards or norms, is now-a-days of the first importance, inasmuch as the substitution of causal description in the place of evahiation of standard has been woefully common in works which profess to found Ethics upon psychology or sociology. § 4. To return to the consideration of the principles which exercise an imperative function. The funda- mental formulae for conjunctive and composite pro- positions have been given in the chapter on compound propositions ; these must be included in the general consideration of the Laws of Thought. Certain of these laws, and in particular the Reiterative, Commutative and Associative laws of Conjunction are — not only the materials which explicitly compose the logical system — but are also implicitly used in the process of building up the system. Thus, for example, we not only explicitly formulate the Reiterative Law, but in making repeated use of this or of any other law, we are implicitly using the Reiterative Principle itself This will be seen to hold in the same way of the Commutative and Associa- tive principles of Conjunction. Finally, inasmuch as the system is developed by means of inference, the essen- tial principles of implication are not only explicitly LAWS OF THOUGHT 227 formulated in the formulae for composite propositions, but also implicitly used in constructing the logical system itself. \ 5, The next set of laws to be considered will be those which express the nature of Identity, since this is a formal conception which applies with absolute uni- versality to all possible objects of thought whatever the category to which they may belong. Identity is a re- lation, and as such has certain properties which are exhibited in what we shall call the Laws of Identity. Relations in general may be classified according to the formal properties they possess, irrespectively of the terms related. It will be necessary here to introduce and define three of these properties, viz., transitiveness, symmetry and reflexiveness. Using the symbols x, y, z, to stand for the terms of any relation, and the symbols r and r for any relation and its converse, then, (i) the relation r is called transitive : when 'x is r toy' and 'y is r io 2' together implies 'x is r to 2' for all cases of .r, y, 2; for example, ancestor, greater than, causing, implying; (2) the relation r is called symmetrical : when ^ x is r to y' implies ' y is r to x' ; in other words, when 'x is r to y' implies 'x is r to y,' for all cases o( x andj^; for example, cousin, incompatible with, other than ; (3) the relation r is called reflexive: when '.r is r to x' for all cases oi x\ for example, co??ipatriot of, simul- taneous with, homogeneous with. Now the three Laws of Identity are most simply expressible by the statement that identity is (i) transi- tive, (2) symmetrical, (3) reflexive ; or otherwise, for 15—2 228 CHAPTER XIV every object of thought (represented by the symbols (i) Transitive Law: \{ x is identical with jj/, andjj/ is identical with z ; then x is identical with z. (2) Symmetrical Law. If :r is identical withjy, then y is identical with x. (3) Reflexive Law. x is identical with x. It will be observed that there are a host of other relations which have these same three properties ; e.g. contem- poraneous, homogeneous, compatriot, numerically equal, equal in magnitude, etc., but analysis of every such rela- tion shows it to contain a reference to some identical element, upon which these formal properties depend. § 6. The phrase ' Law of Identity' has been tra- ditionally used for one of the three fundamental logical principles, known as the Laws of Identity, of Non- Contradiction, and of Excluded Middle, to which the term ' Laws of Thought ' has been usually restricted ; but, since these three laws relate exclusively to pro- positions, whereas the conception of identity applies to all objects of thought, I propose to substitute for the traditional terminology, the Principles 'of Implication,' 'of Disjunction' and 'of Alternation' respectively ; and to insert a fourth, to be called the 'Principle of Counter- implication.' The four together will be entitled ' the Principles of Propositional Determination.' The four laws are thus brought into line with the four forms of composite proposition discussed in a preceding chapter. The composite propositions expressed in their general form, i.e. in terms of two independent components p, g, are of course not guaranteed as true by pure logic ; in LAWS OF THOUGHT 229 other words, they require material or experiential certifi- cation as opposed to merely formal or rational certifica- tion. T\\G^ piHnciples, on the other hand, are those cases of the composite propositions, expressed in their quite general form, the truth of which is guaranteed by pure logic. For the purposes of formulating the principles on the lines of the four composite functions, we may slightly modify the expression of these latter as follows : (i) Implicative Function : If /" is true, then Q is true. (2) Counterimplicative Function : If P is false, then g is false. (3) Disjunctive Function : Not both P true and Q true. (4) Alternative Function : Either P true or Q true. T\\& principles are obtained by substituting P for Q in the implicative and counterimplicative functions, and /'-false for ^-true in the disjunctive and alternative