UNIVERSITY MANUALS

EDITED BY PROFESSOR KNIGHT

LOGIC

INDUCTIVE AND DEDUCTIVE

Published May, 1893
Reprinted December, 1893
" November, 1894
" January, 1899
" August, 1904
" June, 1909
" September, 1912
" July, 1913
" January, 1915

LOGIC

INDUCTIVE AND DEDUCTIVE

BY WILLIAM MINTO, M.A.

HON. LL.D. ST. ANDREWS
Late Professor of Logic in the University of Aberdeen

LONDON
JOHN MURRAY, ALBEMARLE STREET, W.
1915

PREFACE.

In this little treatise two things are attempted that at first might appear incompatible. One of them is to put the study of logical formulæ on a historical basis. Strangely enough, the scientific evolution of logical forms, is a bit of history that still awaits the zeal and genius of some great scholar. I have neither ambition nor qualification for such a magnum opus, and my life is already more than half spent; but the gap in evolutionary research is so obvious that doubtless some younger man is now at work in the field unknown to me. All that I can hope to do is to act as a humble pioneer according to my imperfect lights. Even the little I have done represents work begun more than twenty years ago, and continuously pursued for the last twelve years during a considerable portion of my time.

The other aim, which might at first appear inconsistent with this, is to increase the power of Logic as a practical discipline. The main purpose of this practical science, or scientific art, is conceived to be the organisation of reason against error, and error in its various kinds is made the basis of the division of the subject. To carry out this practical aim along with the historical one is not hopeless, because throughout its long history Logic has been a practical science; and, as I have tried to show at some length in introductory chapters, has concerned itself at different periods with the risks of error peculiar to each.

To enumerate the various books, ancient and modern, to which I have been indebted, would be a vain parade. Where I have consciously adopted any distinctive recent contribution to the long line of tradition, I have made particular acknowledgment. My greatest obligation is to my old professor, Alexander Bain, to whom I owe my first interest in the subject, and more details than I can possibly separate from the general body of my knowledge.

W. M.

Aberdeen, January, 1893.

Since these sentences were written, the author of this book has died; and Professor Minto's Logic is his last contribution to the literature of his country. It embodies a large part of his teaching in the philosophical class-room of his University, and doubtless reflects the spirit of the whole of it.

Scottish Philosophy has lost in him one of its typical representatives, and the University of the North one of its most stimulating teachers. There have been few more distinguished men than William Minto in the professoriate of Aberdeen; and the memory of what he was, of his wide and varied learning, his brilliant conversation, his urbanity, and his rare power of sympathy with men with whose opinions he did not agree, will remain a possession to many who mourn his loss.

It will be something if this little book keeps his memory alive, both amongst the students who owed so much to him, and in the large circle of friends who used to feel the charm of his personality.

WILLIAM KNIGHT.

GENERAL PLAN OF THE SERIES.

This Series is primarily designed to aid the University Extension Movement throughout Great Britain and America, and to supply the need so widely felt by students, of Text-books for study and reference, in connexion with the authorised Courses of Lectures.

The Manuals differ from those already in existence in that they are not intended for School use, or for Examination purposes; and that their aim is to educate, rather than to inform. The statement of details is meant to illustrate the working of general laws, and the development of principles; while the historical evolution of the subject dealt with is kept in view, along with its philosophical significance.

The remarkable success which has attended University Extension in Britain has been partly due to the combination of scientific treatment with popularity, and to the union of simplicity with thoroughness. This movement, however, can only reach those resident in the larger centres of population, while all over the country there are thoughtful persons who desire the same kind of teaching. It is for them also that this Series is designed. Its aim is to supply the general reader with the same kind of teaching as is given in the Lectures, and to reflect the spirit which has characterised the movement, viz., the combination of principles with facts, and of methods with results.

The Manuals are also intended to be contributions to the Literature of the Subjects with which they respectively deal, quite apart from University Extension; and some of them will be found to meet a general rather than a special want.

They will be issued simultaneously in England and America. Volumes dealing with separate sections of Literature, Science, Philosophy, History, and Art have been assigned to representative literary men, to University Professors, or to Extension Lecturers connected with Oxford, Cambridge, London, and the Universities of Scotland and Ireland.

A list of the works in this Series will be found at the end of the volume.

CONTENTS.

INTRODUCTION.

[I.]

[II.]

[III.]

[BOOK I.]

THE LOGIC OF CONSISTENCY—SYLLOGISM AND DEFINITION.

[PART I.]

THE ELEMENTS OF PROPOSITIONS.

[Chapter I.]

[Chapter II.]

[PART II.]

DEFINITION.

[Chapter I.]

[Chapter II.]

[Chapter III.]

[Chapter IV.]

[PART III.]

THE INTERPRETATION OF PROPOSITIONS.

[Chapter I.]

[Chapter II.]

[Chapter III.]

[Chapter IV.]

[PART IV.]

THE INTERDEPENDENCE OF PROPOSITIONS.

[Chapter I.]

[Chapter II.]

[Chapter III.]

[Chapter IV.]

[Chapter V.]

[Chapter VI.]

[Chapter VII.]

[Chapter VIII.]

[Chapter IX.]

[BOOK II.]

INDUCTIVE LOGIC, OR THE LOGIC OF SCIENCE.

[Chapter I.]

[Chapter II.]

[Chapter III.]

[Chapter IV.]

[Chapter V.]

[Chapter VI.]

[Chapter VII.]

[Chapter VIII.]

[Chapter IX.]

[Chapter X.]

INTRODUCTION.

I.—THE ORIGIN AND SCOPE OF LOGIC.

The question has sometimes been asked, Where should we begin in Logic? Particularly within the present century has this difficulty been felt, when the study of Logic has been revived and made intricate by the different purposes of its cultivators.

Where did the founder of Logic begin? Where did Aristotle begin? This seems to be the simplest way of settling where we should begin, for the system shaped by Aristotle is still the trunk of the tree, though there have been so many offshoots from the old stump and so many parasitic plants have wound themselves round it that Logic is now almost as tangled a growth as the Yews of Borrowdale—

An intertwisted mass of fibres serpentine

Upcoiling and inveterately convolved.

It used to be said that Logic had remained for two thousand years precisely as Aristotle left it. It was an example of a science or art perfected at one stroke by the genius of its first inventor. The bewildered student must often wish that this were so: it is only superficially true. Much of Aristotle's nomenclature and his central formulæ have been retained, but they have been very variously supplemented and interpreted to very different purposes—often to no purpose at all.

The Cambridge mathematician's boast about his new theorem—"The best of it all is that it can never by any possibility be made of the slightest use to anybody for anything"—might be made with truth about many of the later developments of Logic. We may say the same, indeed, about the later developments of any subject that has been a playground for generation after generation of acute intellects, happy in their own disinterested exercise. Educational subjects—subjects appropriated for the general schooling of young minds—are particularly apt to be developed out of the lines of their original intention. So many influences conspire to pervert the original aim. The convenience of the teacher, the convenience of the learner, the love of novelty, the love of symmetry, the love of subtlety; easy-going indolence on the one hand and intellectual restlessness on the other—all these motives act from within on traditional matter without regard to any external purpose whatever. Thus in Logic difficulties have been glossed over and simplified for the dull understanding, while acute minds have revelled in variations and new and ingenious manipulations of the old formulæ, and in multiplication and more exact and symmetrical definition of the old distinctions.

To trace the evolution of the forms and theories of Logic under these various influences during its periods of active development is a task more easily conceived than executed, and one far above the ambition of an introductory treatise. But it is well that even he who writes for beginners should recognise that the forms now commonly used have been evolved out of a simpler tradition. Without entering into the details of the process, it is possible to indicate its main stages, and thus furnish a clue out of the modern labyrinthine confusion of purposes.

How did the Aristotelian Logic originate? Its central feature is the syllogistic forms. In what circumstances did Aristotle invent these? For what purpose? What use did he contemplate for them? In rightly understanding this, we shall understand the original scope or province of Logic, and thus be in a position to understand more clearly how it has been modified, contracted, expanded, and supplemented.

Logic has always made high claims as the scientia scientiarum, the science of sciences. The builders of this Tower of Babel are threatened in these latter days with confusion of tongues. We may escape this danger if we can recover the designs of the founder, and of the master-builders who succeeded him.

Aristotle's Logic has been so long before the world in abstract isolation that we can hardly believe that its form was in any way determined by local accident. A horror as of sacrilege is excited by the bare suggestion that the author of this grand and venerable work, one of the most august monuments of transcendent intellect, was in his day and generation only a pre-eminent tutor or schoolmaster, and that his logical writings were designed for the accomplishment of his pupils in a special art in which every intellectually ambitious young Athenian of the period aspired to excel. Yet such is the plain fact, baldly stated. Aristotle's Logic in its primary aim was as practical as a treatise on Navigation, or "Cavendish on Whist". The latter is the more exact of the two comparisons. It was in effect in its various parts a series of handbooks for a temporarily fashionable intellectual game, a peculiar mode of disputation or dialectic,[¹] the game of Question and Answer, the game so fully illustrated in the Dialogues of Plato, the game identified with the name of Socrates.

We may lay stress, if we like, on the intellectuality of the game, and the high topics on which it was exercised. It was a game that could flourish only among a peculiarly intellectual people; a people less acute would find little sport in it. The Athenians still take a singular delight in disputation. You cannot visit Athens without being struck by it. You may still see groups formed round two protagonists in the cafés or the squares, or among the ruins of the Acropolis, in a way to remind you of Socrates and his friends. They do not argue as Gil Blas and his Hibernians did with heat and temper, ending in blows. They argue for the pure love of arguing, the audience sitting or standing by to see fair play with the keenest enjoyment of intellectual thrust and parry. No other people could argue like the Greeks without coming to blows. It is one of their characteristics now, and so it was in old times two thousand years ago. And about a century before Aristotle reached manhood, they had invented this peculiarly difficult and trying species of disputative pastime, in which we find the genesis of Aristotle's logical treatises.

To get a proper idea of this debate by Question and Answer, which we may call Socratic disputation after its most renowned master, one must read some of the dialogues of Plato. I will indicate merely the skeleton of the game, to show how happily it lent itself to Aristotle's analysis of arguments and propositions.

