STUDIES AND EXERCISES
IN
FORMAL LOGIC

INCLUDING A GENERALISATION OF LOGICAL PROCESSES IN THEIR APPLICATION TO COMPLEX INFERENCES

BY
JOHN NEVILLE KEYNES, M.A., Sc.D.
UNIVERSITY LECTURER IN MORAL SCIENCE AND FORMERLY FELLOW OF PEMBROKE COLLEGE IN THE UNIVERSITY OF CAMBRIDGE

FOURTH EDITION RE-WRITTEN AND ENLARGED

𝕷𝖔𝖓𝖉𝖔𝖓

MACMILLAN AND CO., LIMITED

NEW YORK: THE MACMILLAN COMPANY

1906

[The Right of Translation and Reproduction is reserved]

First Edition (Crown 8vo.) printed 1884.

Second Edition (Crown 8vo.) 1887.

Third Edition (Demy 8vo.) 1894.

Fourth Edition (Demy 8vo.) 1906.

PREFACE TO THE FOURTH EDITION.

IN this edition many of the sections have been re-written and a good deal of new matter has been introduced. The following are some of the more important modifications.

In Part I a new definition of “connotative name” is proposed, in the hope that some misunderstanding may thereby be avoided; and the treatment of negative names has been revised.

In Part II the problem of the import of judgments and propositions in its various aspects is dealt with in much more detail than before, and greater importance is attached to distinctions of modality. Partly in consequence of this, the treatment of conditional and hypothetical propositions has been modified. I have partially re-written the chapter on the existential import of propositions in order to meet some recent criticisms and to explain my position more clearly. Many other minor changes in Part II have been made.

Amongst the changes in Part III are a more systematic treatment of the process of the indirect reduction of syllogisms, and the introduction of a chapter on the characteristics of inference.

An appendix on the fundamental laws of thought has been added; and the treatment of complex propositions which previously constituted Part IV of the book has now been placed in an appendix.

The reader of this edition will perceive my indebtedness to Sigwart’s Logic. I have received valuable help from Professor J. S. Mackenzie and from my son, Mr J. M. Keynes; and I cannot express too strongly the debt I once more owe to Mr W. E. Johnson, who by his criticisms has enabled me to improve my exposition in many parts of the book, and also to avoid some errors.

J. N. KEYNES.

6, HARVEY ROAD,
CAMBRIDGE,
4 September 1906.

vi

PREFACE TO THE FIRST EDITION.[1]

[¹] With some omissions.

IN addition to a somewhat detailed exposition of certain portions of what may be called the book-work of formal logic, the following pages contain a number of problems worked out in detail and unsolved problems, by means of which the student may test his command over logical processes.

In the expository portions of Parts I, II, and III, dealing respectively with terms, propositions, and syllogisms, the traditional lines are in the main followed, though with certain modifications; e.g., in the systematisation of immediate inferences, and in several points of detail in connexion with the syllogism. For purposes of illustration Euler’s diagrams are employed to a greater extent than is usual in English manuals.

In Part IV, which contains a generalisation of logical processes in their application to complex inferences, a somewhat new departure is taken. So far as I am aware this part constitutes the first systematic attempt that has been made to deal with formal reasonings of the most complicated character without the aid of mathematical or other symbols of operation, and without abandoning the ordinary non-equational or predicative form of proposition. This attempt has on the whole met with greater success than I had anticipated; and I believe that the methods formulated will be found to be both as easy and as effective as the symbolical methods of Boole and his followers. The book concludes with a general and sure method of solution of what Professor Jevons called the inverse problem, and which he himself seemed to regard as soluble only by a series of guesses.

The writers on logic to whom I have been chiefly indebted are De Morgan, Jevons, and Venn. To Mr Venn I am peculiarly indebted, not merely by reason of his published writings, vii especially his Symbolic Logic, but also for most valuable suggestions and criticisms while this book was in progress. I am glad to have this opportunity of expressing to him my thanks for the ungrudging help he has afforded me. I am also under great obligation to Miss Martin of Newnham College, and to Mr Caldecott of St John’s College, for criticisms which I have found extremely helpful.

CAMBRIDGE,
19 January 1884.

viii

PREFACE TO THE SECOND EDITION.

THIS edition has been carefully revised, and numerous sections have been almost entirely re-written.

In addition to the introduction of some brief prefatory sections, the following are among the more important modifications. In Part I an attempt has been made to differentiate the meanings of the three terms connotation, intension, comprehension, with the hope that such differentiation of meaning may help to remove an ambiguity which is the source of much of the current controversy on the subject of connotation. In Part II a distinction between conditional and hypothetical propositions is adopted for which I am indebted to Mr W. E. Johnson; and the treatment of the existential import of propositions has been both expanded and systematised. In Part IV particular propositions, which in the first edition were practically neglected, are treated in detail; and, while the number of mere exercises has been diminished, many points of theory have received considerable development. Throughout the book the unanswered exercises are now separated from the expository matter and placed together at the end of the several chapters in which they occur. An index has been added.

I have to thank several friends and correspondents, amongst whom I must especially mention Mr Henry Laurie of the University of Melbourne and Mr W. E. Johnson of King’s College, Cambridge, for suggestions and criticisms from which I have derived the greatest assistance. Mr Johnson has kindly read the proof sheets throughout; and I am particularly indebted to him for the generous manner in which he has placed at my disposal not only his time but also the results of his own work on various points of formal logic.

CAMBRIDGE,
22 June 1887.

ix

PREFACE TO THE THIRD EDITION.

THIS edition has been in great part re-written and the book is again considerably enlarged.

In Part I the mutual relations between the extension and the intension of names are examined from a new point of view, and the distinction between real and verbal propositions is treated more fully than in the two earlier editions. In Part II more attention is paid to tables of equivalent propositions, certain developments of Euler’s and Lambert’s diagrams are introduced, the interpretation of propositions in extension and intension is discussed in more detail, and a brief explanation is given of the nature of logical equations. The chapters on the existential import of propositions and on conditional, hypothetical, and disjunctive (or, as I now prefer to call them, alternative) propositions have also been expanded, and the position which I take on the various questions raised in these chapters is I hope more clearly explained. In Parts III and IV there is less absolutely new matter, but the minor modifications are numerous. An appendix is added containing a brief account of the doctrine of division.

In the preface to earlier editions I was glad to have the opportunity of acknowledging my indebtedness to Professor Caldecott, to Mr W. E. Johnson, to Professor Henry Laurie, to Dr Venn, and to Mrs Ward. In the present edition my indebtedness to Mr Johnson is again very great. Many new developments are due to his suggestion, and in every important discussion in the book I have been most materially helped by his criticism and advice.

CAMBRIDGE,
25 July 1894.

TABLE OF CONTENTS.

xxiv

REFERENCE LIST OF INITIAL LETTERS SHEWING THE AUTHORSHIP OR SOURCE OF QUESTIONS AND PROBLEMS.

• B = Professor J. I. Beare, Trinity College, Dublin;
• C = University of Cambridge;
• J = Mr W. E. Johnson, King’s College, Cambridge;
• K = Dr J. N. Keynes, Pembroke College, Cambridge;
• L = University of London;
• M = University of Melbourne;
• N = Professor J. S. Nicholson, University of Edinburgh;
• O = University of Oxford;
• O’S = Mr C. A. O’Sullivan, Trinity College, Dublin;
• R = the late Professor G. Croom Robertson;
• RR = Mr R. A. P. Rogers, Trinity College, Dublin;
• T = Dr F. A. Tarleton, Trinity College, Dublin;
• V = Dr J. Venn, Gonville and Caius College, Cambridge;
• W = Professor J. Ward, Trinity College, Cambridge.

Note. A few problems have been selected from the published writings of Boole, De Morgan, Jevons, Solly, Venn, and Whately, from the Port Royal Logic, and from the Johns Hopkins Studies in Logic. In these cases the source of the problem is appended in full.

STUDIES AND EXERCISES IN FORMAL LOGIC.

INTRODUCTION.

1. The General Character of Logic.—Logic may be defined as the science which investigates the general principles of valid thought. Its object is to discuss the characteristics of judgments, regarded not as psychological phenomena but as expressing our knowledge and beliefs; and, in particular, it seeks to determine the conditions under which we are justified in passing from given judgments to other judgments that follow from them.

As thus defined, logic has in view an ideal; it is concerned fundamentally with how we ought to think, and only indirectly and as a means to an end with how we actually think. It may accordingly be described as a normative or regulative science. This character it possesses in common with ethics and aesthetics. These three branches of knowledge—all of them based on psychology—form a unique trio, to be distinguished from positive sciences on the one hand, and from practical arts on the other. It may be said roughly that they are concerned with the ideal in the domains of thought, action, and feeling respectively. Logic seeks to determine the general principles of valid thought, ethics the general principles of right conduct, aesthetics the general principles of correct taste.

2. Formal Logic.—As regards the scope of logic, one of the principal questions ordinarily raised is whether the science is formal or material, subjective or objective, concerned with 2 thoughts or with things. It is usual to say that logic is formal, in so far as it is concerned merely with the form of thought, that is, with our manner of thinking irrespective of the particular objects about which we are thinking; and that it is material, in so far as it regards as fundamental the objective reference of our thought, and recognises as of essential importance the differences existing in the objects themselves about which we think.

Logic is certainly formal, or at any rate non-material, in the sense that it cannot guarantee the actual objective or material truth of any particular conclusions. Moreover any valid reasoning whatsoever must conform to some definite type, or—in other words—must be reducible to some determinate form; and one of the main objects of logic is by abstraction to discover what are the various types or forms to which all valid reasoning may be reduced.

But, on the other hand, it is essential that logic should recognise an objective reference in every judgment, that is, a reference outside the state of mind which constitutes the judgment itself: apart from this, as we shall endeavour to shew in more detail later on, the true nature of judgment cannot be understood. It is, moreover, possible for logic to examine and formulate certain general conditions which must be satisfied if our thoughts and judgments are to have objective validity; and the science may recognise and discuss certain general presuppositions relating to external nature which are involved in passing from the particular facts of observation to general laws.

Logic fully treated has then both a formal and a material side. The question may indeed be raised whether the distinction between form and matter is not a relative, rather than an absolute, distinction. All sciences are in a sense formal, since they abstract to some extent from the matter of thought. Thus physics abstracts in the main from the chemical properties of bodies, while geometry abstracts also from their physical properties, considering their figure only. In this way we become more and more formal as we become more and more general; and logic may be said to be more abstract, more 3 general, more formal, than any other science, except perhaps pure mathematics.

It is to be added that, within the domain of logic itself, the answer to the question whether two given propositions have or have not the same form may depend upon the particular system of propositions in connexion with which they are considered. Thus, if we carry our analysis no further than is usual in ordinary formal logic, the two propositions, Every angle in a semi-circle is equal to a right angle, Any two sides of a triangle are together greater than the third side, may be considered to be identical in form. Each is universal, and each is affirmative; they differ only in matter. But it will be found that in the logic of relatives, to which further reference will subsequently be made, the two propositions (one expressing an equality and the other an inequality) may be regarded as differing in form as well as in matter; and, moreover, that the difference between them in form is capable of being symbolically expressed.

The difficulty of assigning a distinctive scope to formal logic par excellence is increased by the fact that certain problems falling naturally into the domain of material logic—for example, the inductive methods—admit up to a certain point of a purely formal treatment.

It is not possible then to draw a hard and fast line and to say that a certain determinate portion of logic is formal, and that the rest is not formal. We must content ourselves with the statement that when we speak of formal logic in a distinctive sense we mean the most abstract parts of the science, in which no presuppositions are made relating to external nature, and in which—beyond the recognition of the necessary objective reference contained in all judgments—there is an abstraction from the matter of thought. Because they are so abstract, the problems of formal logic as thus conceived admit usually of symbolic treatment; and it is with problems admitting of such treatment that we shall more particularly concern ourselves in the following pages.

3. Logic and Language.—Some logicians, in their treatment of the problems of formal logic, endeavour to abstract not 4 merely from the matter of thought but also from the language which is the instrument of thought. This method of treatment is not adopted in the following pages. In order to justify the adoption of the alternative method, it is not necessary to maintain that thought is altogether impossible without language. It is enough that all thought-processes of any degree of complexity are as a matter of fact carried on by the aid of language, and that thought-products are normally expressed in language. That language is in this sense the universal instrument of thought will not be denied; and it seems a fair corollary that the principles by which valid thought is regulated, and more especially the application of these principles to the criticism of thought-products, cannot be adequately discussed, unless account is taken of the way in which this instrument actually performs its functions.

Language is full of ambiguities, and it is impossible to proceed far with the problems with which logic is concerned until a precise interpretation has been placed upon certain forms of words as representing thought. In ordinary discourse, to take a simple example, the word some may or may not be used in a sense in which it is exclusive of all ; it may be understood to mean not-all as well as not-none, or its full meaning may be taken to be not-none. The logician must decide in which of these senses the word is to be understood in any given scheme of propositional forms. Now, if thought were considered exclusively in itself, such a question as this could not arise; it has to do with the expression of thought in language. The fact that such questions do arise and cannot help arising shews that actually to eliminate all consideration of language from logic is an impossibility. A not infrequent result of attempting to rise above mere considerations of language is needless prolixity and dogmatism in regard to what are really verbal questions, though they are not recognised as such.

The method of treating logic here advocated is sometimes called nominalist, and the opposed method conceptualist. A word or two of explanation is, however, desirable in order that this use of terms may not prove misleading. Nominalism and conceptualism usually denote certain doctrines concerning the 5 nature of general notions. Nominalism is understood to involve the assertion that generality belongs to language alone and that there is nothing general in thought. But a so-called nominalist treatment of logic does not involve this. It involves no more than a clear recognition of the importance of language as the instrument of thought; and this is a circumstance upon which modern advocates of conceptualism have themselves insisted.

It is perhaps necessary to add that on the view here taken logic in no way becomes a mere branch of grammar, nor does it cease to have a place amongst the mental sciences. Whatever may be the aid derived from language, it remains true that the validity of formal reasonings depends ultimately on laws of thought. Formal logic is, therefore, still concerned primarily with thought, and only secondarily with language as the instrument of thought.

In our subsequent discussion of the relation of terms to concepts, and of propositions to judgments, we shall return to a consideration of the question raised in this section.[2]

[²] See sections [7] and [46].

4. Logic and Psychology.—Since processes of reasoning are mental processes depending upon the constitution of our minds, they fall within the cognizance of psychology as well as of logic. But laws of reasoning are regarded from different points of view by these two sciences. Psychology deals with such laws in the sense of uniformities, that is, as laws in accordance with which men are found by experience normally to think and reason. Logic, on the other hand, deals with laws of reasoning as regulative and authoritative, as affording criteria by the aid of which valid and invalid reasonings may be discriminated, and as determining the formal relations in which different products of thought stand to one another.

Looking at the relations between logic and psychology from a slightly different standpoint, we observe that while the latter is concerned with the actual, the former is concerned with the ideal. Logic does not, like psychology, treat of all the ways in which men actually reach conclusions, or of all the various modes in which, through the association of ideas or otherwise, one belief actually generates another. It is concerned with 6 reasonings only as regards their cogency, and with the dependence of one judgment upon another only in so far as it is a dependence in respect of proof.

There are various other ways in which the contrast between the two sciences may be expressed. We may, for example, say that psychology is concerned with thought-processes, logic with thought-products; or that psychology is concerned with the origin of our beliefs, logic with their validity.

Logic has thus a unique character of its own, and is not a mere branch of psychology. Psychological and logical discussions are no doubt apt to overlap one another at certain points, in connexion, for example, with theories of conception and judgment. In the following pages, however, the psychological side of logic is comparatively little touched upon. The metaphysical questions also to which logic tends to give rise are as far as possible avoided.

5. The Utility of Logic.—We have seen that logic has in view an ideal and treats of what ought to be. Its object is, however, to investigate general principles, and it puts forward no claim to be a practical art. Its utility is accordingly not to be measured by any direct help that it may afford towards the attainment of particular scientific truths. No doubt the procedure in all sciences is subject to the general principles formulated by logic; but, in details, the weighing of evidence will often be better performed by the judgment of the expert than by any formal or systematic observance of logical rules.

It is important to bear in mind that, in the study of logic, our immediate aim is the scientific investigation of general principles recognised as authoritative in relation to thought-products, not the formulation of a system of rules and precepts. It may be said that the art of dealing with particular concrete arguments, with the object of determining their validity, is related to the science of logic in the same way as the art of casuistry (that is, the art of deciding what it is right to do in particular concrete circumstances) is related to the science of ethics. Moreover, just as in the art of casuistry we meet with problems which are elusive and difficult to decide because in the concrete they cannot be brought exactly under the abstract 7 formulae of ethical science, so in the art of detecting fallacies we meet with arguments which cannot easily be brought under the abstract formulae of logical science. As it would be a mistake to subordinate ethics to the treatment of casuistical questions, so it would be a mistake to mould the science of logic with constant reference to concrete arguments which, either because of the ambiguity of the terms employed, or because of the uncertain bearing of the context in which they occur, elude any attempt to reduce them to a form to which general principles are directly applicable.

Wherein then consists the utility of logic? In answer to this question, it may be observed primarily that if logic determines truly the principles of valid thought, then its study is of value simply in that it adds to our knowledge. To justify the study of logic it is, as Mansel has observed, sufficient to shew that what it teaches is true, and that by its aid we advance in the knowledge of ourselves and of our capacities.

To this it must be added (in qualification of what has been said previously) that, while logic is not to be regarded as an art of attaining truth, it still does possess utility as propaedeutic to other studies and independently of the addition that it makes to our knowledge. Fallacious arguments can no doubt usually be recognised as such by an acute intellect apart from any logical study; and, as we have seen, it is not the primary function of logic to deal with particular concrete arguments. At the same time, it is only by the aid of logic that we can analyse a reasoning, explain precisely why a fallacious argument is faulty, and give the fallacy a name. In other words, while logic is not to be identified with the criticism of particular concrete arguments, such criticism when systematically undertaken must be based on logic.

Greater, however, than the indirect value of logic in its subsequent application to the examination of particular reasonings is its value as a general intellectual discipline. The study of logic cultivates the power of abstract thought; and it is not too much to say that, when undertaken with thoroughness, it affords a unique mental training.

PART I.

TERMS.

CHAPTER I.

THE LOGIC OF TERMS.

6. The Three Parts of Logical Doctrine—It has been usual to divide logical doctrine into three parts, dealing with terms (or concepts), propositions (or judgments), and reasonings respectively; and it will be convenient to adopt this arrangement in the present treatise. At the same time, we may in passing touch upon certain objections that have been raised to this mode of treating the subject.

Mr Bosanquet treats of logic in two parts, not in three, giving no separate discussion of names (or concepts). His main ground for taking up this position is that “the name or concept has no reality in living language or living thought, except when referred to its place in a proposition or judgment” (Essentials of Logic, p. 87). He urges that “we ought not to think of propositions as built up by putting words or names together, but of words or names as distinguished though not separable elements in propositions.” There is undoubted force in this argument, and attention should be called to the points raised by Mr Bosanquet, even though we may not be led to quite the same conclusion.

Logic is essentially concerned with truth and falsity as characteristics of thought, and truth and falsity are embodied in judgments and in judgments only. Hence the judgment 9 (or the proposition as expressing the judgment) may be regarded as fundamentally the logical unit. It would, moreover, now be generally agreed that the concept is not by itself a complete mental state, but is realised only as occurring in a context. Correspondingly the name does not by itself express any mental state. If a mere name is pronounced it leaves us in a state of expectancy, except in so far as it is the abbreviated expression of a proposition, as it may be when spoken in answer to a question or when the special circumstances or manner of its utterance connect it with a context that gives it predicative force.

At the same time, in ideal analysis the developed judgment yields the concept as at any rate a distinguishable element of which it is composed, while the proposition similarly yields the term; and in order that the import of judgments and propositions may be properly understood some discussion of concepts and terms is necessary.

The question as to the proper order of treatment remains. In dealing with this question we need not trouble ourselves with the enquiry as to whether the concept or the judgment has psychological priority, that is to say, as to whether in the first instance the process of forming judgments requires that concepts should have been already formed, or whether on the other hand the process of forming conceptions itself involves the formation of judgments, or whether the two processes go on pari passu. It is enough that the developed judgment and the proposition, as we are concerned with them in logic, yield respectively the concept, and the term as elements out of which they are constituted.

We shall then give a separate discussion of terms, and shall enter upon this part of the subject before discussing propositions. But in doing this we shall endeavour constantly to bear in mind that the proposition is the true logical unit, and that the logical import of terms cannot be properly understood except with reference to their employment in propositions.[3]

[³] In this connexion attention may be called to Mill’s well known dictum that “names are names of things, not of our ideas,” Apart from its context, the force of this antithesis may easily be misunderstood. It is clear that every name that is employed in an intelligible sense must have some mental equivalent, must call up some idea or other to our minds, and must therefore in this sense be the name of an idea. It is not, however, Mill’s intention to deny this. Nor, on the other hand, does he intend to assert that things actually exist corresponding to all the names we employ. His dictum really has reference to predication. What he means is that when any name appears as the subject of a proposition, an assertion is made not about the corresponding idea, but about something which is distinct both from the name and the idea, though both are related to it. He is in fact affirming the objective reference that is essential to the conception of truth or falsity. The discussion may, therefore, be said to be properly part of the discussion of the import of propositions rather than of names, and it would certainly be less puzzling if it were introduced in that connexion. Our special object, however, in referring to the matter here is not to criticise Mill, but to illustrate the difficulty of discussing names logically apart from the use that may be made of them for purposes of predication.

10 7. Names and Concepts.—We have in the preceding section spoken more or less indiscriminately of names (or terms) and of concepts, and this has been intentional. We have already expressed our disagreement with those who would exclude from logic all consideration of language. Our judgments cannot have certainty and universal validity unless the ideas which enter into them are fixed and determined; and, apart from the aid that we derive from language, our ideas cannot be thus fixed and determined.

It is, therefore, a mistake to treat of concepts to the exclusion of names. But, on the other hand, we must not forget that the logician is concerned with names only as representive of ideas. His real aim is to treat of ideas, though he may think it wiser to do so not directly, but indirectly by considering the names by which ideas are represented. For this reason it is well, now and then at any rate, to refer explicitly to the concept.

The so-called conceptualist school of logicians are apt in their treatment of the first part of logical doctrine to discuss problems of a markedly psychological character, as, for example, the mode of formation of concepts and the controversy between conceptualism and nominalism. Apart, however, from the fact that the conceptualist logicians do not draw so clear a line of distinction as do the nominalists between logic and psychology, the difference between the two schools is to a large extent 11 a mere difference of phraseology. Practically the same points, for example, are raised whether we discuss the extension and intension of concepts or the denotation and connotation of names. At the same time, it must be said that the attempt to deal with the intension of concepts to the entire exclusion of any consideration of the connotation of names appears to be responsible for a good deal of confusion.

8. The Logic of Terms.—Attention has already been called to the relation of dependence that exists between the logic of terms and the logic of propositions. It will be found that we cannot in general fully determine the logical characteristics of a given name without explicit reference to its employment as a constituent of a proposition. We cannot again properly discuss or understand the import of so-called negative names without reference to negative judgments.

It must be added that in dealing with distinctions between names, it is particularly difficult for the logician who follows at all on the traditional lines to avoid discussing problems that belong more appropriately to psychology, metaphysics, or grammar; and to some of the questions which arise it may hardly be possible to give a completely satisfactory answer from the purely logical point of view. This remark applies especially to the distinction between abstract and concrete terms, a distinction, moreover, which is of little further logical utility or significance. It is introduced in the following pages in accordance with custom; but adequately to discriminate between things and their attributes is the function of metaphysics rather than of logic. The portion of the logic of terms (or concepts) to which by far the greatest importance attaches is that which is concerned with the distinction between extension and intension.

9. General and Singular Names.—A general name is a name which is actually or potentially predicable in the same sense of each of an indefinite number of units; a singular or individual name is a name which is understood in the particular circumstances in which it is employed to denote some one determinate unit only.

The nature and logical importance of this distinction may 12 be illustrated by considering names as the subjects of propositions. A general name is the name of a divisible class, and predication is possible in respect of the whole or a part of the class; a singular name is the name of a unit indivisible. Hence we may take as the test or criterion of a general name, the possibility of prefixing all or some to it with any meaning.

Thus, prime minister of England is a general name, since it is applicable to more than one individual, and statements may be made which are true of all prime ministers of England or only of some. The name God is singular to a monotheist as the name of the Deity, general to a polytheist, or as the name of any object of worship. Universe is general in so far as we distinguish different kinds of universes, e.g., the material universe, the terrestrial universe, &c.; it is singular if we mean the totality of all things. Space is general if we mean any portion of space, singular if we mean space as a whole. Water is general. Professor Bain takes a different view here; he says, “Names of material—earth, atone, salt, mercury, water, flame—are singular. They each denote the entire collection of one species of material” (Logic, Deduction, pp. 48, 49). But when we predicate anything of these terms it is generally of any portion (or of some particular portion) of the material in question, and not of the entire collection of it considered as one aggregate ; thus, if we say, “Water is composed of oxygen and hydrogen,” we mean any and every particle of water, and the name has all the distinctive characters of the general name. Again, we can distinguish this water from that water, and we can say, “some water is not fit to drink”; but the word some cannot, as we have seen above, be attached to a really singular name. Similarly with regard to the other terms in question. It is also to be observed that we distinguish between different kinds of stone, salt, &c.[4]

[⁴] Terms of the kind here under discussion are called by Jevons substantial terms. (See Principles of Science, 2, § 4.) Their peculiarity is that, although they are concrete, the things denoted by them possess a peculiar homogeneity or uniformity of structure; also we do not as a rule use the indefinite article with them as we do with other general names.

A name is to be regarded as general if it is potentially 13 predicable of more than one object, although as a matter of fact it happens that it can be truly affirmed of only one, e.g., an English sovereign six times married. A really singular name is not even potentially applicable to more than one individual; e.g., the last of the Mohicans, the eldest son of King Edward the First. This may be differently expressed by saying that a really singular name implies in its signification the uniqueness of the corresponding object. We may take as examples the summum bonum, the centre of gravity of the material universe. It is not easy to find such names except in cases where uniqueness results from some explicit or implicit limitation in time or space or from some relation to an object denoted by a proper name. Even in such a case as the centre of gravity of the material universe some limitation in time appears to be necessary, for the centre of gravity of the universe may be differently situated at different periods.

Any general name may be transformed into a singular name by means of an individualising prefix, such as a demonstrative pronoun (e.g., this book), or by the use of the definite article, which usually indicates a restriction to some one determinate person or thing (e.g., the Queen, the pole star). Such restriction by means of the definite article may sometimes need to be interpreted by the context, e.g., the garden, the river ; in other cases some limitation of place or time or circumstance is introduced which unequivocally defines the individual reference, e.g., the first man, the present Lord Chancellor, the author of Paradise Lost.

On the other hand, propositions with singular names as subjects may sometimes admit of subdivision into universal and particular. This is the case when, with reference to different times or different conditions, a distinction is made or implied in regard to the manner of existence, actual or potential, of the object denoted by the name: for example, “Homer sometimes nods,” “The present Pope always dwells in the Vatican,” “This country is sometimes subject to earthquakes.”[5]

[⁵] Compare sections [70] and [82].

10. Proper Names.—A proper name is a name assigned as a mark to distinguish an individual person or thing from others, 14 without implying in its signification the possession by the individual in question of any specific attributes. Such names are given to objects which possess interest in respect of their individuality and independently of their specific nature. For the most part they are confined to persons and places; but they are also given to domestic animals, and sometimes to inanimate objects to which affection-value is attached, as, for example, by children to their dolls. Proper names form a sub-class of singular names, being distinguished from the singular names of which examples were given in the preceding section in that they denote individual objects without at the same time necessarily conveying any information as to particular properties belonging to those objects.[6]

[⁶] Proper names are farther discussed in section [25] in connexion with the connotation of names.

Many proper names, e.g., John, Victoria, are as a matter of fact assigned to more than one individual; but they are not therefore general names, since on each particular occasion of their use, with the exception noted below, there is an understood reference to some one determinate individual only. There is, moreover, no implication that different individuals who may happen to be called by the same proper name have this name assigned to them on account of properties which they possess in common.[7] The exception above referred to occurs when we speak of the class composed of those who bear the name, and who are constituted a distinct class by this common feature alone: e.g., “All Victorias are honoured in their name,” “Some Johns are not of Anglo-Saxon origin, but are negroes.” The subjects of such propositions as these must, however, be regarded as elliptical; written out more fully, they become all persons called Victoria, some individuals named John.

[⁷] Professor Bain brings out this distinction in his definition of a general name: “A general name is applicable to a number of things in virtue of their being similar, or having something in common.”

11. Collective Names.—A collective name is one which is applied to a group of similar things regarded as constituting a single whole; e.g., regiment, nation, army. A non-collective name, e.g., stone, may also be the name of something which is 15 composed of a number of precisely similar parts, but this is not in the same way present to the mind in the use of the name.[8]

[⁸] To collective name as above defined there is no distinctive antithetical term in ordinary use. The antithesis between the collective and the distributive use of names arises, as we shall see, in connexion with predication only.

A collective name may be singular or general. It is the name of a group or collection of things, and so far as it is capable of being correctly affirmed in the same sense of only one such group, it is singular; e.g., the 29th regiment of foot, the English nation, the Bodleian Library, But if it is capable of being correctly affirmed in the same sense of each of several such groups it is to be regarded as general; e.g., regiment, nation, library.[9]

[⁹] It is pointed out by Dr Venn that certain proper names may be regarded as collective, though such names are not common. “One instance of them is exhibited in the case of geographical groups. For instance, the Seychelles, and the Pyrenees, are distinctly, in their present usage, proper names, denoting respectively two groups of things. They simply denote these groups, and give us no information whatever about any of their characteristics” (Empirical Logic, p. 172).

Some logicians imply an antithesis between collective and general names, either regarding collectives as a sub-class of singulars, or else recognising a threefold division into singular, collective, and general. There is, properly speaking, no such antithesis; and both the above alternatives must be regarded as misleading, if not actually erroneous; for, as we have just seen, the class of collective names overlaps each of the other classes.

The correct and really important logical antithesis is between the collective and the distributive use of names. A collective name such as nation, or any name in the plural number, is the name of a collection or group of similar things. These we may regard as one whole, and something may be predicated of them that is true of them only as a whole; in this case the name is used collectively. On the other hand, the group may be regarded as a series of units, and something may be predicated of these which is true of them taken individually; in this case the name is used distributively.[10]

[¹⁰] It is held by Dr Venn (Empirical Logic, p. 170) that substantial terms are always used collectively when they appear as subjects of general propositions. If, however, we take such a proposition as “Oil is lighter than water” it seems clear that the subject is used not collectively, but distributively; for the assertion is made of each and every portion of oil, whereas if we used the term collectively our assertion would apply only to all the portions taken together. The same is clearly true in other instances; for example, in the propositions, “Water is composed of oxygen and hydrogen,” “Ice melts when the temperature rises above 32° Fahr.”

16 The above distinction may be illustrated by the propositions, “All the angles of a triangle are equal to two right angles,” “All the angles of a triangle are less than two right angles.” In the first case the predication is true only of the angles all taken together, while in the second it is true only of each of them taken separately; in the first case, therefore, the term is used collectively, in the second distributively. Compare again the propositions, “The people filled the church,” “The people all fell on their knees.”[11]

[¹¹] When in an argument we pass from the collective to the distributive use of a term, or vice versâ, we have what is technically called a fallacy of division or of composition as the case may be. The following are examples: The people who attended Great St Mary’s contributed more than those who attended Little St Mary’s, therefore, A (who attended the former) gave more than B (who attended the latter); All the angles of a triangle are less than two right angles, therefore A, B, and C, which are all the angles of a triangle, are together less than two right angles. The point of the old riddle, “Why do white sheep eat more than black?” consists in the unexpected use of terms collectively instead of distributively.

12. Concrete and Abstract Names.—The distinction between concrete and abstract names, as ordinarily recognised, may be most briefly expressed by saying that a concrete name is the name of a thing, whilst an abstract name is the name of an attribute. The question, however, at once arises as to what is meant by a thing as distinguished from an attribute ; and the only answer to be given is that by a thing we mean whatever is regarded as possessing attributes. It would appear, therefore, that our definitions may be made more explicit by saying that a concrete name is the name of anything which is regarded as possessing attributes, i.e., as a subject of attributes ; while an abstract name is the name of anything which is regarded as an attribute of something else, i.e., as an attribute of subjects.[12]

[¹²] The distinction is sometimes expressed by saying that an abstract name is the name of an attribute, a concrete name the name of a substance. If by substance is merely meant whatever possesses attributes, then this distinction is equivalent to that given in the text; but if, as would ordinarily be the case, a fuller meaning is given to the term, then the division of names into abstract and concrete is no longer an exhaustive one. Take such names as astronomy, proposition, triangle: these names certainly do not denote attributes; but, on the other hand, it seems paradoxical to regard them as names of substances. On the whole, therefore, it is best to avoid the term substance in this connexion.

17 This distinction is in most cases easy of application; for example, plane triangle is the name of all figures that possess the attribute of being bounded by three straight lines, and is a concrete name; triangularity is the name of this distinctive attribute of triangles, and is an abstract name. Similarly, man, living being, generous are concretes; humanity, life, generosity are the corresponding abstracts.[13]

[¹³] It will be observed that, according to the above definitions, a name is not called abstract, simply because the corresponding idea is the result of abstraction, i.e., attending to some qualities of a thing or class of things to the exclusion as far as possible of others. In this sense all general names, such as man, living being, &c., would be abstract.

Abstract and concrete names usually go in pairs as in the above illustrations. A concrete general name is the name of a class of things grouped together in virtue of some quality or set of qualities which they possess in common; the name given to the quality or qualities themselves apart from the individuals to which they belong is the corresponding abstract.[14] Using the terms connote and denote in their technical senses, as defined in the following [chapter], an abstract name denotes the qualities which are connoted by the corresponding concrete name. This relation between concretes and the corresponding abstracts is the one point in connexion with abstract and concrete names that is of real logical importance, and it may be observed that it does not in itself give rise to the somewhat fruitless subtleties with which the distinction is apt to be 18 associated. For when two names are given which are thus related, there will never be any difficulty in determining which is concrete and which is abstract in relation to the other.