functions. Thus : Principles of Propositional Determinatio7i (P being any proposition) (i) Itnplicative: It must be that if P is true, then P is true, (2) Counterimplicative : It must be that if/* is false, then P is false, (3) Disjunctive : P cannot be both true and false, (4) Alternative : P must be either true or false, where the words ' must be ' and ' cannot be ' serve to indicate that the principles are formally or rationally certified. This formulation uses a single proposition P to- gether with the two adjectives true and false, in preference to the more usual mode of expression which employs two propositions, P and not-/*, and a single adjective 'true'; as in the following: 230 CHAPTER XIV (i) If /* is true, then P is true. (2) If not-P is true, then not-/* is true. (3) P and not-/* cannot both be true. (4) Either P or not-/* must be true. There are several reasons for adopting the former of these two modes of formulation in preference to the latter. In the first place it uses the comparatively simple notion of P being false instead of the rather awkward notion of not-/* being true. Secondly it enables us to define 'contradiction' by means of the principles, which would be impossible without a circle if we introduced the contradictories P and not-/* into the formulation. In the third place, the introduction of the phrases /'-true and P-false is in accordance with the fact that the adjectives true and false are the first characteristics by which the nature of the proposition as such is to be understood. A closer analysis of this formulation of the alternative and disjunctive principles will throw further liorht on the nature of the antithesis between the adjectives true and false. We have emphasised the point that these adjectives are predicable only of pro- positions; in other words 'anything that is true or false is a proposition ' ; the principle of alternation adds to this statement its complementary, viz., 'anything that is a proposition is true or false.' It is clear, of course, that these two statements are not synonymous. Again, the principle of disjunction states that the adjectives true and false are incompatible ; and this again goes be- yond what is explicitly involved in the statement that they are predicable exclusively of propositions. The most obvious immediate application of these LAWS OF THOUGHT 231 principles is obtained by taking P to stand for a definite singular proposition : ' 5- is /,' where ' s ' stands for a uniquely determined or singular subject, and 'p ' for any adjective. Then /'-false becomes 's is not-/.' In this application, the four principles may be called the Principles of Adjectival Determination, and assume the following form : Principle of Implication : If j- is/, then j' is/. Principle of Counterimplication : If j is not-/, then s is not-/. Principle of Disjunction : s cannot be both/ and not-/. Principle of Alternation : s must be either/ or not-/. In this application, the principles are expressed in terms of any adjectives/ and not-/ predicated of any subject s ; instead of being expressed in terms of the adjectives true and false predicated of any proposition P. In ordinary logical text-books the ' Laws of Thought ' are almost always expressed in this specialised form ; but, by this mode of enunciation, the generality which characterises the formulation in terms of propositions is lost; for when 'adjectives predicated of any subject' is substituted for 'propositions,' we have only a special case from which the general could not have been de- rived. It is convenient for many purposes to use the term ' predication ' to stand for ' adjective ' or * pro- position ' ; thus we may include both the general and the special formulae of determination in the abbreviated forms ' If/ then/'; ' If not-/ then not-/'; ' Not-both/ and not-/ ' ; 'Either/ or not-/' ; where/ stands for any predication. The two sets of formulae might again be expressed — without any modification of meaning — in the form of universals, since P stands for any proposi- tion, and s for any subject; thus: 232 CHAPTER XIV Generalised Form for Prepositional Determination Generalised Form for Adjectival Determination If any proposition is true, it is true If anything is^, it \s p If any proposition is false, it is false No proposition can be both true and false Any proposition must be either true or false If anything is not-^, it is not-^ Nothing can be both/ and not-/ Anything must be either/ or not/ A comparison between these two generalised formulations of the principles will bring out the impor- tant distinction between false and not-true and again between true and not-false. According to the principles of adjectival determination ' Anything must be either true or not-true ' ; whereas of propositions we can say ' Any proposition must be either true or false.' Now, since it is only propositions of which truth is properly predicable, therefore of anything that is not a proposi- tion the adjective true must be denied; thus we must say ' The table is not true ' on the elementary ground that 'the table' is not a proposition; but we cannot say that 'The table is false,' because it is only propositions which can be said to be false. Thus the principles of propositional determination force upon us the notable consideration that the word false does not really