A thesis or proposition is put up for debate, e.g., that knowledge is nothing else than sensible perception,[²] that it is a greater evil to do wrong than to suffer wrong,[³] that the love of gain is not reprehensible.[⁴] There are two disputants, but they do not speak on the question by turns, so many minutes being allowed to each as in a modern encounter of wits. One of the two, who may be called the Questioner, is limited to asking questions, the other, the Respondent, is limited to answering. Further, the Respondent can answer only "Yes" or "No," with perhaps a little explanation: on his side the Questioner must ask only questions that admit of the simple answer "Yes" or "No". The Questioner's business is to extract from the Respondent admissions involving the opposite of what he has undertaken to maintain. The Questioner tries in short to make him contradict himself. Only a very stupid Respondent would do this at once: the Questioner plies him with general principles, analogies, plain cases; leads him on from admission to admission, and then putting the admissions together convicts him out of his own mouth of inconsistency.[⁵]

Now mark precisely where Aristotle struck in with his invention of the Syllogism, the invention on which he prided himself as specially his own, and the forms of which have clung to Logic ever since, even in the usage of those who deride Aristotle's Moods and Figures as antiquated superstitions. Suppose yourself the Questioner, where did he profess to help you with his mechanism? In effect, as the word Syllogism indicates, it was when you had obtained a number of admissions, and wished to reason them together, to demonstrate how they bore upon the thesis in dispute, how they hung together, how they necessarily involved what you were contending for. And the essence of his mechanism was the reduction of the admitted propositions to common terms, and to certain types or forms which are manifestly equivalent or inter-dependent. Aristotle advised his pupils also in the tactics of the game, but his grand invention was the form or type of admissions that you should strive to obtain, and the effective manipulation of them when you had got them.

An example will show the nature of this help, and what it was worth. To bring the thing nearer home, let us, instead of an example from Plato, whose topics often seem artificial to us now, take a thesis from last century, a paradox still arguable, Mandeville's famous—some would say infamous—paradox that Private Vices are Public Benefits. Undertake to maintain this, and you will have no difficulty in getting a respondent prepared to maintain the negative. The plain men, such as Socrates cross-questioned, would have declared at once that a vice is a vice, and can never do any good to anybody. Your Respondent denies your proposition simply: he upholds that private vices never are public benefits, and defies you to extract from him any admission inconsistent with this. Your task then is to lure him somehow into admitting that in some cases what is vicious in the individual may be of service to the State. This is enough: you are not concerned to establish that this holds of all private vices. A single instance to the contrary is enough to break down his universal negative. You cannot, of course, expect him to make the necessary admission in direct terms: you must go round about. You know, perhaps, that he has confidence in Bishop Butler as a moralist. You try him with the saying: "To aim at public and private good are so far from being inconsistent that they mutually promote each other". Does he admit this?

Perhaps he wants some little explanation or exemplification to enable him to grasp your meaning. This was within the rules of the game. You put cases to him, asking for his "Yes" or "No" to each. Suppose a man goes into Parliament, not out of any zeal for the public good, but in pure vainglory, or to serve his private ends, is it possible for him to render the State good service? Or suppose a milk-seller takes great pains to keep his milk pure, not because he cares for the public health, but because it pays, is this a benefit to the public?

Let these questions be answered in the affirmative, putting you in possession of the admission that some actions undertaken for private ends are of public advantage, what must you extract besides to make good your position as against the Respondent? To see clearly at this stage what now is required, though you have to reach it circuitously, masking your approach under difference of language, would clearly be an advantage. This was the advantage that Aristotle's method offered to supply. A disputant familiar with his analysis would foresee at once that if he could get the Respondent to admit that all actions undertaken for private ends are vicious, the victory was his, while nothing short of this would serve.

Here my reader may interject that he could have seen this without any help from Aristotle, and that anybody may see it without knowing that what he has to do is, in Aristotelian language, to construct a syllogism in Bokardo. I pass this over. I am not concerned at this point to defend the utility of Aristotle's method. All that I want is to illustrate the kind of use that it was intended for. Perhaps if Aristotle had not habituated men's minds to his analysis, we should none of us have been able to discern coherence and detect incoherence as quickly and clearly as we do now.

But to return to our example. As Aristotle's pupil, you would have seen at the stage we are speaking of that the establishment of your thesis must turn upon the definition of virtue and vice. You must proceed, therefore, to cross-examine your Respondent about this. You are not allowed to ask him what he means by virtue, or what he means by vice. In accordance with the rules of the dialectic, it is your business to propound definitions, and demand his Yes or No to them. You ask him, say, whether he agrees with Shaftesbury's definition of a virtuous action as an action undertaken purely for the good of others. If he assents, it follows that an action undertaken with any suspicion of a self-interested motive cannot be numbered among the virtues. If he agrees, further, that every action must be either vicious or virtuous, you have admissions sufficient to prove your original thesis. All that you have now to do to make your triumph manifest, is to display the admissions you have obtained in common terms.

Some actions done with a self-interested motive are public benefits.

All actions done with a self-interested motive are private vices.

From these premisses it follows irresistibly that

Some private vices are public benefits.

This illustration may serve to show the kind of disputation for which Aristotle's logic was designed, and thus to make clear its primary uses and its limitations.

To realise its uses, and judge whether there is anything analogous to them in modern needs, conceive the chief things that it behoved Questioner and Respondent in this game to know. All that a proposition necessarily implies; all that two propositions put together imply; on what conditions and to what extent one admission is inconsistent with another; when one admission necessarily involves another; when two necessarily involve a third. And to these ends it was obviously necessary to have an exact understanding of the terms used, so as to avoid the snares of ambiguous language.

That a Syllogistic or Logic of Consistency should emerge out of Yes-and-No Dialectic was natural. Things in this world come when they are wanted: inventions are made on the spur of necessity. It was above all necessary in this kind of debate to avoid contradicting yourself: to maintain your consistency. A clever interrogator spread out proposition after proposition before you and invited your assent, choosing forms of words likely to catch your prejudices and lure you into self-contradiction. An organon, instrument, or discipline calculated to protect you as Respondent and guide you as Questioner by making clear what an admission led to, was urgently called for, and when the game had been in high fashion for more than a century Aristotle's genius devised what was wanted, meeting at the same time, no doubt, collateral needs that had arisen from the application of Dialectic to various kinds of subject-matter.

The thoroughness of Aristotle's system was doubtless due partly to the searching character of the dialectic in which it had its birth. No other mode of disputation makes such demands upon the disputant's intellectual agility and precision, or is so well adapted to lay bare the skeleton of an argument.

The uses of Aristotle's logical treatises remained when the fashion that had called them forth had passed.[⁶] Clear and consistent thinking, a mastery of the perplexities and ambiguities of language, power to detect identity of meaning under difference of expression, a ready apprehension of all that a proposition implies, all that may be educed or deduced from it—whatever helps to these ends must be of perpetual use. "To purge the understanding of those errors which lie in the confusion and perplexities of an inconsequent thinking," is a modern description of the main scope of Logic.[⁷] It is a good description of the branch of Logic that keeps closest to the Aristotelian tradition.

The limitations as well as the uses of Aristotle's logic may be traced to the circumstances of its origin. Both parties to the disputation, Questioner and Respondent alike, were mainly concerned with the inter-dependence of the propositions put forward. Once the Respondent had given his assent to a question, he was bound in consistency to all that it implied. He must take all the consequences of his admission. It might be true or it might be false as a matter of fact: all the same he was bound by it: its truth or falsehood was immaterial to his position as a disputant. On the other hand, the Questioner could not go beyond the admissions of the Respondent. It has often been alleged as a defect in the Syllogism that the conclusion does not go beyond the premisses, and ingenious attempts have been made to show that it is really an advance upon the premisses. But having regard to the primary use of the syllogism, this was no defect, but a necessary character of the relation. The Questioner could not in fairness assume more than had been granted by implication. His advance could only be an argumentative advance: if his conclusion contained a grain more than was contained in the premisses, it was a sophistical trick, and the Respondent could draw back and withhold his assent. He was bound in consistency to stand by his admissions; he was not bound to go a fraction of an inch beyond them.

We thus see how vain it is to look to the Aristotelian tradition for an organon of truth or a criterion of falsehood. Directly and primarily, at least, it was not so; the circumstances of its origin gave it a different bent. Indirectly and secondarily, no doubt, it served this purpose, inasmuch as truth was the aim of all serious thinkers who sought to clear their minds and the minds of others by Dialectic. But in actual debate truth was represented merely by the common-sense of the audience. A dialectician who gained a triumph by outraging this, however cleverly he might outwit his antagonist, succeeded only in amusing his audience, and dialecticians of the graver sort aimed at more serious uses and a more respectful homage, and did their best to discountenance merely eristic disputation. Further, it would be a mistake to conclude because Aristotle's Logic, as an instrument of Dialectic, concerned itself with the syllogism of propositions rather than their truth, that it was merely an art of quibbling. On the contrary, it was essentially the art of preventing and exposing quibbling. It had its origin in quibbling, no doubt, inasmuch as what we should call verbal quibbling was of the essence of Yes-and-No Dialectic, and the main secret of its charm for an intellectual and disputatious people; but it came into being as a safeguard against quibbling, not a serviceable adjunct.

The mediæval developments of Logic retained and even exaggerated the syllogistic character of the original treatises. Interrogative dialectic had disappeared in the Middle Ages whether as a diversion or as a discipline: but errors of inconsistency still remained the errors against which principally educated men needed a safeguard. Men had to keep their utterances in harmony with the dogmas of the Church. A clear hold of the exact implications of a proposition, whether singly or in combination with other propositions, was still an important practical need. The Inductive Syllogism was not required, and its treatment dwindled to insignificance in mediæval text-books, but the Deductive Syllogism and the formal apparatus for the definition of terms held the field.

It was when observation of Nature and its laws became a paramount pursuit that the defects of Syllogistic Logic began to be felt. Errors against which this Logic offered no protection then called for a safeguard—especially the errors to which men are liable in the investigation of cause and effect. "Bring your thoughts into harmony one with another," was the demand of Aristotle's age. "Bring your thoughts into harmony with authority," was the demand of the Middle Ages. "Bring them into harmony with fact," was the requirement most keenly felt in more recent times. It is in response to this demand that what is commonly but not very happily known as Inductive Logic has been formulated.

In obedience to custom, I shall follow the now ordinary division of Logic into Deductive and Inductive. The titles are misleading in many ways, but they are fixed by a weight of usage which it would be vain to try to unsettle. Both come charging down the stream of time each with its cohort of doctrines behind it, borne forward with irresistible momentum.