[¹⁴] Thus, in the case of every general concrete name there is or may be constructed a corresponding abstract. But this is not true of proper names or other singular names regarded strictly as such. We may indeed have such abstracts as Caesarism and Bismarckism. These names, however, do not denote all the differentiating attributes of Caesar and Bismarck respectively, but only certain qualities supposed to be specially characteristic of these individuals. In forming the above abstracts we generalise, and contemplate a certain type of character and conduct that may possibly be common to a whole class. Compare page [45].

But whilst the distinction is absolute and unmistakeable when names are thus given in pairs, the application of our definitions is by no means always easy when we consider names in themselves and not in this definite relation to other names. We shall find indeed that if we adopt the definitions given above, then the division of names into abstract and concrete is not an exclusive one in the sense that every name can once and for all be assigned exclusively to one or other of the two categories.

We are at any rate driven to this if we once admit that attributes may themselves be the subjects of attributes, and it is difficult to see how this admission can be avoided. If, for example, we say that “unpunctuality is irritating,” we ascribe the attribute of being irritating to unpunctuality, which is itself an attribute. Unpunctuality, therefore, although primarily an abstract name, can also be used in such a way that it is, according to our definition, concrete.

Similarly when we consider that an attribute may appear in different forms or in different degrees, we must regard it as something which can itself be modified by the addition of a further attribute; as, for example, when we distinguish physical courage from moral courage, or the whiteness of snow from the whiteness of smoke, or when we observe that the beauty of a diamond differs in its characteristics from the beauty of a landscape.

Hence, if the definitions under discussion are adopted, we arrive at the conclusion that while some names are concrete and never anything but concrete, names which are primarily formed as abstracts and continue to be used as such are apt also to be used as concretes, that is to say, they are names of attributes which can themselves be regarded as possessing attributes. They are abstract names when viewed in one relation, concrete when viewed in another.[15]

[¹⁵] The use of the same term as both abstract and concrete in the manner above described must be distinguished from the not unfrequent case of quite another kind in which a name originally abstract changes its meaning and comes to be used in the sense of the corresponding concrete; as, for example, when we talk of the Deity meaning thereby God, not the qualities of God. Compare Jevons, Elementary Lessons in Logic, pp. 21, 22.

19 It must be admitted that this result is paradoxical. As yielding a division of names that is non-exclusive, it is also unscientific. There are two ways of avoiding this difficulty.

In the first place, we may further modify our definitions and say that an abstract name is the name of anything which can be regarded as an attribute of something else (whether it is or is not itself a subject of attributes), while a concrete name is the name of that which cannot be regarded as an attribute of something else. This distinction is simple and easy of application, it is in accordance with popular usage, and it satisfies the condition that the members of a division shall be mutually exclusive. But it may be doubted whether it has any logical value.

A second way of avoiding the difficulty is to give up for logical purposes the distinction between concrete and abstract names, and to substitute for it a distinction between the concrete and the abstract use of names. A name is then used in a concrete sense when the thing called by the name is contemplated as a subject of attributes, and in an abstract sense when the thing called by the name is contemplated as an attribute of subjects. It follows from what has been already said that some names can be used as concrete only, while others can be used either as abstract or as concrete. This solution is satisfactory from the logical point of view, since logic is concerned not with names as such, but with the use of names in propositions. It may be added that as logicians we have very little to do with the abstract use of names, A consideration of the import of propositions will shew that when a name appears either as the subject or as the predicate of a non-verbal proposition its use is always concrete.

13. Can Abstract Names be subdivided into General and Singular?—The question whether any abstract names can be considered general has given rise to much difference of opinion amongst logicians. On the one hand, it is argued that all 20 abstract names must necessarily be singular, since an attribute considered purely as such and apart from its concrete manifestations is one and indivisible, and cannot admit of numerical distinction.[16] On the other hand, it is urged that some abstracts must certainly be considered general since they are names of attributes of which there are various kinds or subdivisions; and in confirmation of this view it is pointed out that we frequently write abstracts in the plural number, as when we say, “Redness and yellowness are colours,” “Patience and meekness are virtues.”[17]

[¹⁶] This represents the view taken by Jevons. See Principles of Science, 2, § 3.

[¹⁷] Compare Mill, Logic, i. 2, § 4.

The solution of the question really depends upon our use of the term abstract.

If we adopt the definition given in the last paragraph but one of the preceding [section], and include under abstract names the names of attributes which are themselves the subjects of attributes, these latter attributes possibly varying in different instances, then there can be no doubt that some abstracts are general; for they are the names of a class of things which, while having something in common, are also distinguishable inter se.

So far, however, as the question is raised in regard to the abstract (as distinguished from the concrete) use of names in the manner indicated in the last paragraph of the preceding [section], we are led to the conclusion that it is only when names are used in a concrete sense that they can be considered general. For it is clear that the name of an attribute can be described as general only in so far as the attribute is regarded as exhibiting characteristics which vary in different instances, only in so far, that is to say, as it is itself a subject of attributes; and when the attribute is so regarded, the name is used in a concrete, not an abstract, sense.

Take the propositions, “Some colours are painfully vivid,” “All yellows are agreeable,” “Some courage is the result of ignorance,” “Some cruelty is the result of fear,” “All cruelty is detestable.” The subjects of these propositions are certainly 21 general. According to the definition given in the last paragraph but one of the preceding section they are also abstract. If, however, in place of distinguishing between abstract and concrete names per se, we distinguish between the abstract and the concrete use of names as proposed in the last paragraph of the preceding section, then the terms in question are all used in a concrete, not an abstract, sense.

EXERCISES.

14. Discuss Mill’s statement that “names are names of things, not of our ideas,” with special reference to the following names: dodo, mermaid, chimaera, toothache, jealousy, idea. [C.]

15. Discuss the logical characteristics of adjectives. [K.]

CHAPTER II.

EXTENSION AND INTENSION.

16. The Extension and the Intension of Names.[18]—Every concrete general name is the name of a real or imaginary class of objects which possess in common certain attributes; and there are, therefore, two aspects under which it may be regarded. We may consider the name (i) in relation to the objects which are called by it; or (ii) in relation to the qualities belonging to those objects. It is desirable to have terms by which to refer to this broad distinction without regard to further refinements of meaning; and the terms extension and intension will accordingly be employed to express in the most general way these two aspects of names respectively.[19]

[¹⁸] We may speak also of the extension and the intension of concepts. In the discussion, however, of questions concerning extension and intension, it is essential to recognise the part played by language as the instrument of thought. Hence it seems better to start from names rather than from concepts. Neglect to consider names explicitly in this connexion has been responsible for much confusion.

[¹⁹] It is usual to employ the terms comprehension and connotation as simply synonymous with intension, and denotation as synonymous with extension. We shall, however, presently find it convenient to differentiate the meanings of these terms. The force of the terms extension and intension in the most general sense might perhaps also be expressed by the pair of terms application and implication.

The extension of a name then consists of objects of which the name can be predicated; its intension consists of properties which can be predicated of it. For example, by the extension of plane triangle we mean a certain class of geometrical figures, and by its intension certain qualities belonging to such figures. 23 Similarly, by the extension of man is meant a certain class of material objects, and by its intension the qualities of rationality, animality, &c., belonging to these objects.

17. Connotation, Subjective Intension, and Comprehension.—The term intension has been used in the preceding section to express in the most general way that aspect of general names under which we consider not the objects called by the names but the qualities belonging to those objects. Taking any general name, however, there are at least three different points of view from which the qualities of the corresponding class may be regarded; and it is to a want of discrimination between these points of view that we may attribute many of the controversies and misunderstandings to which the problem of the connotation of names has given rise.

(1) There are those qualities which are essential to the class in the sense that the name implies them in its definition. Were any of this set of qualities absent the name would not be applicable; and any individual thing lacking them would accordingly not be regarded as a member of the class. The standpoint here taken may be said to be conventional, since we are concerned with the set of characteristics which are supposed to have been conventionally agreed upon as determining the application of the name.

(2) There are those qualities which in the mind of any given individual are associated with the name in such a way that they are normally called up in idea when the name is used. These qualities will include the marks by which the individual in question usually recognises or identifies an object as belonging to the class. They may not exhaust the essential qualities of the class in the sense indicated in the preceding paragraph, but on the other hand they will probably include some that are not essential to it. The standpoint here taken is subjective and relative. Even when there is agreement as to the actual meaning of a name, the qualities that we naturally think of in connexion with it may vary both from individual to individual, and, in the case of any given individual, from time to time.

We may consider as a special case under this head the 24 complete group of attributes known at any given time to belong to the class. All these attributes can be called up in idea by any person whose knowledge of the class is fully up to date; and this group may, therefore, be regarded as constituting the most scientific form of intension from the subjective point of view.

(3) There is the sum-total of qualities actually possessed in common by all members of the class. These will include all the qualities included under the two preceding heads,[20] and usually many others in addition. The standpoint here taken is objective.[21]

[²⁰] It is here assumed, as regards the qualities mentally associated with the name, that our knowledge of the class, so far as it extends, is correct.

[²¹] When the objective standpoint is taken, there is an implied reference to some particular universe of discourse, within which the class denoted by the name is supposed to be included. The force of this remark will be made clearer at a subsequent stage.

In seeking to give a precise meaning to connotation, we may start from the above classification. It suggests three distinct senses in which the term might possibly be used, and as a matter of fact all three of these senses have been selected by different logicians, sometimes without any clear recognition of divergence from the usage of other writers. It is desirable that we should be quite clear in our own minds in which sense we intend to employ the term.

(i) According to Mill’s usage, which is that adopted in the following pages, the conventional standpoint is taken when we speak of the connotation of a name. On this view, we do not mean by the connotation of a class-name all the qualities possessed in common by the class; nor do we necessarily mean those particular qualities which may be mentally associated with the name; but we mean just those qualities on account of the possession of which any individual is placed in the class and called by the name. In other words, we include in the connotation of a class-name only those attributes upon which the classification is based, and in the absence of any of which the name would not be regarded as applicable. For example, although all equilateral triangles are equiangular, equiangularity is not on this view included in the connotation of equilateral 25 triangle, since it is not a property upon which the classification of triangles into equilateral and non-equilateral is based; although all kangaroos may happen to be Australian kangaroos, this is not part of what is necessarily implied by the use of the name, for an animal subsequently found in the interior of New Guinea, but otherwise possessing all the properties of kangaroos, would not have the name kangaroo denied to it; although all ruminant animals are cloven-hoofed, we cannot regard cloven-hoofed as part of the meaning of ruminant, and (as Mill observes) if an animal were to be discovered which chewed the cud, but had its feet undivided, it would certainly still be called ruminant.

(ii) Some writers who regard proper names as connotative appear to include in the connotation of a name all those attributes which the name suggests to the mind, whether or not they are actually implied by it. And it is to be observed in this connexion that a name may in the mind of any given individual be closely associated with properties which even the same individual would in no way regard as implied in the meaning of the name, as, for instance, “Trinity undergraduate” with a blue gown. This interpretation of connotation is, therefore, clearly to be distinguished from that given in the preceding paragraph.

We may further distinguish the view, apparently taken by some writers, according to which the connotation of a class-name at any given time would include all the properties known at that time to belong to the class.

(iii) Other writers use the term in still another sense and would include in the connotation of a class-name all the properties, known and unknown, which are possessed in common by all members of the class. Thus, Mr E. C. Benecke writes,—“Just as the word ‘man’ denotes every creature, or class of creatures, having the attributes of humanity, whether we know him or not, so does the word properly connote the whole of the properties common to the class, whether we know them or not. Many of the facts, known to physiologists and anatomists about the constitution of man’s brain, for example, are not involved in most men’s idea of the brain; the possession 26 of a brain precisely so constituted does not, therefore, form any part of their meaning of the word ‘man.’ Yet surely this is properly connoted by the word…. We have thus the denotation of the concrete name on the one side and its connotation on the other, occupying perfectly analogous positions. Given the connotation,—the denotation is all the objects that possess the whole of the properties so connoted. Given the denotation,—the connotation is the whole of the properties possessed in common by all the objects so denoted” (Mind, 1881, p. 532). Jevons uses the term in the same sense. “A term taken in intent (connotation) has for its meaning the whole infinite series of qualities and circumstances which a thing possesses. Of these qualities or circumstances some may be known and form the description or definition of the meaning; the infinite remainder are unknown” (Pure Logic, p. 6).[22]

[²²] Bain appears to use the term in an intermediate sense, including in the connotation of a class-name not all the attributes common to the class but all the independent attributes, that is, all that cannot be derived or inferred from others.

While rejecting the use of the term connotation in any but the first of the above mentioned senses, we shall find it convenient to have distinctive terms which can be used with the other meanings that have been indicated. The three terms connotation, intension, and comprehension are commonly employed almost synonymously, and there will certainly be a gain in endeavouring to differentiate their meanings. Intension, as already suggested, may be used to indicate in the most general way what may be called the implicational aspect of names; the complex terms conventional intension, subjective intension, and objective intension will then explain themselves. Connotation may be used as equivalent to conventional intension ; and comprehension as equivalent to objective intension. Subjective intension is less important from the logical standpoint, and we need not seek to invent a single term to be used as its equivalent.[23]

[²³] For anyone who is given the meaning of a name but knows nothing of the objects denoted by the name, subjective intension coincides with connotation. Were the ideal of knowledge to be reached, subjective intension would coincide with comprehension.

27 Conventional intension or connotation will then include only those attributes which constitute the meaning of a name;[24] subjective intension will include those that are mentally associated with it, whether or not they are actually signified by it; objective intension or comprehension will include all the attributes possessed in common by all members of the class denoted by the name. We might perhaps speak more strictly of the connotation of the name itself, the subjective intension of the notion which is the mental equivalent of the name, and the comprehension of the class which is denoted by the name.[25]

[²⁴] It is to be observed that in speaking of the connotation of a name we may have in view either the signification that the name bears in common acceptation, or some special meaning assigned to it by explicit definition for some scientific or other specific purpose.

[²⁵] The distinctions of meaning indicated in this section will be found essential for clearness of view in discussing certain questions to which we shall pass on immediately; in particular, the questions whether connotation and denotation necessarily vary inversely, and whether proper names are connotative.

18. Sigwart’s distinction between Empirical, Metaphysical, and Logical Concepts.—Sigwart observes that in speaking of concepts we ought to distinguish between three meanings of the word. These three meanings of “concept” he describes as follows.[26]

[²⁶] Logic, I. p. 245. This and future references to Sigwart are to the English translation of his work by Mrs Bosanquet.

(1) By a concept may be meant a natural psychological production,—the general idea which has been developed in the natural course of thought. Such ideas are different for different people, and are continually in process of formation; even for the individual himself they change, so that a word does not always keep the same meaning even for the same person. Strictly speaking, it is only by a fiction which neglects individual peculiarities that we can speak of the concepts corresponding to the terms used in ordinary language.

(2) In contrast with this empirical meaning the concept may be viewed as an ideal; it is then the mark at which we aim in our endeavour to attain knowledge, for we seek to find in it an adequate copy of the essence of things. 28

(3) Between these two meanings of the word, which may be called the empirical and the metaphysical, there lies the logical. This has its origin in the logical demand for certainty and universal validity in our judgments. All that is required is that our ideas should be absolutely fixed and determined, and that all who make use of the same system of denotation should have the same ideas.

This threefold distinction may be usefully compared with that drawn in the preceding section. Sigwart is approaching the question from a different point of view, but it will be observed that his three “meanings of concept” correspond broadly with subjective intension, objective intension, and conventional intension respectively.

It may be added that Mr Bosanquet’s distinction (Logic, I. pp. 41 to 46) between the objective reference of a name (its logical meaning) and its content for the individual mind (the psychical idea) appears to some extent to correspond to the distinction between connotation and subjective intension.

19. Connotation and Etymology.—The connotation of a name must not be confused with its etymology. In dealing with names from the etymological or historical point of view we consider the circumstances in which they were first imposed and the reasons for their adoption; also the successive changes, if any, in their meaning that have subsequently occurred. In making precise the connotation to be attached to a name we may be helped by considering its etymology. But we must clearly distinguish between the two; in finally deciding upon the connotation to be assigned to a name for any particular scientific purpose, we may indeed find it necessary to depart not merely from its original meaning, but also from its current meaning in everyday discourse.

20. Fixity of Connotation.—It has been already pointed out that subjective intension is variable. A given name will almost certainly call up in the minds of different persons different ideas; and even in the case of the same person it will probably do so at different times. The question may be raised how far the same is true of connotation. It has been implied in the preceding section that the scientific use of a name may differ 29 from its use in everyday discourse; and there can be no doubt that as a matter of fact different people may by the same name intend to signify different things, that is to say, they would include different attributes in the connotation of the name. It is, moreover, not unfrequently the case that some of us may be unable to say precisely what is the meaning that we ourselves attach to the words we use.

At the same time a clear distinction ought to be drawn between subjective intension and connotation in respect of their variability. Subjective intension is necessarily variable; it can never be otherwise. Connotation, on the other hand, is only variable by accident; and in so far as there is variation language fails of its purpose. “Identical reference,” as Mr Bosanquet puts it, “is the root and essence of the system of signs which we call language” (Logic, I. p. 16). It is only by some conventional agreement which shall make language fixed that scientific discussions can be satisfactorily carried on; and there would be no variation in the connotation of names in the case of an ideal language properly employed. In dealing with reasonings from the point of view of logical doctrine, it is, therefore, no unreasonable assumption to make that in any given argument the connotation of the names employed is fixed and definite; in other words, that every name employed is either used in its ordinary sense and that this is precisely determined, or else that, the name being used with a special meaning, such meaning is adhered to consistently and without equivocation.

21. Extension and Denotation.—The terms extension and denotation are usually employed as synonymous, but there will be some advantage in drawing a certain distinction between them. We shall find that when names are regarded as the subjects of propositions there is an implied reference to some universe of discourse, which may be more or less limited. For example, we should naturally understand such propositions as all men are mortal, no men are perfect, to refer to all men who have actually existed on the earth, or are now existing, or will exist hereafter, but we should not understand them to refer to all fictitious persons or all beings possessing the essential characteristics of men whom we are able to conceive or imagine. 30 The meaning of universe of discourse will be further illustrated [subsequently]. The only reason for introducing the conception at this point is that we propose to use the term denotation or objective extension rather than the term extension simply when there is an explicit or implicit limitation to the objects actually to be found in some restricted universe. By the subjective extension of a general name, on the other hand, we shall understand the whole range of objects real or imaginary to which the name can be correctly applied, the only limitation being that of logical conceivability. Every name, therefore, which can be used in an intelligible sense will have a positive subjective extension, but its denotation in a universe which is in some way restricted by time, place, or circumstance may be zero.[27]

[²⁷] The value of the above distinction may be illustrated by reference to the divergence of view indicated in the following quotation from Mr Monck, who uses the terms denotation and extension as synonymous: “It is a matter of accident whether a general name will have any extension (or denotation) or not. Unicorn, griffin, and dragon are general names because they have a meaning, and we can suppose another world in which such beings exist; but the terms have no extension, because there are no such animals in this world. Some logicians speak of these terms as having an extension, because we can suppose individuals corresponding to them. In this way every general term would have an extension which might be either real or imaginary. It is, however, more convenient to use the word extension for a real extension (past, present, or future) only” (Introduction to Logic, p. 10). It should be added, in order to prevent possible misapprehension, that by universe of discourse, as used in the text, we do not necessarily mean the material universe; we may, for example, mean the universe of fairy-land, or of heraldry, and in such a case, unicorn, griffin, and dragon may have denotation (in our special sense), as well as subjective extension, greater than zero. What is the particular universe of reference in any given proposition will generally be determined by the context. For logical purposes we may assume that it is conventionally understood and agreed upon, and that it remains the same throughout the course of any given argument. As Dr Venn remarks, “We might include amongst the assumptions of logic that the speaker and hearer should be in agreement, not only as to the meaning of the words they use, but also as to the conventional limitations under which they apply them in the circumstances of the case” (Empirical Logic, p. 180).

In the sense here indicated, denotation is in certain respects the correlative of comprehension rather than of connotation. For in speaking of denotation we are, as in the case of comprehension, taking an objective standpoint; and there is, moreover, in the case of comprehension, as in that of denotation, a 31 tacit reference to some particular universe of discourse. Since, however, denotation is generally speaking determined by connotation, there is one very important respect in which connotation and denotation are still correlatives.

22. Dependence of Extension and Intension upon one another.[28]—Taking any class-name X, let us first suppose that there has been a conventional agreement to use it wherever a certain selected set of properties P₁, P₂, … P_m, are present. This set of properties will constitute the connotation of X, and will, with reference to a given universe of discourse,[29] determine the denotation of the name, say, Q₁, Q₂, … Q_y ; that is, Q₁, Q₂, … Q_y, are all the individuals possessing in common the properties P₁, P₂, … P_m.

[²⁸] This section may be omitted on a first reading.

[²⁹] It will be assumed in the remainder of this section that we are throughout speaking with reference to a given universe of discourse.

These properties may not, and almost certainly will not, exhaust the properties common to Q₁, Q₂, … Q_y. Let all the common properties be P₁, P₂, … P_x ; they will include P₁, P₂, … P_m, and in all probability more besides, and will constitute the comprehension of the class-name.

Now it will always be possible in one or more ways to make out of Q₁, Q₂, … Q_y, a selection Q₁, Q₂, … Q_n, which shall be precisely typical of the whole class;[30] that is to say, Q₁, Q₂, … Q_n will possess in common those attributes and only those attributes (namely, P₁, P₂, … P_x) which are possessed in common by Q₁, Q₂, … Q_y.[31] Q₁, Q₂, … Q_n may be called the exemplification or 32 extensive definition of X. The reason for selecting the name extensive definition will appear in a moment. It will sometimes be convenient correspondingly to speak of the connotation of a name as its intensive definition.

[³⁰] It may chance to be necessary to make Q₁, Q₂, … Q_n coincide with Q₁, Q₂, … Q_y. But this must be regarded as the limiting case; usually a smaller number of individuals will be sufficient.

[³¹] Mr Johnson points out to me that in pursuing this line of argument certain restrictions of a somewhat subtle kind are necessary in regard to what may be called our “universe of attributes.” The “universe of objects” which is what we mean by the “universe of discourse,” implies individuality of object and limitation of range of objects ; and if we are to work out a thoroughgoing reciprocity between attributes and objects, we must recognise in our “universe of attributes” restrictions analogous to the above, namely, simplicity of attribute and limitation of range of attributes. By “simplicity of attribute” is meant that the universe of attributes must not contain any attribute which is a logical function of any other attribute or set of attributes. Thus, if A, B are two attributes recognised in our universe, we must not admit such attributes as X (= A and B), or Y (= A or B), or Z (= not-A). We may indeed have a negatively defined attribute, but it must not be the formal contradictory of another or formally involve the contradictory of another. The following example will shew the necessity of this restriction. Let P₁, P₂, P₃, be selected as typical of the whole class P₁, P₂, P₃ P₄, P₅, P₆; and let A₁ be an attribute possessed by P₁ alone, A₂ an attribute possessed by P₂ alone, and so on. Then if we recognise A₁ or A₂ or A₃ as a distinct attribute, it is at once clear that P₁, P₂, P₃ will no longer be typical of the whole class; and the same result follows if not-A₄ is recognised as a distinct attribute. Similarly, without the restriction in question any selection (short of the whole) would necessarily fail to be typical of the whole class. As a concrete example, suppose that we select from the class of professional men a set of examples that have in common no attribute except those that are common to the whole class. It may turn out that our examples are all barristers or doctors, but none of them solicitors. Now if we recognise as a distinct attribute being “either a barrister or a doctor,” our selected group will thereby have an extra common attribute not possessed by every professional man. The same result will follow if we recognise the attribute “non-solicitor.” Not much need be added as regards the necessity of some limitation in the range of attributes which are recognised. The mere fact of our having selected a certain group would indeed constitute an additional attribute, which would at once cause the selection to fail in its purpose, unless this were excluded as inessential. Similarly, such attributes as position in space or in time &c. must in general be regarded as inessential. For example, I might draw on a sheet of paper a number of triangles sufficiently typical of the whole class of triangles, but for this it would be necessary to reject as inessential the common property which they would possess of all being drawn on a particular sheet of paper.

We have then, with reference to X,
(1)  Connotation: P₁ … P_m ;
(2)  Denotation: Q₁ … Q_n … Q_y ;
(3)  Comprehension: P₁ … P_m … P_x ;
(4)  Exemplification: Q₁ … Q_n.

Of these, either the connotation or the exemplification will suffice to mark out or completely identify the class, although they do not exhaust either all its common properties or all the individuals contained in it. In other words, whether we start from the connotation or from the exemplification, the denotation and the comprehension will be the same.[32]

[³²] It will be observed that connotation and exemplification are distinguished from comprehension and denotation in that they are selective, while the latter pair are exhaustive. In making our selection our aim will usually be to find the minimum list which will suffice for our purpose.

33 For a concrete illustration of the above, the term metal may be taken. From the chemical point of view a metal may be defined as an element which can replace hydrogen in an acid and thus form a salt. This then is the connotation of the name. Its denotation consists of the complete list of elements fulfilling the above condition now known to chemists, and possibly of others not yet discovered.[33] The members of the whole class thus constituted are, however, found to possess other properties in common besides those contained in the definition of the name, for example, fusibility, the characteristic lustre termed metallic, a high degree of opacity, and the property of being good conductors of heat and electricity. The complete list of these properties forms the comprehension of the name. Now a chemist would no doubt be able from the full denotation of metal to make a selection of a limited number of metals which would be precisely typical of the whole class;[34] that is to say, his selected list would possess in common only such properties as are common to the whole class. This selected class would constitute the exemplification of the name.

[³³] It is necessary to distinguish between the known extension of a term and its full denotation, just as we distinguish between the known intension of a term and its full comprehension.

[³⁴] He would take metals as different from one another as possible, such as aluminium, antimony, copper, gold, iron, mercury, sodium, zinc.

We have so far assumed that (1) connotation or intensive definition has first been arbitrarily fixed, and that this has successively determined (2) denotation, (3) comprehension, and—with a certain range of choice—(4) exemplification. But it is clear that theoretically we might start by arbitrarily fixing (i) the exemplification or extensive definition ; and that this would successively determine (ii) comprehension, (iii) denotation, and then—again with a certain range of choice[35]—(iv) connotation.

[³⁵] It is ordinarily said that “of the denotation and connotation of a term one may, both cannot, be arbitrary,” and this is broadly true. It is possible, however, to make the statement rather more exact. Given either intensive or extensive definition, then both denotation and comprehension are, with reference to any assigned universe of discourse, absolutely fixed. But different intensive definitions, and also different extensive definitions, may sometimes yield the same results; and it is therefore possible that, everything else being given, connotation or exemplification may still be within certain limits indeterminate. For example, given the class of parallel straight lines, the connotation may be determined in two or three different ways; or, given the class of equilateral equiangular triangles, we may select as connotation either having three equal sides or having three equal angles. Again, given the connotation of metal, it would no doubt be possible to select in more ways than one a limited number of metals not possessing in common any attributes which are not also possessed by the remaining members of the class.

34 It is interesting from a theoretical point of view to note the possibility of this second order of procedure; and this order may, moreover, be said to represent what actually occurs—at any rate in the first instance—in certain cases, as, for example, in the case of natural groups in the animal, vegetable, and mineral kingdoms. Men form classes out of vaguely recognised resemblances long before they are able to give an intensive definition of the class-name, and in such a case if they are asked to explain their use of the name, their reply will be to enumerate typical examples of the class. This would no doubt ordinarily be done in an unscientific manner, but it would be possible to work it out scientifically. The extensive definition of a name will take the form: X is the name of the class of which Q₁, Q₂, … Q_n are typical. This primitive form of definition may also be called definition by type.[36]

[³⁶] It is not of course meant that when we start from an extensive definition, we are classing things together at random without any guiding principle of selection. No doubt we shall be guided by a resemblance between the objects which we place in the same class, and in this sense intension may be said always to have the priority. But the resemblance may be unanalysed, so that we may be far more familiar with the application of the class-name than with its implication; and even when a connotation has been assigned to the name, it may be extensively controlled, and constantly subject to modification, just because we are much more concerned to keep the denotation fixed than the connotation.

In this connexion the names of simple feelings which are incapable of analysis may be specially considered. For the names of ultimate elements, there is, says Sigwart,[37] no definition; we must assume that everyone attaches the same meaning to them. To such names we may indeed be able to assign a proximate genus, as when we say “red is a colour”; but we 35 cannot add a specific difference. It is, however, only an intensive definition that is wanting in these cases; and the deficiency is supplied by means of an extensive definition. The way in which we make clear to others our use of such a term as “red” is by pointing out or otherwise indicating various objects which give rise in us to the feeling. Thus “red” is the colour of field poppies, hips and haws, ordinary sealing-wax, bricks made from certain kinds of clay, &c. This is nothing more or less than an extensive definition as above defined.

[³⁷] Logic, I. p. 289.

In the case of most names, however, where formal definition is attempted, it is more usual, as well as really simpler, to start from an intensive definition, and this in general corresponds with the ultimate procedure of science. For logical purposes, it is accordingly best to assume this order of procedure, unless an explicit statement is made to the contrary.[38]

[³⁸] It is worth noticing that in practice an intensive definition is often followed by an enumeration of typical examples, which, if well selected, may themselves almost amount to an extensive definition. In this case, we may be said to have the two kinds of definition supplementing one another.

23. Inverse Variation of Extension and Intension.[39]—In general, as intension is increased or diminished, extension is diminished or increased accordingly, and vice versâ. If, for example, rational is added to the connotation of animal, the denotation is diminished, since all non-rational animals are now excluded, whereas they were previously included. On the other hand, if the denotation of animal is to be extended so as to include the vegetable kingdom, it can only be by omitting sensitive from the connotation. Hence the following law has been formulated: “In a series of common terms standing to one another in a relation of subordination[40] the extension and the intension vary inversely.” Is this law to be accepted? It must be observed at the outset that the notion of inverse variation is at any rate not to be interpreted in any strict mathematical or numerical sense. It is certainly not true that whenever the number of 36 attributes included in the intension is altered in any manner, then the number of individuals included in the extension will be altered in some assigned numerical proportion. If, for example, to the connotation of a given name different single attributes are added, the denotation will be affected in very different degrees in different cases. Thus, the addition of resident to the connotation of member of the Senate of the University of Cambridge will reduce its denotation in a much greater degree than the addition of non-resident. There is in short no regular law of variation. The statement must not then be understood to mean more than that any increase or diminution of the intension of a name will necessarily be accompanied by some diminution or increase of the extension as the case may be, and vice versâ.[41] We will discuss the alleged law in this form, considering, first, connotation and denotation, exemplification and comprehension; and, secondly, denotation and comprehension.[42]

[³⁹] This section may be omitted on a first reading.

[⁴⁰] As in the Tree of Porphyry: Substance, Corporeal Substance (Body), Animate Body (Living Being), Sensitive Living Being (Animal), Rational Animal (Man). In this series of terms the intension is at each step increased, and the extension diminished.

[⁴¹] It has been said that while the extension of a term is capable of quantitative measurement, the same is not equally true of intension. “The parts of extension may be counted, but it is inept to count the parts in intension. For they are not external to each other, and they form a whole such as cannot be divided into units except by the most arbitrary dilaceration. And if it were so divided, all its parts would vary in value, and there would be no reason to expect that ten of them (that is, ten attributes) should have twice the amount or value of five” (Bosanquet, Logic, I. p. 59). There is some force in this, and it is decisive against interpreting inverse variation in the present connexion in any strict numerical sense. But, at the same time, no error is committed and no difficulty of interpretation arises, if we content ourselves with speaking merely of the enlargement or restriction of the intension of a term. There can be no doubt that intension is increased when we pass from animal to man, or from man to negro; or again when we pass from triangle to isosceles triangle, or from isosceles triangle to right-angled isosceles triangle.

[⁴²] The discussion is purposely made as formal and exact as possible. If indeed the doctrine of inverse variation cannot be treated with precision, it is better not to attempt to deal with it at all.

A. (1) Let connotation be supposed arbitrarily fixed, and used to determine denotation in some assigned universe of discourse. Then it will not be true that connotation and denotation will necessarily vary inversely. For suppose the connotation of a name, i.e., the attributes signified by it, to be a, b, c. It may happen that in fact wherever the attributes a and b are present, the attributes c and d are also present. 37 In this case, if c is dropped from the connotation, or d added to it, the denotation of the name will remain unaffected. We have concrete examples of this, if we suppose equiangularity added to the connotation of equilateral triangle, or cloven-hoofed to that of ruminant, or having jaws opening up and down to that of vertebrate, or if we suppose invalid dropped from the connotation of invalid syllogism with undistributed middle. It is clear, however, that if any alteration in denotation takes place when connotation is altered, it must necessarily be in the opposite direction. Some individuals possessing the attributes a and b may lack the attribute c or the attribute d ; but no individuals possessing the attributes a, b, c, or a, b, c, d can fail to possess the attributes a, b, or a, b, c. For example, if to the connotation of metal we add fusible, it makes no difference to the denotation; but if we add having great weight, we exclude potassium, sodium, &c.

The law of variation of denotation with connotation may then be stated as follows:—If the connotation of a term is arbitrarily enlarged or restricted, the denotation in an assigned universe of discourse will either remain unaltered or will change in the opposite direction.[43]

[⁴³] Since reference is here made to the actual denotation of a term in some assigned universe of discourse, the above law may be said to turn partly on material, and not on purely formal, considerations. It should, therefore, be added that although an alteration in the connotation of a term will not always alter its actual denotation in an assigned universe of discourse, it will always affect potentially its subjective extension. If, for example, the connotation of a term X is a, b, c, and we add d ; then the (real or imaginary) class of X’s that are not d is necessarily excluded from, while it was previously included in, the subjective extension of the term X. Hence, if the connotation of a term is arbitrarily enlarged or restricted, the subjective extension will be potentially restricted or enlarged accordingly. Cf. Jevons, Principles of Science, 30, § 13.