mean the same as not-true. To have expressed the principle of alternation in the form ' Anything must be either true or false' without the necessary restriction to a pro- position would have been actually wrong. On the other hand, the form ' Any proposition must be either true or not-true' is not sufficiently determinate, for this alter- native would hold of any subject whatever and fails to express the alternative peculiar to the proposition itself. LAWS OF THOUGHT 233 On the same ground, the disjunctive principle is not properly expressed in the form ' No proposition can be both true and not-true.' This affords another, and, in my view, the most important justification for formulating the principles in terms of the adjectives true and false instead of in terms of the propositions P and not-/*. In illustration of the above discussion we may point to the analogy between the four adjectives true, false, not- true, not-false and the four adjectives male, female, not- male, not- female. The antithesis between male and not-male or again between female and not-female is applicable to any subject whatever, but that between male and female is applicable exclusively to organisms. Analogously the antithesis between true and not-true or again between false and not-false is applicable to any subject-term whatever, but that between true and false is applicable exclusively to propositions. § 7. Now the so-called Law of Identity — which I have expressed in the form 'If /* is true then P is true' where P stands illustratively for any proposition — has been the favourite object of attack by critics of the prin- ciples of formal logic, on the score of its insignificance or triviality. The reason why the formula appears to have little or no significance Is that its implicans Is lite7'ally identical with its implicate. It will be found, however, that the necessary condition for the explicit use of the relation of Identity is that the Identified ele- ment should have entered into different contexts. It must be noted that this necessary reference to difference of context does not render the relation of Identity other than absolute, i.e. It In no way implies that in its two occurrences the identified element is partly identical 234 CHAPTER XIV and partly different. The application of this general condition to the Principle of Implication requires us to contemplate the proposition ' P is true' as one that may- have been asserted in different connections or on different occasions or by different persons. Then, since the formula '\i P is true then P is true' is to be under- stood as logically general, its full import can be expressed in the form: ' If the asserting of P in any one context is true, then the asserting of P in any context whatever is true.' If this analysis be accepted, it will be found that the principle could not have been enunciated except for the possibility of identifying an assertum or pro- position as distinct from the various attitudes (belief, interrogation, doubt, denial) which might have been adopted towards it on different occasions by the same or different persons. One important element of meaning, therefore, implicit in the formula is that it tacitly implies the identifiability of a proposition as such. Turning now to the adjective ' true ' as it occurs in our analysis of the formula, let us contrast it with cer- tain adjectives that are predicable of things in general. The principle — that what can be asserted in one con- text as true must be asserted in any other context as true — is- more familiarly particularised in the form 'any proposition that is once true is always true ' ; that is to say that ' true ' as predicable of any proposition is un- alterable ; whereas there are certain adjectives and relations predicable of things in general which may characterise them only temporarily. Contrasting, for instance, the Principle of Propositional Determination 'If any proposition is true it is true' with the Principle of Adjectival Determination ' If anything is p it is /,' LAWS OF THOUGHT 235 we find that in the former the copula 'is' is to be inter- preted without reference to the present or any other assigned time ; whereas in the latter the adjective/ may be alterable, so that the copula 'is' must here be under- stood as referring to definitely assigned time. In the case of anything that is at an assigned moment of time p, the principles of logic do not entitle us to assert that it will be or has always been p. Taking as examples 'The water has a temperature of 30° C or 'Mr B. is at home,' we must say on the one hand that if these pro- positions are true at any time, they are true at all times. But we must not say that if the predicate ' having a temperature of 30° C or the relation ' being at home ' is true of a given subject at one time, it will be true at all times. This obvious comment would not have been required if language had distinguished in the mode of the verb 'to be ' between a timeless predication and a tense (present, past or future). Certain logicians have, however, deliberately denied the dictum that what is once true is always true, and their denial appears to be due to a confusion between the time at which an asser- tion is made, and the time to which an assertion refers ; or as Mr Bosanquet has neatly put it — 'between the time