The best way of preventing confusion now is to retain the established titles, recognise that the doctrines behind each have a radically different aim or end, and supply the interpretation of this end from history. What they have in common may be described as the prevention of error, the organisation of reason against error. I have shown that owing to the bent impressed upon it by the circumstances of its origin, the errors chiefly safeguarded by the Aristotelian logic were the errors of inconsistency. The other branch of Logic, commonly called Induction, was really a separate evolution, having its origin in a different practical need. The history of this I will trace separately after we have seen our way through the Aristotelian tradition and its accretions. The Experimental Methods are no less manifestly the germ, the evolutionary centre or starting-point, of the new Logic than the Syllogism is of the old, and the main errors safeguarded are errors of fact and inference from fact.

At this stage it will be enough to indicate briefly the broad relations between Deductive Logic and Inductive Logic.

Inductive Logic, as we now understand it—the Logic of Observation and Explanation—was first formulated and articulated to a System of Logic by J. S. Mill. It was he that added this wing to the old building. But the need of it was clearly expressed as early as the thirteenth century. Roger Bacon, the Franciscan friar (1214-1292), and not his more illustrious namesake Francis, Lord Verulam, was the real founder of Inductive Logic. It is remarkable that the same century saw Syllogistic Logic advanced to its most complete development in the system of Petrus Hispanus, a Portuguese scholar who under the title of John XXI. filled the Papal Chair for eight months in 1276-7.

A casual remark of Roger Bacon's in the course of his advocacy of Experimental Science in the Opus Majus draws a clear line between the two branches of Logic. "There are," he says, "two ways of knowing, by Argument and by Experience. Argument concludes a question, but it does not make us feel certain, unless the truth be also found in experience."

On this basis the old Logic may be clearly distinguished from the new, taking as the general aim of Logic the protection of the mind against the errors to which it is liable in the acquisition of knowledge.

All knowledge, broadly speaking, comes either from Authority, i.e., by argument from accepted premisses, or from Experience. If it comes from Authority it comes through the medium of words: if it comes from Experience it comes through the senses. In taking in knowledge through words we are liable to certain errors; and in taking in knowledge through the senses we are liable to certain errors. To protect against the one is the main end of "Deductive" Logic: to protect against the other is the main end of "Inductive" Logic. As a matter of fact the pith of treatises on Deduction and Induction is directed to those ends respectively, the old meanings of Deduction and Induction as formal processes (to be explained afterwards) being virtually ignored.

There is thus no antagonism whatever between the two branches of Logic. They are directed to different ends. The one is supplementary to the other. The one cannot supersede the other.

Aristotelian Logic can never become superfluous as long as men are apt to be led astray by words. Its ultimate business is to safeguard in the interpretation of the tradition of language. The mere syllogistic, the bare forms of equivalent or consistent expression, have a very limited utility, as we shall see. But by cogent sequence syllogism leads to proposition, and proposition to term, and term to a close study of the relations between words and thoughts and things.

[Footnote 1:] We know for certain—and it is one of the evidences of the importance attached to this trivial-looking pastime—that two of the great teacher's logical treatises, the Topics and the Sophistical Refutations, were written especially for the guidance of Questioners and Respondents. The one instructs the disputant how to qualify himself methodically for discussion before an ordinary audience, when the admissions extracted from the respondent are matters of common belief, the questioner's skill being directed to make it appear that the respondent's position is inconsistent with these. The other is a systematic exposure of sophistical tricks, mostly verbal quibbles, whereby a delusive appearance of victory in debate may be obtained. But in the concluding chapter of the Elenchi, where Aristotle claims not only that his method is superior to the empirical methods of rival teachers, but that it is entirely original, it is the Syllogism upon which he lays stress as his peculiar and chief invention. The Syllogism, the pure forms of which are expounded in his Prior Analytics, is really the centre of Aristotle's logical system, whether the propositions to which it is applied are matters of scientific truth as in the Posterior Analytics, or matters of common opinion as in the Topics. The treatise on Interpretation, i.e., the interpretation of the Respondent's "Yes" and "No," is preliminary to the Syllogism, the reasoning of the admissions together. Even in the half-grammatical half-logical treatise on the Categories, the author always keeps an eye on the Syllogistic analysis.

[Footnote 2:] Theætetus, 151 E.

[Footnote 3:] Gorgias, 473 D.

[Footnote 4:] Hipparchus, 225 A.

[Footnote 5:] In its leading and primary use, this was a mode of debate, a duel of wits, in which two men engaged before an audience. But the same form could be used, and was used, notably by Socrates, not in an eristic spirit but as a means of awakening people to the consequences of certain admissions or first principles, and thus making vague knowledge explicit and clear. The mind being detained on proposition after proposition as assent was given to it, dialectic was a valuable instrument of instruction and exposition. But whatever the purpose of the exercise, controversial triumph, or solid grounding in the first principles—"the evolution of in-dwelling conceptions"—the central interest lay in the syllogising or reasoning together of the separately assumed or admitted propositions.

[Footnote 6:] Like every other fashion, Yes-and-No Dialectic had its period, its rise and fall. The invention of it is ascribed to Zeno the Eleatic, the answering and questioning Zeno, who flourished about the middle of the fifth century B.C. Socrates (469-399) was in his prime at the beginning of the great Peloponnesian War when Pericles died in 429. In that year Plato was born, and lived to 347, "the olive groves of Academe" being established centre of his teaching from about 386 onwards. Aristotle (384-322), who was the tutor of Alexander the Great, established his school at the Lyceum when Alexander became king in 336 and set out on his career of conquest. That Yes-and-No Dialectic was then a prominent exercise, his logical treatises everywhere bear witness. The subsequent history of the game is obscure. It is probable that Aristotle's thorough exposition of its legitimate arts and illegitimate tricks helped to destroy its interest as an amusement.

[Footnote 7:] Hamilton's Lectures, iii. p. 37.

II.—LOGIC AS A PREVENTIVE OF ERROR OR FALLACY.—THE INNER SOPHIST.

Why describe Logic as a system of defence against error? Why say that its main end and aim is the organisation of reason against confusion and falsehood? Why not rather say, as is now usual, that its end is the attainment of truth? Does this not come to the same thing?

Substantially, the meaning is the same, but the latter expression is more misleading. To speak of Logic as a body of rules for the investigation of truth has misled people into supposing that Logic claims to be an art of Discovery, that it claims to lay down rules by simply observing which investigators may infallibly arrive at new truths. Now, this does not hold even of the Logic of Induction, still less of the older Logic, the precise relation of which to truth will become apparent as we proceed. It is only by keeping men from going astray and by disabusing them when they think they have reached their destination that Logic helps men on the road to truth. Truth often lies hid in the centre of a maze, and logical rules only help the searcher onwards by giving him warning when he is on the wrong track and must try another. It is the searcher's own impulse that carries him forward: Logic does not so much beckon him on to the right path as beckon him back from the wrong. In laying down the conditions of correct interpretation, of valid argument, of trustworthy evidence, of satisfactory explanation, Logic shows the inquirer how to test and purge his conclusions, not how to reach them.

To discuss, as is sometimes done, whether Fallacies lie within the proper sphere of Logic, is to obscure the real connexion between Fallacies and Logic. It is the existence of Fallacies that calls Logic into existence; as a practical science Logic is needed as a protection against Fallacies. Such historically is its origin. We may, if we like, lay down an arbitrary rule that a treatise on Logic should be content to expound the correct forms of interpretation and reasoning and should not concern itself with the wrong. If we take this view we are bound to pronounce Fallacies extra-logical. But to do so is simply to cripple the usefulness of Logic as a practical science. The manipulation of the bare logical forms, without reference to fallacious departures from them, is no better than a nursery exercise. Every correct form in Logic is laid down as a safeguard against some erroneous form to which men are prone, whether in the interpretation of argument or the interpretation of experience, and the statement and illustration of the typical forms of wrong procedure should accompany pari passu the exposition of the right procedure.

In accordance with this principle, I shall deal with special fallacies, special snares or pitfalls—misapprehension of words, misinterpretation of propositions, misunderstanding of arguments, misconstruction of facts, evidences, or signs—each in connexion with its appropriate safeguard. This seems to me the most profitable method. But at this stage, it may be worth while, by way of emphasising the need for Logic as a science of rational belief, to take a survey of the most general tendencies to irrational belief, the chief kinds of illusion or bias that are rooted in the human constitution. We shall then better appreciate the magnitude of the task that Logic attempts in seeking to protect reason against its own fallibility and the pressure of the various forces that would usurp its place.

It is a common notion that we need Logic to protect us against the arts of the Sophist, the dishonest juggler with words and specious facts. But in truth the Inner Sophist, whose instruments are our own inborn propensities to error, is a much more dangerous enemy. For once that we are the victims of designing Sophists, we are nine times the victims of our own irrational impulses and prejudices. Men generally deceive themselves before they deceive others.

Francis Bacon drew attention to these inner perverting influences, these universal sources of erroneous belief, in his De Augmentis and again in his Novum Organum, under the designation of Idola, (εἴδωλα) deceptive appearances of truth, illusions. His classification of Idola—Idola Tribus, illusions common to all men, illusions of the race; Idola Specus, personal illusions, illusions peculiar to the "den" in which each man lives; Idola Fori, illusions of conversation, vulgar prejudices embodied in words; Idola Theatri, illusions of illustrious doctrine, illusions imposed by the dazzling authority of great names—is defective as a classification inasmuch as the first class includes all the others, but like all his writings it is full of sagacious remarks and happy examples. Not for the sake of novelty, but because it is well that matters so important should be presented from more than one point of view, I shall follow a division adapted from the more scientific, if less picturesque, arrangement of Professor Bain, in his chapter on the Fallacious Tendencies of the Human Mind.[¹]

The illusions to which we are all subject may best be classified according to their origin in the depths of our nature. Let us try to realise how illusory beliefs arise.

What is a belief? One of the uses of Logic is to set us thinking about such simple terms. An exhaustive analysis and definition of belief is one of the most difficult of psychological problems. We cannot enter upon that: let us be content with a few simple characters of belief.

First, then, belief is a state of mind. Second, this state of mind is outward-pointing: it has a reference beyond itself, a reference to the order of things outside us. In believing, we hold that the world as it is, has been, or will be, corresponds to our conceptions of it. Third, belief is the guide of action: it is in accordance with what we believe that we direct our activities. If we want to know what a man really believes, we look at his action. This at least is the clue to what he believes at the moment. "I cannot," a great orator once said, "read the minds of men." This was received with ironical cheers. "No," he retorted, "but I can construe their acts." Promoters of companies are expected to invest their own money as a guarantee of good faith. If a man says he believes the world is coming to an end in a year, and takes a lease of a house for fifteen years, we conclude that his belief is not of the highest degree of strength.