(2) Let exemplification be supposed arbitrarily fixed, and used to determine comprehension. It is unnecessary to shew in detail that a corresponding law of variation of comprehension with exemplification will hold good, namely:—If the exemplification (extensive definition) of a term is arbitrarily enlarged or restricted, the comprehension in an assigned universe of discourse will either remain unaltered or will change in the opposite direction. 38

B. We may now consider the relation between the comprehension and the denotation of a term. Let P₁, P₂, … P_x be the totality of attributes possessed by the class X, and Q₁, Q₂, … Q_y the totality of objects included in the class X. Both these groups are objectively, not arbitrarily,[44] determined; and the relation between them is reciprocal. P₁, P₂, … P_x are the only attributes possessed in common by the objects Q₁, Q₂, … Q_y ; and Q₁, Q₂, … Q_y are the only objects possessing all the attributes P₁, P₂, … P_x.

[⁴⁴] What may be arbitrary is the intensive definition (P₁, P₂, … P_m) or the extensive definition (Q₁, Q₂, … Q_n) which determines them both.

We cannot suppose any direct arbitrary alteration either in comprehension or in denotation. We can, however, establish the following law of inverse variation, namely, that any arbitrary alteration in either intensive definition or extensive definition which results in an alteration of either denotation or comprehension will also result in an alteration in the opposite direction of the other.

Let X and Y be two terms which are so related that the definition (either intensive or extensive, as the case may be) of Y includes all that is included in the definition of X and more besides. We have to shew that either the denotations and comprehensions of X and Y will be identical or if the denotation of one includes more than the denotation of the other then its comprehension will include less, and vice versâ.

(a) Let X and Y be determined by connotation or intensive definition. Thus, let X be determined by the set of properties P₁ … P_m and Y by the set P₁ … P_m+1, which includes the additional property P_m+1.
Then P_m+1 either does or does not always accompany P₁ … P_m.
If the former, no object included in the denotation of X is excluded from that of Y, so that the denotations of X and Y are the same; and it follows that the comprehensions of X and Y are also the same.
If the latter, then the denotation of Y is less than that of X by all those objects that possess P₁ … P_m without also possessing P_m+1. At the same time, the comprehension of Y includes at 39 least P_m+1 in addition to the properties included in the comprehension of X.

(b) Let X and Y be determined by exemplification or extensive definition. Thus, let X be determined by the set of examples Q₁ … Q_n, and Y by the set Q₁ … Q_n+1 which includes the additional object Q_n+1.
Then Q_n+1 either does or does not possess all the properties common to Q₁ … Q_n.
If the former, no property included in the comprehension of X is excluded from that of Y, so that the comprehensions of X and Y are the same; and it follows that the denotations of X and Y are also the same.
If the latter, then the comprehension of Y is less than that of X by all those properties that belong to Q₁ … Q_n without also belonging to Q_n+1. At the same time, the denotation of Y includes at least Q_n+1 in addition to the objects included in the denotation of X.

All cases have now been considered, and it has been shewn that the law above formulated holds good universally. This law and the two laws given on page [37] must together be substituted for the law of inverse relation between extension and intension in its usual form if full precision of statement is desired.

It should be observed that in speaking of variations in comprehension or denotation, no reference is intended to changes in things or in our knowledge of them. The variation is always supposed to have originated in some arbitrary alteration in the intensive or extensive definition of a given term, or in passing from the consideration of one term to that of another with a different extensive or intensive definition. Thus fresh things may be discovered to belong to a class, and the comprehension of the class-name may not thereby be affected. But in this case the denotation has not itself varied; only our knowledge of it has varied. Or we may discover fresh attributes previously overlooked; in which case similar remarks will apply. Again, new things may be brought into existence which come under the denotation of the name, and still its comprehension may remain unchanged. Or possibly new qualities may be developed by 40 the whole of the class. In these cases, however, there is no arbitrary alteration in the application or implication of the name, and hence no real exception to what has been laid down above.

24. Connotative Names.—Mill’s use of the word connotative, which is that generally adopted in modern works on logic, is as follows: “A non-connotative term is one which signifies a subject only, or an attribute only. A connotative term is one which denotes a subject, and implies an attribute” (Logic, I. 2, § 5). According to this definition, a connotative name must not only possess extension, but must also have a conventional intension assigned to it.

Mill considers that the following kinds of names are connotative in the above sense:—(1) All concrete general names. (2) Some singular names. For example, city is a general name, and as such no one would deny it to be connotative. Now if we say the largest city in the world, we have individualised the name, but it does not thereby cease to be connotative. Proper names are, however, according to Mill, non-connotative, since they merely denote a subject and do not imply any attributes. To this point, which is a subject of controversy, we shall return in the following [section]. (3) While admitting that most abstract names are non-connotative, since they merely signify an attribute and do not denote a subject, Mill maintains that some abstracts may justly be “considered as connotative; for attributes themselves may have attributes ascribed to them; and a word which denotes attributes may connote an attribute of those attributes” (Logic, I. 2, § 5).

The wording of Mill’s definition is unfortunate and is probably responsible for a good deal of the controversy that has centred round the question as to whether certain classes of names are or are not connotative.

All names that we are able to use in an intelligible sense must have subjective intension for us. For we must know to what objects or what kinds of objects the names are applicable, and we cannot but associate some properties with these objects and therefore with the names.

Moreover all names that have denotation in any given 41 universe of discourse must have comprehension also; for no object can exist without possessing properties of some kind.

If then any name can properly be described as non-connotative, it cannot be in the sense that it has no subjective intension or no comprehension. This is at least obscured when Mill speaks of non-connotative names as not implying any attributes; and if misunderstanding is to be avoided, his definitions must be amended, so as to make it quite clear that in a non-connotative name it is connotation only that is lacking, and not either subjective intension or comprehension.

A connotative name may be defined as a name whose application is determined by connotation or intensive definition, that is, by a conventionally assigned attribute or set of attributes. A non-connotative name is an exemplificative name, a name whose application is determined by exemplification or extensive definition in the sense explained in section [22]; in other words, it is a name whose application is determined by pointing out or indicating, by means of a description or otherwise, the particular individual (if the name is singular), or typical individuals (if the name is general), to which the name is attached.

If it is allowed that the application of any names can be determined in the latter way, as distinguished from the former, then it must be allowed that some names are non-connotative.

25. Are proper names connotative?—To this question absolutely contradictory answers are given by ordinarily clear thinkers as being obviously correct. To some extent, however, the divergence is merely verbal, the terms “connotation” and “connotative name” being used in different senses.

It is necessary at the outset to guard against a misconception which quite obscures the real point at issue. Thus, with reference to Mill, Jevons says, “Logicians have erroneously asserted, as it seems to me, that singular terms are devoid of meaning in intension, the fact being that they exceed all other terms in that kind of meaning” (Principles of Science, 2, § 2, with a reference to Mill in a foot-note). But Mill distinctly states that some singular names are connotative, e.g., the 42 sun,[45] the first emperor of Rome (Logic, I. 2, § 5). We may certainly narrow down the extension of a term till it becomes individualised without destroying its connotation; “the present Professor of Pure Mathematics in University College, London” is a singular term—its extension cannot be further diminished—but it is certainly connotative.

[⁴⁵] The question has been asked on what grounds the sun can be regarded as connotative, while John is considered non-connotative; compare T. H. Green, Philosophical Works, ii. p. 204. The answer is that sun is a general name with a definite signification which determines its application, and that it does not lose its connotation when individualised by the prefix the ; while John, on the other hand, is a name given to an object merely as a mark for purposes of future reference, and without signifying the possession by that object of any conventionally selected attributes.

It must then be understood that only one class of singular names, namely, proper names, are affirmed to be non-connotative; and that no more is meant by this than that their application is not determined by a conventionally assigned set of attributes.[46] The ground may be further cleared by our explicitly recognising that, although proper names have no connotation, they nevertheless have both subjective intension and comprehension. An individual object can be recognised only through its attributes; and a proper name when understood by me to be a mark of a certain individual undoubtedly suggests to my mind certain qualities.[47] The qualities thus suggested by the name constitute its subjective intension. The comprehension of the name will include a good deal more than its subjective intension, namely, 43 the whole of the properties that belong to the individual denoted.

[⁴⁶] The treatment of the question adopted in this work has been criticised on the ground that it is question-begging, since in section [10] proper names have really been defined as non-connotative. This criticism cannot, however, be pressed unless it is at the same time maintained that the definition given in section [10] yields a denotation different from that ordinarily understood to belong to proper names.

[⁴⁷] A proper name may have suggestive force even for those who are not actually acquainted with the person or thing denoted by it. Thus William Stanley Jevons may suggest any or all of the following to one who never heard the name before: an organised being, a human being, a male, an Anglo-Saxon, having some relative named Stanley, having parents named Jevons. But at the same time, the name cannot be said necessarily to signify any of these things, in the sense that if they were wanting it would be misapplied. Consider, for example, such a name as Victoria Nyanza. Some further remarks bearing on this point will be found later on in this section.

It will be found that most writers who regard proper names as possessing connotation really mean thereby either subjective intension or comprehension. Thus Jevons puts his case as follows:—“Any proper name such as John Smith, is almost without meaning until we know the John Smith in question. It is true that the name alone connotes the fact that he is a Teuton, and is a male; but, so soon as we know the exact individual it denotes the name surely implies, also, the peculiar features, form, and character, of that individual. In fact, as it is only by the peculiar qualities, features, or circumstances of a thing, that we can ever recognise it, no name could have any fixed meaning unless we attached to it, mentally at least, such a definition of the kind of thing denoted by it, that we should know whether any given thing was denoted by it or not. If the name John Smith does not suggest to my mind the qualities of John Smith, how shall I know him when I meet him? For he certainly does not bear his name written upon his brow” (Elementary Lessons in Logic, p. 43). A wrong criterion of connotation in Mill’s sense is here taken. The connotation of a name is not the quality or qualities by which I or any one else may happen to recognise the class which it denotes. For example, I may recognise an Englishman abroad by the cut of his clothes, or a Frenchman by his pronunciation, or a proctor by his bands, or a barrister by his wig; but I do not mean any of these things by these names, nor do they (in Mill’s sense) form any part of the connotation of the names. Compare two such names as Henry Montagu Butler and the Master of Trinity College, Cambridge. At the present time they denote the same person; but the names are not equivalent,—the one is given to a certain individual as a mark to distinguish him from others, and has no further signification; the other is given because of the performance of certain functions, on the cessation of which the name would cease to apply. Surely there is a distinction here, and one which it is important that we should not overlook.

It may indeed fairly be said that many, if not most, proper 44 names do signify something, in the sense that they were chosen in the first instance for a special reason. For example, Strongi’th’arm, Smith, Jungfrau. But such names even if in a certain sense connotative when first imposed soon cease to be so, since their subsequent application to the persons or things designated is not dependent on the continuance of the attribute with reference to which they were originally given. As Mill puts it, the name once given is independent of the reason. In other words, we ought carefully to distinguish between the connotation of a name and its history. Thus, a man may in his youth have been strong, but we should not continue to calling strong in his dotage; whilst the name Strongi’th’arm once given would not be taken from him. Again, the name Smith may in the first instance have been given because a man plied a certain handicraft, but he would still be called by the same name if he changed his trade, and his descendants continue to be called Smith whatever their occupations may be.[48]

[⁴⁸] It cannot, however, be said that the name necessarily implies ancestors of the same name. As Dr Venn remarks, “he who changes his family name may grossly deceive genealogists, but he does not tell a falsehood” (Empirical Logic, p. 185).

It has been argued that proper names must be connotative because the use of a proper name conveys more information than the use of a general name. “Few persons,” says Mr Benecke,[49] “will deny that if I say the principal speaker was Mr Gladstone, I am giving not less but more information than if, instead of Mr Gladstone, I say either a member of Parliament, or an eminent man, or a statesman, or a Liberal leader. It will be admitted that the predicate Mr Gladstone tells us all that is told us by all these other connotative predicates put together, and more; and, if so, I cannot see how it can be denied that it also connotes more.” It is clear, however, that the information given when a thing is called by any name depends not on the connotation of the name, but on its intension for the person addressed. To anyone who knows that Mr Gladstone was Prime Minister in 1892 the same information is afforded whether a speaker is referred to as Mr Gladstone or as Prime Minister of 45 Great Britain and Ireland in 1892. But it certainly cannot be maintained that the connotation of these two names is identical.

[⁴⁹] In a paper on the Connotation of Proper Names read before the Aristotelian Society.

In criticism of the position that the application of a proper name such as Gladstone is determined by some attribute or set of attributes, we may naturally ask, what attribute or set of attributes? The answer cannot be that the connotation consists of the complete group of attributes possessed by the individual designated; for it is absurd to require any such enumeration as this in order to determine the application of the name. It is, however, impossible to select some particular attributes of the individual in question, and point to them as a group that would be accepted as constituting the definition of the name; and if it is said that the application of the name is determined by any set of attributes that will suffice for identification, the case is given up. For this amounts to identifying the individual by a description (that is, practically by exemplification), not by a particular set of attributes conventionally attached to the name as such. The truth is that no one would ever propose to give an intensive definition of a proper name. All names, however, that are connotative must necessarily admit of intensive definition.[50]

[⁵⁰] Mr Bosanquet arrives at the conclusion that “a proper name has a connotation, but not a fixed general connotation. It is attached to a unique individual, and connotes whatever may be involved in his identity, or is instrumental in bringing it before the mind” (Essentials of Logic, p. 93). So far as I can understand this statement, it amounts to saying that proper names have comprehension and subjective intension, but not connotation, in the senses in which I have defined these terms.

Proper names of course become connotative when they are used to designate a certain type of person; for example, a Diogenes, a Thomas, a Don Quixote, a Paul Pry, a Benedick, a Socrates. But, when so used, such names have really ceased to be proper names at all; they have come to possess all the characteristics of general names.[51]

[⁵¹] Compare Gray’s lines,—

“Some village Hampden, that, with dauntless breast,
The little tyrant of his fields withstood,
Some mute inglorious Milton here may rest,
Some Cromwell guiltless of his country’s blood.”

Attention may be called to a class of singular names, such as 46 Miss Smith, Captain Jones, President Roosevelt, the Lake of Lucerne, the Falls of Niagara, which may be said to be partially but only partially connotative. Their peculiarity is that they are partly made up of elements that have a general and permanent signification, and that consequently some change in the object denoted might render them no longer applicable, as, for example, if Captain Jones received promotion and were made a major; while, at the same time, such connotation as they possess is by itself insufficient to determine completely their application. It may be said that their application is limited, but not determined, by reference to specific assignable attributes. They occupy an intermediate position, therefore, between connotative singular names, such as the first man, and strictly proper names.

We may in this connexion touch upon Jevons’s argument that such a name as “John Smith” connotes at any rate “Teuton” and “male.” This is not strictly the case, since “John Smith” might be a dahlia, or a racehorse, or a negro, or the pseudonym of a woman, as in the case of George Eliot. In none of these cases could the name be said to be misapplied as it would be if a dahlia or a horse were called a man, or a negro a Teuton, or a woman a male. At the same time, it cannot be denied that certain proper names are in practice so much limited to certain classes of objects, that some incongruity would be felt if they were applied to objects belonging to any other class. It is, for example, unlikely that a parent would deliberately have his daughter christened “John Richard.” So far as this is the case, the names in question may be said to be partially connotative in the same way as the names referred to in the preceding paragraph, though to a less extent; that is to say, their application is limited, though not determined, by reference to specific attributes. We should have a still clearer case of a similar kind if the right to bear a certain name carried with it specific legal or social privileges.[52]

[⁵²] Compare Bosanquet, Logic, i. p. 53.

The position has been taken that every proper name is at least partially connotative inasmuch as it necessarily implies individuality and the property of being called by the name in question. If we refer to anything by any name whatsoever, it 47 must at any rate have the quality of being called by that name. If we call a man John when he really passes by the name of James, we make a mistake; we attribute to him a quality which he does not possess,—that of passing by the name of John. This argument, although it does not appear to establish the conclusion that proper names are in any degree connotative, nevertheless calls attention to a distinctive peculiarity of proper names that is worthy of notice. The denotation of connotative names may, and usually does, vary from time to time; and this is true of connotative singular names as well as of general names. But it is clearly essential in the case of a proper name that (in any given use) the name shall be consistently affixed to the same individual object. It is, however, one thing to say that the identity of the object called by the name with that to which the name has previously been assigned is a condition essential to the correct use of a proper name, and another thing to say that this is connoted by a proper name. If indeed by connotation we mean the attributes by reason of the possession of which by any object the name is applicable to that object, it seems a case of ὕστερον πρότερον to include in the connotation the property of being called by the name.

EXERCISES.

26. Are such concepts as “equilateral triangle” and “equiangular triangle” identical or different? [K.]
[This question should be considered with reference to the discussion in sections [17] and [18].]

27. Let X₁, X₂, X₃, X₄, and X₅ constitute the whole of a certain universe of discourse: also let a, b, c, d, e, f exhaust the properties of X₁; a, b, c, d, e, g, those of X₂; b, c, d, f, g, those of X₃; a, b, d, e, f, those of X₄; and a, c, e, f, g those of X₅.
(i) Given that, under these conditions, a term has the connotation a, b, find its denotation and its comprehension, and determine an exemplification that would yield the same result.
(ii) Given that, under the same conditions, a term has the exemplification X₄, X₅, find its comprehension and its denotation, and determine a connotation that would yield the same result. [K.]

48 28. On what grounds may it be held that names may possess (a) denotation without connotation, (b) connotation without denotation?
Give illustrations shewing that the denotation of a term of which the connotation is known must be regarded as relative to the proposition in which it is used as subject and to the context in which the proposition occurs. [J.]

29. What do you consider to be the question really at issue when it is asked whether proper names are connotative?
Enquire whether the following names are respectively connotative or non-connotative: Caesar, Czar, Lord Beaconsfield, the highest mountain in Europe, Mont Blanc, the Weisshorn, Greenland, the Claimant, the pole star, Homer, a Daniel come to judgment. [K.]

30. Bring out any special points that arise in the discussion of the extensional and intensional aspects of the following terms respectively: the Rosaceae, equilateral triangle, colour, giant. [C.]

CHAPTER III.

REAL, VERBAL, AND FORMAL PROPOSITIONS.

31. Real (Synthetic), Verbal (Analytic or Synonymous), and Formal Propositions.—(1) A real proposition is one which gives information of something more than the meaning or application of the term which constitutes its subject; as when a proposition predicates of a connotative subject some attribute not included in its connotation, or when a connotative term is predicated of a non-connotative subject. For example, All bodies have weight, The angles of any triangle are together equal to two right angles, Negative propositions distribute their predicates, Wordsworth is a great poet.

Real propositions are also described as synthetic, ampliative, accidental.

(2) A verbal proposition is one which gives information only in regard to the meaning or application of the term which constitutes its subject.[53]

[⁵³] Although verbal propositions may be distinguished from real propositions in accordance with the above definitions, it may be argued that every verbal proposition implies a real proposition of a certain sort behind it. For the question as to what meaning is attached to a given term in ordinary discourse, or by a given individual, is a question of matter of fact, and a statement respecting it may be true or false. Thus, X means abc is a verbal proposition; but such propositions as The meaning commonly attached to the term X is abc, The meaning attached in this work to the term X is abc, The meaning with which it would be most convenient to employ the term X is abc, are real. Looked at from this point of view the distinction between verbal and real propositions may perhaps be thought to be a rather subtle one. It remains true, however, that the proposition X means abc is verbal relatively to its subject X. Out of the given material we cannot by any manipulation obtain a real predication about X, that is, about the thing signified by the term X, but only about the meaning of the term X. The real proposition involved can thus only be obtained by substituting for the original subject another subject.

50 Two classes of verbal propositions are to be distinguished, which may be called respectively analytic and synonymous. In the former the predicate gives a partial or complete analysis of the connotation of the subject; e.g., Bodies are extended, An equilateral triangle is a triangle having three equal sides, A negative proposition has a negative copula.[54] Definitions are included under this division of verbal propositions; and the importance of definitions is so great, that it is clearly erroneous to speak of verbal propositions as being in all cases trivial. In general they are trivial only in so far as their true nature is misunderstood; when, for example, people waste time in pretending to prove what has been already assumed in the meaning assigned to the terms employed.[55]

[⁵⁴] Since we do not here really advance beyond an analysis of the subject-notion, Dr Bain describes the verbal proposition as the “notion under the guise of the proposition.” Hence the appropriateness of treating verbal propositions under the general head of Terms.

[⁵⁵] By a verbal dispute is meant a dispute that turns on the meaning of words. Dr Venn observes that purely verbal disputes are very rare, since “a different usage of words almost necessarily entails different convictions as to facts” (Empirical Logic, p. 296). This is true and important; it ought indeed always to be borne in mind that the problem of scientific definition is not a mere question of words, but a question of things. At the same time, disputes which are partly verbal are exceedingly common, and it is also very common for their true character in this respect to be unrecognised. When this is the case, the controversy is more likely than not to be fruitless. The questions whether proper names are connotative, and whether every syllogism involves a petitio principii, may be taken as examples. We certainly go a long way towards the solution of these questions by clearly differentiating between different meanings which may be attached to the terms employed.

Besides propositions giving a more or less complete analysis of the connotation of names, the following—which we may speak of as synonymous propositions—are to be included under the head of verbal propositions: (a) where the subject and predicate are both proper names, e.g., Tully is Cicero ; (b) where they are dictionary synonyms, e.g., Wealth is riches, A story is a tale, Charity is love. In these cases information is given only in regard to the application or meaning of the terms which appear as the subjects of the propositions.

Analytic propositions are also described as explicative and as essential. Very nearly the same distinction, therefore, as 51 that between verbal and real propositions is expressed by the pairs of terms—analytic and synthetic, explicative and ampliative, essential and accidental. These terms do not, however, cover quite the same ground as verbal and real, since they leave out of account synonymous propositions, which cannot, for example, be properly described as either analytic or synthetic.[56]

[⁵⁶] Thus, Mansel calls attention to “a class of propositions which are not, in the strict sense of the word, analytical, viz., those in which the predicate is a single term synonymous with the subject” (Mansel’s Aldrich, p. 170).

The distinction between real and verbal propositions as above given assumes that the use of terms is fixed by their connotation and that this connotation is determinate.[57] Whether any given proposition is as a matter of fact verbal or real will depend on the meaning attached to the terms which it contains; and it is clear that logic cannot lay down any rule for determining under which category any given proposition should be placed.[58] Still, while we cannot with certainty distinguish a real proposition by its form, it may be observed that the attachment of a sign of quantity, such as all, every, some, &c., to the subject of a proposition may in general be regarded as an indication that in the view of the person laying down the 52 proposition a fact is being stated and not merely a term explained. Verbal propositions, on the other hand, are usually unquantified or indesignate (see section [69]). For example, in order to give a partially correct idea of the meaning of such a name as square, we should not say “all squares are four-sided figures,” or “every square is a four-sided figure,” but “a square is a four-sided figure.”[59]

[⁵⁷] We can, however, adapt the distinction to the case in which the use of terms is fixed by extensive definition. We may say that whilst a proposition (expressed affirmatively and with a copula of inclusion) is intensively verbal when the connotation of the predicate is a part or the whole of the connotation of the subject, it is extensively verbal when the subject taken in extension is a part or the whole of the extensive definition of the predicate. Thus, if the use of the term metal is fixed by an extensive definition, that is to say, by the enumeration of certain typical metals, of which we may suppose iron to be one, then it is a verbal proposition to say that iron is a metal. If, however, tin is not included amongst the typical metals, then it is a real proposition to say that tin is a metal.

[⁵⁸] It does not follow from this that the distinction between verbal and real propositions is of no logical importance. Although the logician cannot quâ logician determine in doubtful cases to which category a given proposition belongs, he can point out what are the conditions upon which this depends, and he can shew that in any discussion or argument no progress is possible until it is clearly understood by all who are taking part whether the propositions laid down are to be interpreted as being real or merely verbal. To refer to an analogous case, it will not be said that the distinction between truth and falsity is of no logical importance because the logician cannot quâ logician determine whether a given proposition is true or false.

[⁵⁹] It should be added that we may formally distinguish a full definition from a real proposition by connecting the subject and the predicate by the word “means” instead of the word “is.”

(3) There are propositions usually classed as verbal which ought rather to be placed in a class by themselves, namely, those which are valid whatever may be the meaning of the terms involved; e.g., All A is A, No A is not-A, All Z is either B or not-B, If all A is B then no not-B is A, If all A is B and all B is C then all A is C. These may be called formal propositions, since their validity is determined by their bare form.[60]

[⁶⁰] Propositions which are in appearance purely tautologous have sometimes an epigrammatic force and are used for rhetorical purposes, e.g., A man’s a man (for a’ that). In such cases, however, there is usually an implication which gives the proposition the character of a real proposition; thus, in the above instance the true force of the proposition is that Every man is as such entitled to respect. “In the proposition, Children are children, the subject-term means only the age characteristic of childhood; the predicate-term the other characteristics which are connected with it. By the proposition, War is war, we mean to say that when once a state of warfare has arisen, we need not be surprised that all the consequences usually connected with it appear also. Thus the predicate adds new determinations to the meaning in which the subject was first taken” (Sigwart, Logic, I. p. 86).

Formal propositions are the only propositions whose validity is examined and guaranteed by logic itself irrespective of other sources of knowledge, and many of the results reached in formal logic may be summed up in such propositions; for any formally valid reasoning can be expressed by a formal hypothetical proposition as in the last two of the examples given above.

A formal proposition as here defined must not be confused with a proposition expressed in symbols. A formal proposition need not indeed be expressed in symbols at all. Thus, the proposition An animal is an animal is a formal proposition; 53 All S is P is not. Strictly speaking, a symbolic expression, such as All S is P, is to be regarded as a propositional form, rather than as a proposition per se. For it cannot be described as in itself either true or false. What we are largely concerned with in logic are relations between propositional forms; because these involve corresponding relations between all propositions falling into the forms in question.

We have then three classes of propositions—formal, verbal, and real—the validity or invalidity of which is determined respectively by their bare form, by the mere meaning or application of the terms involved, by questions of fact concerning the things denoted by these terms.[61]

[⁶¹] Real propositions are divided into true and false according as they do or do not accurately correspond with facts. By verbal and formal propositions we usually mean propositions which from the point of view taken are valid. A proposition which from either of these points of view is invalid is spoken of as a contradiction in terms. Properly speaking we ought to distinguish between a verbal contradiction in terms and a formal contradiction in terms, the contradiction depending in the first case upon the force of the terms employed and in the second case upon the mere form of the proposition; e.g., Some men are not animals, A is not-A. Any purely formal fallacy may be said to resolve itself into a formal contradiction in terms. It should be added that a mere term, if it is complex, may involve a contradiction in terms; e.g., Roman Catholic (if the separate terms are interpreted literally), A not-A.

32. Nature of the Analysis involved in Analytic Propositions.—Confusion is not unfrequently introduced into discussions relating to analytic propositions by a want of agreement as to the nature of the analysis involved. If identified, as above, with a division of the verbal proposition, an analytic proposition gives an analysis, partial or complete, of the connotation of the subject-term. Some writers, however, appear to have in view an analysis of the subjective intension of the subject-term. There is of course nothing absolutely incorrect in this interpretation, if consistently adhered to, but it makes the distinction between analytic and synthetic propositions logically valueless and for all practical purposes nugatory. “Both intension and extension,” says Mr Bradley, “are relative to our knowledge. And the perception of this truth is fatal to a well-known Kantian distinction. A judgment is not fixed as ‘synthetic’ or ‘analytic’: its character varies with the knowledge 54 possessed by various persons and at different times. If the meaning of a word were confined to that attribute or group of attributes with which it set out, we could distinguish those judgments which assert within the whole one part of its contents from those which add an element from outside; and the distinction thus made would remain valid for ever. But in actual practice the meaning itself is enlarged by synthesis. What is added to-day is implied to-morrow. We may even say that a synthetic judgment, so soon as it is made, is at once analytic.”[62]

[⁶²] Principles of Logic, p. 172. Professor Veitch expresses himself somewhat similarly. “Logically all judgments are analytic, for judgment is an assertion by the person judging of what he knows of the subject spoken of. To the person addressed, real or imaginary, the judgment may contain a predicate new—a new knowledge. But the person making the judgment speaks analytically, and analytically only; for he sets forth a part of what he knows belongs to the subject spoken of. In fact, it is impossible anyone can judge otherwise. We must judge by our real or supposed knowledge of the thing already in the mind” (Institutes of Logic, p. 237).

If by intension is meant subjective intension, and by an analytic judgment one which analyses the intension of the subject, the above statements are unimpeachable. It is indeed so obviously true that in this sense synthetic judgments are only analytic judgments in the making, that to dwell upon the distinction itself at any length would be only waste of time. It is, however, misleading to identify subjective intension with meaning ;[63] and this is especially the case in the present connexion, since it may be maintained with a certain degree of plausibility that some synthetic judgments are only analytic judgments in the making, even when by an analytic judgment is meant one which analyses the connotation of the subject. For undoubtedly the connotation of names is not in practice unalterably fixed. As our knowledge progresses, many of our 55 definitions are modified, and hence a form of words which is synthetic at one period may become analytic at another.

[⁶³] Compare the following criticism of Mill’s distinction between real and verbal propositions: “If every proposition is merely verbal which asserts something of a thing under a name that already presupposes what is about to be asserted, then every statement by a scientific man is for him merely verbal” (T. H. Green, Works, ii. p. 233). This criticism seems to lose its force if we bear in mind the distinction between connotation and subjective intension.

But, in the first place, it is very far indeed from being a universal rule that newly-discovered properties of a class are taken ultimately into the connotation or intensive definition of the class-name. Dr Bain (Logic, Deduction, pp. 69 to 73) seems to imply the contrary; but his doctrine on this point is not defensible on the ground either of logical expediency or of actual practice. As to logical expediency, it is a generally recognised principle of definition that we ought to aim at including in a definition the minimum number of properties necessary for identification rather than the maximum which it is possible to include.[64] And as to what actually occurs, it is easy to find cases where we are able to say with confidence that certain common properties of a class never will as a matter of fact be included in the definition of the class-name; for example, equiangularity will never be included in the definition of equilateral triangle, or having cloven hoofs in the definition of ruminant animal.

[⁶⁴] If we include in the definition of a class-name all the common properties of the class, how are we to make any universal statement of fact about the class at all? Given that the property P belongs to the whole of the class S, then by hypothesis P becomes part of the meaning of S, and the proposition All S is P merely makes this verbal statement, and is no assertion of any matter of fact at all. We are, therefore, involved in a kind of vicious circle.

In the second place, even when freshly discovered properties of things come ultimately to be included in the connotation of their names, the process is at any rate gradual, and it would, therefore, be incorrect to say—in the sense in which we are now using the terms—that a synthetic judgment becomes in the very process of its formation analytic. On the other hand, it may reasonably be assumed that in any given discussion the meaning of our terms is fixed, and the distinction between analytic and synthetic propositions then becomes highly significant and important. It may be added that when a name changes its meaning, any proposition in which it occurs does not strictly speaking remain the same proposition as before. We ought 56 rather to say that the same form of words now expresses a different proposition.[65]

[⁶⁵] This point is brought out by Mr Monck in the admirable discussion of the above question contained in his Introduction to Logic, pp. 130 to 134.

EXERCISES.

33. State which of the following propositions you consider real, and which verbal, giving your reasons in each case:

[C.]

34. Enquire whether the following propositions are real or verbal: (a) Homer wrote the Iliad, (b) Milton wrote Paradise Lost. [C.]

35. How would you characterise a proposition which is formally inferred from the conjunction of a verbal proposition with a real material proposition? Explain your view by the aid of an illustration. [J.]

36. If all x is y, and some x is z, and p is the name of those z’s which are x ; is it a verbal proposition to say that all p is y? [V.]

37. Is it possible to make any term whatever the subject (a) of a verbal proposition, (b) of a real proposition? [J.]

CHAPTER IV.

NEGATIVE NAMES AND RELATIVE NAMES.

38. Positive and Negative Names.—A pair of names of the forms A and not-A are commonly described as positive and negative respectively. The true import of the negative name not-A, including the question whether it really has any signification at all, has, however, given rise to much discussion.

Strictly speaking neither affirmation nor negation has any meaning except in reference to judgments or propositions. A concept or a term cannot be itself either affirmed or denied. If I affirm, it must be a judgment or a proposition that I affirm; if I deny, it must be a judgment or a proposition that I deny.

Starting from this position, Sigwart is led to the conclusion that, “taken literally, the formula not-A, where A denotes any idea, has no meaning whatever” (Logic, I. p. 134). Apart from the fact that the mere absence of an idea is not itself an idea, not-A cannot be interpreted to mean the absence of A in thought; for, on the contrary, it implies the presence of A in thought. We cannot, for instance, think of not-white except by thinking of white. Nor again can we interpret not-A as denoting whatever does not necessarily accompany A in thought. For, if so, A and not-A would not as a rule be exclusive or incompatible. For example, square, solid, do not necessarily accompany white in thought; but there is no opposition between these ideas and the idea of white. In order to interpret not-A as a real negation we must, says Sigwart, tacitly introduce a judgment or rather a series of judgments, 58 meaning by not-A “whatever is not A,” that is, everything whatsoever of which A must be denied. “I must review in thought all possible things in order to deny A of them, and these would be the positive objects denoted by not-A. But even if there were any use in this, it would be an impossible task” (p. 135).

Whilst agreeing with much that Sigwart says in this connexion, I cannot altogether accept his conclusion. We shall return to the question from the more controversial point of view in the following [section]. In the meantime we may indicate the result to which Sigwart’s general argument really seems to lead us.

We must agree that not-A cannot be regarded as representing any independent concept; that is to say, we cannot form any idea of not-A that negates the notion A. It is, therefore, true that, taken literally (that is, as representing an idea which is the pure negation of the idea A), the formula not-A is unintelligible. Regarding not-A, however, as equivalent to whatever is not A, we may say that its justification and explanation is to be found primarily by reference to the extension of the name. The thinking of anything as A involves its being distinguished from that which is not A. Thus on the extensive side every concept divides the universe with reference to which it is thought (whatever that may be) into two mutually exclusive subdivisions, namely, a portion of which A can be predicated and a portion of which A cannot be predicated. These we designate A and not-A respectively. While it may be said that A and not-A involve intensively only one concept, they are extensively mutually exclusive.