The close connexion of belief with our activities, enables us to understand how illusions, false conceptions of reality, arise. The illusions of Feeling and the illusions of Custom are well understood, but other sources of illusion, which may be designated Impatient Impulse and Happy Exercise, are less generally recognised. An example or two will show what is meant. We cannot understand the strength of these perverting influences till we realise them in our own case. We detect them quickly enough in others. Seeing that in common speech the word illusion implies a degree of error amounting almost to insanity, and the illusions we speak of are such as no man is ever quite free from, it is perhaps less startling to use the word bias.

The Bias of Impatient Impulse.

As a being formed for action, not only does healthy man take a pleasure in action, physical and mental, for its own sake, irrespective of consequences, but he is so charged with energy that he cannot be comfortable unless it finds a free vent. In proportion to the amount and excitability of his energy, restraint, obstruction, delay is irksome, and soon becomes a positive and intolerable pain. Any bar or impediment that gives us pause is hateful even to think of: the mere prospect annoys and worries.

Hence it arises that belief, a feeling of being prepared for action, a conviction that the way is clear before us for the free exercise of our activities, is a very powerful and exhilarating feeling, as much a necessity of happy existence as action itself. We see this when we consider how depressing and uncomfortable a condition is the opposite state to belief, namely, doubt, perplexity, hesitation, uncertainty as to our course. And realising this, we see how strong a bias we have in this fact of our nature, this imperious inward necessity for action; how it urges us to act without regard to consequences, and to jump at beliefs without inquiry. For, unless inquiry itself is our business, a self-sufficient occupation, it means delay and obstruction.

This ultimate fact of our nature, this natural inbred constitutional impatience, explains more than half of the wrong beliefs that we form and persist in. We must have a belief of some kind: we cannot be happy till we get it, and we take up with the first that seems to show the way clear. It may be right or it may be wrong: it is not, of course, necessarily always wrong: but that, so far as we are concerned, is a matter of accident. The pressing need for a belief of some sort, upon which our energies may proceed in anticipation at least, will not allow us to stop and inquire. Any course that offers a relief from doubt and hesitation, any conviction that lets the will go free, is eagerly embraced.

It may be thought that this can apply only to beliefs concerning the consequences of our own personal actions, affairs in which we individually play a part. It is from them, no doubt, that our nature takes this set: but the habit once formed is extended to all sorts of matters in which we have no personal interest. Tell an ordinary Englishman, it has been wittily said, that it is a question whether the planets are inhabited, and he feels bound at once to have a confident opinion on the point. The strength of the conviction bears no proportion to the amount of reason spent in reaching it, unless it may be said that as a general rule the less a belief is reasoned the more confidently it is held.

"A grocer," writes Mr. Bagehot in an acute essay on "The Emotion of Conviction,"[²] "has a full creed as to foreign policy, a young lady a complete theory of the Sacraments, as to which neither has any doubt. A girl in a country parsonage will be sure that Paris never can be taken, or that Bismarck is a wretch." An attitude of philosophic doubt, of suspended judgment, is repugnant to the natural man. Belief is an independent joy to him.

This bias works in all men. While there is life, there is pressure from within on belief, tending to push reason aside. The force of the pressure, of course, varies with individual temperament, age, and other circumstances. The young are more credulous than the old, as having greater energy: they are apt, as Bacon puts it, to be "carried away by the sanguine element in their temperament". Shakespeare's Laertes is a study of the impulsive temperament, boldly contrasted with Hamlet, who has more discourse of reason. When Laertes hears that his father has been killed, he hurries home, collects a body of armed sympathisers, bursts into the presence of the king, and threatens with his vengeance—the wrong man. He never pauses to make inquiry: like Hotspur he is "a wasp-stung and impatient fool"; he must wreak his revenge on somebody, and at once. Hamlet's father also has been murdered, but his reason must be satisfied before he proceeds to his revenge, and when doubtful proof is offered, he waits for proof more relative.

Bacon's Idola Tribus and Dr. Bain's illustrations of incontinent energy, are mostly examples of unreasoning intellectual activity, hurried generalisations, unsound and superficial analogies, rash hypotheses. Bacon quotes the case of the sceptic in the temple of Poseidon, who, when shown the offerings of those who had made vows in danger and been delivered, and asked whether he did not now acknowledge the power of the god, replied: "But where are they who made vows and yet perished?" This man answered rightly, says Bacon. In dreams, omens, retributions, and such like, we are apt to remember when they come true and to forget the cases when they fail. If we have seen but one man of a nation, we are apt to conclude that all his countrymen are like him; we cannot suspend our judgment till we have seen more. Confident belief, as Dr. Bain remarks, is the primitive attitude of the human mind: it is only by slow degrees that this is corrected by experience. The old adage, "Experience teaches fools," has a meaning of its own, but in one sense it is the reverse of the truth. The mark of a fool is that he is not taught by experience, and we are all more or less intractable pupils, till our energies begin to fail.

The Bias of Happy Exercise.

If an occupation is pleasant in itself, if it fully satisfies our inner craving for action, we are liable to be blinded thereby to its consequences. Happy exercise is the fool's Paradise. The fallacy lies not in being content with what provides a field for the full activity of our powers: to be content in such a case may be the height of wisdom: but the fallacy lies in claiming for our occupation results, benefits, utilities that do not really attend upon it. Thus we see subjects of study, originally taken up for some purpose, practical, artistic, or religious, pursued into elaborate detail far beyond their original purpose, and the highest value, intellectual, spiritual, moral, claimed for them by their votaries, when in truth they merely serve to consume so much vacant energy, and may be a sheer waste of time that ought to be otherwise employed.

But as I am in danger of myself furnishing an illustration of this bias—it is nowhere more prevalent than in philosophy—I will pass to our next head.

The Bias of the Feelings.

This source of illusion is much more generally understood. The blinding and perverting influence of passion on reason has been a favourite theme with moralists ever since man began to moralise, and is acknowledged in many a popular proverb. "Love is blind;" "The wish is father to the thought;" "Some people's geese are all swans;" and so forth.

We need not dwell upon the illustration of it. Fear and Sloth magnify dangers and difficulties; Affection can see no imperfection in its object: in the eyes of Jealousy a rival is a wretch. From the nature of the case we are much more apt to see examples in others than in ourselves. If the strength of this bias were properly understood by everybody, the mistake would not so often be committed of suspecting bad faith, conscious hypocrisy, when people are found practising the grossest inconsistencies, and shutting their eyes apparently in deliberate wilfulness to facts held under their very noses. Men are inclined to ascribe this human weakness to women. Reasoning from feeling is said to be feminine logic. But it is a human weakness.

To take one very powerful feeling, the feeling of self-love or self-interest—this operates in much more subtle ways than most people imagine, in ways so subtle that the self-deceiver, however honest, would fail to be conscious of the influence if it were pointed out to him. When the slothful man saith, There is a lion in the path, we can all detect the bias to his belief, and so we can when the slothful student says that he will work hard to-morrow, or next week, or next month; or when the disappointed man shows an exaggerated sense of the advantages of a successful rival or of his own disadvantages. But self-interest works to bias belief in much less palpable ways than those. It is this bias that accounts for the difficulty that men of antagonistic interests have in seeing the arguments or believing in the honesty of their opponents. You shall find conferences held between capitalists and workmen in which the two sides, both represented by men incapable of consciously dishonest action, fail altogether to see the force of each other's arguments, and are mutually astonished each at the other's blindness.

The Bias of Custom.

That custom, habits of thought and practice, affect belief, is also generally acknowledged, though the strength and wide reach of the bias is seldom realised. Very simple cases of unreasoning prejudice were adduced by Locke, who was the first to suggest a general explanation of them in the "Association of Ideas" (Human Understanding, bk. ii. ch. xxxiii.). There is, for instance, the fear that overcomes many people when alone in the dark. In vain reason tells them that there is no real danger; they have a certain tremor of apprehension that they cannot get rid of, because darkness is inseparably connected in their minds with images of horror. Similarly we contract unreasonable dislikes to places where painful things have happened to us. Equally unreasoning, if not unreasonable, is our attachment to customary doctrines or practices, and our invincible antipathy to those who do not observe them.

Words are very common vehicles for the currency of this kind of prejudice, good or bad meanings being attached to them by custom. The power of words in this way is recognised in the proverb: "Give a dog a bad name, and then hang him". These verbal prejudices are Bacon's Idola Fori, illusions of conversation. Each of us is brought up in a certain sect or party, and accustomed to respect or dishonour certain sectarian or party names, Whig, Tory, Radical, Socialist, Evolutionist, Broad, Low, or High Church. We may meet a man without knowing under what label he walks and be charmed with his company: meet him again when his name is known, and all is changed.

Such errors are called Fallacies of Association to point to the psychological explanation. This is that by force of association certain ideas are brought into the mind, and that once they are there, we cannot help giving them objective reality. For example, a doctor comes to examine a patient, and finds certain symptoms. He has lately seen or heard of many cases of influenza, we shall say; influenza is running in his head. The idea once suggested has all the advantage of possession.

But why is it that a man cannot get rid of an idea? Why does it force itself upon him as a belief? Association, custom, explains how it got there, but not why it persists in staying.

To explain this we must call in our first fallacious principle, the Impatience of Doubt or Delay, the imperative inward need for a belief of some sort.

And this leads to another remark, that though for convenience of exposition, we separate these various influences, they are not separated in practice. They may and often do act all together, the Inner Sophist concentrating his forces.

Finally, it may be asked whether, seeing that illusions are the offspring of such highly respectable qualities as excess of energy, excess of feeling, excess of docility, it is a good thing for man to be disillusioned. The rose-colour that lies over the world for youth is projected from the abundant energy and feeling within: disillusion comes with failing energies, when hope is "unwilling to be fed". Is it good then to be disillusioned? The foregoing exposition would be egregiously wrong if the majority of mankind did not resent the intrusion of Reason and its organising lieutenant Logic. But really there is no danger that this intrusion succeeds to the extent of paralysing action and destroying feeling, and uprooting custom. The utmost that Logic can do is to modify the excess of these good qualities by setting forth the conditions of rational belief. The student who masters those conditions will soon see the practical wisdom of applying his knowledge only in cases where the grounds of rational belief are within his reach. To apply it to the consequences of every action would be to yield to that bias of incontinent activity which is, perhaps, our most fruitful source of error.