Confining ourselves to connotative names, we may express the distinction between positive and negative names somewhat differently by saying that a positive name implies the presence in the things called by the name of a certain specified attribute or set of attributes, while a negative name implies the absence of one or other of certain specified attributes. A negative name, therefore, has its denotation determined indirectly. The class denoted by the positive name is determined positively, and then the negative name denotes what is left.

59 39. Indefinite Character of Negative Names.—Infinite and indefinite are designations that have been applied to negative names when interpreted in such a way as not to involve restriction to a limited universe of discourse. For without such restriction (explicit or implicit) a negative name, for example, not-white, must be understood to denote the whole infinite or indefinite class of things of which white cannot truly be affirmed, including such entities as virtue, a dream, time, a soliloquy, New Guinea, the Seven Ages of Man.

Many logicians hold that no significant term can be really infinite or indefinite in this way.[66] They say that if a term like not-white is to have any meaning at all, it must be understood as denoting, not all things whatsoever except white things, but only things that are black, red, green, yellow, etc., that is, all coloured things except such as are white. In other words, the universe of discourse which any pair of contradictory terms A and not-A between them exhaust is considered to be necessarily limited to the proximate genus of which A is a species; as, for example, in the case of white and not-white, the universe of colour.

[⁶⁶] This is at the root of Sigwart’s final difficulty with regard to negative names, as indicated in the preceding section. Later on he points out that in division we are justified in including negative characteristics of the form not-A in a concept, although we cannot regard not-A itself as an independent concept. Thus we may divide the concept organic being into feeling and not-feeling, a specific difference being here constituted by the absence of a characteristic which is compatible with the remaining characteristics, but is not necessarily connected with them (Logic, I. p. 278). Compare also Lotze, Logic, § 40.

It is doubtless the case that we seldom or never make use of negative names except with reference to some proximate genus. For instance, in speaking of non-voters we are probably referring to the inhabitants of some town or locality whom we subdivide into those who have votes and those who have not. In a similar way we ordinarily deny red only of things that are coloured, squareness only of things that have some figure, etc., so that there is an implicit limitation of sphere. It may be granted further that a proposition containing a negative name interpreted as infinite can have little or no practical value. But it does not follow that some limitation 60 of sphere is necessary in order that a negative term may have meaning. The argument is used that it is an utterly impossible feat to hold together in any one idea a chaotic mass of the most different things. But the answer to this argument is that we do not profess to hold together the things denoted by a negative name by reference to any positive elements which they may have in common: they are held together simply by the fact that they all lack some one or other of certain determinate elements. In other words, the argument only shews that a negative name has no positive concept corresponding to it.[67] It may be added that if this argument had force, it would apply also to the subdivision of a genus with reference to the presence or absence of a certain quality. If we divide coloured objects into red and not-red, we may say equally that we cannot hold together coloured objects other than red by any positive element that they have in common: the fact that they are all coloured is obviously insufficient for the purpose.

[⁶⁷] For a good statement of the counter-argument, compare Mrs Ladd Franklin in Mind, January, 1892, pp. 130, 1.

A somewhat different argument is implied by Sigwart when he says, “If A = mortal, where will justice, virtue, law, order, distance find a place? They are neither mortal beings, nor yet not-mortal beings, for they are not beings at all.” The answer seems clear. They are not-(mortal beings), and therefore not-A. As a rule, it is needless to exclude explicitly from a species what does not even belong to some higher genus. But the fact of the exclusion remains.

Granting then that in practice we rarely, if ever, employ a negative name except with reference to some proximate genus, we nevertheless hold that not-A is perfectly intelligible whatever the universe of discourse may be and however wide it may be. For it denotes in that universe whatever is not denoted by the corresponding positive name. Moreover in formal processes we should be unnecessarily hampered if not allowed to pass unreservedly from X is not A to X is not-A.[68]

[⁶⁸] Writers who take the view which we are here criticising must in consistency deny the universal validity of the process of immediate inference called obversion. Thus Lotze, rightly on his own view, will not allow us to pass from spirit is not matter to spirit is not-matter ; in fact he rejects altogether the form of judgment S is not-P (Logic, § 40). Some writers, who follow Lotze on the general question here raised, appear to go a good deal further than he does, not merely disallowing such a proposition as virtue is not-blue but also such a proposition as virtue is not blue, on the ground that if we say “virtue is not blue,” there is no real predication, since the notion of colour is absolutely foreign to an unextended and abstract concept such as “virtue.” Lotze, however, expressly draws a distinction between the two forms S is non-Q and S is not Q, and tells us that “everything which it is wished to secure by the affirmative predicate non-Q is secured by the intelligible negation of Q” (Logic, § 72; cf. § 40). On the more extreme view it is wrong to say that Virtue is either blue or it is not blue ; but Lotze himself does not thus deny the universality of the law of excluded middle.

61 From this point of view attention may be called to the difference in ordinary use between such forms as unholy, immoral, discourteous and such forms as non-holy, non-moral, non-courteous. The latter may be used with reference to any universe of discourse, however extensive. But not so the former; in their case there is undoubtedly a restriction to some universe of discourse that is more or less limited in its range. We can, for example, speak of a table as non-moral, although we cannot speak of it as immoral. A want of recognition of this distinction may be partly responsible for the denial that any terms can properly be described as infinite or indefinite.[69]

[⁶⁹] It should be added that in the ordinary use of language the negative prefix does not always make a term negative as here defined. Thus, as Mill points out, “the word unpleasant, notwithstanding its negative form, does not connote the mere absence of pleasantness, but a less degree of what is signified by the word painful, which, it is hardly necessary to say, is positive.” On the other hand, some names positive in form may, with reference to a limited universe of discourse, be negative in force; e.g., alien, foreign. Another example is the term Turanian, as employed in the science of language. This term has been used to denote groups lying outside the Aryan and Semitic groups, but not distinguished by any positive characteristics which they possess in common.

40. Contradictory Terms.—A positive name and the corresponding negative are spoken of as contradictory. We may define contradictory terms as a pair of terms so related that between them they exhaust the entire universe to which reference is made, whilst in that universe there is no individual of which both can be affirmed at the same time. It is desirable to repeat here that contradiction can exist primarily between 62 judgments or propositions only, so that as applied to terms or ideas the notion of contradiction must be interpreted with reference to predication. A and not-A are spoken of as contradictory because they cannot without contradiction be predicated together of the same subject. Thus it is in their exclusive character that they are termed contradictory; as between them exhausting the universe of discourse they might rather be called complementary.[70]

[⁷⁰] Dr Venn (Empirical Logic, p. 191) distinguishes between formal contradictories and material contradictories, according as the relation in which the pair of terms stand to one another is or is not apparent from their mere form. Thus A and not-A are formal contradictories; so are human and non-human. Material contradictories, on the other hand, are not constructed “for the express purpose of indicating their mutual relation.” No formal contradiction, for example, is apparent between British and Foreign, or between British and Alien ; and yet “within their range of appropriate application—which in the latter case includes persons only, and in the former case is extended to produce of most kinds—these two pairs of terms fulfil tolerably well the conditions of mutual exclusion and collective exhaustion.”

41. Contrary Terms.—Two terms are usually spoken of as contrary[71] to one another when they denote things which can be regarded as standing at opposite ends of some definite scale in the universe to which reference is made; e.g., first and last, black and white, wise and foolish, pleasant and painful.[72] Contraries differ from contradictories in that they admit of a mean, and therefore do not between them exhaust the entire universe of discourse. It follows that, although two contraries cannot both be true of the same thing at the same time, they may both be false. Thus, a colour may be neither black nor 63 white, but blue; a feeling may be neither pleasant nor painful, but indifferent.

[⁷¹] De Morgan uses the terms contrary and contradictory as equivalent, his definition of them corresponding to that given in the preceding section.

[⁷²] It has been already pointed out that the negative prefix does not always make a term really negative in force. Thus pleasant and unpleasant are not contradictories, for they admit of a mean; when we say that anything is unpleasant, we intend something more than the mere denial that it is pleasant. It should be added that a pair of terms of this kind may also fail to be contraries as above defined, since while admitting of a mean they may at the same time not denote extremes. Unpleasant, for example, denotes only that which is mildly painful: unless intended ironically, it would be a misuse of terms to speak of the tortures of the Inquisition as merely unpleasant. Compare Carveth Read, Logic, p. 49.

It will be observed that not every term has a contrary as above defined, for the thing denoted by a term may not be capable of being regarded as representing the extreme in any definite scale. Thus blue can hardly be said to have a contrary in the universe of colour, or indifferent in the universe of feeling.

By some writers, the term contrary is used in a wider sense than the above, contrariety being identified with simple incompatibility (a mean between the two incompatibles being possible); thus, blue and yellow equally with black, would in this sense be called contraries of white.[73] Other writers use the term repugnant to express the mere relation of incompatibility; thus red, blue, yellow are in this sense repugnant to one another.[74]

[⁷³] There is much to be said in favour of this wider use of the term contrary. Compare the discussion of contrary propositions in section [81].

[⁷⁴] So long as we are confined to simple terms the relations of contrariety and repugnancy cannot be expressed formally or in mere symbols. But it is otherwise when we pass on to the consideration of complex terms. Thus, while XY and not-X or not-Y are formal contradictories, XY and X not-Y may be said to be formal repugnants, XY and not-X not-Y formal contraries (in the narrower of the two senses indicated above).

42. Relative Names.—A name is said to be relative, when, over and above the object that it denotes, it implies in its signification another object, to which in explaining its meaning reference must be made. The name of this other object is called the correlative of the first. Non-relative names are sometimes called absolute.

Jevons considers that in certain respects all names are relative. “The fact is that everything must really have relations to something else, the water to the elements of which it is composed, the gas to the coal from which it is manufactured, the tree to the soil in which it is rooted “ (Elementary Lessons in Logic, p. 26). Again, by the law of relativity, consciousness is possible only in circumstances of change. We cannot think of any object except as distinguished from something else. Every term, therefore, implies its negative as an object 64 of thought. Take the term man. It is an ambiguous term, and in many of its meanings is clearly relative,—for example, as opposed to master, to officer, to wife. If in any sense it is absolute it is when opposed to not-man; but even in this case it may be said to be relative to not-man. To avoid this difficulty, Jevons remarks, “Logicians have been content to consider as relative terms those only which imply some peculiar and striking kind of relation arising from position in time or space, from connexion of cause and effect, &c.; and it is in this special sense, therefore, that the student must use the distinction.”

A more satisfactory solution of the difficulty may be found by calling attention to the distinction already drawn between the point of view of connotation (which has to do with the signification of names) and the subjective and objective points of view respectively. From the subjective point of view all notions are relative by the law of relativity above referred to. Again, from the objective point of view all things, at any rate in the phenomenal world, are relative in the sense that they could not exist without the existence of something else; e.g., man without oxygen, or a tree without soil. But when we say that a name is relative, we do not mean that what it denotes cannot exist or be thought about without something else also existing or being thought about; we mean that its signification cannot be explained without reference to something else which is called by a correlative name, e.g., husband, parent. It cannot be said that in this sense all names are relative.

The fact or facts constituting the ground of both correlative names is called the fundamentum relationis. For example, in the case of partner, the fact of partnership; in the case of husband and wife, the facts which constitute the marriage tie; in the case of ruler and subject, the control which the former exercises over the latter.

Sometimes the relation which each correlative bears to the other is the same; for example, in the case of partner, where the correlative name is the same name over again. Sometimes it is not the same; for example, father and son, slave-owner and slave. 65

The consideration of relative names is not of importance except in connexion with the logic of relatives, to which further reference will be made [subsequently].

EXERCISES.

43. Give one example of each of the following,—(i) a collective general name, (ii) a singular abstract name, (iii) a connotative singular name, (iv) a connotative abstract name. Add reasons justifying your example in each case. [K.]

44. Discuss the logical characteristics of the following names:—beauty, fault, Mrs Grundy, immortal, nobility, slave, sovereign, the Times, truth, ungenerous. [K.]

[In discussing the character of any name it is necessary first of all to determine whether it is univocal, that is, used in one definite sense only, or equivocal (or ambiguous), that is, used in more senses than one. In the latter case, its logical characteristics may vary according to the sense in which it is used.]

45. It has been maintained that the doctrine of terms is extra-logical. Justify or controvert this position. [J.]

PART II.

PROPOSITIONS.

CHAPTER I.

IMPORT OF JUDGMENTS AND PROPOSITIONS.

46. Judgments and Propositions.—In passing to the next division of our subject we are confronted, first of all, with a question which is partly, but not entirely, a question of phraseology. Shall we speak of the judgment or of the proposition? The usage of logicians differs widely. Some treat almost exclusively of judgments; others almost exclusively of propositions. It will be found that for the most part the former are those who tend to emphasise the psychological or the metaphysical aspects of logic, while the latter are those who are more inclined to develop the symbolic or the material aspects.

To a certain extent it is a matter of little importance which of the alternatives is ostensively adopted. Those who deal with judgments from the logical standpoint must when pressed admit that they can deal with them only as expressed in language, and all their illustrations necessarily consist of judgments expressed in language. But a judgment expressed in language is precisely what is meant by a proposition. Hence in treating of judgments it is impossible not to treat also of propositions. 67

On the other hand, so far as we treat of propositions in logic, we treat of them not as grammatical sentences, but as assertions, as verbal expressions of judgments. The logical proposition is the proposition as understood; and a proposition as understood is a judgment. Hence in treating of propositions in logic we necessarily treat also of judgments.

In a large degree, then, the problem does resolve itself into a merely verbal question. At the same time, reasons and counter-reasons may be adduced in favour of the one alternative and in favour of the other.

On the one side, it is said that the use of the term proposition tends to confuse the sentence as a grammatical combination of words with the proposition as apprehended and intellectually affirmed; and it is urged that in treating of propositions the logician tends to become a mere grammarian.

On the other side, it is submitted that the logician is primarily concerned, not with the process of judgment, the discussion of which belongs to the sphere of psychology, but with judgment as a product, and moreover that he is concerned with this product only in so far as it assumes a fixed and definite form, which it cannot do until it receives verbal expression; and it is urged that if we concentrate our attention on judgments without explicit regard to their expression in language, our treatment tends to become too psychological.

It has been said above that logically we can deal with judgments only as expressed in propositions; and no doubt all judgments can with more or less effort be so expressed. But as a matter of fact we constantly judge in a vague sort of way without the precision that is necessary even in loose modes of expression, and we find that to give expression to our judgments may sometimes require very considerable effort. It must be remembered that logic has in view an ideal. Its object is to determine the conditions to which valid judgments must conform, and it is concerned with the characteristics of actual judgments only in subordination to this end. From this point of view it is specially important that we should deal with judgments in the only form in which it is possible for them to attain precision; and this consideration appears to be conclusive in favour of our 68 treating explicitly of propositions in some part at any rate of a logical course.

No doubt in dealing with propositions we have to raise certain questions that relate to the usage of language. Unfortunately the same propositional form may be understood as expressing very different judgments. It is therefore requisite that in any scientific treatment of logic we should discuss the interpretation of the propositional forms that we recognise. This problem is akin to the problem of definition which has to be faced sooner or later in every science; and, as is also true of a definition, the solution in any particular case is largely of the nature of a convention. But this does not detract from its importance as conducing to clearness of thought.

The question of the interpretation of propositional forms is as a matter of fact one that cannot be altogether avoided on any treatment of logic; and it is of importance to recognise explicitly that in discussing this question we are not dealing with judgments pure and simple. Words are like mathematical symbols, and the meaning of a given form of words is not something inherent either in the words themselves or in the thoughts that they may represent, but is dependent on a convention established by those who employ the words. In the force of a given judgment, however, there can be nothing that is dependent on convention. This distinction is not always remembered by those who confine their attention mainly to judgments, and they are consequently sometimes led to express themselves with an appearance of dogmatism on questions that do not really admit of dogmatic treatment.

But while in certain aspects of logical enquiry it is requisite to deal explicitly with propositions, it must never be forgotten that as logicians we are concerned with propositions only as the expressions of judgments; and there are numerous occasions when we have to go behind propositional forms and ask what are the fundamental characteristics of the judgments that they express.

47. The Abstract Character of Logic.—Reference has been made in the preceding section to the necessity for logical purposes of making our judgments precise. For only if they 69 are precise is it possible to determine with accuracy what are their logical implications considered either individually or in conjunction with one another. It has also been pointed out that we can make our judgments precise only by expressing them in propositional forms, the interpretation of which has been agreed upon.

But this is not without its disadvantages. Sometimes the full force of an actual judgment hardly admits of being expressed in words, and even the force of a proposition as understood may not be found exclusively in the words of which it composed, but may depend partly on the context in which it is placed. Hence the isolated proposition must frequently be regarded as in a sense an abstraction, leaving behind it some portion of the actual judgment for which it stands.

This is indeed much less true of the propositions of science than of those of everyday life; and the more fully a statement is independent of context the more fully may it be regarded as fulfilling its purpose from the scientific standpoint. Still the abstract character of logic must be frankly recognised. “Just as thought is abstract in its dealings with reality, so logic is abstract in its dealings with ordinary thought.”[75]

[⁷⁵] Hobhouse, Theory of Knowledge, p. 7.

That they are in some degree abstractions is true not only of propositions, but also of inferences, as we have to deal with them in logic. Much of the reasoning of everyday life does not admit of expression in the form of definite premisses and conclusions such as would satisfy the canons of logic. The grounds upon which our conclusions are based are often so complex, and the influence which some of them exert upon our beliefs is so subtle and delicate, that they cannot be completely set forth. This will be realised at once if an attempt is made to apply the rules of logic to any ordinary inference; and an explanation is herein found why the illustrations given in logical text-books frequently appear so artificial and unreal.

It must be admitted that the abstract character of logic detracts to some extent from its utility as an art, though the extent of this drawback may easily be exaggerated. Regarded as a science, however, the value, of logic remains unimpaired. 70 Other sciences besides logic have to proceed by abstractions and separations that do not fully correspond to the complexities of nature; and this often becomes the more true the higher the stage that the science has reached. Its necessary abstractness does not prevent logic from analysing successfully the characteristics of the developed judgment or from determining the principles of valid reasoning. If we were to seek to treat logical problems without abstraction we should be in danger of destroying the scientific character of logic without achieving any valuable result even from the purely utilitarian point of view. It is of little value to criticise received systems without providing any new constructive system in their place.

48. Nature of the Enquiry into the Import of Propositions.—Under the general head of the import of propositions it is usual to include problems that are really very different in character.[76]

[⁷⁶] Compare Mr W. E. Johnson in Mind, April, 1895, p. 242.

(1) There is, in the first place, the fundamental problem or series of problems as to what are the essential characteristics of judgments, and therefore of propositions as expressing judgments. The discussion of questions of this character must be based directly on psychological or philosophical considerations, and in the solutions nothing arbitrary or conventional can find a place.

Under this head are to be included such problems as the following: Do all judgments contain a reference to reality? In what sense, if any, can all judgments claim to possess universality or necessity? What is the nature of significant denial? Are distinctions of modality subjective or objective?

(2) In the interpretation of propositional forms we have an enquiry of a very different character, an enquiry which relates distinctively to propositions, and not to judgments considered apart from their expression. The problem is indeed to determine what is the precise judgment that a given proposition shall be understood to express; and, in consequence of the uncertainty and ambiguity of ordinary language, the solution of the problem includes an optional or selective element.

71 As a simple illustration of the kind of problem that we here have in view, we may note that in the traditional scheme of propositions, All S is P, No S is P, Some S is P, Some S is not P, the signs of quantity have to be interpreted. The existential and modal import of these propositions is also partly a question of interpretation.

In connexion with the interpretation of propositions, the distinction between meaning and implication has to be considered. What we do in interpreting propositions is to assign to them a meaning; and when the meaning has once been fixed, the implications are determined in accordance with logical principles.

The dividing line between meaning and implication is not in practice always easy to draw, and some writers seek to ignore it by including within the scope of meaning all the implications of a proposition. But this is a fatal error. The assignment of meaning is within certain limits arbitrary and selective. But if element a necessarily involves element b, then a having been assigned as part of the meaning of a given propositional form, it is no question of meaning as to whether the form in question does or does not imply b, and there is nothing arbitrary or selective in the solution of this question.

Sometimes the elements a and b mutually involve one another. It may then be a question of interpretation whether a shall be included in meaning, b thus becoming an implication, or whether b shall be included in meaning, a becoming an implication.

A failure to recognise what is really the point at issue in a case like this has sometimes caused discussions to take a wrong turn. Thus the question is raised whether the import of the proposition All S is P is that the class S is included in the class P, or that the set of attributes S is invariably accompanied by the set of attributes P ; and these are regarded as antagonistic theories. If the implications of a proposition are regarded as part of its import, then the proposition may be said to import both these things. But if by the import of a proposition we intend to signify its meaning only, then we may adopt an interpretation that will make either of them (but not both) part of its import, or our interpretation may be such 72 that the proposition imports neither of them. The question here raised is dealt with in more detail [later on].

(3) A third problem, distinct from both those described above, arises in connexion with the expression of judgments in propositional form.

In ordinary discourse we meet with an infinite variety of forms of statement. To recognise and deal separately with all these forms in our treatment of logical problems would, however, be impracticable. We have, therefore, in some at any rate of our discussions, to limit ourselves to a certain number of selected forms; and in such discussions we have to assume that the judgments with which we are dealing are at the outset expressed in one or other or a combination of these selected forms.

This reduction of a statement to some canonical form has been called by Mr Johnson its formulation.

A given statement, since it involves many different relations which mutually implicate one another, may be formulated in a number of different ways; and it is needless to say that there is no one scheme of formulating propositions that we are bound to accept to the exclusion of others. Different schemes are useful for different purposes, and several schedules of propositions (for example, equational and existential schedules) will presently be considered in addition to the traditional fourfold schedule. It should be added that a given scheme may profess to cover part only of the field. Thus the traditional schedule (All S is P, etc.) professes to be a scheme for categorical judgments only, and (as traditionally interpreted) for assertoric judgments only.

With reference to the reduction of a statement to a form in which it belongs to a given schedule two points call for notice.

(a) There is danger lest some part of the force of the original statement may be lost.

To a certain extent this is inevitable, especially if the original statement contains suggestion or innuendo in addition to what it definitely affirms; and this must be taken in connexion with what has already been said about the abstract character of logic. If, however, there is any substantial loss of 73 import, the scheme stands condemned so far as it professes to be a complete scheme of formulation. It may, as we have seen, not profess to be a complete scheme, but only to formulate statements falling within a certain category, for example, assertoric statements or categorical statements.

It is to be added that a statement which does not admit of being translated into any one of the simple forms included in a given scheme may still be capable of being expressed by a conjunctive or disjunctive combination of such simple forms. Thus, if the statement Some S is P is made with an emphasis on some, implying not all, then the statement cannot be expressed in any one of the forms of the traditional schedule of propositions, but it is equivalent to Some S is P and some S is not P.

(b) In the reduction of a statement to a form in which it belongs to a given schedule there may be involved what must be admitted to be inference. As, for instance, if statements are given in the ordinary predicative form and have to be expressed in an equational scheme.

It may perhaps be urged that this is legitimate, simply on the ground that one of the postulates of logic is that we be allowed to substitute for any given form of words the technical form (and in an equational system this will be an equation) which is equivalent to it. Have we not, however, in reality a vicious circle if a process which involves inference is to be regarded as a postulate of logic?

The difficulty here raised is a serious one only if we suppose ourselves rigidly limited in logic to a single scheme of formulation; and the solution is to be found in our not confining ourselves to any one scheme, but in our recognising several and investigating the logical relations between them. We can then refuse to regard any substitution of one set of words for another as pre-logical except in so far as it consists of a merely verbal transformation: and our postulate will merely be that we are free to make verbal changes as we please; it will not by itself authorise any change of an inferential character. For a change of this kind, appeal must be made to logical principles.

74 We have then in this section distinguished between three problems any or all of which may be involved in discussions concerning the import of propositions. We have
(1)  the discussion of the essential nature of judgments and of the fundamental distinctions between judgments;
(2)  the interpretation of propositional forms;
(3)  the discussion and comparison of logical schedules or schemes of propositions, drawn up with a view to the expression of judgments in a limited number of propositional forms.

These problems are inter-related and do not admit of being discussed in complete isolation. It is clear, for instance, that the drawing up of a schedule of propositions needs to be supplemented by the exact interpretation of the different forms which it is proposed to recognise; and both in the drawing up of the schedule and in the interpretation we shall be guided and controlled by a consideration of fundamental distinctions between judgments.

The problems are, however, in themselves distinct; and some misunderstanding may be avoided if we can make it clear what is the actual problem that we are discussing at any given point.

In particular, it is important to recognise that in the formulation and interpretation of propositions there is an arbitrary and selective element which is absent from the more fundamental problem. Systems of formulation and interpretation, therefore, if only they are intelligible and self-consistent, can hardly be condemned as radically wrong, though they may be rejected as inconvenient or unsuitable. When, however, we are dealing with the fundamental import of judgments, the questions raised do become questions of absolute right or wrong.

It should be added that in the present treatise, since it is concerned with logic in its more formal aspects, questions of interpretation and formulation occupy a position of greater relative importance than they would in a treatment of the science more fully developed on the philosophical side.

49. The Objective Reference in Judgments.—A judgment can be formed or understood only through the occurrence of certain psychical events in the minds of those who form or 75 understand it; and in this sense it may be included amongst subjective states. It is, however, distinguished from all other subjective states by the fact that it claims to be true.

This claim to be true implies an objective reference. For a merely subjective state is not, as such, either true or false; it is simply an occurrence. Thus, the distinction between truth and falsity is inapplicable to an emotion or a volition. An emotion may be pleasurable or painful; it may be strong or weak; it may or may not impel to action; but we cannot describe it as true or false.

And the same applies to a judgment regarded as no more than a subjective connexion of ideas. The claim to truth necessarily involves more than this, namely, a reference to something external to the psychical occurrence involved in the formation of the judgment. Every judgment implies, therefore, on the part of the judging mind, the recognition of an objective system of reality of some sort. The validity that is claimed for judgment is an objective validity.

The word “objective” is always a dangerous word to use, and some further explanation may be given of the meaning to be attached to it here. When we say that a judgment refers to an objective system, we mean a system that subsists independently of the act of judgment itself, and that is not dependent on the passing fancy of the person who forms the judgment. An objective system of reality in this sense may, however, include subjective states, that is, states of consciousness. A body of psychological doctrine consists of judgments relating to states of mind. But such judgments have an external reference (that is, external to the judgments themselves) just as much as a body of judgments relating to material phenomena. Indeed the doctrine of judgment here laid down is not inconsistent with the theory of subjective idealism that resolves all phenomena into states of consciousness.

Even when a judgment relates to purely fictitious objects there is still an external reference,—in this case, to the world of convention.

The particular aspect or portion of the total system of reality referred to in any judgment may sometimes be 76 conveniently spoken of as the universe of discourse. The limits, if any, intended to be placed upon the universe of discourse in any given proposition are usually not explicitly stated; but they must be considered to be implicit in the judgment which the proposition is meant to express, and to be capable of being themselves expressed should there be any danger of misunderstanding. At the same time, it is only fair to add that attempts to define the universe of discourse are likely to raise metaphysical difficulties as to the ultimate nature of reality. What is of main importance from the logical standpoint is the recognition that there is a reference to some system of reality which is to be distinguished from the uncontrolled course of our own ideas. And so far as a distinction can be drawn between different systems of reality, there is need of the assumption that, when we combine judgments or view them in their mutual relations, the universe of discourse is the same throughout.

50. The Universality of Judgments.—The fundamental characteristic then of judgments is their objective reference, their claim to objective validity. It follows that all judgments claim universality, that is to say, they claim to be acknowledged as true not for a given person only, or for a limited number of persons, but for everyone; and again, not for a given time only, or for a limited time, but for all time. In other words, the import of a judgment is not merely to express some connexion of ideas in my own mind; but to express something that claims to be true. And truth is not relative to the individual, nor is it when fully set forth limited by considerations of time.

We shall have subsequently to deal with the ordinary distinction between universal and particular propositions; but it will be clear that the claim to universality which we are now considering is one that must be made on behalf of so-called particular, as well as of so-called universal, propositions. The judgment that some men are six feet in height claims universal acceptance just as much as the judgment that all men are mortal.

Some judgments again contain an explicit or implicit reference to time. But this is really part of the judgment. As 77 soon as the judgment is fully stated it becomes independent of time. It may perhaps be said that the judgment France is under Bourbon rule was true two centuries ago, but is not true now. But the judgment as it stands, without context, is incompletely stated. That France is (or was) under Bourbon rule in the year 1906 A.D. is for all time false; that France is (or was) under Bourbon rule in the year 1706 A.D. is for all time true.

In regard to the nature and significance of the reference to time in judgments, Mr Bosanquet draws a useful distinction between the time of predication and the time in predication.[77] By the time of predication is meant the time at which some thinking being makes the judgment; and this in no way affects the truth of the judgment. But, as Sigwart points out, everything which exists as a particular thing occupies a definite position in time. Hence all judgments relating to particular things, including singular judgments and so-called narrative judgments, relate to some definite time, past, present, or future, with reference to which alone the statements made are valid. This is the time in predication, and the reference to it must be regarded as an intrinsic part of the judgment itself, although it is not always explicitly mentioned.

[⁷⁷] Logic, I. p. 215. Compare Sigwart, Logic, § 15.

It will be seen that the recognition of the universality of all judgments in the sense here indicated is but the recognition in another aspect of their objective character.

51. The Necessity of Judgments.—A further characteristic that has been ascribed to all judgments, when considered in relation to the judging mind, is necessity. This too is connected with the claim to objective validity. When we judge, we are not free to judge as we will. No doubt by controlling the intellectual influences to which we subject ourselves we may indirectly and in the long run modify within certain limits our beliefs. This is a question belonging to psychology into which we need not now enter. But at any given moment the judgments we form are determined by our mental history and the circumstances in which we are placed. We are bound to judge as we do judge; so far as we feel a question to be an 78 open one our judgment regarding it is suspended. It must be granted that we not unfrequently make statements which do not betray the doubts which as a matter of fact we feel with regard to the point at issue; but such statements do not represent our real judgments. The propositions we utter are the expressions of possible judgments, but not of our judgments.

In any discussion of the modality of judgments, other senses in which the term “necessary” may be applied to judgments have to be considered. In here affirming necessity as a characteristic of all judgments, we are merely declaring over again in another aspect their objective character. The merely subjective sequence of ideas in our minds is more or less under our own control. At any rate we can at will bring given ideas together in our mind. But a judgment is more than a relation between ideas. It claims to be true of some system of reality; and hence it is not so much determined by us, as for us by the knowledge which we have come to possess or think we have come to possess about that system of reality.

EXERCISE.

52. “What is once true is always true.”
“What is true to-day may be false to-morrow.”
Examine these statements. [L.]

CHAPTER II.

KINDS OF JUDGMENTS AND PROPOSITIONS.

53. The Classification of Judgments.—It is customary for logicians to offer a classification of judgments or propositions. There is, however, so much variation in the objects they have in view in drawing up their classifications, that very often their results are not really comparable.

(1) Our object in classifying propositions may, in the first place, be to produce a working scheme for the formulation of judgments. An illustration of this is afforded by the traditional scheme of propositions (All S is P, No S is P, etc.), or by the Hamiltonian scheme based upon the quantification of the predicate. A classification of this kind is essentially formal. The different propositional forms that are recognised must receive clearly defined interpretations; and the resulting scheme, if it is worth anything at all, will be orderly and compact. On the other hand, it is not likely to be comprehensive or exhaustive; for many natural modes of judgment will not find a place in it, at any rate until they have been expressed in a modified, though as nearly as possible equivalent, form.

There are many ways of formulating judgments, each of which has its special merits and is from some particular point of view specially appropriate. We must, however, give up the idea that any one of these ways can hold the field as a fundamental and essentially suitable classification of judgments looked at from the psychological point of view.

(2) From the psychological standpoint our endeavour must be to give rather what may be called a natural history 80 classification of judgments. Primitive types of judgment, which in a logical scheme of formulation are not likely to find a place at all, will now be regarded as of equal importance with more developed and scientific types. Our object may indeed be (as with Mr Bosanquet) to sketch the development of judgments from the most primitive types to those which give expression to the ideal of knowledge.

In a classification of this kind the dividing lines are not so clear and sharply defined as in a scheme framed for the logical formulation of judgments. The different types, moreover, do not stand out in marked distinction from one another, and it is difficult to arrange the different classes in due subordination, and with complete avoidance of cross divisions. The underlying plan is indeed apt to be obscured by details, so that the whole discussion tends to become somewhat cumbrous.

(3) A classification of propositions of still another kind is given by Mill in the later part of his chapter on the Import of Propositions. The conclusion at which he arrives is that every proposition affirms, or denies, either simple existence, or else some sequence, coexistence, causation, or resemblance. This classification is certainly not a formal one; it is not a scheme for the logical formulation of judgments. Nor, on the other hand, can it be regarded as a psychological classification of types of judgment, designed to illustrate the nature and growth of thought. Mill’s point of view is objective and material. In one place he describes his scheme as a classification of matters of fact, of all things that can be believed; and the main use that he subsequently makes of it is in connexion with the enquiry as to the methods of proof that are appropriate according to the nature of the matter of fact that is asserted.

In the pages that follow various schemes for formulating judgments will be considered. For reasons already stated, however, no scheme of this kind can be regarded as constituting an exhaustive classification of judgments. The traditional scheme, for example, is ludicrously unsatisfactory and incomplete if put forward as affording such a classification.

We shall not attempt to give what has been spoken of above as a natural history classification of judgments. The really 81 important distinctions involved in such a classification can be raised independently, and the general plan of this work is to dwell principally on the more formal aspects of logic. It may be added that even from a broader point of view the problem of the evolution of thought is hardly to be regarded as primarily a logical problem.

Again, such a classification as Mill’s involves material considerations that are outside the scope of this treatise.

Without, however, professing to give any complete scheme of classification, we shall endeavour to touch upon the most fundamental differences that may exist between judgments.

54. Kant’s Classification of Judgments.—Kant classified judgments according to four different principles (Quantity, Quality, Relation, and Modality) each yielding three subdivisions, as follows:

This arrangement is open to criticism from several points of view; and its symmetry, although attractive, is not really defensible. At the same time it has the great merit of making prominent what really are the fundamental distinctions between judgments.