[Footnote 1:] Bain's Logic, bk. vi. chap. iii. Bacon intended his Idola to bear the same relation to his Novum Organum that Aristotle's Fallacies or Sophistical Tricks bore to the old Organum. But in truth, as I have already indicated, what Bacon classifies is our inbred tendencies to form idola or false images, and it is these same tendencies that make us liable to the fallacies named by Aristotle. Some of Aristotle's, as we shall see, are fallacies of Induction.

[Footnote 2:] Bagehot's Literary Studies, ii. 427.

III.—THE AXIOMS OF DIALECTIC AND OF SYLLOGISM.

There are certain principles known as the Laws of Thought or the Maxims of Consistency. They are variously expressed, variously demonstrated, and variously interpreted, but in one form or another they are often said to be the foundation of all Logic. It is even said that all the doctrines of Deductive or Syllogistic Logic may be educed from them. Let us take the most abstract expression of them, and see how they originated. Three laws are commonly given, named respectively the Law of Identity, the Law of Contradiction and the Law of Excluded Middle.

1. The Law of Identity. A is A. Socrates is Socrates. Guilt is guilt.

2. The Law of Contradiction. A is not not-A. Socrates is not other than Socrates. Guilt is not other than guilt. Or A is not at once b and not-b. Socrates is not at once good and not-good. Guilt is not at once punishable and not-punishable.

3. The Law of Excluded Middle. Everything is either A or not-A; or, A is either b or not-b. A given thing is either Socrates or not-Socrates, either guilty or not-guilty. It must be one or the other: no middle is possible.

Why lay down principles so obvious, in some interpretations, and so manifestly sophistical in others? The bare forms of modern Logic have been reached by a process of attenuation from a passage in Aristotle's Metaphysics[¹] (iii. 3, 4, 1005b – 1008). He is there laying down the first principle of demonstration, which he takes to be that "it is impossible that the same predicate can both belong, and not belong, to the same subject, at the same time, and in the same sense".[²] That Socrates knows grammar, and does not know grammar—these two propositions cannot both be true at the same time, and in the same sense. Two contraries cannot exist together in the same subject. The double answer Yes and No cannot be given to one and the same question understood in the same sense.

But why did Aristotle consider it necessary to lay down a principle so obvious? Simply because among the subtle dialecticians who preceded him the principle had been challenged. The Platonic dialogue Euthydemus shows the farcical lengths to which such quibbling was carried. The two brothers vanquish all opponents, but it is by claiming that the answer No does not preclude the answer Yes. "Is not the honourable honourable, and the base base?" asks Socrates. "That is as I please," replies Dionysodorus. Socrates concludes that there is no arguing with such men: they repudiate the first principles of dialectic.

There were, however, more respectable practitioners who canvassed on more plausible grounds any form into which ultimate doctrines about contraries and contradictions, truth and falsehood, could be put, and therefore Aristotle considered it necessary to put forth and defend at elaborate length a statement of a first principle of demonstration. "Contradictions cannot both be true of the same subject at the same time and in the same sense." This is the original form of the Law of Contradiction.

The words "of the same subject," "at the same time," and "in the same sense," are carefully chosen to guard against possible quibbles. "Socrates knows grammar." By Socrates we must mean the same individual man. And even of the same man the assertion may be true at one time and not at another. There was a time when Socrates did not know grammar, though he knows it now. And the assertion may be true in one sense and not in another. It may be true that Socrates knows grammar, yet not that he knows everything that is to be known about grammar, or that he knows as much as Aristarchus.

Aristotle acknowledges that this first principle cannot itself be demonstrated, that is, deduced from any other. If it is denied, you can only reduce the denier to an absurdity. And in showing how to proceed in so doing, he says you must begin by coming to an agreement about the words used, that they signify the same for one and the other disputant.[³] No dialectic is possible without this understanding. This first principle of Dialectic is the original of the Law of Identity. While any question as to the truth or falsehood of a question is pending, from the beginning to the end of any logical process, the words must continue to be accepted in the same sense. Words must have an identical reference to things.

Incidentally in discussing the Axiom of Contradiction (ἀξίωμα τἢς ἀντιφάσεως),[⁴] Aristotle lays down what is now known as the Law of Excluded Middle. Of two contradictories one or other must be true: we must either affirm or deny any one thing of any other: no mean or middle is possible.

In their origin, then, these so-called Laws of Thought were simply the first principles of Dialectic and Demonstration. Consecutive argument, coherent ratiocination, is impossible unless they are taken for granted.

If we divorce or abstract them from their original application, and consider them merely as laws of thinking or of being, any abstract expression, or illustration, or designation of them may easily be pushed into antagonism with other plain truths or first principles equally rudimentary. Without entering into the perplexing and voluminous discussion to which these laws have been subjected by logicians within the last hundred years, a little casuistry is necessary to enable the student to understand within what limits they hold good.

Socrates is Socrates. The name Socrates is a name for something to which you and I refer when we use the name. Unless we have the same reference, we cannot hold any argument about the thing, or make any communication one to another about it.

But if we take Socrates is Socrates to mean that, "An object of thought or thing is identical with itself," "An object of thought or thing cannot be other than itself," and call this a law of thought, we are met at once by a difficulty. Thought, properly speaking, does not begin till we pass beyond the identity of an object with itself. Thought begins only when we recognise the likeness between one object and others. To keep within the self-identity of the object is to suspend thought. "Socrates was a native of Attica," "Socrates was a wise man," "Socrates was put to death as a troubler of the commonweal"—whenever we begin to think or say anything about Socrates, to ascribe any attributes to him, we pass out of his self-identity into his relations of likeness with other men, into what he has in common with other men.

Hegelians express this plain truth with paradoxical point when they say: "Of any definite existence or thought, therefore, it may be said with quite as much truth that it is not, as that it is, its own bare self".[⁵] Or, "A thing must other itself in order to be itself". Controversialists treat this as a subversion of the laws of Identity and Contradiction. But it is only Hegel's fun—his paradoxical way of putting the plain truth that any object has more in common with other objects than it has peculiar to itself. Till we enter into those aspects of agreement with other objects, we cannot truly be said to think at all. If we say merely that a thing is itself, we may as well say nothing about it. To lay down this is not to subvert the Law of Identity, but to keep it from being pushed to the extreme of appearing to deny the Law of Likeness, which is the foundation of all the characters, attributes, or qualities of things in our thoughts.

That self-same objects are like other self-same objects, is an assumption distinct from the Law of Identity, and any interpretation of it that excludes this assumption is to be repudiated. But does not the law of Identity as well as the law of the likeness of mutually exclusive identities presuppose that there are objects self-same, like others, and different from others? Certainly: this is one of the presuppositions of Logic.[⁶] We assume that the world of which we talk and reason is separated into such objects in our thoughts. We assume that such words as Socrates represent individual objects with a self-same being or substance; that such words as wisdom, humour, ugliness, running, sitting, here, there, represent attributes, qualities, characters or predicates of individuals; that such words as man represent groups or classes of individuals.

Some logicians in expressing the Law of Identity have their eye specially upon the objects signified by general names or abstract names, man, education.[⁷] "A concept is identical with the sum of its characters," or, "Classes are identical with the sum of the individuals composing them". The assumptions thus expressed in technical language which will hereafter be explained are undoubtedly assumptions that Logic makes: but since they are statements of the internal constitution of some of the identities that words represent, to call them the Law of Identity is to depart confusingly from traditional usage.[⁸]

That throughout any logical process a word must signify the same object, is one proposition: that the object signified by a general name is identical with the sum of the individuals to each of whom it is applicable, or with the sum of the characters that they bear in common, is another proposition. Logic assumes both: Aristotle assumed both: but it is the first that is historically the original of all expressions of the Law of Identity in modern text-books.

Yet another expression of a Law of Identity which is really distinct from and an addition to Aristotle's original. Socrates was an Athenian, a philosopher, an ugly man, an acute dialectician, etc. Let it be granted that the word Socrates bears the same signification throughout all these and any number more of predicates, we may still ask: "But what is it that Socrates signifies?" The title Law of Identity is sometimes given[⁹] to a theory on this point. Socrates is Socrates. "An individual is the identity running through the totality of its attributes." Is this not, it may be asked, to confuse thought and being, to resolve Socrates into a string of words? No: real existence is one of the admissible predicates of Socrates: one of the attributes under which we conceive him. But whether we accept or reject this "Law of Identity," it is an addition to Aristotle's dialectical "law of identity"; it is a theory of the metaphysical nature of the identity signified by a Singular name. And the same may be said of yet another theory of Identity, that, "An individual is identical with the totality of its predicates," or (another way of putting the same theory), "An individual is a conflux of generalities".

To turn next to the Laws of Contradiction and Excluded Middle. These also may be subjected to Casuistry, making clearer what they assert by showing what they do not deny.

They do not deny that things change, and that successive states of the same thing may pass into one another by imperceptible degrees. A thing may be neither here nor there: it may be on the passage from here to there: and, while it is in motion, we may say, with equal truth, that it is neither here nor there, or that it is both here and there. Youth passes gradually into age, day into night: a given man or a given moment may be on the borderland between the two.

Logic does not deny the existence of indeterminate margins: it merely lays down that for purposes of clear communication and coherent reasoning the line must be drawn somewhere between b, and not-b.

A difference, however, must be recognised between logical negation and the negations of common thought and common speech. The latter are definite to a degree with which the mere Logic of Consistency does not concern itself. To realise this is to understand more clearly the limitations of Formal Logic.

In common speech, to deny a quality of anything is by implication to attribute to it some other quality of the same kind. Let any man tell me that "the streets of such and such a town are not paved with wood," I at once conclude that they are paved with some other material. It is the legitimate effect of his negative proposition to convey this impression to my mind. If, proceeding on this, I go on to ask: "Then they are paved with granite or asphalt, or this or that?" and he turns round and says: "I did not say they were paved at all," I should be justified in accusing him of a quibble. In ordinary speech, to deny one kind of pavement is to assert pavement of some kind. Similarly, to deny that So-and-so is not in the Twenty-first Regiment, is to imply that he is in another regiment, that he is in the army in some regiment. To retort upon this inference: "He is not in the army at all," is a quibble: as much so as it would be to retort: "There is no such person in existence".

Now Logic does not take account of this implication, and nothing has contributed more to bring upon it the reproach of quibbling. In Logic, to deny a quality is simply to declare a repugnance between it and the subject; negation is mere sublation, taking away, and implies nothing more. Not-b is entirely indefinite: it may cover anything except b.

Is Logic then really useless, or even misleading, inasmuch as it ignores the definite implication of negatives in ordinary thought and speech? In ignoring this implication, does Logic oppose this implication as erroneous? Does Logic shelter the quibbler who trades upon it? By no means: to jump to this conclusion were a misunderstanding. The fact only is that nothing beyond the logical Law of Contradiction needs to be assumed for any of the processes of Formal Logic. Aristotle required to assume nothing more for his syllogistic formulæ, and Logic has not yet included in its scope any process that requires any further assumption. "If not-b represent everything except b, everything outside b, then that A is b, and that A is not-b, cannot both be true, and one or other of them must be true."

Whether the scope of Logic ought to be extended is another question. It seems to me that the scope of Logic may legitimately be extended so as to take account both of the positive implication of negatives and the negative implication of positives. I therefore deal with this subject in a separate chapter following on the ordinary doctrines of Immediate Inference, where I try to explain the simple Law of Thought involved. When I say that the extension is legitimate, I mean that it may be made without departing from the traditional view of Logic as a practical science, conversant with the nature of thought and its expression only in so far as it can provide practical guidance against erroneous interpretations and inferences. The extension that I propose is in effect an attempt to bring within the fold of Practical Logic some of the results of the dialectic of Hegel and his followers, such as Mr. Bradley and Mr. Bosanquet, Professor Caird and Professor Wallace.[¹⁰]

The logical processes formulated by Aristotle are merely stages in the movement of thought towards attaining definite conceptions of reality. To treat their conclusions as positions in which thought may dwell and rest, is an error, against which Logic itself as a practical science may fairly be called upon to guard. It may even be conceded that the Aristotelian processes are artificial stages, courses that thought does not take naturally, but into which it has to be forced for a purpose. To concede this is not to concede that the Aristotelian logic is useless, as long as we have reason on our side in holding that thought is benefited and strengthened against certain errors by passing through those artificial stages.

[Footnote 1:] The first statement of the Law of Identity in the form Ens est ens is ascribed by Hamilton (Lectures, iii. 91) to Antonius Andreas, a fourteenth century commentator on the Metaphysics. But Andreas is merely expounding what Aristotle sets forth in iii. 4, 1006 a, b. Ens est ens does not mean in Andreas what A is A means in Hamilton.

[Footnote 2:] τὸ γὰρ αὐτὸ ἅμα ὑπάρχειν τε καὶ μὴ ὑπάρχειν ἀδύνατον τῷ αὐτῷ καὶ κατὰ τὸ αὐτὸ, . . . αὕτη δὴ πασῶν ἐστὶ βεβαιοτάτη τῶν ἀρχῶν. iii. 3, 1005b, 19-23.

[Footnote 3:] Hamilton credits Andreas with maintaining, "against Aristotle," that "the principle of Identity, and not the principle of Contradiction, is the one absolutely first". Which comes first, is a scholastic question on which ingenuity may be exercised. But in fact Aristotle put the principle of Identity first in the above plain sense, and Andreas only expounded more formally what Aristotle had said.

[Footnote 4:] Μεταξὑ ὰντιφάσεως ἐνδέχεται εἶναι οὐθέν, ἀλλ᾿ ἀνάγκη ἢ φάναι ἢ ὰποφάναι ἒν καθ᾿ ἑνὸς ὁτιοῦν. Metaph. iii. 7, 1011b, 23-4.

[Footnote 5:] Prof. Caird's Hegel, p. 138.

[Footnote 6:] See Venn, Empirical Logic, 1-8.

[Footnote 7:] E.g., Hamilton, lect. v.; Veitch's Institutes of Logic, chaps, xii., xiii.

[Footnote 8:] The confusion probably arises in this way. First, these "laws" are formulated as laws of thought that Logic assumes. Second, a notion arises that these laws are the only postulates of Logic: that all logical doctrines can be "evolved" from them. Third, when it is felt that more than the identical reference of words or the identity of a thing with itself must be assumed in Logic, the Law of Identity is extended to cover this further assumption.

[Footnote 9:] E.g., Bosanquet's Logic, ii. 207.

[Footnote 10:] Bradley, Principles of Logic; Bosanquet, Logic or The Morphology of Knowledge; Caird, Hegel (in Blackwood's Philosophical Classics); Wallace, The Logic of Hegel.

BOOK I.

THE LOGIC OF CONSISTENCY. SYLLOGISM AND DEFINITION.

PART I.

THE ELEMENTS OF PROPOSITIONS.

Chapter I.

GENERAL NAMES AND ALLIED DISTINCTIONS.

To discipline us against the errors we are liable to in receiving knowledge through the medium of words—such is one of the objects of Logic, the main object of what may be called the Logic of Consistency.

Strictly speaking, we may receive knowledge about things through signs or single words, as a nod, a wink, a cry, a call, a command. But an assertory sentence, proposition, or predication, is the unit with which Logic concerns itself—a sentence in which a subject is named and something is said or predicated about it. Let a man once understand the errors incident to this regular mode of communication, and he may safely be left to protect himself against the errors incident to more rudimentary modes.

A proposition, whether long or short, is a unit, but it is an analysable unit. And the key to syllogistic analysis is the General Name. Every proposition, every sentence in which we convey knowledge to another, contains a general name or its equivalent. That is to say, every proposition may be resolved into a form in which the predicate is a general name. A knowledge of the function of this element of speech is the basis of all logical discipline. Therefore, though we must always remember that the proposition is the real unit of speech, and the general name only an analytic element, we take the general name and its allied distinctions in thought and reality first.

How propositions are analysed for syllogistic purposes will be shown by-and-by, but we must first explain various technical terms that logicians have devised to define the features of this cardinal element. The technical terms Class, Concept, Notion, Attribute, Extension or Denotation, Intension or Connotation, Genus, Species, Differentia, Singular Name, Collective Name, Abstract Name, all centre round it.

A general name is a name applicable to a number of different things on the ground of some likeness among them, as man, ratepayer, man of courage, man who fought at Waterloo.

From the examples it will be seen that a general name logically is not necessarily a single word. Any word or combination of words that serves a certain function is technically a general name. The different ways of making in common speech the equivalent of a general name logically are for the grammarian to consider.

In the definition of a general name attention is called to two distinct considerations, the individual objects to each of which the name is applicable, and the points of resemblance among them, in virtue of which they have a common name. For those distinctions there are technical terms.

Class is the technical term for the objects, different yet agreeing, to each of which a general name may be applied.

The points of resemblance are called the common attributes of the class.

A class may be constituted on one attribute or on several. Ratepayer, woman ratepayer, unmarried woman ratepayer; soldier, British soldier, British soldier on foreign service. But every individual to which the general name can be applied must possess the common attribute or attributes.

These common attributes are also called the Notion of the class, inasmuch as it is these that the mind notes or should note when the general name is applied. Concept is a synonym perhaps in more common use than notion; the rationale of this term (derived from con and capere, to take or grasp together) being that it is by means of the points of resemblance that the individuals are grasped or held together by the mind. These common points are the one in the many, the same amidst the different, the identity signified by the common name. The name of an attribute as thought of by itself without reference to any individual or class possessing it, is called an Abstract name. By contradistinction, the name of an individual or a class is Concrete.

Technical terms are wanted also to express the relation of the individuals and the attributes to the general name. The individuals jointly are spoken of as the Denotation, or Extension or Scope of the name; the common attributes as its Connotation, Intension, Comprehension, or Ground. The whole denotation, etc., is the class; the whole connotation, etc., is the concept.[¹] The limits of a "class" in Logic are fixed by the common attributes. Any individual object that possesses these is a member. The statement of them is the Definition.

To predicate a general name of any object, as, "This is a cat," "This is a very sad affair," is to refer that object to a class, which is equivalent to saying that it has certain features of resemblance with other objects, that it reminds us of them by its likeness to them. Thus to say that the predicate of every proposition is a general name, expressed or implied, is the same as to say that every predication may be taken as a reference to a class.

Ordinarily our notion or concept of the common features signified by general names is vague and hazy. The business of Logic is to make them clear. It is to this end that the individual objects of the class are summoned before the mind. In ordinary thinking there is no definite array or muster of objects: when we think of "dog" or "cat," "accident," "book," "beggar," "ratepayer," we do not stop to call before the mind a host of representatives of the class, nor do we take precise account of their common attributes. The concept of "house" is what all houses have in common. To make this explicit would be no easy matter, and yet we are constantly referring objects to the class "house". We shall see presently that if we wish to make the connotation or concept clear we must run over the denotation or class, that is to say, the objects to which the general name is applied in common usage. Try, for example, to conceive clearly what is meant by house, tree, dog, walking-stick. You think of individual objects, so-called, and of what they have in common.

A class may be constituted on one property or on many. There are several points common to all houses, enclosing walls, a roof, a means of exit and entrance. For the full concept of the natural kinds, men, dogs, mice, etc., we should have to go to the natural historian.

Degrees of generality. One class is said to be of higher generality than another when it includes that other and more. Thus animal includes man, dog, horse, etc.; man includes Aryan, Semite, etc.; Aryan includes Hindoo, Teuton, Celt, etc.

The technical names for higher and lower classes are Genus and Species. These terms are not fixed as in Natural History to certain grades, but are purely relative one to another, and movable up and down a scale of generality. A class may be a species relatively to one class, which is above it, and a genus relatively to one below it. Thus Aryan is a species of the genus man, Teuton a species of the genus Aryan.

In the graded divisions of Natural History genus and species are fixed names for certain grades. Thus: Vertebrates form a "division"; the next subdivision, e.g., Mammals, Birds, Reptiles, etc., is called a "class"; the next, e.g., Rodents, Carnivora, Ruminants, an "order"; the next, e.g., Rats, Squirrels, Beavers, a "genus"; the next, e.g., Brown rats, Mice, a "species".

Vertebrates (division).

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Mammals, Birds, Reptiles, etc. (class).

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Rodents, Ruminants, Carnivors, etc. (order).

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Rats, Squirrels, Beavers, etc. (genus).

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Brown rats, Mice, etc. (species).

If we subdivide a large class into smaller classes, and, again, subdivide these subdivisions, we come at last to single objects.

Men

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Europeans, Asiatics, etc.

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Englishmen, Frenchmen, etc.

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John Doe, Richard Roe, etc.