The first distinction that we shall consider is that between simple and compound judgments (replacing Kant’s distinction according to relation). We shall then consider in turn distinctions of modality, of quantity, and of quality. 82

55. Simple Judgments and Compound Judgments.—Under the head of relation, Kant gave the well-known threefold division of judgments into categorical, where the affirmation or denial is absolute (S is P); hypothetical (or conditional), where the affirmation or denial is made under a condition (If A is B then S is P); and disjunctive, where the affirmation or denial is made with an alternative (Either S is P or Q is R).

These three kinds of judgment cannot, however, properly be co-ordinated as on an equality with one another in a threefold division. For the categorical judgment appears as an element in both the others, and hence the distinction between the categorical, on the one hand, and the hypothetical and the disjunctive, on the other, appears to be on a different level from that between the two latter. Moreover, the hypothetical and the disjunctive do not exhaust the modes in which categorical judgments may be combined so as to form further judgments. It is, therefore, better not to start from the above threefold division, but from a twofold, namely, into simple and compound.

A compound judgment may be defined as a judgment into the composition of which other judgments enter as elements.[78] There are three principal ways in which judgments may be combined, and in each case the denial of the validity of the combination yields a further form of judgment, so that there are six kinds of compound judgments to be considered.

[⁷⁸] The distinction here implied has been criticised on the ground that (a) if the so-called elements are really judgments, the combination of them yields no fresh judgment; while (b) if the combination is really an independent judgment, the elements into which it can be analysed are not themselves judgments. It will be seen that (a) is intended to apply to conjunctive syntheses, and (b) to hypotheticals and disjunctives. We shall consider the argument under these heads severally.

(1) We may affirm two or more simple judgments together. Thus, given that P and Q stand separately for judgments, we may affirm “P and Q.”

It has been held that a synthesis of two independent judgments in this way does not really yield any fresh judgment distinct from the two judgments themselves.[79] In a sense this is true. Anyone may, however, be challenged for holding two 83 judgments together on grounds which would have no application to either taken separately. Hence it is convenient to regard the combination as constituting a distinct logical whole, which demands some kind of separate treatment; and on this ground the description of “P and Q” as a compound judgment may be justified.

[⁷⁹] Compare Sigwart, Logic, i. p. 214.

The synthesis involved is conjunctive. Hence P and Q may be spoken of more distinctively as a conjunctive judgment. Its denial yields “Not both P and Q” and this form is more truly disjunctive than the form (P or Q) to which that designation is more commonly applied.

(2) Without committing ourselves to the affirmation of either P or Q we may hold them to be so related that the truth of the former involves that of the latter. This yields the hypothetical judgment, “If P then Q.”

It has been held that to regard this as a combination of judgments, and to speak of it as in this sense a compound judgment, is misleading, since P and Q are here not judgments at all, that is to say, they are not at the moment intended as statements. Neither P nor Q is affirmed to be true. What is affirmed to be true is a certain relation between them.[80]

[⁸⁰] Compare Sigwart, Logic, i. p. 219.

It is certainly the case that when I judge “If P then Q,” P need not be my judgment, nor need Q ; my object may even be to establish the falsity of P on the ground of the known falsity of Q. A more impersonal view, however, being taken, P and Q are suppositions, that is, possible judgments, so that they have meaning as judgments; and If P then Q may fairly be said to express a relation between judgments in the sense of its force being that the acceptance of P as a true judgment involves the acceptance of Q as a true judgment also. The description of the hypothetical judgment as compound appears therefore to be in this sense justified. Such a judgment as If P then Q cannot be interpreted except on the supposition that P and Q taken separately have meaning as judgments.

As we get a compound judgment when we declare two judgments to be so related that if one is accepted the other must be accepted also, so we get a compound judgment when 84 we deny that this relation subsists between them. Thus in addition to the judgment “If P then Q,” we have its denial, namely, “If P then not necessarily Q.”[81] The best mode of describing this form of proposition will be considered in a subsequent [chapter].

[⁸¹] In giving this as the contradictory of If P then Q, we are assuming a particular doctrine of the import of the hypothetical judgment. The question will be discussed more fully later on.

(3) We have another form of compound judgment when we affirm that one or other of two given judgments is true. This form of judgment, “P or Q,” is usually called disjunctive, though alternative would be a better name. It has been already pointed out that Not both P and Q is the more distinctively disjunctive form.

It may be denied that P or Q is a compound judgment on the same grounds as those on which this is denied of If P then Q. Since, however, the points at issue are practically the same as before, the discussion need not be repeated.

The denial of “P or Q” yields “Neither P nor Q.” This may be called a remotive judgment if a distinctive name is wanted for it.

It should be added that not all forms of proposition which would ordinarily be described as hypothetical or disjunctive are really the expressions of compound judgments as above described. Thus the forms If any S is P it is Q (If a triangle is isosceles the angles at its base are equal). Every S is either P or Q (Every blood vessel is either a vein or an artery), do not—like the forms If P is true Q is true (If there is a righteous God the wicked will not escape their just punishment), Either P or Q is true (Either free will is a fact or the sense of obligation is an illusion)—express any relation between two independent judgments or propositions. This point will be developed [subsequently] in a distinction that will be drawn between the true hypothetical (If P is true Q is true) and the conditional (If any S is P it is Q).

56. The Modality of Judgments.—Very different accounts of the modality of judgments or propositions are given by different writers, and the problems to which distinctions of modality give 85 rise are as a rule not easy of solution. At the same time such distinctions are of a fundamental character, and they are apt to present themselves in a disguised form, thus obscuring many questions that at first sight appear to have no connexion with modality at all. It is a drawback to have to deal with so difficult a problem nearly at the commencement of our treatment of judgments, and the space at our disposal will not admit of our dealing with it in great detail. Moreover, it can hardly be hoped that the solution offered will be accepted by all readers. Still a brief consideration of modal distinctions at this stage will help to make some subsequent discussions easier.

The main point at issue is whether distinctions of modality are subjective or objective. In attempting to decide this question it will be convenient to deal separately with simple judgments and compound judgments.

57. Modality in relation to Simple Judgments.—The Aristotelian doctrine of modals, which was also the scholastic doctrine, gave a fourfold division into (a) necessary, (b) contingent, (c) possible, and (d) impossible, according as a proposition expresses (a) that which is necessary and unchangeable, and which cannot therefore be otherwise; or (b) that which happens to be at any given time, but might have been otherwise; or (c) that which is not at any given time, but may be at some other time; or (d) that which cannot be. The point of view here taken is objective, not subjective; that is to say, the distinctions indicated depend upon material considerations, and do not relate to the varying degrees of belief with which different propositions are accepted.[82]

[⁸²] The consideration of modality as above conceived has sometimes been regarded as extra-logical on the ground that necessity, contingency, possibility, and impossibility depend upon matters of fact with which the logician as such has no concern. But it also depends upon matters of fact whether any given predicate can rightly be predicated affirmatively or negatively, universally or particularly, of any given subject. Distinctions of quality and quantity can nevertheless be formally expressed, and if distinctions of modality can also be formally expressed, there is no initial reason why they should not be recognised by the logician, even though he is not competent to determine the validity of any given modal. In so far, however, as the modality of a proposition is something that does not admit of formal expression, so that propositions of the same form may have a different modality, then the argument that the doctrine of modals is extra-logical is more worthy of consideration.

86 Kant’s doctrine of modality is distinguished from the scholastic doctrine in that the point of view taken is subjective, not objective, according to one of the senses in which Kant uses these terms. Kant divides judgments according to modality into (a) apodeictic judgments—S must be P, (b) assertoric judgments—S is P, and (c) problematic judgments—S may be P ; and the distinctions between these three classes have come to be interpreted as depending upon the character of the belief with which the judgments are accepted.

The distinction between these two doctrines is fundamental; for, as Sigwart puts it,[83] the statement that a judgment is possible or necessary is not the same as the statement that it is possible or necessary for a predicate to belong to a subject. The former (which is the Kantian doctrine) refers to the subjective possibility or necessity of judgment; the latter (which is the Aristotelian doctrine) refers to the objective possibility or necessity of what is stated in the judgment.

[⁸³] Logic, i. p. 176.

58. Subjective Distinctions of Modality.—We must reject the view that subjective distinctions of modality can be drawn in relation to simple judgments.[84] For all judgments, as we have [seen], possess the characteristic of necessity, and hence this characteristic cannot be made the distinguishing mark of a particular class of judgments, the apodeictic.

[⁸⁴] What follows in this section is based mainly on Sigwart’s treatment of the subject (Logic, § 31).

We may touch on two ways in which it has been attempted to draw a distinction, from the subjective point of view, between assertoric and apodeictic judgments.

The assertoric judgment has been regarded as expressing what has only subjective validity, that is, what can be affirmed to be true only for the person forming the judgment, while the apodeictic judgment expresses what has universal validity and can be affirmed to be true for everyone.

This again conflicts with the general doctrine of judgment already laid down. We hold that every judgment claims to be true, and that truth cannot be relative to the individual. The assertoric judgment, therefore, as thus defined is no true 87 judgment at all, and we find that all judgments are really apodeictic.

Another suggested ground of distinction is that between immediate knowledge and knowledge that is based on inference, the former being expressed by the assertoric judgment, and the latter by the apodeictic.

There is no doubt that we often say a thing is so and so when this is a matter of direct perception, while we say it must be so and so when we cannot otherwise account for certain perceived facts. Thus, if I have been out in the rain, I say it has rained ; if, without having observed any rain fall, I notice that the roads and roofs are wet, I say it must have rained.

It is obvious, however, that this distinction is quite inconsistent with the ascription of any superior certainty to the apodeictic judgment. For that which we know mediately must always be based on that which we know immediately; and, since in the process of inference error may be committed, it follows that that which we know mediately must have inferior certainty to that of which we have immediate knowledge. Accordingly in ordinary discourse the statement that anything must be so and so would generally be understood as expressing a certain degree of doubt.

We cannot then justify the recognition of the apodeictic judgment as expressing a higher degree of certainty than the merely assertoric.

On the other hand, the so-called problematic judgment, interpreted as expressing mere uncertainty,[85] cannot be regarded as in itself expressing a judgment at all. It may imply a judgment in regard to the validity of arguments brought forward in support or in disproof of a given thesis; and it implies also a judgment as to the state of mind of the person who is in a state of uncertainty; but it is in itself a mere suspension of judgment.

[⁸⁵] The problematic judgment as interpreted in the following section does more than express mere uncertainty. The form of proposition S may be P is no doubt ambiguous.

59. Objective Distinctions of Modality.—We have next to consider whether, having regard not to the judgment as a 88 subjective product, but to the objective fact expressed in a judgment, any valid distinction can be drawn between the necessary, the actual (or contingent), and the possible ; and our answer must be in the affirmative, provided that we are prepared to admit the conception of the operation of law.

Thus the judgment Planets move in elliptic orbits is in this sense a judgment of necessity. It expresses something which we regard as the manifestation of a law, and it has an indefinitely wide application. For we believe it to hold good not only of the planets with which we are acquainted, but also of other planets (if such there be) which have not yet been discovered.

Now take the judgment, All the kings who ruled in France in the eighteenth century were named Louis. This is a statement of fact, but clearly is not the expression of any law. The proposition relates to a limited number of individuals who happened to have the same name given to them; but we recognise that their names might have been different, and that their being kings of France was not dependent on their possessing the name in question. This then we may call a judgment of actuality.

We have a judgment of possibility when we make such a statement as that a seedling rose may be produced different in colour from any roses with which we are at present acquainted, meaning that there is nothing in the inherent nature of roses (or in the laws regulating the production of roses) to render this impossible.

We have then a judgment of necessity (an apodeictic judgment) when the intention is to give expression to some law relating to the class of objects denoted by the subject-term; we have a judgment of actuality (an assertoric judgment) when the intention is to state a fact, as distinguished from the affirmation or denial of a law; we have a judgment of possibility (a problematic judgment) when the intention is to deny the operation of any law rendering some complex of properties impossible.[86]

[⁸⁶] The case of a proposition which may be regarded as expressing a particular instance of the operation of a law needs to be specially considered. Granting, for instance, that the proposition Every triangle has its angles equal to two right angles is apodeictic, are we to describe the proposition This triangle has its angles equal to two right angles as apodeictic or as assertoric? The right answer seems to be that, as thus barely stated, the proposition may be merely assertoric; for it may do no more than express a fact that has been ascertained by measurement. If, however, the proposition is interpreted as meaning This figure, being a triangle, has its angles equal to two right angles, then it is apodeictic.

89 I shall not attempt to give here any adequate philosophic analysis of the conception of objective necessity. It must suffice to say that we all have the conception of the operation of law, and that for our present purpose the validity of this conception is assumed.

With regard to this treatment of modality the objection may perhaps be raised that, whatever their value in themselves, the distinctions involved are not of a kind with which formal logic has any concern. It is true that, in a sense, judgments of necessity are the peculiar concern of inductive, as distinguished from formal, logic. The main function of inductive logic is indeed to determine how apodeictic judgments (as above defined) are to be established on the basis of individual observations; for what we mean by induction is the process of passing from particulars to the laws by which they are governed. Granting this, however, there are also many problems, with which logic in its more formal aspects has to deal, in the solution of which some recognition of the distinctions under discussion is desirable, if not essential.

But it will be said that the distinctions cannot be applied formally: that, for example, given a proposition in the bare form S is P, or given an ordinary universal affirmative proposition All S is P, it cannot be determined, apart from the matter of the proposition, whether it is apodeictic (in the sense in which that term is used in this section) or merely assertoric. This is true if we are limited to the traditional schedule of propositions. But it is to be remembered that the formulation and the interpretation of propositions are within certain limits under our own control, and that it is within our power so to interpret propositional forms for logical purposes as to bring out distinctions that are not made clear in ordinary discourse or in the traditional logic. Thus, the form S as such is P might be used for giving formal expression to the apodeictic judgment, S is P being interpreted as merely assertoric.

90 Another solution, however, and one that may be made to yield a symmetrical scheme, is to utilise the conditional (as distinguished from the true hypothetical,[87]) proposition, and to differentiate it from the categorical, by interpreting it as modal,[88] while the categorical remains merely assertoric.

[⁸⁷] See section [173].

[⁸⁸] Here and elsewhere in speaking of a proposition as modal (in contradistinction to assertoric) we mean a proposition that is either apodeictic or problematic.

Thus, we should have,—
If anything is S it is P,—apodeictic;
All S is P,—assertoric;
If anything is S it may be P,—problematic.[89]

[⁸⁹] It will be observed that in this scheme (leaving on one side the question of existential import) the categorical proposition All S is P is inferable from the conditional If anything is S it is P, but not vice versâ.

It is of course not pretended that the differentiation here proposed is adopted in the ordinary use of the propositional forms in question; we shall, for example, have presently to point out that in the customary usage of categoricals the universal affirmative has frequently an apodeictic force. We shall return to a discussion of the suggested scheme [later on].

60. Modality in relation to Compound Judgments.—We may now consider the application of distinctions of modality to compound judgments, that is, to judgments which express a relation in which simple judgments stand one to another. It is one thing to say that as a matter of fact two judgments are not both true; it is another thing to say that two judgments are so related to one another that they cannot both be true. We may describe the one statement as assertoric, the other as apodeictic. An apodeictic judgment thus conceived expresses a relation of ground and consequence; an obligation, therefore, to affirm the truth of a certain proposition when the truth of a certain other proposition or combination of propositions is admitted. The obligation may sometimes depend upon the assistance of certain other propositions which are left unexpressed.[90]

[⁹⁰] In an apodeictic compound judgment, the necessity may (at any rate in certain cases) be described as subjective. This is so in the case of a formal hypothetical; as, for example, in the proposition If all S is P then all not-P is not-S, or in the proposition If all S is M and all M is P then all S is P.

91 In section [55] a threefold classification of compound judgments was given; the distinction now under consideration points, however, to a more fundamental twofold classification. From this point of view a scheme may be suggested in which conjunctives (P and Q) and so-called disjunctives (P or Q) would be regarded as assertoric, while hypotheticals (If P then Q) would be regarded as modal. The enquiry as to how far this is in accordance with the ordinary usage of the propositional forms in question must be deferred. It may, however, be desirable to point out at once that, if this scheme is adopted, certain ordinarily recognised logical relations are not valid. For the hypothetical If P then Q is ordinarily regarded as equivalent to the disjunctive Either not-P or Q, and this as equivalent to the denial of the conjunctive Both P and not-Q. If, however, the conjunctive (and, therefore, its denial) and also the disjunctive are merely assertoric, while the hypothetical is apodeictic, it is clear that this equivalence no longer holds good. The disjunctive can indeed still be inferred from the hypothetical, but not the hypothetical from the disjunctive. This result will be considered further at a [later] stage.

So far we have spoken only of the apodeictic form, If P then Q. The corresponding problematic form is, If P then possibly Q ; for example, If all S is P it is still possible that some P is not S. This denies the obligation to admit that all P is S when it has been admitted that all S is P. It is to be observed that in any treatment of modality, the apodeictic and the problematic involve one another, since the one form is always required to express the contradictory of the other.

61. The Quantity and the Quality of Propositions.—Propositions are commonly divided into universal and particular, according as the predication is made of the whole or of a part of the subject. This division of propositions is said to be according to their quantity.

Kant added a third subdivision, namely, singular ; and other logicians have added a fourth, namely, indefinite. Under the head of quantity there have also to be considered what are called plurative and numerically definite propositions; and the possibility of multiple quantification has to be recognised. The 92 question may also be raised whether there are not some propositions, e.g., hypothetical propositions, which do not admit of division according to quantity at all. The discussion of the various points here indicated may, however, conveniently be deferred until the traditional scheme of categorical propositions, which is based on the definitive division into universal and particular, has been briefly touched upon.

Another primary division of propositions is into affirmative and negative, according as the predicate is affirmed or denied of the subject. This division of propositions is said to be according to their quality.

Here, again, Kant added a third subdivision, namely, infinite. This threefold division and the more fundamental question as to the true significance of logical denial, will also be deferred until some account has been given of the traditional scheme of propositions.

62. The traditional Scheme of Propositions.—The traditional scheme of formulating propositions is intended primarily for categoricals, and it is based on distinctions of quantity and quality only, distinctions of modality not being taken into account. For the purposes of the traditional scheme the following analysis of the categorical proposition may be given.

A categorical proposition consists of two terms (which are respectively the subject and the predicate), united by a copula, and usually preceded by a sign of quantity. It thus contains four elements, two of which—the subject and the predicate—constitute its matter, while the remaining two—the copula and the sign of quantity—constitute its form.[91]

[⁹¹] The logical analysis of a proposition must be distinguished from its grammatical analysis. Grammatically only two elements are recognised, namely, the subject and the predicate. Logically we further analyse the grammatical subject into quantity and logical subject, and the grammatical predicate into copula and logical predicate.

The subject is that term about which affirmation or denial is made. The predicate is that term which is affirmed or denied of the subject.

When propositions are brought into one of the forms recognised in the traditional scheme the subject precedes the predicate. In ordinary discourse, however, this order is sometimes 93 inverted for the sake of literary effect, for example, in the proposition—Sweet are the uses of adversity.

The sign of quantity attached to the subject indicates the extent to which the individuals denoted by the subject-term are referred to. Thus, in the proposition All S is P the sign of quantity is all, and the affirmation is understood to be made of each and every individual denoted by the term S.

The copula is the link of connexion between the subject and the predicate, and indicates whether the latter is affirmed or denied of the former.

The different elements of the proposition as here distinguished are by no means always separately expressed in the propositions of ordinary discourse; but by analysis and expansion they may be made to appear without any change of meaning. Some grammatical change of form is, therefore, often necessary before propositions can be dealt with in the traditional scheme. Thus in such a proposition as “All that love virtue love angling,” the copula is not separately expressed. The proposition may, however, be written—

and in this form the four different elements are made distinct. The older logicians distinguished between propositions secundi adjacentis and propositions tertii adjacentis. In the former, the copula and the predicate are not separated, e.g., The man runs, All that love virtue love angling; in the latter, they are made distinct, e.g., The man is running, All lovers of virtue are lovers of angling.

The traditional scheme of propositions is obtained by a combination of the division (according to quantity) into universal and particular, and the division (according to quality) into affirmative and negative. This combination yields four fundamental forms of proposition as follows:—
(1) the universal affirmative—All S is P (or Every S is P, or Any S is P, or All S’s are P’s)—usually denoted by the symbol A; 94
(2) the particular affirmative—Some S is P (or Some S’s are P’s)—usually denoted by the symbol I;
(3) the universal negative—No S is P (or No S’s are P’s)—usually denoted by the symbol E;
(4) the particular negative—Some S is not P (or Not all S is P, or Some S’s are not P’s, or Not all S’s are P’s)—usually denoted by the symbol O.

These symbols A, I, E, O, are taken from the Latin words affirmo and nego, the affirmative symbols being the first two vowels of the former, and the negative symbols the two vowels of the latter.

Besides these symbols, it will sometimes be found convenient to use the following,—

SaP = All S is P ;

SiP = Some S is P ;

SeP = No S is P ;

SoP = Some S is not P.

These forms are useful when it is desired that the symbol which is used to denote the proposition as a whole should also indicate what symbols have been chosen for the subject and the predicate respectively. Thus,

MaP = All M is P ;

PoQ = Some P is not Q.

It will further be found convenient sometimes to denote not-S by Sʹ, not-P by Pʹ, and so on. Thus we shall have

SʹaPʹ = All not-S is not-P ;

PiQʹ = Some P is not-Q.

It is better not to write the universal negative in the form All S is not P ;[92] for this form is ambiguous and would usually be interpreted as being merely particular, the not being taken to qualify the all, so that we have All S is not P = Not-all S is P. Thus, “All that glitters is not gold” is intended for an O proposition, and is equivalent to “Some things that glitter are not gold.”

[⁹²] Similar remarks apply to the form Every S is not P.

95 The traditional scheme of formulation is somewhat limited in its scope, and from more points of view than one it is open to criticism. It has, however, the merit of simplicity, and it has met with wide acceptation. For these reasons it is as a rule convenient to adopt it as a basis of discussion, though it is also not infrequently necessary to look beyond it.

63. The Distribution of Terms in a Proposition.—A term is said to be distributed when reference is made to all the individuals denoted by it; it is said to be undistributed when they are only referred to partially, that is, when information is given with regard to a portion of the class denoted by the term, but we are left in ignorance with regard to the remainder of the class. It follows immediately from this definition that the subject is distributed in a universal, and undistributed in a particular,[93] proposition. It can further be shewn that the predicate is distributed in a negative, and undistributed in an affirmative proposition. Thus, if I say All S is P, I identify every member of the class S with some member of the class P, and I therefore imply that at any rate some P is S, but I make no implication with regard to the whole of P. It is left an open question whether there is or is not any P outside the class S. Similarly if I say Some S is P. But if I say No S is P, in excluding the whole of S from P, I am also excluding the whole of P from S, and therefore P as well as S is distributed. Again, if I say Some S is not P, although I make an assertion with regard to a part only of S, I exclude this part from the whole of P, and therefore the whole of P from it. In this case, then, the predicate is distributed, although the subject is not.[94]

[⁹³] Some being used in the sense of some, it may be all. If by some we understand some, but not all, then we are not really left in ignorance with regard to the remainder of the class which forms the subject of our proposition.

[⁹⁴] Hence we may say that the quantity of a proposition, so far as its predicate is concerned, is determined by its quality. The above results, however, no longer hold good if we explicitly quantify the predicate as in Hamilton’s doctrine of the quantification of the predicate. According to this doctrine, the predicate of an affirmative proposition is sometimes expressly distributed, while the predicate of a negative proposition is sometimes given undistributed. For example, such forms are introduced as Some S is all P, No S is some P. This doctrine will be discussed in [chapter 7].

96 Summing up our results, we find that
A distributes its subject only,
I distributes neither its subject nor its predicate,
E distributes both its subject and its predicate,
O distributes its predicate only.

64. The Distinction between Subject and Predicate in the traditional Scheme of Propositions.—The nature of the distinction ordinarily drawn between the subject and the predicate of a proposition may be expressed by saying that the subject is that of which something is affirmed or denied, the predicate that which is affirmed or denied of the subject; or we may say that the subject is that which we regard as the determined or qualified notion, while the predicate is that which we regard as the determining or qualifying notion.

It follows that the subject must be given first in idea, since we cannot assert a predicate until we have something about which to assert it. Can it, however, be said that because the subject logically comes first in order of thought, it must necessarily do so in order of statement, the subject always preceding the copula, and the predicate always following it? In other words, can we consider the order of the terms in a proposition to suffice as a criterion? If the subject and the predicate are pure synonyms[95] or if the proposition is practically reduced to an equation, as in the doctrine of the quantification of the predicate, it is difficult to see what other criterion can be taken; or it may rather be said that in these cases the distinction between subject and predicate loses all importance. The two are placed on an equality, and nothing is left by which to distinguish them except the order in which they are stated. This view is indicated by Professor Baynes in his Essay on the New Analytic of Logical Forms. In such a proposition, for example, as “Great is Diana of the Ephesians,” he would call “great” the subject, reading the proposition, “(Some) great is (all) Diana of the Ephesians.”

[⁹⁵] For illustrations of this point, and on the general question raised in this section, compare Venn, Empirical Logic, pp. 208 to 214.

With reference to the traditional scheme of propositions, however, it cannot be said that the order of terms is always a 97 sufficient criterion. In the proposition just quoted, “Diana of the Ephesians” would generally be accepted as the subject. What further criterion then can be given? In the case of E and I propositions (propositions, as will be shewn, which can be simply converted) we must appeal to the context or to the question to which the proposition is an answer. If one term clearly conveys information regarding the other term, it is the predicate. It will be shewn also that it is more usual for the subject to be read in extension and the predicate in intension.[96] If these considerations are not decisive, then the order of the terms must suffice. In the case of A and O propositions (propositions, as will be shewn, which cannot be simply converted) a further criterion may be added. From the rules relating to the distribution of terms in a proposition it follows that in affirmative propositions the distributed term (if either term is distributed) is the subject; whilst in negative propositions, if only one term is distributed, it is the predicate. It is doubtful if the inversion of terms ever occurs in the case of an O proposition; but in A propositions it is not infrequent. Applying the above considerations to such a proposition as “Workers of miracles were the Apostles,” it is clear that the latter term is distributed while the former is not; the latter term is, therefore, the subject. Since a singular term is equivalent to a distributed term, it follows further as a corollary that in an affirmative proposition if one and only one term is singular it is the subject. This decides such a case as “Great is Diana of the Ephesians.”

[⁹⁶] The subject is often a substantive and the predicate an adjective. Compare section [135].

65. Universal Propositions.—In discussing the import of the universal proposition All S is P, attention must first be called to a certain ambiguity resulting from the fact that the word all may be used either distributively or collectively. In the proposition, All the angles of a triangle are less than two right angles, it is used distributively, the predicate applying to each and every angle of a triangle taken separately. In the proposition. All the angles of a triangle are equal to two right angles, it is used collectively, the predicate applying to all the 98 angles taken together, and not to each separately. This ambiguity attaches to the symbolic form All S is P, but not to the form All S’s are P’s. Ambiguity may also be avoided by using every instead of all, as the sign of quantity. In any case the ambiguity is not of a dangerous character, and it may be assumed that all is to be interpreted distributively, unless by the context or in some other way an indication is given to the contrary.

A more important distinction between propositions expressed in the form All S is P remains to be considered. For such propositions may be merely assertoric or they may be apodeictic, in the sense given to these terms in section [59].

It will be convenient here to commence with a threefold distinction.

(1) The proposition All S is P may, in the first place, make a predication of a limited number of particular objects which admit of being enumerated: e.g., All the books on that shelf are novels, All my sons are in the army, All the men in this year’s eleven were at public schools. A proposition of this kind may be called distinctively an enumerative universal. It is clear that such a proposition cannot claim to be apodeictic.

(2) The proposition All S is P may, in the second place, express what is usually described as an empirical law or uniformity: e.g., All lions are tawny, All scarlet flowers are without sweet scent, All violets are white or yellow or have a tinge of blue in them. Many propositions relating to the use of drugs, to the succession of certain kinds of weather to certain appearances of sky, and so on, fall into this class. A proposition of this kind expresses a uniformity which has been found to hold good within the range of our experience, but which we should hesitate to extend much beyond that range either in space or in time. The predication which it makes is not limited to a definite number of objects which can be enumerated, but at the same time it cannot be regarded as expressing a necessary relation between subject and predicate. Such a proposition is, therefore, assertoric, not apodeictic.

(3) The proposition All S is P may, in the third place, express a law in the strict sense, that is to say, a uniformity 99 that we believe to hold good universally and unconditionally: e.g., All equilateral triangles are equiangular, All bodies have weight, All arsenic is poisonous. A proposition of this kind is to be regarded as expressing a necessary relation between subject and predicate, and it is, therefore, apodeictic.

Propositions falling under the first two of the above categories may be described as empirically universal, and those falling under the third as unconditionally universal.[97]

[⁹⁷] I have borrowed these terms from Sigwart, Logic, § 27; but I cannot be sure that my usage of them corresponds exactly with his. In section 27 he appears to include under empirically universal judgments only such judgments as belong to the first of the three classes distinguished from one another above. At the same time, his description of the unconditionally universal judgment applies to the third class only: such a judgment, he says, expresses a necessary connexion between the predicate P and the subject S ; it means, If anything is S it must also be P. And it seems clear from his subsequent treatment (in § 96) of judgments belonging to the second class that he does not regard them as unconditionally universal.

Lotze (Logic, § 68) indicates the distinction we are discussing by the terms universal and general. But again there seems some uncertainty as to which term he would apply to judgments belonging to our second class. In the universal judgment, he says, we have merely a summation of what is found to be true in every individual instance of the subject; in the general judgment the predication is of the whole of an indefinite class, including both examined and unexamined cases. From this it would appear that the universal judgment corresponds to (1) only, while the general judgment includes both (2) and (3). Lotze, however, continues, “The universal judgment is only a collection of many singular judgments, the sum of whose subjects does as a matter of fact fill up the whole extent of the universal concept; … the universal proposition, All men are mortal, leaves it still an open question whether, strictly speaking, they might not all live for ever, and whether it is not merely a remarkable concatenation of circumstances, different in every different case, which finally results in the fact that no one remains alive. The general judgment, on the other hand, Man is mortal, asserts by its form that it lies in the character of mankind that mortality is inseparable from everyone who partakes in it.” The illustration here given seems to imply that a judgment may be regarded as universal, though it relates to a class of objects, not all of which can be enumerated.

If this distinction is regarded merely as a distinction between different ways in which judgments may be obtained (for example, by enumeration or empirical generalisation on the one hand, or by abstract reasoning or the aid of the principle of causality on the other hand), without any real difference of content, it becomes merely genetic and can hardly be retained as a 100 distinction between judgments considered in and by themselves. If we are so to retain it, it must be as a distinction between the merely assertoric and the apodeictic in the sense already explained. In order to be able to deal with it as a formal distinction, we must further be prepared to assign distinctive forms of expression to the two kinds of universal judgments respectively. Lotze appears to regard the forms All S is P and S is P as sufficiently serving this purpose. But this is hardly borne out by the current usage of these forms. All the S’s are P might serve for the enumerative universal and S as such is P for the unconditionally universal. These forms do not, however, fit into any generally recognised schedules; and our second class of universal would be left out. Another solution, which has been already indicated in section [59], would be to use the categorical form for the empirically universal judgment only, adopting the conditional form for the unconditionally universal judgment.

The most important outcome of the above discussion is that a proposition ordinarily expressed in the form All S is P may be either assertoric or apodeictic. It will be found that this distinction has an important bearing on several questions subsequently to be raised.

66. Particular Propositions.—In dealing with particular propositions it is necessary to assign a precise signification to the sign of quantity some.

In its ordinary use, the word some is always understood to be exclusive of none, but in its relation to all there is ambiguity. For it is sometimes interpreted as excluding all as well as none, while sometimes it is not regarded as carrying this further implication. The word may, therefore, be defined in two conflicting senses: first, as equivalent simply to one at least, that is, as the pure contradictory of none, and hence as covering every case (including all) which is inconsistent with none ; secondly, as any quantity intermediate between none and all and hence carrying with it the implication not all as well as not none. In ordinary speech the latter of these two meanings is probably the more usual.[98] It has, however, been customary with 101 logicians in interpreting the traditional scheme to adopt the other meaning, so that Some S is P is not inconsistent with All S is P. Using the word in this sense, if we want to express Some, but not all, S is P, we must make use of two propositions—Some S is P, Some S is not P. The particular proposition as thus interpreted is indefinite, though with a certain limit; that is, it is indefinite in so far that it may apply to any number from a single one up to all, but on the other hand it is definite in so far as it excludes none. We shall henceforth interpret some in this indefinite sense unless an explicit indication is given to the contrary.

[⁹⁸] We might indeed go further and say that in ordinary speech some usually means considerably less than all, so that it becomes still more limited in its signification. In common language, as is remarked by De Morgan, “some usually means a rather small fraction of the whole; a larger fraction would be expressed by a good many ; and somewhat more than half by most ; while a still larger proportion would be a great majority or nearly all” (Formal Logic, p. 58).

Mr Bosanquet regards the particular proposition as unscientific, on the ground that it always depends either upon imperfect description or upon incomplete enumeration.[99] I may, for instance, know that all S’s of some particular description are P, but not caring or not troubling to define them I content myself with saying Some S is P, for example, Some truth is better kept to oneself.[100] Contrasted with this, we have the particular proposition of incomplete enumeration where our ground for asserting it is simply the observation of individual instances in which the proposition is found to hold good.

[⁹⁹] Essentials of Logic, pp. 116, 117.

[¹⁰⁰] It is implied that a proposition of this kind might be expanded into the proposition All S that is A is P, that is, All AS is P. Mr Bosanquet gives, as an example, Some engines can drag a train at a mile a minute for a long distance. “This does not mean a certain number of engines, though of course there are a certain number. It means certain engines of a particular make, not specified in the judgment.”