A table of higher and lower classes arranged in order has been known from of old as a tree of division or classification. The following is Porphyry's "tree":—

The single objects are called Individuals, because the division cannot be carried farther. The highest class is technically the Summum Genus, or Genus generalissimum; the next highest class to any species is the Proximum Genus; the lowest group before you descend to individuals is the Infima Species, or Species specialissima.

The attribute or attributes whereby a species is distinguished from other species of the same genus, is called its differentia or differentiæ. The various species of houses are differentiated by their several uses, dwelling-house, town-house, ware-house, public-house. Poetry is a species of Fine Art, its differentia being the use of metrical language as its instrument.

A lower class, indicated by the name of its higher class qualified by adjectives or adjective phrases expressing its differential property or properties, is said to be described per genus et differentiam. Examples: "Black-bird," "note-book," "clever man," "man of Kent," "eminent British painter of marine subjects". By giving a combination of attributes common to him with nobody else, we may narrow down the application of a name to an individual: "The Commander-in-Chief of the British forces at the battle of Waterloo".

Other attributes of classes as divided and defined, have received technical names.

An attribute common to all the individuals of a class, found in that class only, and following from the essential or defining attributes, though not included among them, is called a Proprium.

An attribute that belongs to some, but not to all, or that belongs to all, but is not a necessary consequence of the essential attributes, is called an Accident.

The clearest examples of Propria are found in mathematical figures. Thus, the defining property of an equilateral triangle is the equality of the sides: the equality of the angles is a proprium. That the three angles of a triangle are together equal to two right angles is a proprium, true of all triangles, and deducible from the essential properties of a triangle.

Outside Mathematics, it is not easy to find propria that satisfy the three conditions of the definition. It is a useful exercise of the wits to try for such. Educability—an example of the proprium in mediæval text-books—is common to men, and results from man's essential constitution; but it is not peculiar; other animals are educable. That man cooks his food is probably a genuine proprium.

That horses run wild in Thibet: that gold is found in California: that clergymen wear white ties, are examples of Accidents. Learning is an accident in man, though educability is a proprium.

What is known technically as an Inseparable Accident, such as the black colour of the crow or the Ethiopian, is not easy to distinguish from the Proprium. It is distinguished only by the third character, deducibility from the essence.[²]

Accidents that are both common and peculiar are often useful for distinguishing members of a class. Distinctive dresses or badges, such as the gown of a student, the hood of a D.D., are accidents, but mark the class of the individual wearer. So with the colours of flowers.

Genus, Species, Differentia, Proprium, and Accidens have been known since the time of Porphyry as the Five Predicables. They are really only terms used in dividing and defining. We shall return to them and endeavour to show that they have no significance except with reference to fixed schemes, scientific or popular, of Division or Classification.

Given such a fixed scheme, very nice questions may be raised as to whether a particular attribute is a defining attribute, or a proprium, or an accident, or an inseparable accident. Such questions afford great scope for the exercise of the analytic intellect.

We shall deal more particularly with degrees of generality when we come to Definition. This much has been necessary to explain an unimportant but much discussed point in Logic, what is known as the inverse variation of Connotation and Denotation.

Connotation and Denotation are often said to vary inversely in quantity. The larger the connotation the smaller the denotation, and vice versâ. With certain qualifications the statement is correct enough, but it is a rough compendious way of expressing the facts and it needs qualification.

The main fact to be expressed is that the more general a name is, the thinner is its meaning. The wider the scope, the shallower the ground. As you rise in the scale of generality, your classes are wider but the number of common attributes is less. Inversely, the name of a species has a smaller denotation than the name of its genus, but a richer connotation. Fruit-tree applies to fewer objects than tree, but the objects denoted have more in common: so with apple and fruit-tree, Ribston Pippin and apple.

Again, as a rule, if you increase the connotation you contract the area within which the name is applicable. Take any group of things having certain attributes in common, say, men of ability: add courage, beauty, height of six feet, chest measurement of 40 inches, and with each addition fewer individuals are to be found possessing all the common attributes.

This is obvious enough, and yet the expression inverse variation is open to objection. For the denotation may be increased in a sense without affecting the connotation. The birth of an animal may be said to increase the denotation: every year thousands of new houses are built: there are swarms of flies in a hot summer and few in a cold. But all the time the connotation of animal, house, or fly remains the same: the word does not change its meaning.

It is obviously wrong to say that they vary in inverse proportion. Double or treble the number of attributes, and you do not necessarily reduce the denotation by one-half or one-third.

It is, in short, the meaning or connotation that is the main thing. This determines the application of a word. As a rule if you increase meaning, you restrict scope. Let your idea, notion, or concept of culture be a knowledge of Mathematics, Latin and Greek: your men of culture will be more numerous than if you require from each of them these qualifications plus a modern language, an acquaintance with the Fine Arts, urbanity of manners, etc.

It is just possible to increase the connotation without decreasing the denotation, to thicken or deepen the concept without diminishing the class. This is possible only when two properties are exactly co-extensive, as equilaterality and equiangularity in triangles.

Singular and Proper Names. A Proper or Singular name is a name used to designate an individual. Its function, as distinguished from that of the general name, is to be used purely for the purpose of distinctive reference.

A man is not called Tom or Dick because he is like in certain respects to other Toms or other Dicks. The Toms or the Dicks do not form a logical class. The names are given purely for purposes of distinction, to single out an individual subject. The Arabic equivalent for a Proper name, alam, "a mark," "a sign-post," is a recognition of this.

In the expressions "a Napoleon," "a Hotspur," "a Harry," the names are not singular names logically, but general names logically, used to signify the possession of certain attributes.

A man may be nicknamed on a ground, but if the name sticks and is often used, the original meaning is forgotten. If it suggests the individual in any one of his qualities, any point in which he resembles other individuals, it is no longer a Proper or Singular name logically, that is, in logical function. That function is fulfilled when it has called to mind the individual intended.

To ask, as is sometimes done, whether Proper names are connotative or denotative, is merely a confusion of language. The distinction between connotation and denotation, extension and intension, applies only to general names. Unless a name is general, it has neither extension nor intension:[³] a Proper or Singular name is essentially the opposite of a general name and has neither the one nor the other.

A nice distinction may be drawn between Proper and Singular names, though they are strict synonyms for the same logical function. It is not essential to the discharge of that function that the name should be strictly appropriated to one object. There are many Toms and many Dicks. It is enough that the word indicates the individual without confusion in the particular circumstances.

This function may be discharged by words and combinations of words that are not Proper in the grammatical sense. "This man," "the cover of this book," "the Prime Minister of England," "the seer of Chelsea," may be Singular names as much as Honolulu or Lord Tennyson.

In common speech Singular names are often manufactured ad hoc by taking a general name and narrowing it down by successive qualifications till it applies only to one individual, as "The leading subject of the Sovereign of England at the present time". If it so happens that an individual has some attribute or combination peculiar to himself, he may be suggested by the mention of that attribute or combination:—"the inventor of the steam-engine," "the author of Hudibras".

Have such names a connotation? The student may exercise his wits on the question. It is a nice one, an excellent subject of debate. Briefly, if we keep rigid hold of the meaning of connotation, this Singular name has none. The combination is a singular name only when it is the subject of a predication or an attribution, as in the sentences, "The position of the leading subject of etc., is a difficult one," or "The leading subject of etc., wears an eyeglass". In such a sentence as "So-and-so is the leading subject of etc.," the combined name has a connotation, but then it is a general and not a singular name.

Collective Names, as distinguished from General Names. A collective name is a name for a number of similar units taken as a whole—a name for a totality of similar units, as army, regiment, mob, mankind, patrimony, personal estate.

A group or collection designated by a collective name is so far like a class that the individual objects have something in common: they are not heterogeneous but homogeneous. A mob is a collection of human beings; a regiment of soldiers; a library of books.

The distinction lies in this, that whatever is said of a collective name is said about the collection as a whole, and does not apply to each individual; whatever is said of a general name applies to each individual. Further, the collective name can be predicated only of the whole group, as a whole; the general name is predicable of each, distributively. "Mankind has been in existence for thousands of years;" "The mob passed through the streets." In such expressions as "An honest man's the noblest work of God," the subject is functionally a collective name.

A collective name may be used as a general name when it is extended on the ground of what is common to all such totalities as it designates. "An excited mob is dangerous;" "An army without discipline is useless." The collective name is then "connotative" of the common characters of the collection.

Material or Substantial Names. The question has been raised whether names of material, gold, water, snow, coal, are general or collective singular. In the case of pieces or bits of a material, it is true that any predicate made concerning the material, such as "Sugar is sweet," or "Water quenches thirst," applies to any and every portion. But the separate portions are not individuals in the whole signified by a material name as individuals are in a class. Further, the name of material cannot be predicated of a portion as a class name can be of an individual. We cannot say, "This is a sugar". When we say, "This is a piece of sugar," sugar is a collective name for the whole material. There are probably words on the borderland between general names and collective names. In such expressions as "This is a coal," "The bonnie water o' Urie," the material name is used as a general name. The real distinction is between the distributive use and the collective use of a name; as a matter of grammatical usage, the same word may be used either way, but logically in any actual proposition it must be either one or the other.

Abstract Names are names for the common attributes or concepts on which classes are constituted. A concrete name is a name directly applicable to an individual in all his attributes, that is, as he exists in the concrete. It may be written on a ticket and pinned to him. When we have occasion to speak of the point or points in which a number of individuals resemble one another, we use what is called an abstract name. "Generous man," "clever man," "timid man," are concrete names; "generosity," "cleverness," "timidity," are abstract names.

It is disputed whether abstract names are connotative. The question is a confused one: it is like asking whether the name of a town is municipal. An abstract name is the name of a connotation as a separate object of thought or reference, conceived or spoken of in abstraction from individual accidents. Strictly speaking it is notative rather than connotative: it cannot be said to have a connotation because it is itself the symbol of what is called the connotation of a general name.[⁴]

The distinction between abstract names and concrete names is virtually a grammatical distinction, that is, a distinction in mode of predication. We may use concrete names or abstract names at our pleasure to express the same meaning. To say that "John is a timid man" is the same thing as saying that "Timidity is one of the properties or characteristics or attributes of John". "Pride and cruelty generally go together;" "Proud men are generally cruel men."

General names are predicable of individuals because they possess certain attributes: to predicate the possession of those attributes is the same thing as to predicate the general name.

Abstract forms of predication are employed in common speech quite as frequently as concrete, and are, as we shall see, a great source of ambiguity and confusion.