It is true that the particular proposition is not in itself of much scientific importance; and its indefinite character naturally limits its practical utility. It seems, however, hardly correct to describe it as unscientific, since—as will subsequently be shewn in more detail—it may be regarded as possessing distinctive functions. Two such functions may be distinguished, though they are often implicated the one in the other. In the first place, the utility of the particular proposition often depends 102 rather on what it denies than on what it affirms, and the proposition that it denies is not indefinite. One of the principal functions of the particular affirmative is to deny the universal negative, and of the particular negative to deny the universal affirmative. In the second place, the distinctive purpose of the particular proposition may be to affirm existence; and this is probably as a rule the case with propositions which are described as resulting from incomplete description. If, for example, we say that “some engines can drag a train at a mile a minute for a long distance,” our object is primarily to affirm that there are such engines; and this would not be so clearly expressed in the universal proposition of which the particular is said to be the incomplete and imperfect expression.

The relation of the particular proposition, Some S is P, to the problematic proposition, S may be P, will be considered subsequently.

67. Singular Propositions.—By a singular or individual proposition is meant a proposition in which the affirmation or denial is made of a single individual only: for example, Brutus is an honourable man ; Much Ado about Nothing is a play of Shakespeare’s ; My boat is on the shore.

Singular propositions may be regarded as forming a sub-class of universals, since in every singular proposition the affirmation or denial is of the whole of the subject.[101] More definitely, the singular proposition may be said to fall into line, as a rule, with the enumerative universal proposition.

[¹⁰¹] It is argued by Father Clarke that singulars ought to be included under particulars, on the ground that when a predicate is asserted of one member only of a class, it is asserted of a portion only of the class. “Now if I say, This Hottentot is a great rascal, my assertion has reference to a smaller portion of the Hottentot nation than the proposition Some Hottentots are great rascals. The same is the case even if the subject be a proper name. London is a large city must necessarily be a more restricted proposition than Some cities are large cities ; and if the latter should be reckoned under particulars, much more the former” (Logic, p. 274). This view fails to recognise that what is really characteristic of the particular proposition is not its restricted character—since the particular is not inconsistent with the universal—but its indefinite character.

Hamilton distinguishes between universal and singular propositions, the predication being in the former case of a whole undivided, and in the latter case of a unit indivisible. The 103 distinction here indicated is sometimes useful; but it can with advantage be expressed somewhat differently. A singular proposition may generally without risk of confusion be denoted by one of the symbols A or E; and in syllogistic inferences a singular may ordinarily be treated as equivalent to a universal proposition. The use of independent symbols for singular propositions (affirmative and negative) would introduce considerable additional complexity into the treatment of the syllogism; and for this reason it seems desirable as a rule to include singulars under universals. Universal propositions may, however, be divided into general and singular, and there will then be terms whereby to call attention to the distinction whenever it may be necessary or useful to do so.

There is also a certain class of propositions which, while singular, inasmuch as they relate but to a single individual, possess also the indefinite character which belongs to the particular proposition: for example, A certain man had two sons ; A great statesman was present ; An English officer was killed. Having two such propositions in the same discourse we cannot, apart from the context, be sure that the same individual is referred to in both cases. Carrying the distinction indicated in the preceding paragraph a little further, we have a fourfold division of propositions:—general definite, “All S is P”; general indefinite, “Some S is P”; singular definite, “This S is P”; singular indefinite, “A certain S is P.” This classification admits of our working with the ordinary twofold distinction into universal and particular—or, as it is here expressed, definite and indefinite—wherever this is adequate, as in the traditional doctrine of the syllogism; while at the same time it introduces a further distinction which may in certain connexions be of importance.

68. Plurative Propositions and Numerically Definite Propositions.—Other signs of quantity besides all and some are sometimes recognised by logicians. Thus, propositions of the forms Most S’s are P’s, Few S’s are P’s, are called plurative propositions. Most may be interpreted as equivalent to at least one more than half. Few has a negative force; and Few S’s are P’s may be regarded as equivalent to Most S’s are not 104 P’s.[102] Formal logicians (excepting De Morgan and Hamilton) have not as a rule recognised these additional signs of quantity; and it is true that in many logical combinations they cannot be regarded as yielding more than particular propositions, Most S’s are P’s being reduced to Some S’s are P’s, and Few S’s are P’s to Some S’s are not P’s. Sometimes, however, we are able to make use of the extra knowledge given us; e.g., from Most M’s are P’s, Most M’s are S’s, we can infer Some S’s are P’s, although from Some M’s are P’s, Some M’s are S’s, we can infer nothing.

[¹⁰²] With perhaps the further implication “although some S’s are P’s”; thus, Few S’s are P’s is given by Kant as an example of the exponible proposition (that is, a proposition which, though not compound in form, can nevertheless be resolved into a conjunction of two or more simpler propositions, which are independent of one another), on the ground that it contains both an affirmation and a negation, though one of them in a concealed way. It should be added that a few has not the same signification as few, but must be regarded as affirmative, and generally, as simply equivalent to some ; e.g., A few S’s are P’s = Some S’s are P’s. Sometimes, however, it means a small number, and in this case the proposition is perhaps best regarded as singular, the subject being collective. Thus, “a few peasants successfully defended the citadel” may be rendered “a small band of peasants successfully defended the citadel,” rather than “some peasants successfully defended the citadel,” since the stress is intended to be laid at least as much on the paucity of their numbers as on the fact that they were peasants. Whilst the proposition interpreted in this way is singular, not general, it is singular indefinite, not singular definite; for what small band is alluded to is left indeterminate.

Numerically definite propositions are those in which a predication is made of some definite proportion of a class; e.g., Two-thirds of S are P. A certain ambiguity may lurk in numerically definite propositions; e.g., in the above proposition is it meant that exactly two-thirds of S neither more nor less are P, so that we are also given implicitly one-third of S are not P, or is it merely meant that at least two-thirds of S but perhaps more are P? In ordinary discourse we should no doubt mean sometimes the one and sometimes the other. If we are to fix our interpretation, it will probably be best to adopt the first alternative, on the ground that if figures are introduced at all we should aim at being quite determinate.[103] Some such words 105 as at least can then be used when it is not professed to state more than the minimum proportion of S’s that are P’s.

[¹⁰³] De Morgan remarks that “a perfectly definite particular, as to quantity, would express how many X’s are in existence, how many Y’s, and how many of the X’s are or are not Y’s; as in 70 of the 100 X’s are among the 200 Y’s” (Formal Logic, p. 58). He contrasts the definite particular with the indefinite particular which is of the form Some X’s are Y’s. It will be noticed that De Morgan’s definite particular, as here defined, is still more explicit than the numerically definite proposition, as defined in the text.

69. Indefinite Propositions.—According to quantity, propositions have by some logicians been divided into (1) Universal, (2) Particular, (3) Singular, (4) Indefinite. Singular propositions have already been discussed.

By an indefinite proposition is meant one “in which the quantity is not explicitly declared by one of the designatory terms all, every, some, many, &c.”; e.g., S is P, Cretans are liars. We may perhaps say with Hamilton, that indesignate would be a better term to employ. At any rate the so-called indefinite proposition is not the expression of a distinct form of judgment. It is a form of proposition which is the imperfect expression of a judgment. For reasons already stated, the particular has more claim to be regarded as an indefinite judgment.

When a proposition is given in the indesignate form, we can generally tell from our knowledge of the subject-matter or from the context whether it is meant to be universal or particular. Probably in the majority of cases indesignate propositions are intended to be understood as universals, e.g., “Comets are subject to the law of gravitation”; but if we are really in doubt with regard to the quantity of the proposition, it must logically be regarded as particular.[104]

[¹⁰⁴] In the Port Royal Logic a distinction is drawn between metaphysical universality and moral universality. “We call metaphysical universality that which is perfect and without exception; and moral universality that which admits of some exception, since in moral things it is sufficient that things are generally such” (Port Royal Logic, Professor Baynes’s translation, p. 150). The following are given as examples of moral universals: All women love to talk ; All young people are inconstant ; All old people praise past times. Indesignate propositions may almost without exception be regarded as universals either metaphysical or moral. But it seems clear that moral universals have in reality no valid claim to be called universals at all. Logically they ought not to be treated as more than particulars, or at any rate pluratives.

70. Multiple Quantification.—The application of a predicate to a subject is sometimes limited with reference to times or conditions, and this may be treated as yielding a secondary quantification of the proposition; for example, All men are 106 sometimes unhappy, In some countries all foreigners are unpopular. This differentiation may be carried further so as to yield triple or any higher order of quantification. Thus, we have triple quantification in the proposition, In all countries all foreigners are sometimes unpopular.[105]

[¹⁰⁵] For a further development of the notion of multiple quantification see Mr Johnson’s articles on The Logical Calculus in Mind, 1892.

In this way a proposition with a singular term for subject may, with reference to some secondary quantification, be classified as universal or particular as the case may be; for example, Gladstone is always eloquent, Browning is sometimes obscure.

71. Infinite or Limitative Propositions.—In place of the ordinary twofold division of propositions in respect of quality, Kant gave a threefold division, recognising a class of infinite (or limitative) judgments, which are neither affirmative nor negative. Thus, S is P being affirmative, and S is not P negative, S is not-P is spoken of as infinite or limitative.[106] It is, however, difficult to justify the separate recognition of this third class, whether we take the purely formal stand-point, or have regard to the real content of the propositions. From the formal stand-point we might substitute some other symbol, say Q, for not-P, and from this point of view Some S is not-P must be regarded as simply affirmative. On the other hand, Some S is not-P is equivalent in meaning to Some S is not P, and (assuming P to be a positive term) these two propositions must, having regard to their real content, be equally negative in force.

[¹⁰⁶] An infinite judgment, in the sense in which the term is here used, may be described as the affirmative predication of a negative. Some writers, however, include under propositiones infinitae those whose subject, as well as those whose predicate, is negative. Thus Father Clarke defines propositiones infinitae as propositions in which “the subject or predicate is indefinite in extent, being limited only in its exclusion from some definite class or idea: as, Not to advance is to recede” (Logic, p. 268).

Some writers go further and appear to deny that the so-called infinite judgment has any meaning at all. This point is closely connected with a question that we have already discussed, namely, whether the negative term not-P has any meaning. If we recognise the negative term—and we have endeavoured to 107 shew that we [ought] to do so—then the proposition S is not-P is equivalent to the proposition S is not P, and the former proposition must, therefore, have just as much meaning as the latter.

The question of the utility of so called infinite propositions has been further mixed up with the question as to the nature of significant denial. But it is better to keep the two questions distinct. Whatever the true character of denial may be, it is not dependent on the use of negative terms.

EXERCISES.

72. Determine the quality of each of the following propositions, and the distribution of its terms: (a) A few distinguished men have had undistinguished sons; (b) Few very distinguished men have had very distinguished sons; (c) Not a few distinguished men have had distinguished sons. [J.]

73. Examine the significance of few, a few, most, any, in the following propositions; Few artists are exempt from vanity; A few facts are better than a great deal of rhetoric; Most men are selfish; If any philosophers have been wise, Socrates and Plato must be numbered among them. [M.]

74. Everything is either X or Y ; X and Y are coextensive ; Only X is Y ; The class X comprises the class Y and something more. Express each of these statements by means of ordinary A, I, E, O categorical propositions. [C.]

75. Express each of the following statements in one or more of the forms recognised in the traditional scheme of categorical propositions: (i) No one can be rich and happy unless he is also temperate and prudent, and not always then; (ii) No child ever fails to be troublesome if ill taught and spoilt; (iii) It would be equally false to assert that the rich alone are happy, or that they alone are not. [V.]

76. Express, as nearly as you can, each of the following statements in the form of an ordinary categorical proposition, and determine its quality and the distribution of its terms:
(a) It cannot be maintained that pleasure is the sole good; 108
(b) The trade of a country does not always suffer, if its exports are hampered by foreign duties;
(c) The man who shews fear cannot be presumed to be guilty;
(d) One or other of the members of the committee must have divulged the secret. [C.]

77. Find the categorical propositions, expressed in terms of cases of Q or non-Q and of R or non-R, which are directly or indirectly implied by each of the following statements:
(a) The presence of Q is a necessary, but not a sufficient, condition for the presence of R ;
(b) The absence of Q is a necessary, but not a sufficient, condition for the presence of R ;
(c) The presence of Q is a necessary, but not a sufficient, condition for the absence of R.
In what respects, if any, does the categorical form fail to express the full significance of such propositions as the above? [J.]

78. “Honesty of purpose is perfectly compatible with blundering ignorance.”
“The affair might have turned out otherwise than it did.”
“It may be that Hamlet was not written by the actor known by his contemporaries as Shakespeare.”
Employ the above propositions to illustrate your views in regard to the modality of propositions; and examine the relations between each of the propositions and any assertoric proposition which may be taken to be its ground or to be partially equivalent to it. [C.]

CHAPTER III.

THE OPPOSITION OF PROPOSITIONS.[107]

[¹⁰⁷] This chapter will be mainly concerned with the opposition of categorical propositions; and, as regards categoricals, complications arising in connexion with their existential interpretation will for the present be postponed.

79. The Square of Opposition.—In dealing with the subject of this chapter it will be convenient to begin with the ancient square of opposition which relates exclusively to the traditional schedule of propositions. It will, however, ultimately be found desirable to give more general accounts of what is to be understood by the terms contradictory, contrary, &c., so that they may be adapted to other schedules of propositions.

Two propositions are technically said to be opposed to each other when they have the same subject and predicate respectively, but differ in quantity or quality or both.[108]

[¹⁰⁸] This definition, according to which opposed propositions are not necessarily incompatible with one another, is given by Aldrich (p. 53 in Mansel’s edition). Ueberweg (Logic, § 97) defines opposition in such a way as to include only contradiction and contrariety; and Mansel remarks that “subalterns are improperly classed as opposed propositions” (Aldrich, p. 59). Modern logicians, however, usually adopt Aldrich’s definition, and this seems on the whole the best course. Some term is wanted to signify the above general relation between propositions; and though it might be possible to find a more convenient term, no confusion is likely to result from the use of the term opposition if the student is careful to notice that it is here employed in a technical sense.

Taking the propositions SaP, SiP, SeP, SoP, in pairs, we find that there are four possible kinds of relation between them.

(1) The pair of propositions may be such that they can neither both be true nor both false. This is called contradictory opposition, and subsists between SaP and SoP, and between SeP and SiP. 110

(2) They may be such that whilst both cannot be true, both may be false. This is called contrary opposition. SaP and SeP.

(3) They may be such that they cannot both be false, but may both be true. Subcontrary opposition. SiP and SoP.

(4) From a given universal proposition, the truth of the particular having the same quality follows, but not vice versâ.[109] This is subaltern opposition, the universal being called the subalternant, and the particular the subalternate or subaltern. SaP and SiP. SeP and SoP.

[¹⁰⁹] This result and some of our other results may need to be modified when, later on, account is taken of the existential interpretation of propositions. But, as stated in the note at the beginning of the chapter, all complications resulting from considerations of this kind are for the present put on one side.

All the above relations are indicated in the ancient square of opposition.

The doctrine of opposition may be regarded from two different points of view, namely, as a relation between two given propositions; and, secondly, as a process of inference by which one proposition being given either as true or as false, the truth or falsity of certain other propositions may be determined. Taking the second of these points of view, we have the following table:— 111
A being given true, E is false, I true, O false ;
E being given true, A is false, I false, O true ;
I being given true, A is unknown, E false, O unknown;
O being given true, A is false, E unknown, I unknown;
A being given false, E is unknown, I unknown, O true ;
E being given false, A is unknown, I true, O unknown;
I being given false, A is false, E true, O true ;
O being given false, A is true, E false, I true.

80. Contradictory Opposition.—The doctrine of opposition in the preceding section is primarily applicable only to the fourfold schedule of propositions ordinarily recognised. We must, however, look at the question from a wider point of view. It is, in particular, important that we should understand clearly the nature of contradictory opposition whatever may be the schedule of propositions with which we are dealing.

The nature of significant denial will be considered in some detail in the concluding [section] of this chapter. At this point it will suffice to say that to deny the truth of a proposition is equivalent to affirming the truth of its contradictory ; and vice versâ. The criterion of contradictory opposition is that of the two propositions, one must be true and the other must be false ; they cannot be true together, but on the other hand no mean is possible between them. The relation between two contradictories is mutual; it does not matter which is given true or false, we know that the other is false or true accordingly. Every proposition has its contradictory, which may however be more or less complicated in form.

It will be found that attention is almost inevitably called to any ambiguity in a proposition when an attempt is made to determine its contradictory. It has been truly said that we can never fully understand the meaning of a proposition until we know precisely what it denies; and indeed the problem of the import of propositions sometimes resolves itself at least partly into the question how propositions of a given form are to be contradicted.

The nature of contradictory opposition may be illustrated by reference to a discussion entered into by Jevons (Studies in 112 Deductive Logic, p. 116) as to the precise meaning of the assertion that a proposition—say, All grasses are edible—is false. After raising this question, Jevons begins by giving an answer, which may be called the orthodox one, and which, in spite of what he goes on to say, must also be considered the correct one. When I assert that a proposition is false, I mean that its contradictory is true. The given proposition is of the form A, and its contradictory is the corresponding O proposition—Some grasses are not edible. When, therefore, I say that it is false that all grasses are edible, I mean that some grasses are not edible. Jevons, however, continues, “But it does not seem to have occurred to logicians in general to enquire how far similar relations could be detected in the case of disjunctive and other more complicated kinds of propositions. Take, for instance, the assertion that ‘all endogens are all parallel-leaved plants.’ If this be false, what is true? Apparently that one or more endogens are not parallel-leaved plants, or else that one or more parallel-leaved plants are not endogens. But it may also happen that no endogen is a parallel-leaved plant at all. There are three alternatives, and the simple falsity of the original does not shew which of the possible contradictories is true.”

This statement is open to criticism in two respects. In the first place, in saying that one or more endogens are not parallel-leaved plants, we do not mean to exclude the possibility that no endogen is a parallel-leaved plant at all. Symbolically, Some S is not P does not exclude No S is P. The three alternatives are, therefore, at any rate reduced to the two first given. But in the second place, it is incorrect to speak of either of these alternatives as being by itself a contradictory of the original proposition. The true contradictory is the affirmation of the truth of one or other of these alternatives. If the original proposition is false, we certainly know that the new proposition limiting us to such alternatives is true, and vice versâ.

The point at issue may be made clearer by taking the proposition in question in a symbolic form. All S is all P is a condensed expression, resolvable into the form, All S is P and 113 all P is S. It has but one contradictory, namely, Either some S is not P or some P is not S.[110] If either of these alternatives holds good, the original statement must in its entirety be false; and, on the other hand, if the latter is false, one at least of these alternatives must be true. Some S is not P is not by itself a contradictory of All S is all P. These two propositions are indeed inconsistent with one another; but they may both be false.

[¹¹⁰] The contradictory of All S is all P may indeed be expressed in a different form, namely, S and P are not coextensive, but this has precisely the same force as the contradictory given in the text. We go on to shew that two different forms of the contradictory of the same proposition must necessarily be equivalent to one another.

It follows that we must reject Jevons’s further statement that “a proposition of moderate complexity has an almost unlimited number of contradictory propositions, which are more or less in conflict with the original. The truth of any one or more of these contradictories establishes the falsity of the original, but the falsity of the original does not establish the truth of any one or more of its contradictories.”[111] No doubt a proposition which is complicated in form may yield an indefinite number of other non-equivalent propositions the truth of any one of which is inconsistent with its own. It will also be true that its contradictory can be expressed in more than one form. But these forms will necessarily be equivalent to one another, since it is impossible for a proposition to have two or more non-equivalent contradictories. This position may be formally established as follows. Let Q and R be both contradictories of P. They will be equivalent if it can 114 be shewn that if Q then R, and if R then Q. Since P and Q are contradictories, we have If Q then not P, and since P and R are contradictories we have If not P then R. Combining these two propositions we have the conclusion If Q then R. If R then Q follows similarly. Hence we have established the desired result.

[¹¹¹] It must be admitted that it has not been uncommon for logicians to use the word contradict somewhat loosely. For example, in the Port Royal Logic, we find the following: “Except the wise man (said the Stoics) all men are truly fools. This may be contradicted (1) by maintaining that the wise man of the Stoics was a fool as well as other men; (2) by maintaining that there were others, besides their wise man, who were not fools; (3) by affirming that the wise man of the Stoics was a fool, and that other men were not” (p. 140). The affirmation of any one of these three propositions certainly renders it necessary to deny the truth of the given proposition, but no one of them is by itself the contradictory of the given proposition. The true contradictory is the alternative proposition: Either the wise man of the Stoics is a fool or some other men are not fools.

In connexion with the same point, Jevons raises another question, in regard to which his view is also open to criticism. He says, “But the question arises whether there is not confusion of ideas in the usual treatment of this ancient doctrine of opposition, and whether a contradictory of a proposition is not any proposition which involves the falsity of the original, but is not the sole condition of it. I apprehend that any assertion is false which is made without sufficient grounds. It is false to assert that the hidden side of the moon is covered with mountains, not because we can prove the contradictory, but because we know that the assertor must have made the assertion without evidence. If a person ignorant of mathematics were to assert that ‘all involutes are transcendental curves,’ he would be making a false assertion, because, whether they are so or not, he cannot know it.” We should, however, involve ourselves in hopeless confusion were we to consider the truth or falsity of a proposition to depend upon the knowledge of the person affirming it, so that the same proposition would be now true, now false. It will be observed further that on Jevons’s view both the propositions S is P and S is not P would be false to a person quite ignorant of the nature of S. This would mean that we could not pass from the falsity of a proposition to the truth of its contradictory; and such a result as this would render any progress in thought impossible.

81. Contrary Opposition.—Seeking to generalise the relation between A and E, we might naturally be led to characterize the contrary of a given proposition by saying that it goes beyond mere denial, and sets up a further assertion as far as possible removed from the original assertion; so that, whilst the contradictory of a proposition denies its entire truth, its contrary may be said to assert its entire falsehood. A pair of contraries as thus defined may be regarded as standing at the opposite 115 ends of a scale on which there are a number of intermediate positions.

On this definition, however, the notion of contrariety cannot very satisfactorily be extended much beyond the particular case contemplated in the ordinary square of opposition. For if we have a proposition which cannot itself be regarded as standing at one end of a scale, but only as occupying an intermediate position, such proposition cannot be regarded as forming one of a pair of contraries. Plurative and numerically definite propositions may be taken as illustrations.

Hence if it is desired to define contrariety so that the conception may be generally applicable, the idea of two propositions standing, as it were, furthest apart from each other must be given up, and any two propositions may be described as contraries if they are inconsistent with one another without at the same time exhausting all possibilities. Contraries must on this definition always admit of a mean, but they may not always be what we should speak of as diametrical opposites, and any given proposition is not limited to a single contrary, but may have an indefinite number of non-equivalent contraries. At the same time, it will be observed that this definition still suffices to identify A and E as a pair of contraries, and as the only pair in the traditional scheme of opposition.

82. The Opposition of Singular Propositions.—Taking the proposition Socrates is wise, its contradictory is Socrates is not wise ;[112] and so long as we keep to the same terms, we cannot go beyond this simple denial. The proposition has, therefore, no formal contrary.[113] This opposition of singulars has been called secondary opposition (Mansel’s Aldrich, p. 56).

[¹¹²] This must be regarded as the correct contradictory from the point of view reached in the present chapter. The question becomes a little more difficult when the existential interpretation of propositions is taken into account.

[¹¹³] We can obtain what may be called a material contrary of the given proposition by making use of the contrary of the predicate instead of its mere contradictory; thus, Socrates has not a grain of sense. This is spoken of as material contrariety because it necessitates the introduction of a fresh term that could not be formally obtained out of the given proposition. It should be added that the distinction between formal and material contrariety might also be applied in the case of general propositions.

116 If, however, there is secondary quantification in a proposition having a singular subject, then we may obtain the ordinary square of opposition. Thus, if our original proposition is Socrates is always (or in all respects) wise, it is contradicted by the statement that Socrates is sometimes (or in some respects) not wise, while it has for its contrary, Socrates is never (or in no respects) wise, and for its subaltern, Socrates is sometimes (or in some respects) wise. It may be said that when we thus regard Socrates as having different characteristics at different times or under different conditions, our subject is not strictly singular, since it is no longer a whole indivisible. This is in a sense true, and we might no doubt replace our proposition by one having for its subject “the judgments or the acts of Socrates.” But it does not appear that this resolution of the proposition is necessary for its logical treatment.

The possibility of implicit secondary quantification, although no such quantification is explicitly indicated, is a not unfruitful source of fallacy in the employment of propositions having singular subjects. If we take such propositions as Browning is obscure, Epimenides is a liar, This flower is blue, and give as their contradictories Browning is not obscure, Epimenides is not a liar, This flower is not blue, shall we say that the original proposition or its contradictory is true in case Browning is sometimes (but not always) obscure, or in case Epimenides sometimes (but not often) speaks the truth, or in case the flower is partly (but not wholly) blue? There is certainly a considerable risk in such instances as these of confusing contradictory and contrary opposition, and this will be avoided if we make the secondary quantification of the propositions explicit at the outset by writing them in the form Browning is always (or sometimes) obscure, &c.[114] The contradictory will then be particular or universal accordingly.

[¹¹⁴] Or we might reduce them to the forms,—All (or some) of the poems of Browning are obscure, All (or some) of the statements of Epimenides are false, All (or some) of the surface of this flower is blue.

83. The Opposition of Modal Propositions.—So far in this chapter our attention has been confined to assertoric propositions. For the present, a very brief reference to the opposition 117 of modals will suffice. The main points involved will come up for further consideration [later on].

We have seen that the unconditionally universal proposition, whether expressed in the ordinary categorical form All S is P, or as a conditional If anything is S it is P, affirms a necessary connexion, by which is meant not merely that all the S’s are as a matter of fact P’s, but that it is inherent in their nature that they should be so. The statement that some S’s are not P’s is inconsistent with this proposition, but is not its contradictory, since both the propositions might be false: the S’s might all happen to be P’s, and yet there might be no law of connexion between S and P. The proposition in question being apodeictic will have for its contradictory a modal of another description, namely, a problematic proposition; and this may be written in the form S need not be P, or If anything is S still it need not be P, according as our original proposition is expressed as a categorical or as a conditional

Similarly, the contradictory of the hypothetical If P is true then Q is true, this proposition being interpreted modally, is If P is true still Q need not be true.

84. Extension of the Doctrine of Opposition.[115]—If we do not confine ourselves to the ordinary square of opposition, but consider any pair of propositions (whatever may be the schedule to which they belong), it becomes necessary to amplify the list of formal relations recognised in the square of opposition, and also to extend the meaning of certain terms. We may give the following classification:

[¹¹⁵] The illustrations given in this section presuppose a knowledge of immediate inferences. The section may accordingly on a first reading be postponed until part of the following chapter has been read.

(1) Two propositions may be equivalent or equipollent, each proposition being formally inferable from the other. Hence if either one of the propositions is true, the other is also true; and if either is false, the other is also false. For example, as will presently be shewn, All S is P and All not-P is not-S stand to each other in this relation.

(2) and (3) One of the two propositions may be formally inferable from the other, but not vice versâ. If we are 118 considering two given propositions Q and R, this yields two cases: for Q may carry with it the truth of R, but not conversely; or R may carry with it the truth of Q, but not conversely. Ordinary subaltern propositions with their subalternants fall into this class; and it will be convenient to extend the meaning of the term subaltern, so as to apply it to any pair of propositions thus related, whether they belong to the ordinary square of opposition or not. It will indeed be found that any pair of simple propositions of the forms A, E, I, O, that are subaltern in the extended sense, are equivalent to some pair that are subaltern in the more limited sense.[116] Thus All S is P and Some P is S, which are subaltern in the extended sense, are equivalent to All S is P and Some S is P. All S is P and Some not-S is not P are another pair of subalterns. Here it is not so immediately obvious in what direction we are to look for a pair of equivalent propositions belonging to the ordinary square of opposition. No not-P is S and Some not-P is not S will, however, be found to satisfy the required conditions.

[¹¹⁶] This will of course not hold good when we apply the term subaltern to compound propositions, e.g., to the pair Some S is not P and some P is not S, Some S is not P or some P is not S.

(4) The propositions may be such that they can both be true together, or both false, or either one true and the other false. For example, All S is P and All P is S. Such propositions may be called independent in their relation to one another.

(5) The propositions may be such that one or other of them must be true while both may be true. A pair of propositions which are thus related—for example, Some S is P and Some not-S is P—may, by an extension of meaning as in the case of the term subaltern, be said to be subcontrary. It can be shewn that any pair of subcontraries of the forms A, E, I, O are equivalent to some pair of subcontraries belonging to the ordinary square of opposition; thus, the above pair are equivalent to Some P is S and Some P is not S.

(6) The two propositions may be contrary to one another, in the sense that they cannot both be true, but can both be false. It can as before be shewn that any pair of contraries of 119 the forms A, E, I, O are equivalent to some pair of contraries in the more ordinary sense. For example, the contraries All S is P and All not-S is P are equivalent to No not-P is S and All not-P is S.

(7) The two propositions may be contradictory to one another according to the definition given in section [80], that is, they can neither both be true nor both false. All S is P and Some not-P is S afford an example outside the ordinary square of opposition. It will be observed that these two propositions are equivalent to the pair All S is P and Some S is not P.

Two propositions, then, may, in respect of inferability, consistency, or inconsistency, be formally (1) equivalent, (2) and (3) subaltern, (4) independent, (5) subcontrary, (6) contrary, (7) contradictory, the terms subaltern, &c., being used in the most extended sense. What pairs of categorical propositions (into which only the same terms or their contradictories enter) actually fall into these categories respectively will be shewn in sections [106] and [107].

These seven possible relations between propositions (taken in pairs) will be found to be precisely analogous to the seven possible relations between classes (taken in pairs) as brought out in a subsequent chapter (section [130]).

85. The Nature of Significant Denial.—It is desirable that, before concluding this chapter, we should briefly discuss a more fundamental question than any that has yet been raised, namely, the meaning and nature of negation and denial.

We observe, in the first place, that negation always finds expression in a judgment, and that it always involves the denial of some other judgment. The question therefore arises whether negation always presupposes an antecedent affirmation. This question must be answered in the negative if it is understood to mean that in order to be able to deny a proposition we must begin by regarding it as true. The proposition which we deny may be asserted or suggested by someone else; or it may occur to us as one of several possible alternatives; or it may be put in the form of a question.

It is, however, to be added that if a denial is to have any value as a statement of matter of fact, the corresponding 120 affirmation must be consistent with the meaning of the terms employed. Thus if A connotes m, n, p, and B connotes not-p, q, r, then the denial that A is B gives no real information respecting A. For the affirmation that A is B cannot be made by anyone who knows what is meant by A and B respectively. The same point may be otherwise expressed by saying that just as the affirmation of a verbal proposition is insignificant regarded as a real affirmation concerning the subject (and not merely as an affirmation concerning the meaning to be attached to the subject-term), so the denial of a contradiction in terms is insignificant from the same point of view. Such a denial yields merely what is tautologous and practically useless.

For example, the denial that the soul is a ship in full sail is insignificant regarded as a statement of matter of fact; for such denial gives no information to anyone who is already acquainted with the meaning of the terms involved.

The nature of logical negation is of so fundamental and ultimate a character that any attempt to explain it is apt to obscure rather than to illumine. It cannot be expressed more simply and clearly than by the laws of contradiction and excluded middle: a judgment and its contradictory cannot both be true; nor can they both be false.

Because every negative judgment involves the denial of some other judgment, it has been argued that a negative judgment such as S is not P is primarily a judgment concerning the positive judgment S is P, not concerning the subject S ; and hence that a negative judgment is not co-ordinate with a positive judgment, but dependent upon it.[117]

[¹¹⁷] Compare Sigwart, Logic, i. pp. 121, 2.

Passing by the point that a positive judgment also involves the denial of some other judgment, we may observe that a distinction must be drawn between “S is P” is not true (which is a judgment about S is P), and S is not P (which is a judgment about S). Denial no doubt presents itself to the mind most simply in the first of these two forms. But in contradicting a given judgment our method usually is to establish another judgment involving the same terms which stands to the given judgment in the relation expressed by the laws of contradiction 121 and excluded middle; and when we oppose the judgment S is not P to the judgment S is P we have reached the less direct mode of denial in which we have again a judgment concerning our original subject.

The example here taken tends perhaps to obscure the point at issue because the distinction between “S is P” is not true and S is not P may appear to be so slight as to be immaterial. That there is a real distinction will, however, appear clear if we take such pairs of propositions as “All S is P” is not true, Some S is not P ; “All S is all P” is not true, Either some S is not P or some P is not S ; “If any P is Q it is R” is not true, P might be Q without being R.

It will be convenient if in general we understand by the contradictory of a proposition P not its simple denial “P is not true,” but the proposition Q involving the same terms, which is formally so related to P, that P and Q cannot both be true or both false.

Sigwart observes that the ground of a denial may be either (a) a deficiency, or (b) an opposition.[118] I may, for example, pronounce that a certain thing does not possess a given attribute either (a) because I fail to discover the presence of the attribute, or (b) because I recognise the presence of some other attribute which I know to be incompatible with the one suggested.

[¹¹⁸] Logic, i. p. 127.

This distinction may be illustrated by one or two further examples. Thus, I may deny that a man travelled by a certain train either (a) because I searched the train through just before it started and found he was not there, or (b) because I know he was elsewhere when the train started,—I may, for instance, have seen him leave the station at the same moment in another train in the opposite direction. Similarly, I may deny a universal proposition either (a) because I have discovered certain instances of its not holding good, or (b) because I accept another universal proposition which is inconsistent with it. Again, I may deny that a given metal, or the metal contained in a certain salt, is copper (a) on the ground of deficiency, namely, that it does not answer to a certain test, or (b) on the ground 122 of opposition, namely, that I recognise it to be another metal, say, zinc.

The ground of denial always involves something positive, for example, the search through the train, or the discovery of individual exceptions. But it is clear that when we establish an opposition we get a result that is itself positive in a way that is not the case when we merely establish a deficiency. This may lead up to a brief examination of a doctrine of the nature of significant denial that is laid down by Mr Bosanquet.

Mr Bosanquet holds that bare denial has in itself no significance, and he apparently denies that the contradictory of a judgment, apart from the grounds on which it is based, conveys any information.[119] For the meaning of significant negation we must, he says, look to the grounds of the negation; or else for contradictory denial we must substitute contrary denial. As a consequence, a judgment can, strictly and properly, “only be denied by another judgment of the same nature; a singular by a singular judgment, a generic by a generic, a hypothetical by a hypothetical”;[120] and, presumably, a particular by a particular, an apodeictic by an apodeictic.

[¹¹⁹] Logic, i. p. 305.

[¹²⁰] Ibid, p. 383.

It is of course true that every denial must have some kind of positive basis, but it is also necessary that a judgment should be distinguished from the grounds on which it is based. We cannot say that a judgment of given content is different for two people because they accept it on different grounds; and if it is said that this is to beg the question, since a difference in ground constitutes in itself a difference in content, the reply is that such a doctrine must render the content of every judgment so elusive and uncertain as to make it impossible of analysis.

The view that identifies the denial of a judgment with its contrary not only mixes up a judgment with its grounds, but also overlooks one of the two principal grounds of denial. When the ground of negation is an opposition, we may no doubt be said to reach denial through the contrary, though we should still hold that the denial is in itself something less than the contrary; but when the ground of denial is a deficiency, even this cannot be allowed. If, for example, I have arrived 123 at the conclusion that a man did not start by a given train because I searched the train through before its departure and did not find him there; or if I conclude that a given metal is not copper because it does not satisfy a given test; I have obtained no contrary judgment, and yet my denial is justified.

These would be cases of bare denial. I have gained no positive knowledge of the whereabouts of the man in question, nor can I identify the given metal. But surely it cannot be seriously maintained that the denial is meaningless or useless, say, to a detective in the first instance, or to an analytical chemist in the second.

Of course we seldom or never rest content with bare denial. The contrary rather than the contradictory represents our ultimate aim. But it is often the case that, temporarily at any rate, we cannot get beyond bare denial; and we ought not to consider that we have altogether failed to make progress when all that we have achieved is the exclusion of a possible alternative or the overthrow of a false theory. Recent researches, for example, into the origin of cancer have led to no positive results; but it is claimed for them that by destroying preconceived ideas on the subject they have cleared the way for future advance. Will anyone affirm that this was not worth doing or that the time spent on the researches was wasted?

Looking at the question from another point of view, it is surely absurd to say that we cannot deny a universal unless we are able to substitute another universal in its place. Various algebraical formulae have from time to time been suggested as necessarily yielding a prime number. They have all been overthrown, and no valid formula has been established in their place. But knowledge that these formulae are false is not quite appropriately described as ignorance.

Elsewhere Mr Bosanquet says that mere enumerative exceptions are futile and cannot touch the essence of the unconditionally universal judgments they apparently oppose.[121] He appears to have in view cases where nothing more than some modification of the original judgment is shewn to be 124 necessary. But even so the enumerative exceptions have overthrown the original judgment. No doubt a scientific law which has had a great amount of evidence in its favour is likely to contain elements of truth even if it is not altogether true; and the object of a man of science who overthrows a law will be to set up some other law in its place. But, says Mr Bosanquet, even if the first generic judgment were a sheer blunder and confusion, as has been the case from time to time with judgments propounded in science, it is scarcely possible to rectify the confusion except by substituting for it the true positive conceptions that arise out of the cases which overthrew it.” Here it is admitted that the exceptions do overthrow the law, and the rest of the argument is surely an instance of ignoratio elenchi. It is moreover a pure, and in many cases an unjustifiable, assumption that the cases which suffice to overthrow a false law will also suffice as the basis for the establishment of a true law in its place.

[¹²¹] Logic, i. p. 313.

EXERCISES.

86. Examine the nature of the opposition between each pair of the following propositions:—None but Liberals voted against the motion; Amongst those who voted against the motion were some Liberals; It is untrue that those who voted against the motion were all Liberals. [K.]

87. If some were used in its ordinary colloquial sense, how would the scheme of opposition between propositions have to be modified? [J.]

88. Explain the technical terms “contradictory” and “contrary” applying them to the following propositions: Few S are P ; He was not the only one who cheated ; Two-thirds of the army are abroad. [V.]

89. Give the contradictory of each of the following propositions:—Some but not all S is P ; All S is P and some P is not R ; Either all S is P or some P is not R ; Wherever the property A is found, either the property B or the property C will be found with it, but not both of them together. [K.]

125 90. Give the contradictory, and also a contrary, of each of the following propositions:
Half the candidates failed;
Wellington was always successful both in beating the enemy and in utilising his victory;
All men are either not knaves or not fools;
All but he had fled;
Few of them are honest;
Sometimes all our efforts fail;
Some of our efforts always fail. [L.]

91. Give the contradictory, and also a contrary, of each of the following propositions:
I am certain you are wrong;
Sometimes when it rains I find myself without an umbrella;
Whatever you say, I shall not believe you. [C.]

92. Define the terms subaltern, subcontrary, contrary, contradictory, in such a way that they may be applicable to pairs of propositions generally, and not merely to those included in the ordinary square of opposition. Do the above exhaust the formal relations (in respect of inferability, consistency, or inconsistency) that are possible between pairs of propositions?
Illustrate your answer by considering the relation (in respect of inferability, consistency, or inconsistency) between each of the following propositions and each of the remainder: S and P are coincident ; Some S is P ; Not all S is P ; Either some S is not P or some P is not S ; Anything that is not P is S. [K.]

93. Given that the propositions X and Z are contradictory, Y and V contradictory, and X and Y contrary, shew (without assuming that X, Y, V, Z belong to the ordinary schedule of propositions) that the relations of V to X, Z to Y, V to Z are thereby deducible. [J.]

94. Prove formally that if two propositions are equivalent, their contradictories will also be equivalent. [K.]

95. Examine the doctrine that a judgment can properly be denied only by another judgment of the same type. Illustrate by reference to (a) universal judgments, (b) particular judgments (c) disjunctive judgments, (d) apodeictic judgments. [K.]

CHAPTER IV.

IMMEDIATE INFERENCES.[122]

[¹²²] In this chapter we concern ourselves mainly with the traditional scheme of propositions, and except where an explicit statement is made to the contrary we proceed on the assumption that each class represented by a simple term exists in the universe of discourse, while at the same time it does not exhaust that universe. This assumption appears to have been made implicitly in the traditional treatment of logic.

96. The Conversion of Categorical Propositions.—By conversion, in a broad sense, is meant a change in the position of the terms of a proposition.[123] Logic, however, is concerned with conversion only in so far as the truth of the new proposition obtained by the process is a legitimate inference from the truth of the original proposition. For example, the change from All S is P to All P is S is not a legitimate logical conversion, since the truth of the latter proposition does not follow from the truth of the former. In other words, logical conversion is a case of immediate inference, which may be defined as the inference of a proposition from a single other proposition.[124]

[¹²³] Ueberweg (Logic, § 84) defines conversion thus. Compare also De Morgan, Formal Logic, p. 58. In geometry, all equiangular triangles are equilateral would be regarded as the converse of all equilateral triangles are equiangular. In this sense of the term conversion, which is its ordinary non-technical sense, we may say—as we frequently do say—“Yes, such and such a proposition is true; but its converse is not true.”

[¹²⁴] In discussing immediate inferences we “pursue the content of an enunciated judgment into its relations to judgments not yet uttered” (Lotze). Instead of “immediate inferences” Professor Bain prefers to speak of “equivalent propositional forms.” It will be found, however, that the new propositions obtained by immediate inference are not always equivalent to the original proposition, e.g., in conversion per accidens. Miss Jones suggests the term eduction as a synonym for immediate inference (General Logic, p. 79); and she then distinguishes between eversions and transversions, an eversion being an eduction from categorical form to categorical, or from hypothetical to hypothetical, &c., and transversion an eduction from categorical form to conditional, or from conditional to categorical, &c. For the present we shall be concerned with eversions only.

127 The simplest form of logical conversion, and that which is understood in logic when we speak of conversion without further qualification, may be defined as a process of immediate inference in which from a given proposition we infer another, having the predicate of the original proposition for subject, and its subject for predicate. Thus, given a proposition having S for its subject and P for its predicate, our object in the process of conversion is to obtain by immediate inference a new proposition having P for its subject and S for its predicate. The original proposition may be called the convertend, and the inferred proposition the converse.

The process will be valid if the two following rules are observed:
(1) The converse must be the same in quality as the convertend (Rule of Quality);
(2) No term must be distributed in the converse unless it was distributed in the convertend (Rule of Distribution).

Applying these rules to the four fundamental forms of proposition, we have the following table:—

It is desirable at this stage briefly to call attention to a point which will receive fuller consideration later on in connexion with the reading of propositions in extension and intension, namely, that, generally speaking, in any judgment we have naturally before the mind the objects denoted by the 128 subject, but the qualities connoted by the predicate. In the process of converting a proposition, however, the extensive force of the predicate is made prominent, and an import is given to the predicate similar to that of the subject. At the same time the distribution of the predicate has to be made explicit in thought. It is in passing from the predicative to the class reading (e.g. from all men are mortal to all men are mortals), that the difficulty sometimes found in correctly converting propositions probably consists. We shall at any rate do well to recognise that conversion and other immediate inferences usually involve a distinct mental act of the above nature.

It follows from what has been said above that some propositions lend themselves to the process of conversion much more readily than others. When the predicate of a proposition is a substantive little or no effort is required in order to convert the proposition; more effort is necessary when the predicate is an adjective; and still more when in the original proposition the logical predicate is not expressed separately at all, as in propositions secundi adjacentis. Compare for purposes of conversion the propositions, Whales are mammals, Lions are carnivorous, A stitch in time saves nine. In some cases, in consequence of the awkwardness of changing adjectives and verbal predicates into substantives, the conversion of a proposition appears to be a very artificial production.[125]

[¹²⁵] Compare Sigwart, Logic, i. p. 340.

97. Simple Conversion and Conversion per accidens.—It will be observed that in the case of I and E, the converse is of the same form as the original proposition; moreover we do not lose any part of the information given us by the convertend, and we can pass back to it by re-conversion of the converse. The convertend and its converse are accordingly equivalent propositions. The conversion under these conditions is said to be simple.

In the case of A, it is different; we cannot pass by immediate inference from All S is P to All P is S, inasmuch as P is distributed in the latter of these propositions but undistributed in the former. Hence, although we start with a universal proposition, we obtain by conversion a particular 129 proposition only,[126] and by no means of operating upon the converse can we regain the original proposition. The convertend and its converse are accordingly non-equivalent propositions. The conversion in this case is called conversion per accidens,[127] or conversion by limitation.[128]

[¹²⁶] The failure to recognise or to remember that universal affirmative propositions are not simply convertible is a fertile source of fallacy.

[¹²⁷] The conversion of A is said by Mansel to be called conversion per accidens ‘because it is not a conversion of the universal per se, but by reason of its containing the particular. For the proposition ‘Some B is A’ is primarily the converse of ‘Some A is B,’ secondarily of ‘All A is B’” (Mansel’s Aldrich, p. 61). Professor Baynes seems to deny that this is the correct explanation of the use of the term (New Analytic of Logical Forms, p. 29); but however this may be, we certainly need not regard the converse of A as necessarily obtained through its subaltern. It is possible to proceed directly from All A is B to Some B is A without the intervention of Some A is B.

[¹²⁸] Simple conversion and conversion per accidens are also called respectively conversio pura and conversio impura. Compare Lotze, Logic, § 79.

For concrete illustrations of the process of conversion we may take the propositions,—A stitch in time saves nine; None but the brave deserve the fair. The first of these may be written in the form,—All stitches in time are things that save nine stitches. This, being an A proposition, is only convertible per accidens, and we have for our converse,—Some things that save nine stitches are stitches in time. The second of the given propositions may be written,—No one who is not brave is deserving of the fair. This, being an E proposition, may be converted simply, giving, No one deserving of the fair is not brave. Our results may be expressed in a more natural form as follows: One way of saving nine stitches is by a stitch in time; No one deserving of the fair can fail to be brave.

No difficulty ought ever to be found in converting or performing other immediate inferences upon any given proposition when once it has been brought into the traditional logical form, its quantity and quality being determined, its subject, copula, and predicate being definitely distinguished from one another, and its predicate as well as its subject being read in extension. If, however, this rule is neglected, mistakes are pretty sure to follow.

130 98. Inconvertibility of Particular Negative Propositions.—It follows immediately from the rules of conversion given in section [96] that Some S is not P does not admit of ordinary conversion; for S which is undistributed in the convertend would become the predicate of a negative proposition in the converse, and would therefore be distributed.[129] It will be shewn [presently], however, that although we are unable to infer anything about P in this case, we are able to draw an inference concerning not-P.

[¹²⁹] As regards the inconvertibility of O see also sections [99] and [126].

Jevons considers that the fact that the particular negative proposition is incapable of ordinary conversion “constitutes a blot in the ancient logic” (Studies in Deductive Logic, p. 37). There is, however, no sufficient justification for this criticism. We shall find subsequently that just as much can be inferred from the particular negative as from the particular affirmative (since the latter unlike the former does not admit of contraposition). No logic, symbolic or other, can actually obtain more from the given information than the ancient logic does. It has been suggested that what Jevons means is that the inconvertibility of O results in a want of symmetry and that logicians ought specially to aim at symmetry. With this last contention we may heartily agree. The want of symmetry, however, in the case before us is apparent only and results from taking an incomplete view. It will be found that symmetry reappears later on.[130]

[¹³⁰] See sections [105], [106].

99. Legitimacy of Conversion.—Aristotle proves the conversion of E indirectly, as follows;[131] No S is P, therefore, No P is S ; for if not, Some individual P, say Q, is S ; and hence Q is both S and P ; but this is inconsistent with the original proposition.

[¹³¹] “By the method called ἔκθεσις, i.e., by the exhibition of an individual instance.” See Mansel’s Aldrich, pp. 61, 2.

Having shewn that the simple conversion of E is legitimate, we can prove that the conversion per accidens of A is also legitimate. All S is P, therefore, Some P is S ; for, if not, No P is S, and therefore (by conversion) No S is P ; but this 131 is inconsistent with the original supposition. The legitimacy of the simple conversion of I follows similarly.

The above proof appears to involve nothing beyond the principles of contradiction and excluded middle. The proof itself, however, is not satisfactory; for it practically assumes the validity of the very process that it seeks to justify, that is to say, it assumes the equivalence of the propositions S is Q and Q is S.

A better justification of the process of conversion may be obtained by considering the class relations involved in the propositions concerned. Thus, taking an E proposition, it is self-evident that if one class is entirely excluded from another class, this second class is entirely excluded from the first.[132] In the case of an A proposition it is clear on reflection that the statement All S is P is consistent with either of two relations of the classes S and P, namely, S and P coincident, or P containing S and more besides, and further that these are the only two possible relations with which it is consistent. It is self-evident that in each of these cases Some P is S ; and hence the inference by conversion from an A proposition is shewn to be justified.[133] In the case of an O proposition, if we consider all the relationships of classes in which it holds good, we find that nothing is true of P in terms of S in all of them. Hence O is inconvertible.[134] The inconvertibility of O can also be established 132 by shewing that Some S is not P is compatible with every one of the following propositions—All P is S, Some P is S, No P is S, Some P is not S.

[¹³²] It is impossible to agree with Professor Bain, who would establish the rules of conversion by a kind of inductive proof. He writes as follows:—“When we examine carefully the various processes in Logic, we find them to be material to the very core. Take Conversion. How do we know that, if No X is Y, No Y is X? By examining cases in detail, and finding the equivalence to be true. Obvious as the inference seems on the mere formal ground, we do not content ourselves with the formal aspect. If we did, we should be as likely to say, All X is Y gives All Y is X ; we are prevented from this leap merely by the examination of cases” (Logic, Deduction, p. 251). But no one would on reflection maintain it to be self-evident that the simple conversion of A is legitimate; for when the case is put to us we recognise immediately that the contradictory of All P is S is compatible with All S is P. On the other hand, no one can deny that in the case of E the legitimacy of the process of conversion is self-evident.

[¹³³] Compare section [126], where this and other similar inferences are illustrated by the aid of the Eulerian diagrams.

[¹³⁴] Again, compare section [126].

100. Table of Propositions connecting any two terms.—There are—connecting any two terms S and P—eight propositions of the forms A, E, I, O, namely, four with S as subject, and four with P as subject. The results at which we have arrived concerning the conversion of propositions shew that of these eight, the two E propositions are equivalent to one another, and that the same is true of the two I propositions, E and I being simply convertible; also that these are the only equivalences obtainable. We have, therefore, the following table of propositions connecting any two terms S and P:—

The pair of propositions SaP and PaS are independent (see section [84]); and the same is true of the pairs SoP and PoS, SaP and PoS, PaS and SoP. The first pair taken together indicate that the classes S and P are coextensive, and they may be called complementary propositions. The second pair taken together indicate that the classes S and P are neither coextensive nor either included within the other; they may be called sub-complementary propositions. The third pair taken together indicate that the class S is included within the class P but that it does not exhaust that class; they may be called contra-complementary propositions. The fourth pair taken together indicate that the class P is included within the class S but that it does not exhaust that class; they are, therefore, also contra-complementary.[135]

[¹³⁵] The new technical terms here introduced have been suggested by Mr Johnson.

The above table will be supplemented in section [106] by a table of propositions connecting any two terms and their 133 contradictories, S, P, not-S, not-P. It will then be found that we have a symmetry that is at present wanting.

101. The Obversion of Categorical Propositions.[136]—Obversion is a process of immediate inference in which the inferred proposition (or obverse), whilst retaining the original subject, has for its predicate the contradictory of the predicate of the original proposition (or obvertend). This process is legitimate for a proposition of any form if at the same time the quality of the proposition is changed. The inferred proposition is, moreover, in all cases equivalent to the original proposition, so that we can always pass back from the obverse to the obvertend.

[¹³⁶] The process of immediate inference discussed in this section has been called by a good many different names. The term obversion, which is used by Professor Bain, is the most convenient. Other names which have been used are permutation (Fowler), aequipollence (Ueberweg), infinitation (Bowen), immediate inference by private conception (Jevons), contraversion (De Morgan), contraposition (Spalding). Professor Bain distinguishes between formal obversion and material obversion. By formal obversion is meant the kind of obversion discussed in the above section, and this is the only kind of obversion that can properly be recognised by the formal logician. Material obversion is described as the process of making “obverse inferences which are justified only on an examination of the matter of the proposition” (Logic, vol. i., p. 111); and the following are given as examples—“Warmth is agreeable; therefore, cold is disagreeable. War is productive of evil; therefore, peace is productive of good. Knowledge is good; therefore, ignorance is bad.” It is very doubtful if these are legitimate inferences, formal or otherwise. The conclusions appear to require quite independent investigations to establish them. Apart from this, however, it is a mistake to regard the process as analogous to formal obversion. In the latter, the inferred proposition has the same subject as the original proposition, whilst its quality is different; but neither of these conditions is fulfilled in the above examples. The process is really more akin to the immediate inference presently to be discussed under the name of inversion.

We have the following table:—

134 It will be observed that the obversion of All S is P depends upon the principle of contradiction, which tells us that if anything is P then it is not not-P; but that we pass back from No S is not-P to All S is P by the principle of excluded middle, which tells us that if anything is not not-P then it is P. The remaining inferences by obversion also depend upon one or other of these two principles.

102. The Contraposition of Categorical Propositions.[137]—Contraposition may be defined as a process of immediate inference in which from a given proposition another proposition is inferred having for its subject the contradictory of the original predicate. Thus, given a proposition having S for its subject and P for its predicate, we seek to obtain by immediate inference a new proposition having not-P for its subject.

[¹³⁷] This form of immediate inference is called by some logicians conversion by negation ; Miss Jones suggests the name contraversion. More strictly we might speak of conversion by contraposition. The word contrapositive was used by Boethius for the opposite of a term (e.g., not-A), the word contradictory being confined to propositional forms; and the passage from All S is P to All not-P is not-S was called Conversio per contrapositionem terminorum. In this usage Boethius was followed by the medieval logicians. Compare Minto, Logic, pp. 151, 153.

It will be observed that in the above definition it is left an open question whether the contrapositive of a proposition has the original subject or the contradictory of the original subject for its predicate; and every proposition which admits of contraposition will accordingly have two contrapositives, each of which is the obverse of the other. For example, in the case of All S is P there are the two forms No not-P is S and All not-P is not-S. For many purposes the distinction may be practically neglected without risk of confusion. It will be observed, however, that when not-S is taken as the predicate of the contrapositive, the quality of the original proposition is preserved and there is greater symmetry.[138] On the other hand, 135 if we regard contraposition as compounded out of obversion and conversion in the manner indicated in the following paragraph, the form with S as predicate is the more readily obtained. Perhaps the best solution (in cases in which it is necessary to mark the distinction) is to speak of the form with not-S as predicate as the full contrapositive, and the form with S as predicate as the partial contrapositive.[139]

[¹³⁸] The following is from Mansel’s Aldrich, p. 61,—“Conversion by contraposition, which is not employed by Aristotle, is given by Boethius in his first book, De Syllogismo Categorico. He is followed by Petrus Hispanus. It should be observed, that the old logicians, following Boethius, maintain that in conversion by contraposition, as well as in the others, the quality should remain unchanged. Consequently the converse of ‘All A is B’ is ‘All not-B is not-A,’ and of ‘Some A is not B,’ ‘Some not-B is not not-A.’ It is simpler, however, to convert A into E, and O into I, (‘No not-B is A,’ ‘Some not-B is A’), as is done by Wallis and Archbishop Whately; and before Boethius by Apuleius and Capella, who notice the conversion, but do not give it a name. The principle of this conversion may be found in Aristotle, Top. II. 8. 1, though he does not employ it for logical purposes.”

[¹³⁹] In previous editions the form with S as predicate was called the contrapositive, and the form with not-S as predicate was called the obverted contrapositive.

The following rule may be adopted for obtaining the full contrapositive of a given proposition:—Obvert the original proposition, then convert the proposition thus obtained, and then once more obvert. For given a proposition with S as subject and P as predicate, obversion will yield an equivalent proposition with S as subject and not-P as predicate; the conversion of this will make not-P the subject and S the predicate; and a repetition of the process of obversion will yield a proposition with not-P as subject and not-S as predicate.

Applying this rule, we have the following table:—

It will be observed that in the case of A and O, the contrapositive is equivalent to the original proposition, the quantity 136 being unchanged, whereas in the case of E we pass from a universal to a particular.[140] In order to emphasize this difference, and following the analogy of ordinary conversion, the contraposition of A and O has been called simple contraposition, and that of E contraposition per accidens.[141]

[¹⁴⁰] In most text-books, no definition of contraposition is given at all, and it may be pointed out that, in the attempt to generalise from special examples, Jevons in his Elementary Lessons in Logic involves himself in difficulties. For the contrapositive of A he gives All not-P is not-S ; O he says has no contrapositive (but only a converse by negation, Some not-P is S); and for the contrapositive of E he gives No P is S. It is impossible to discover any definition of contraposition that can yield these results. Assuming that in contraposition the quality of the proposition is to remain unchanged as in Jevons’s contrapositive of A, then the contrapositive of both E and O is Some not-P is not not-S.

[¹⁴¹] Compare Ueberweg, Logic, § 90.

That I has no contrapositive follows from the inconvertibility of O. For when Some S is P is obverted it becomes a particular negative, and the conversion of this proposition would be necessary in order to render the contraposition of the original proposition possible.

As regards the utility of the investigation as to the inferences that can be drawn from given propositions by the aid of contraposition, De Morgan[142] points out that the recognition that Every not-P is not-S follows from Every S is P, whatever S and P may stand for, renders unnecessary the special proofs that Euclid gives of certain of his theorems.[143]

[¹⁴²] Syllabus of Logic, p. 32.

[¹⁴³] It will be found that, taking Euclid’s first book, proposition 6 is obtainable by contraposition from proposition 18, and 19 from 5 and 18 combined; or that 5 can be obtained by contraposition from 19, and 18 from 6 and 19. Similar relations subsist between propositions 4, 8, 24, and 25; and, again, between axiom 12 and propositions 16, 28, and 29. Other examples might be taken from Euclid’s later books. In some of the cases the logical relations in which the propositions stand to one another are obvious; in other cases some supplementary steps are necessary.

In consequence of his dislike of negative terms Sigwart regards the passage from All S is P to No not-P is S as an artificial perversion. But he recognises the value of the inference from If anything is S it is P to If anything is not P it is not S. This distinction seems to be little more than verbal. It is to 137 be observed that we can avoid the use of negative terms without having recourse to the conditional form of proposition: for example, Whatever is S is P, therefore, Whatever is not P is not S ; Anything that is S is P, therefore, Anything that is not P is not S.

103. The Inversion of Categorical Propositions.—In discussing conversion and contraposition we have enquired in what cases it is possible, having given a proposition with S as subject and P as predicate, to infer (a) a proposition with P as subject, (b) a proposition with not-P as subject. We may now enquire further in what cases it is possible to infer (c) a proposition with not-S as subject.

If such a proposition can be inferred at all, it will be obtainable by a certain combination of the more elementary processes of ordinary conversion and obversion.[144] We will, therefore, take each of the fundamental forms of proposition and see what can be inferred (1) by first converting it, and then performing alternately the operations of obversion and conversion; (2) by first obverting it, and then performing alternately the operations of conversion and obversion. It will be found that in each case the process can be continued until a particular negative proposition is reached whose turn it is to be converted.

[¹⁴⁴] It might also be obtained directly; by the aid, for example, of Euler’s circles. See the following [chapter].

(1) The results of performing alternately the processes of conversion and obversion, commencing with the former, are as follows:—
(i) All S is P,
therefore (by conversion), Some P is S,
therefore (by obversion), Some P is not not-S.
Here comes the turn for conversion; but as we have to deal with an O proposition, we can proceed no further.

(ii) Some S is P,
therefore (by conversion), Some P is S,
therefore (by obversion), Some P is not not-S ;
and again we can go no further. 138

(iii) No S is P,
therefore (by conversion), No P is S,
therefore (by obversion), All P is not-S,
therefore (by conversion), Some not-S is P,
therefore (by obversion), Some not-S is not not-P.
In this case either of the propositions in italics is the immediate inference that was sought.

(iv) Some S is not P.
In this case we are not able even to commence our series of operations.

(2) The results of performing alternately the processes of conversion and obversion, commencing with the latter, are as follows:—
(i) All S is P,
therefore (by obversion), No S is not-P,
therefore (by conversion), No not-P is S,
therefore (by obversion), All not-P is not-S,
therefore (by conversion), Some not-S is not-P,
therefore (by obversion), Some not-S is not P.
Here again we have obtained the desired form.

(ii) Some S is P,
therefore (by obversion), Some S is not not-P.

(iii) No S is P,
therefore (by obversion), All S is not-P,
therefore (by conversion), Some not-P is S,
therefore (by obversion), Some not-P is not not-S.

(iv)  Some S is not P,
therefore (by obversion), Some S is not-P,
therefore (by conversion), Some not-P is S,
therefore (by obversion), Some not-P is not not-S.

We can now answer the question with which we commenced this enquiry. The required proposition can be obtained only if the given proposition is universal; we then have, according as it is affirmative or negative,—
All S is P, therefore, Some not-S is not P (= Some not-S is not-P); 139
No S is P, therefore, Some not-S is P (= Some not-S is not not-P).

This form of immediate inference has been more or less casually recognised by various logicians, without receiving any distinctive name. Sometimes it has been vaguely classed under contraposition (compare Jevons, Elementary Lessons in Logic, pp. 185, 6), but it is really as far removed from the process to which that designation has been given as the latter is from ordinary conversion. The term inversion was suggested in an earlier edition of this work, and has since been adopted by some other writers. Inversion may be defined as a process of immediate inference in which from a given proposition another proposition is inferred having for its subject the contradictory of the original subject. Thus, given a proposition with S as subject and P as predicate, we obtain by inversion a new proposition with not-S as subject. The original proposition may be called the invertend, and the inferred proposition the inverse.

In the above definition it is not specified whether the inverse is to have for its predicate P or not-P. Hence two forms (each being the obverse of the other) have been obtained as in the case of contraposition. So far as it is necessary to mark the distinction, we may speak of the form in which P is the predicate as the partial inverse, and of that in which not-P is the predicate as the full inverse.

104. The Validity of Inversion.—It will be remembered that we are at present working on the assumption that each class represented by a simple term exists in the universe of discourse, while at the same time it does not exhaust that universe; in other words, we assume that S, not-S, P, not-P, all represent existing classes. This assumption is perhaps specially important in the case of inversion, and it is connected with certain difficulties that may have already occurred to the reader. In passing from All S is P to its inverse Some not-S is not P there is an apparent illicit process, which it is not quite easy either to account for or explain away. For the term P, which is undistributed in the premiss, is distributed in the conclusion, and yet if the universal validity of obversion and 140 conversion is granted, it is impossible to detect any flaw in the argument by which the conclusion is reached. It is in the assumption of the existence of the contradictory of the original predicate that an explanation of the apparent anomaly may be found. That assumption may be expressed in the form Some things are not P. The conclusion Some not-S is not P may accordingly be regarded as based on this premiss combined with the explicit premiss All S is P ; and it will be observed that, in the additional premiss, P is distributed.[145]

[¹⁴⁵] The question of the validity of inversion under other assumptions will be considered in [chapter 8].

105. Summary of Results.—The results obtained in the preceding sections are summed up in the following table:—

[¹⁴⁶] In previous editions what are here called the partial contrapositive and the full contrapositive respectively were called the contrapositive and the obverted contrapositive; and what are here called the partial inverse and the full inverse were called the inverse and the obverted inverse.

It may be pointed out that the following rules apply to all the above immediate inferences:— 141
Rule of Quality.—The total number of negatives admitted or omitted in subject, predicate, or copula must be even.
Rules of Quantity.—If the new subject is S, the quantity may remain unchanged; if Sʹ, the quantity must be depressed;[147] if P, the quantity must be depressed in A and O; if Pʹ, the quantity must be depressed in E and I.

[¹⁴⁷] In speaking of the quantity as depressed, it is meant that a universal yields a particular, and a particular yields nothing.

106. Table of Propositions connecting any two terms and their contradictories.—Taking any two terms and their contradictories, S, P, not-S, not-P, and combining them in pairs, we obtain thirty-two propositions of the forms A, E, I, O. The following table, however, shews that only eight of these thirty-two propositions are non-equivalent.

In this table, columns (i) and (ii) contain the propositions in which S or Sʹ is subject, and columns (iii) and (iv) the propositions in which P or Pʹ is subject. In columns (i) and (iv) we have the forms which admit of simple contraposition (i.e., A and O), and in columns (ii) and (iii) those which admit of simple conversion (i.e., E and I). Contradictories are shewn by identical places in the universal and particular rows. We pass from column (i) to column (ii) by obversion; from column (ii) to column (iii) by simple conversion; and from column (iii) to column (iv) by obversion.

The forms in black type shew that we may take for our 142 eight non-equivalent propositions the four propositions connecting S and P, and a similar set connecting not-S and not-P.[148] To establish their non-equivalence we may proceed as follows: SaP and SeP are already known to be non-equivalent, and the same is true of SʹaPʹ and SʹePʹ ; but no universal proposition can yield a universal inverse; therefore, no one of these four propositions is equivalent to any other. Again, SiP and SoP are already known to be non-equivalent, and the same is true of SʹiPʹ and SʹoPʹ ; but no particular proposition has any inverse; therefore, no one of these propositions is equivalent to any other. Finally, no universal proposition can be equivalent to a particular proposition.[149]

[¹⁴⁸] The former set being denoted by A, E, I, O, the latter set may be denoted by Aʹ, Eʹ, Iʹ, Oʹ.

[¹⁴⁹] Mrs Ladd Franklin, in an article on The Proposition in Baldwin’s Dictionary of Philosophy and Psychology, reaches the result arrived at in this section from a different point of view. Mrs Franklin shews that, if we express everything that can be said in the form of existential propositions (that is, propositions affirming or denying existence), it is at once evident that the actual number of different statements possible in terms of X and Y and their contradictories x and y is eight. For the combinations of X and Y and their contradictories are XY, Xy, xY, xy, and we can affirm each of these combinations to exist or to be non-existent. Hence it is clear that eight different statements of fact are possible, and that these eight must remain different, no matter what the form in which they may be expressed.

It may be worth adding that the conditional and disjunctive forms as well as the categorical may here be included on the understanding that all the propositions are interpreted assertorically. Thus, the four following propositions are, on the above understanding, equivalent to one another: All X is Y (categorical); If anything is X, it is Y (conditional); Nothing is Xy (existential); Everything is x or Y (disjunctive).

107. Mutual Relations of the non-equivalent Propositions connecting any two terms and their contradictories.[150]—We may now investigate the mutual relations of our eight non-equivalent propositions. SaP, SeP, SiP, SoP form an ordinary square of opposition; and so do SʹaPʹ, SʹePʹ, SʹiPʹ, SʹoPʹ. Reference to columns (iii) and (iv) in the table will shew further that SaP, SʹePʹ, SʹiPʹ, SoP are equivalent to another square of opposition; and that the same is true of SʹaPʹ, SeP, SiP, SʹoPʹ. This leaves only the following pairs unaccounted for: 143 SaP, SʹaPʹ ; SeP, SʹePʹ ; SoP, SʹoPʹ ; SiP, SʹiPʹ ; SaP, SʹoPʹ ; SʹaPʹ, SoP ; SeP, SʹiPʹ ; SʹePʹ, SiP ; and it will be found that in each of these cases we have an independent pair.

[¹⁵⁰] This section may be omitted on a first reading.

SaP and SʹaPʹ (which are equivalent to SaP, PaS, and also to PʹaSʹ, SʹaPʹ) taken together serve to identify the classes S and P, and also the classes Sʹ and Pʹ. They are therefore complementary propositions, in accordance with the definition given in section [100]. Similarly, SeP and SʹePʹ (which are equivalent to SaPʹ, PʹaS, and also to PaSʹ, SʹaP) are complementary; they serve to identify the classes S and Pʹ, and also the classes Sʹ and P. It will be observed that the complementary of any universal proposition may be obtained by replacing the subject and predicate respectively by their contradictories. A not uncommon fallacy is the tacit substitution of the complementary of a proposition for the proposition itself.

The complementary relation holds only between universals. Particulars between which there is an analogous relation (the subject and predicate of the one being respectively the contradictories of the subject and predicate of the other) will be found to be sub-complementary in accordance with the definition in section [100]; this relation holds between SoP and SʹoPʹ, and between SiP and SʹiPʹ. SoP and SʹoPʹ (which are equivalent to SoP, PoS, and also to PʹoSʹ, SʹoPʹ) indicate that the classes S and P are neither coextensive nor either included within the other, and also that the same is true of Sʹ and Pʹ ; SiP and SʹiPʹ (which are equivalent to SoPʹ, PʹoS, and also to PoSʹ, SʹoP) indicate the same thing as regards S and Pʹ, Sʹ and P.

The four remaining pairs are contra-complementary, each pair serving conjointly to subordinate a certain class to a certain other class; or, rather, since each such subordination implies a supplementary subordination, we may say that each pair subordinates two classes to two other classes. Thus, SaP and SʹoPʹ (which are equivalent to SaP, PoS, and also to PʹaSʹ, SʹoPʹ) taken together shew that the class S is contained in but does not exhaust the class P, and also that the class Pʹ is contained in but does not exhaust the class Sʹ ; SʹaPʹ and SoP (which are equivalent to SʹaPʹ, PʹoSʹ, and also to PaS, SoP) yield the same results as regards the classes Sʹ and Pʹ, and the classes P and S ; SeP and SʹiPʹ (which are equivalent 144 to SaPʹ, PʹoS, and also to PaSʹ, SʹoP) as regards S and Pʹ, and P and Sʹ ; and SʹePʹ and SiP (which are equivalent to SʹaP, PoSʹ, and also to PʹaS, SoPʹ) as regards Sʹ and P, Pʹ and S.

Denoting the complementaries of A and E by Aʹ and Eʹ, and the sub-complementaries of I and O by Iʹ and Oʹ, the various relations between the non-equivalent propositions connecting any two terms and their contradictories may be exhibited in the following octagon of opposition:

Each of the dotted lines in the above takes the place of four connecting lines which are not filled in; for example, the dotted line marked as connecting contraries indicates the relation between A and E, A and Eʹ, Aʹ and E, Aʹ and Eʹ.[151]

[¹⁵¹] For the octagon of opposition in the form in which it is here given I am indebted to Mr Johnson.

108. The Elimination of Negative Terms.[152]—The process of obversion enables us by the aid of negative terms to reduce all propositions to the affirmative form; and the question may be 145 raised whether the various processes of immediate inference and the use, where necessary, of negative propositions will not equally enable us to eliminate negative terms.

[¹⁵²] This section may be omitted on a first reading.

It is of course clear that by means of obversion we can get rid of a negative term occurring as the predicate of a proposition. The problem is more difficult when the negative term occurs as subject, but in this case elimination may still be possible; for example, SʹiP = PoS. We may even be able to get rid of two negative terms; for example, SʹaPʹ = PaS. So long, however, as we are limited to categorical propositions of the ordinary type we cannot eliminate a negative term (without introducing another in its place) where such a term occurs as subject either (a) in a universal affirmative or a particular negative with a positive term as predicate, or (b) in a universal negative or a particular affirmative with a negative term as predicate.

The validity of the above results is at once shewn by reference to the table of equivalences given in section [106]. At least one proposition in which there is no negative term will be found in each line of equivalences except the fourth and the eighth, which are as follows:

In these cases we may indeed get rid of Sʹ (as, for example, from SʹaP), but it is only by introducing Pʹ (thus, SʹaP = PʹaS); there is no getting rid of negative terms altogether. We may here refer back to the results obtained in sections [100] and [106]; with two terms six non-equivalent propositions were obtained, with two terms and their contradictories eight non-equivalent propositions. The ground of this difference is now made clear.

If, however, we are allowed to enlarge our scheme of propositions by recognising certain additional types, and if we work on the assumption that universal propositions are existentially negative while particular propositions are existentially affirmative,[153] then negative terms may always be eliminated.146 Thus, No not-S is not-P is equivalent to the statement Nothing is both not-S and not-P, and this becomes by obversion Everything is either S or P. Again, Some not-S is not-P is equivalent to the statement Something is both not-S and not-P, and this becomes by obversion Something is not either S or P, or, as this proposition may also be written, There is something besides S and P. The elimination of negative terms has now been accomplished in all cases. It will be observed further that we now have eight non-equivalent propositions containing only S and P—namely, All S is P, No S is P, Some S is P, Some S is not P, All P is S, Some P is not S, Everything is either S or P, There is something besides S and P.

[¹⁵³] It is necessary here to anticipate the results of a discussion that will come at a later stage. See [chapter 8].

Following out this line of treatment, the table of equivalences given in section [106] may be rewritten as follows [columns (ii) and (iii) being omitted, and columns (v) and (vi) taking their places]:

Taking the propositions in two divisions of four sets each, the two diagonals from left to right give propositions containing S and P only.[154]

[¹⁵⁴] The first four propositions in column (v) may be expressed symbolically SPʹ = 0, &c.; the second four SPʹ > 0, &c.; the first four in column (vi) Sʹ + P = 1, &c.; and the second four Sʹ + P < 1, &c.; where 1 = the universe of discourse, and 0 = nonentity, i.e., the contradictory of the universe of discourse. Compare section [138].

147 The scheme of propositions given in this section may be brought into interesting relation with the three fundamental laws of thought.[155] The scheme is based upon the recognition of the following propositional forms and their contradictories:

Every S is P ;

Every not-P is not-S ;

Nothing is both S and not-P ;

Everything is either P or not-S ;
and these four propositions have been shewn to be equivalent to one another.

[¹⁵⁵] Compare Mrs Ladd Franklin in Mind, January, 1890, p. 87.

If in the above propositions we now write S for P, we have the following:

Every S is S ;

Every not-S is not-S ;

Nothing is both S and not-S ;

Everything is either S or not-S.

But the first two of these propositions express the law of identity, with positive and negative terms respectively, the third is an expression of the law of contradiction, and the fourth of the law of excluded middle. The scheme of propositions with which we have been dealing may, therefore, be said to be based upon the recognition of just those propositional forms which are required in order to express the fundamental laws of thought.

Since the propositional forms in question have been shewn to be mutually equivalent to one another, the further argument may suggest itself that if the validity of the immediate inferences involved be granted, then it follows that the fundamental laws of thought have been shewn to be mutually inferable from one another. But it may, on the other hand, be held that this argument is open to the charge of involving a circulus in probando on the ground that the validity of the immediate inferences themselves requires that the laws of thought be first postulated as an antecedent condition.

109. Other Immediate Inferences.—Some other commonly recognised forms of immediate inference may be briefly touched upon. 148

(1) Immediate inferences based on the square of opposition have been discussed in the preceding [chapter].

(2) Immediate inference by change of relation is the process whereby we pass from a categorical proposition to a conditional or a disjunctive, or from a conditional to a disjunctive or a categorical, or from a disjunctive to a categorical or a conditional.[156] For example, All S is P, therefore, If anything is S it is P ; Every S is P or Q, therefore, Any S that is not P is Q. References have been already made to inferences such as these, and they will be further discussed later on.

[¹⁵⁶] Miss Jones speaks of an inference of this kind as a transversion. See [note 3] on page 126.

(3) Immediate inference by added determinants is a process of immediate inference which consists in limiting both the subject and the predicate of the original proposition by means of the same determinant. For example,—All P is Q, therefore, All AP is AQ ; A negro is a fellow creature, therefore, A suffering negro is a suffering fellow creature. The formal validity of the reasoning may be shewn as follows: AP is a subdivision of the class P, namely, that part of it which also belongs to the class A ; and, therefore, whatever is true of the whole of P must be true of AP ; hence, given that All P is Q, we can infer that All AP is Q ; moreover, by the law of identity, All AP is A ; therefore, All AP is AQ.[157]

[¹⁵⁷] It must be observed, however, that the validity of this argument requires an assumption in regard to the existential import of propositions, which differs from that which we have for the most part adopted up to this point. It has to be assumed that universals do not imply the existence of their subjects. Otherwise this inference would not be valid in the case of no P being A. P might exist, and all P might be Q, but we could not pass to AP is AQ, since this would imply the existence of AP, which would be incorrect. It is necessary briefly to call attention to the above at this point, but our aim through all these earlier chapters has been to avoid as far as possible the various complications that arise in connexion with the difficult problem of existential import.

The formal validity of immediate inference by added determinants has been denied on the ground of the obvious fallacy of arguing from such a premiss as an elephant is an animal to the conclusion a small elephant is a small animal, or from such a premiss as cricketers are men to the conclusion poor cricketers are poor men. In these cases, however, the fallacy really results from the ambiguity of language, the added determinant 149 receiving a different interpretation when it qualifies the subject from that which it has when it qualifies the predicate. A term of comparison like small can indeed hardly be said to have an independent interpretation, its force always being relative to some other term with which it is conjoined. While then the inference in its symbolic form (P is Q, therefore, AP is AQ) is perfectly valid, it is specially necessary to guard against fallacy in its use when significant terms are employed. All that we have to insist upon is that the added determinant shall receive the same interpretation in both subject and predicate. There is, for example, no fallacy in the following: An elephant is an animal, therefore, A small elephant is an animal which is small compared with elephants generally; Cricketers are men, therefore, Poor cricketers are men who in their capacity as cricketers are poor.

(4) Immediate inference by complex conception is a process of immediate inference which consists in employing the subject and the predicate of the original proposition as parts of a more complex conception. Symbolically we can only express it somewhat as follows: P is Q, therefore, Whatever stands in a certain relation to P stands in the same relation to Q. The following is a concrete example: An elephant is an animal, therefore, the ear of an elephant is the ear of an animal. A systematic treatment of this kind of inference belongs to the special branch of formal logic known as the logic of relatives, any detailed consideration of which is beyond the scope of the present work. Attention may, however, be called to the danger of our committing a fallacy, if we perform the process carelessly. For example, Protestants are Christians, therefore, A majority of Protestants are a majority of Christians; A negro is a man, therefore, the best of negroes is the best of men. The former of these fallacies is akin to the fallacy of composition (see section [11]), since we pass from the distributive to the collective use of a term.

(5) Immediate inference by converse relation is a process of immediate inference analogous to ordinary conversion but belonging to the logic of relatives. It consists in passing from a statement of the relation in which P stands to Q to a 150 statement of the relation in which Q consequently stands to P. The two terms are transposed and the word by which their relation is expressed is replaced by its correlative. For example, A is greater than B, therefore, B is less than A ; Alexander was the son of Philip, therefore, Philip was the father of Alexander; Freedom is synonymous with liberty, therefore, Liberty is synonymous with freedom.

Mansel gives the first two of the above as examples of material consequence as distinguished from formal consequence. “A Material Consequence is defined by Aldrich to be one in which the conclusion follows from the premisses solely by the force of the terms. This in fact means from some understood Proposition or Propositions, connecting the terms, by the addition of which the mind is enabled to reduce the Consequence to logical form…… The failure of a Material Consequence takes place when no such connexion exists between the terms as will warrant us in supplying the premisses required; i.e., when one or more of the premisses so supplied would be false. But to determine this point is obviously beyond the province of the Logician. For this reason, Material Consequence is rightly excluded from Logic…… Among these material, and therefore extralogical, Consequences, are to be classed those which Reid adduces as cases for which Logic does not provide; e.g., ‘Alexander was the son of Philip, therefore, Philip was the father of Alexander’; ‘A is greater than B, therefore, B is less than A.’ In both these it is our material knowledge of the relations ‘father and son,’ ‘greater and less,’ that enables us to make the inference” (Aldrich, p. 199).

The distinction between what is formal and what is material is not in reality so simple or so absolute as is here implied.[158] It is usual to recognise as formal only those relations which can be expressed by the ordinary copula is or is not ; and there is very good reason for proceeding upon this basis in the greater part of our logical discussions. No other relation is of the same fundamental importance or admits of an equally developed logical superstructure. But it is important to recognise that there are other relations which may remain the 151 same while the things related vary; and wherever this is the case we may regard the relation as constituting the form and the things related the matter. Accordingly with each such relation we may have a different formal system. The logic of relatives deals with such systems as are outside the one ordinarily recognised. Each immediate inference by converse relation will, therefore, be formal in its own particular system. This point is admirably put by Miss Jones: “A proposition containing a relative term furnishes—besides the ordinary immediate inferences—other immediate inferences to any one acquainted with the system to which it refers. These inferences cannot be educed except by a person knowing the ‘system’; on the other hand, no knowledge is needed of the objects referred to, except a knowledge of their place in the system, and this knowledge is in many cases coextensive with ordinary intelligence; consider, e.g., the relations of magnitude of objects in space, of the successive parts of time, of family connexions, of number” (General Logic, p. 34).

[¹⁵⁸] Compare section [2].

(6) Immediate inference by modal consequence or, as it is also called, inference by change of modality, is somewhat analogous to subaltern inference. It consists in nothing more than weakening a statement in respect of its modality; and hence it is never possible to pass back from the inferred to the original proposition. Thus, from the validity of the apodeictic judgment we can pass to the validity of the assertoric, and from that to the validity of the problematic; but not vice versâ. On the other hand, from the invalidity of the problematic judgment we can pass to the invalidity of the assertoric, and from that to the invalidity of the apodeictic; but again not vice versâ.[159]

[¹⁵⁹] Compare Ueberweg, Logic, § 98.

110. Reduction of immediate inferences to the mediate form[160]—Immediate inference has been defined as the inference of a proposition from a single other proposition; mediate inference, on the other hand, is the inference of a proposition from at least two other propositions.

[¹⁶⁰] Students who have not already a technical knowledge of the syllogism may omit this section until they have read the earlier chapters of Part III.

We may briefly consider various ways of establishing the validity of immediate inferences by means of mediate inferences.

152 (1) One of the old Greek logicians, Alexander of Aphrodisias, establishes the conversion of E by means of a syllogism in Ferio.

for, if not, then by the law of contradiction, Some P is S ; and we have this syllogism,—

a reductio ad absurdum.[161]

[¹⁶¹] Compare Mansel’s Aldrich, p. 62. The conversion of A and the conversion of I may be established similarly.

(2) It may be plausibly maintained that in Aristotle’s proof of the conversion of E (given in section [99]), there is an implicit syllogism: namely,—Q is P, Q is S, therefore, Some S is P.

(3) The contraposition of A may be established by means of a syllogism in Camestres as follows:—

[¹⁶²] Similarly, granting the validity of obversion, the contraposition of O may be established by a syllogism in Datisi as follows:—

Given Some S is not P, then we have

It will be found that, adopting the same method, the contraposition of E is yielded by a syllogism in Darapti.

(4) We might also obtain the contrapositive of All S is P as follows:—

By the law of excluded middle, All not-P is S or not-S, and, by hypothesis, All S is P,

[¹⁶³] Compare Jevons, Principles of Science, chapter 6, § 2; and Studies in Deductive Logic, p. 44.

153 (5) The contraposition of A may also be established indirectly by means of a syllogism in Darii:—

for, if not, Some not-P is S ; and we have the following syllogism,—

which is absurd.[164]

[¹⁶⁴] Compare De Morgan, Formal Logic, p. 25. Granting the validity of obversion, the contraposition of E and the contraposition of O may be established similarly.

All the above are interesting, as illustrating the processes of immediate inference; but regarded as proofs they labour under the disadvantage of deducing the less complex by means of the more complex.

EXERCISES.

111. Give all the logical opposites of the proposition,—Some rich men are virtuous; and also the converse of the contrary of its contradictory. How is the latter directly related to the given proposition?
Does it follow that a proposition admits of simple conversion because its predicate is distributed? [K.]

112. Point out any ambiguities in the following propositions, and give the contradictory and (where possible) the converse of each of them:—(i) Some of the candidates have been successful; (ii) All are not happy that seem so; (iii) All the fish weighed five pounds. [K.]

113. State in logical form and convert the following propositions:—(a) He jests at scars who never felt a wound; (b) Axioms are self-evident; (c) Natives alone can stand the climate of Africa; (d) Not one of the Greeks at Thermopylae escaped; (e) All that glitters is not gold. [O.]

114. “The angles at the base of an isosceles triangle are equal.” What can be inferred from this proposition by obversion, conversion, and contraposition respectively? [L.]

154 115. Give the obverse, the contrapositive, and the inverse of each of the following propositions:—The virtuous alone are truly noble; No Athenians are Helots. [M.]

116. Give the contrapositive and (where possible) the inverse of the following propositions:—(i) A stitch in time saves nine; (ii) None but the brave deserve the fair; (iii) Blessed are the peacemakers; (iv) Things equal to the same thing are equal to one another; (v) Not every tale we hear is to be believed. [K.]

117. If it is false that “Not only the virtuous are happy,” what can we infer (a) with regard to the non-virtuous, (b) with regard to the non-happy? [J.]

118. Write down the contradictory, and also—where possible—the converse, the contrapositive, and the inverse of each of the following propositions:
A bird in the hand is worth two in the bush;
No unjust acts are expedient;
All are not saints that go to church. [K.]

119. Give the contrapositive and the inverse of each of the following propositions,—They never fail who die in a great cause; Whom the Gods love die young.
If A is either B or else both C and D, what do we know about that which is not D? [K.]

120. Take the following propositions in pairs, and in regard to each pair state whether the two propositions are consistent or inconsistent with each other; in the former case, state further whether either proposition can be inferred from the other, and, if it can be, point out the nature of the inference; in the latter case, state whether it is possible for both the propositions to be false:—(a) All S is P ; (b) All not-S is P ; (c) No P is S ; (d) Some not-P is S. [K.]

121. Transform the following propositions in such a way that, without losing any of their force, they may all have the same subject and the same predicate:—No not-P is S ; All P is not-S ; Some P is S ; Some not-P is not not-S. [K.]

122. Describe the logical relations, if any, between each of the following propositions, and each of the others:—
(i) There are no inorganic substances which do not contain carbon; 155
(ii) All organic substances contain carbon;
(iii) Some substances not containing carbon are organic;
(iv) Some inorganic substances do not contain carbon. [C.]

123. “All that love virtue love angling.”
Arrange the following propositions in the three following groups:—(α) those which can be inferred from the above proposition; (β) those which are consistent with it, but which cannot be inferred from it; (γ) those which are inconsistent with it.
(i) None that love not virtue love angling.
(ii) All that love angling love virtue.
(iii) All that love not angling love virtue.
(iv) None that love not angling love virtue.
(v) Some that love not virtue love angling.
(vi) Some that love not virtue love not angling
(vii) Some that love not angling love virtue.
(viii) Some that love not angling love not virtue. [K.]

124. Determine the logical relation between each pair of the following propositions:—
(1) All crystals are solids.
(2) Some solids are not crystals.
(3) Some not crystals are not solids.
(4) No crystals are not solids.
(5) Some solids are crystals.
(6) Some not solids are not crystals.
(7) All solids are crystals. [L.]

CHAPTER V.

THE DIAGRAMMATIC REPRESENTATION OF PROPOSITIONS.

125. The use of Diagrams in Logic.—In representing propositions by geometrical diagrams, our object is not that we may have a new set of symbols, but that the relation between the subject and predicate of a proposition may be exhibited by means of a sensible representation, the signification of which is clear at a glance. Hence the first requirement that ought to be satisfied by any diagrammatic scheme is that the interpretation of the diagrams should be intuitively obvious, as soon as the principle upon which they are based has been explained.[165]

[¹⁶⁵] Hamilton’s “geometric scheme,” which he himself describes as “easy, simple, compendious, all-sufficient, consistent, manifest, precise, complete” (Logic, II. p. 475), fails to satisfy this condition. He represents an affirmative copula by a horizontal tapering line (

), the broad end of which is towards the subject. Negation is marked by a perpendicular line crossing the horizontal one (

). A colon (:) placed at either end of the copula indicates that the corresponding term is distributed; a comma (,) that it is undistributed. Thus, for All S is P we have,—

S :

, P ;

and similarly for the other propositions.

Dr Venn rightly observes that this scheme is purely symbolical, and does not deserve to rank as a diagrammatic scheme at all. There is clearly nothing in the two ends of a wedge to suggest subjects and predicates, or in a colon and comma to suggest distribution and non-distribution” (Symbolic Logic, p. 432). Hamilton’s scheme may certainly be rejected as valueless. The schemes of Euler and Lambert belong to an altogether different category.

A second essential requirement is that the diagrams should be adequate; that is to say, they should give a complete, and 157 not a partial, representation of the relations which they are intended to indicate. Hamilton’s use of Euler’s diagrams, as described in the following section, will serve to illustrate the failure to satisfy this requirement.

In the third place, the diagrams should be capable of representing all the propositional forms recognised in the schedule of propositions which are to be illustrated, e.g., particulars as well as universal. One scheme of diagrams may, however, be better suited for one purpose, and another scheme for another purpose. It will be found that Dr Venn’s diagrams, to be described presently, are not quite so well adapted to the representation of particulars as of universals.

Lastly, it is advantageous that a diagrammatic scheme should be as little cumbrous as possible when it is desired to represent two or more propositions in combination with one another. This is the weak point of Euler’s method. A scheme of diagrams may, however, serve a very useful function in making clear the full force of individual propositions, even when it is not well adapted for the representation of combined propositions.

A further requirement is sometimes added, namely, that each propositional form should be represented by a single diagram, not by a set of alternative diagrams. This is, however, by no means essential. For if we adopt a schedule of propositions some of which yield only an indeterminate relation in respect of extension between the terms involved, it is important that this should be clearly brought out, and a set of alternative diagrams may be specially helpful for the purpose. This point will be illustrated, with reference to Euler’s diagrams, in the following section.[166]

[¹⁶⁶] It must be borne in mind that in all the schemes described in this chapter the terms of the propositions which are represented diagrammatically are taken in extension, not in intension.

126. Euler’s Diagrams.—We may begin with the well-known scheme of diagrams, which was first expounded by the Swiss mathematician and logician, Leonhard Euler, and which is usually called after his name.[167]

[¹⁶⁷] Euler lived from 1707 to 1783. His diagrammatic scheme is given in his Lettres à une Princesse d’Allemagne (Letters 102 to 105).

158 Representing the individuals included in any class, or denoted by any name, by a circle, it will be obvious that the five following diagrams represent all possible relations between any two classes:—

The force of the different propositional forms is to exclude one or more of these possibilities.
All S is P limits us to one of the two α, β ;
Some S is P to one of the four α, β, γ, δ ;
No S is P to ε ;
Some S is not P to one of the three γ, δ, ε.

It will be observed that there is great want of symmetry in the number of circles corresponding to the different propositional forms; also that there is an apparent inequality in the amount of information given by A and by E, and again by I and by O. We shall find that these anomalies disappear when account is taken of negative terms.

It is most misleading to attempt to represent All S is P by a single pair of circles, thus

or Some S is P by a single pair, thus

159 for in each case the proposition really leaves us with other alternatives. This method of employing the diagrams has, however, been adopted by a good many logicians who have used them, including Sir William Hamilton (Logic, I. p. 255), and Professor Jevons (Elementary Lessons in Logic, pp. 72 to 75); and the attempt at such simplification has brought their use into undeserved disrepute. Thus, Dr Venn remarks, “The common practice, adopted in so many manuals, of appealing to these diagrams—Eulerian diagrams as they are often called—seems to me very questionable. The old four propositions A, E, I, O, do not exactly correspond to the five diagrams, and consequently none of the moods in the syllogism can in strict propriety be represented by these diagrams” (Symbolic Logic, pp. 15, 16; compare also pp. 424, 425). This criticism, while perfectly sound as regards the use of Euler’s circles by Hamilton and Jevons, loses most of its force if the diagrams are employed with due precautions. It is true that the diagrams become somewhat cumbrous in relation to the syllogism; but the logical force of propositions and the logical relations between propositions can in many respects be well illustrated by their aid. Thus, they may be employed:—

(1) To illustrate the distribution of the predicate in a proposition. In the case of each of the four fundamental propositions we may shade the part of the predicate concerning which information is given us.

We then have,—

160

We see that with A and I, only part of P is in some of the cases shaded; whereas with E and O, the whole of P is in every case shaded; and it is thus made clear that negative propositions distribute, while affirmative propositions do not distribute, their predicates.

(2) To illustrate the opposition of propositions. Comparing two contradictory propositions, e.g., A and O, we see that they have no case in common, but that between them they exhaust all possible cases. Hence the truth, that two contradictory propositions cannot be true together but that one of them must be true, is brought home to us under a new aspect. Again, comparing two subaltern propositions, e.g., A and I, we notice that the former gives us all the information given by the latter and something more, since it still further limits the possibilities. The other relations involved in the doctrine of opposition may be illustrated similarly.

(3) To illustrate the conversion of propositions. Thus it is made clear by the diagrams how it is that A admits only of conversion per accidens. All S is P limits us to one or other of the following,—

What then do we know of P? In the first case we have All P is S, in the second Some P is S ; and since we are ignorant as to which of the two cases holds good, we can only state what is common to them both, namely, Some P is S.

Again, it is made clear how it is that O is inconvertible. Some S is not P limits us to one or other of the following,—

161 What then do we know concerning P? The three cases give us respectively,—(i) All P is S ; (ii) Some P is S and Some P is not S ; (iii) No P is S. But (i) and (iii) are inconsistent with one another. Hence nothing can be affirmed of P that is true in all three cases indifferently.

(4) To illustrate the more complicated forms of immediate inference. Taking, for example, the proposition All S is P, we may ask, What does this enable us to assert about not-P and not-S respectively? We have one or other of these cases,—

As regards not-P, these yield respectively (i) No not-P is S ; (ii) No not-P is S. And thus we obtain the contrapositive of the given proposition.

As regards not-S, we have (i) No not-S is P, (ii) Some not-S is P and some not-S is not P.[168] Hence in either case we may infer Some not-S is not P.

[¹⁶⁸] It is assumed in the use of Euler’s diagrams that S and P both exist in the universe of discourse, while neither of them exhausts that universe. This assumption is the same as that upon which our treatment of immediate inferences in the preceding chapter has been based.

E, I, O may be dealt with similarly.

(5) To illustrate the joint force of a pair of complementary or contra-complementary or sub-complementary propositions (compare section [100]). Thus, the pair of complementary propositions, SaP and PaS, taken together, limit us to

Similarly the pair of contra-complementary propositions, SaP and PoS, limit us to the relation marked β on page [158]; and the pair of contra-complementary propositions, SoP and 162 PaS, to γ ; while the pair of sub-complementary propositions, SoP and PoS, give us a choice between δ and ε.

The application of the diagrams to syllogistic reasonings will be considered in a subsequent [chapter].

With regard to all the above, it may be said that the use of the circles gives us nothing that could not easily have been obtained independently. This is of course true; but no one, who has had experience of the difficulty that is sometimes found by students in properly understanding the elementary principles of formal logic, and especially in dealing with immediate inferences, will despise any means of illustrating afresh the old truths, and presenting them under a new aspect.

The fact that we have not a single pair of circles corresponding to each fundamental form of proposition is fatal if we wish to illustrate any complicated train of reasoning in this way; but in indicating the real nature of the information given by the propositions themselves, it is rather an advantage than otherwise, inasmuch as it shews how limited in some cases this information actually is.[169]

[¹⁶⁹] Dr Venn writes in criticism of Euler’s scheme, “A fourfold scheme of propositions will not very conveniently fit in with a fivefold scheme of diagrams… What the five diagrams are competent to illustrate is the actual relation of the classes, not our possibly imperfect knowledge of that relation” (Empirical Logic, p. 229). The reply to this criticism is that inasmuch as the fourfold scheme of propositions gives but an imperfect knowledge of the actual relation of the classes denoted by the terms, the Eulerian diagrams are specially valuable in making this clear and unmistakeable. By the aid of dotted lines it is indeed possible to represent each proposition by a single Eulerian figure; but the diagrams then become so much more difficult to interpret that the loss is considerably greater than the gain. The first and second of the following diagrams are borrowed from Ueberweg (Logic, § 71). In the case of O, Ueberweg’s diagram is rather complicated; and I have substituted a simpler one.

In the last of these diagrams we get the three cases yielded by an O proposition by (1) filling in the dotted line to the left and striking out the other, (2) filling in both dotted lines, (3) filling in the dotted line to the right and striking out the other. These three cases are respectively those marked γ, δ, ε, on page [158].

163 127. Lambert’s Diagrams.—A scheme of diagrams was employed by Lambert[170] in which horizontal straight lines take the place of Euler’s circles. The extension of a term is represented by a horizontal straight line, and so far as two such lines overlap it is indicated that the corresponding classes are coincident, while so far as they do not overlap these classes are shewn to be mutually exclusive. Both the absolute and the relative length of the lines is of course arbitrary and immaterial.

[¹⁷⁰] Johann Heinrich Lambert was a German philosopher and mathematician who lived from 1728 to 1777. His Neues Organon was published at Leipzig in 1768. Lambert’s own diagrammatic scheme differs somewhat from both of those given in the text; but it very closely resembles the one in which portions of the lines are dotted. The modifications in the text have been introduced in order to obviate certain difficulties involved in Lambert’s own diagrams. See note [2] on page 165.

We may first shew how Lambert’s lines may be used in such a manner as to be precisely analogous to Euler’s circles. 164 Thus, the four fundamental propositions may be represented as follows:—

These diagrams occupy less space than Euler’s circles. But they seem also to be less intuitively clear and less suggestive. The different cases too are less markedly distinct from one another. It is probable that one would in consequence be more liable to error in employing them.

The different cases may, however, be combined by the use of dotted lines so as to yield but a single diagram for each proposition much more satisfactorily than in Euler’s scheme. Thus, All S is P may be represented by the diagram

where the dotted line indicates that we are uncertain as to whether there is or is not any P which is not S. We obviously get two cases according as we strike out the dots or fill them in, and these are the two cases previously shewn to be compatible with an A proposition.

Again, Some S is P may be represented by the diagram

and here we get the four cases previously given for an I 165 proposition by (a) filling in the dots to the left and striking out those to the right, (b) filling in all the dots, (c) striking them all out, (d) filling in those to the right and striking out those to the left.

Two complete schemes of diagrams may be constructed on this plan, in one of which no part of any S line, and in the other no part of any P line, is dotted.[171] These two schemes are given below to the left and right respectively of the propositional forms themselves.

[¹⁷¹] It is important to give both these schemes as it will be found that neither one of them will by itself suffice when this method is used for illustrating the syllogism. For obvious reasons the E diagram is the same in both schemes.

It must be understood that the two diagrams given above in the cases of A, I, and O are alternative in the sense that we may select which we please to represent our proposition; but either represents it completely.

We shall find later on that for the purpose of illustrating the syllogistic moods, Lambert’s method is a good deal less cumbrous than Euler’s.[172] An adaptation of Lambert’s diagrams in which the contradictories of S and P are introduced as well 166 as S and P themselves will be given in section [131]. This more elaborated scheme will be found useful for illustrating the various processes of immediate inference.

[¹⁷²] Dr Venn (Symbolic Logic, p. 432) remarks, “As a whole Lambert’s scheme seems to me distinctly inferior to the scheme of Euler, and has in consequence been very little employed by other logicians.” The criticism offered in support of this statement is directed chiefly against Lambert’s own representation of the particular affirmative proposition, namely,

This diagram certainly seems as appropriate to O as it does to I; but the modification introduced in the text, and indeed suggested by Dr Venn himself, is not open to a similar objection.

128. Dr Venn’s Diagrams.—In the diagrammatic scheme employed by Dr Venn (Symbolic Logic, chapter 5) the diagram

does not itself represent any proposition, but the framework into which propositions may be fitted. Denoting not-S by Sʹ and what is both S and P by SP, &c., it is clear that everything must be contained in one or other of the four classes SP, SPʹ, SʹP, SʹPʹ ; and the above diagram shews four compartments (one being that which lies outside both the circles) corresponding to these four classes. Every universal proposition denies the existence of one or more of such classes, and it may therefore be diagrammatically represented by shading out the corresponding compartment or compartments. Thus, All S is P, which denies the existence of SPʹ, is represented by

No S is P by

167 With three terms we have three circles and eight compartments, thus,—

All S is P or Q is represented by

All S is P and Q by

It is in cases involving three or more terms that the advantage of this scheme over the Eulerian scheme is most manifest. The diagrams are not, however, quite so well adapted to the case of particular propositions. Dr Venn (in Mind, 1883, pp. 599, 600) suggests that we might draw a bar across the compartment declared to be saved by a particular proposition;[173] thus, Some S is P would be represented by drawing a bar across the SP compartment. This plan can be worked out satisfactorily; but in representing a combination of propositions in this way special care is needed in the interpretation of the diagrams. For example, if we have the diagram for three terms S, P, Q, and are given Some S is P, 168 we do not know that both the compartments SPQ, SPQʹ, are to be saved, and in a case like this a bar drawn across the SP compartment is in some danger of misinterpretation.

[¹⁷³] Dr Venn’s scheme differs from the schemes of Euler and Lambert, in that it is not based upon the assumption that our terms and their contradictories all represent existing classes. It involves, however, the doctrine that particulars are existentially affirmative, while universals are existentially negative.

129. Expression of the possible relations between any two classes by means of the propositional forms A, E, I, O.—Any information given with respect to two classes limits the possible relations between them to something less than the five à priori possibilities indicated diagrammatically by Euler’s circles as given at the beginning of section [126]. It will be useful to enquire how such information may in all cases be expressed by means of the propositional forms A, E, I, O.

The five relations may, as before, be designated respectively α, β, γ, δ, ε.[174] Information is given when the possibility of one or more of these is denied; in other words, when we are limited to one, two, three, or four of them. Let limitation to α, or β, the exclusion of γ, δ, ε be denoted by α, β ; limitation to α, β, or γ (i.e., the exclusion of δ and ε) by α, β, γ ; and so on.

[¹⁷⁴] Thus, the classes being S and P, α denotes that S and P are wholly coincident; β that P contains S and more besides; β that S contains P and more besides; δ that S and P overlap each other, but that each includes something not included by the other; ε that S and P have nothing whatever in common.

In seeking to express our information by means of the four ordinary propositional forms, we find that sometimes a single proposition will suffice for our purpose; thus α, β is expressed by All S is P. Sometimes we require a combination of propositions; thus α is expressed by the pair of complementary propositions All S is P and all P is S, (since all S is P excludes γ, δ, ε, and all P is S further excludes β). Some other cases are more complicated; thus the fact that we are limited to α or δ cannot be expressed more simply than by saying, Either All S is P and all P is S, or else Some S is P, some S is not P, and some P is not S.

Let A = All S is P, A₁ = All P is S, and similarly for the other propositions. Also let AA₁ = All S is P and all P is S, &c. Then the following is a scheme for all possible cases:— 169