[Footnote 1:] It has been somewhat too hastily assumed on the authority of Mansel (Note to Aldrich, pp. 16, 17) that Mill inverted the scholastic tradition in his use of the word Connotative. Mansel puts his statement doubtfully, and admits that there was some licence in the use of the word Connotative, but holds that in Scholastic Logic an adjective was said to "signify primarily the attribute, and to connote or signify secondarily (προσσημαίνειν ) the subject of inhesion". The truth is that Mansel's view was a theory of usage not a statement of actual usage, and he had good reason for putting it doubtfully.

As a matter of fact, the history of the distinction follows the simple type of increasing precision and complexity, and Mill was in strict accord with standard tradition. By the Nominalist commentators on the Summulæ of Petrus Hispanus certain names, adjectives grammatically, are called Connotativa as opposed to Absoluta, simply because they have a double function. White is connotative as signifying both a subject, such as Socrates, of whom "whiteness" is an attribute, and an attribute "whiteness": the names "Socrates" and "whiteness" are Absolute, as having but a single signification. Occam himself speaks of the subject as the primary signification, and the attribute as the secondary, because the answer to "What is white?" is "Something informed with whiteness," and the subject is in the nominative case while the attribute is in an oblique case (Logic, part I. chap. x.). Later on we find that Tataretus (Expositio in Summulas, A.D. 1501), while mentioning (Tract. Sept. De Appellationibus) that it is a matter of dispute among Doctores whether a connotative name connotat the subject or the attribute, is perfectly explicit in his own definition, "Terminus connotativus est qui præter illud pro quo supponit connotat aliquid adjacere vel non adjacere rei pro qua supponit" (Tract. Sept. De Suppositionibus). And this remained the standard usage as long as the distinction remained in logical text-books. We find it very clearly expressed by Clichtoveus, a Nominalist, quoted as an authority by Guthutius in his Gymnasium Speculativum, Paris, 1607 (De Terminorum Cognitione, pp. 78-9). "Terminus absolutus est, qui solum illud pro quo in propositione supponit, significat. Connotativus autem, qui ultra idipsum, aliud importat." Thus man and animal are absolute terms, which simply stand for (supponunt pro) the things they signify. White is a connotative name, because "it stands for (supponit pro) a subject in which it is an accident: and beyond this, still signifies an accident, which is in that subject, and is expressed by an abstract name". Only Clichtoveus drops the verb connotat, perhaps as a disputable term, and says simply ultra importat.

So in the Port Royal Logic (1662), from which possibly Mill took the distinction: "Les noms qui signifient les choses comme modifiées, marquant premièrement et directement la chose, quoique plus confusément, et indirectement le mode, quoique plus distinctement, sont appelés adjectifs ou connotatifs; comme rond, dur, juste, prudent" (part i. chap ii.).

What Mill did was not to invert Scholastic usage but to revive the distinction, and extend the word connotative to general names on the ground that they also imported the possession of attributes. The word has been as fruitful of meticulous discussion as it was in the Renaissance of Logic, though the ground has changed. The point of Mill's innovation was, premising that general names are not absolute but are applied in virtue of a meaning, to put emphasis on this meaning as the cardinal consideration. What he called the connotation had dropped out of sight as not being required in the Syllogistic Forms. This was as it were the point at which he put in his horn to toss the prevalent conception of Logic as Syllogistic.

The real drift of Mill's innovation has been obscured by the fact that it was introduced among the preliminaries of Syllogism, whereas its real usefulness and significance belongs not to Syllogism in the strict sense but to Definition. He added to the confusion by trying to devise forms of Syllogism based on connotation, and by discussing the Axiom of the Syllogism from this point of view. For syllogistic purposes, as we shall see, Aristotle's forms are perfect, and his conception of the proposition in extension the only correct conception. Whether the centre of gravity in Consistency Logic should not be shifted back from Syllogism to Definition, the latter being the true centre of consistency, is another question. The tendency of Mill's polemic was to make this change. And possibly the secret of the support it has recently received from Mr. Bradley and Mr. Bosanquet is that they, following Hegel, are moving in the same direction.

In effect, Mill's doctrine of Connotation helped to fix a conception of the general name first dimly suggested by Aristotle when he recognised that names of genera and species signify Quality, in showing what sort a thing is. Occam carried this a step farther towards clear light by including among Connotative Terms such general names as "monk," name of classes that at once suggest a definite attribute. The third step was made by Mill in extending the term Connotation to such words as "man," "horse," the Infimæ Species of the Schoolmen, the Species of modern science.

Whether connotation was the best term to use for this purpose, rather than extension, may be questioned: but at least it was in the line of tradition through Occam.

[Footnote 2:] The history of the definition of the Proprium is an example of the tendency of distinctions to become more minute and at the same time more purposeless. Aristotle's ῐδιον was an attribute, such as the laugh of the man or the bark of the dog, common to all of a class and peculiar to the class (quod convenit omni soli et semper) yet not comprised in the definition of the class. Porphyry recognised three varieties of ῐδια besides this, four in all, as follows:—(1) an attribute peculiar to a species but not possessed by all, as knowledge of medicine or geometry; (2) possessed by a whole species but not peculiar to it, as being a biped in man; (3) peculiar to a species, and possessed by all at a certain time, as turning grey in old age; (4) Aristotle's "proprium," peculiar and possessed by all, as risibility. The idea of the Proprium as deducible from or consequent on the essence would seem to have originated in the desire to find something common to all Poryphyry's four varieties.

[Footnote 3:] It is a plausible contention that in the case of the Singular name the extension is at a minimum and the intension at a maximum, the extension being one individual, and the intension the totality of his attributes. But this is an inexact and confused use of words. A name does not extend beyond the individual except when it is used to signify one or more of his prominent qualities, that is, is used with the function of a general name. The extension of a Singular name is zero: it has no extension. On the other hand, it suggests, in its function as a Singular name, no properties or qualities; it suggests only a subject; i.e., it has no intension. The ambiguity of the term Denotation helps the confusion in the case of Singular names. "Denote" in common speech means to indicate, to distinguish. But when in Logic we say that a general name denotes individuals, we have no thought of indicating or distinguishing: we mean only that it is applicable to any one, without respect of individuals, either in predication or epithetic description.

[Footnote 4:] Strictly speaking, as I have tried to indicate all along, the words Connotation and Denotation, or Extension and Intension, apply only to general names. Outside general names, they have no significance. An adjective with its noun is a general name, of which the adjective gives part of the Connotation. If we apply the word connotation to signify merely the suggestion of an attribute in whatever grammatical connexion, then an abstract name is undoubtedly as much connotative as an adjective. The word Sweetness has the same meaning as Sweet: it indicates or signifies, conveys to the mind of the reader the same attribute: the only difference is that it does not at the same time indicate a subject in which the attribute is found, as sweet apple. The meaning is not connoted.

Chapter II.

THE SYLLOGISTIC ANALYSIS OF PROPOSITIONS INTO TERMS.

I.—The Bare Analytic Forms.

The word "term" is loosely used as a mere synonym for a name: strictly speaking, a term (ὅρος, a boundary) is one of the parts of a proposition as analysed into Subject and Predicate. In Logic, a term is a technical word in an analysis made for a special purpose, that purpose being to test the mutual consistency of propositions.

For this purpose, the propositions of common speech may be viewed as consisting of two Terms, a linkword called the copula (positive or negative) expressing a relation between them, and certain symbols of quantity used to express that relation more precisely.

Let us indicate the Subject term by S, and the Predicate term by P.

All propositions may be analysed into one or other of four forms:—

All S is P,

No S is P,

Some S is P,

Some S is not P.

All S is P is called the Universal Affirmative, and is indicated by the symbol A (the first vowel of Affirmo).

No S is P is called the Universal Negative, symbol E (the first vowel of Nego).

Some S is P is called the Particular Affirmative, symbol I (the second vowel of affIrmo).

Some S is not P is called the Particular Negative, symbol O (the second vowel of negO).

The distinction between Universal and Particular is called a distinction in Quantity; between Affirmative and Negative, a distinction in Quality. A and E, I and O, are of the same quantity, but of different quality: A and I, E and O, same in quality, different in quantity.

In this symbolism, no provision is made for expressing degrees of particular quantity. Some stands for any number short of all: it may be one, few, most, or all but one. The debates in which Aristotle's pupils were interested turned mainly on the proof or disproof of general propositions; if only a proposition could be shown to be not universal, it did not matter how far or how little short it came. In the Logic of Probability, the degree becomes of importance.

Distinguish, in this Analysis, to avoid subsequent confusion, between the Subject and the Subject Term, the Predicate and the Predicate Term. The Subject is the Subject Term quantified: in A and E,[¹] "All S"; in I and O, "Some S". The Predicate is the Predicate Term with the Copula, positive or negative: in A and I, "is P"; in E and O, "is not P".

It is important also, in the interest of exactness, to note that S and P, with one exception, represent general names. They are symbols for classes. P is so always: S also except when the Subject is an individual object. In the machinery of the Syllogism, predications about a Singular term are treated as Universal Affirmatives. "Socrates is a wise man" is of the form All S is P.

S and P being general names, the signification of the symbol "is" is not the same as the "is" of common speech, whether the substantive verb or the verb of incomplete predication. In the syllogistic form, "is" means is contained in, "is not," is not contained in.

The relations between the terms in the four forms are represented by simple diagrams known as Euler's circles.

Diagram 5 is a purely artificial form, having no representative in common speech. In the affirmations of common speech, P is always a term of greater extent than S.

No. 2 represents the special case where S and P are coextensive, as in All equiangular triangles are equilateral.

S and P being general names, they are said to be distributed when the proposition applies to them in their whole extent, that is, when the assertion covers every individual in the class.

In E, the Universal Negative, both terms are distributed: "No S is P" wholly excludes the two classes one from the other, imports that not one individual of either is in the other.

In A, S is distributed, but not P. S is wholly in P, but nothing is said about the extent of P beyond S.

In O, S is undistributed, P is distributed. A part of S is declared to be wholly excluded from P.

In I, neither S nor P is distributed.

It will be seen that the Predicate term of a Negative proposition is always distributed, of an Affirmative, always undistributed.

A little indistinctness in the signification of P crept into mediæval text-books, and has tended to confuse modern disputation about the import of Predication. Unless P is a class name, the ordinary doctrine of distribution is nonsense; and Euler's diagrams are meaningless. Yet many writers who adopt both follow mediæval usage in treating P as the equivalent of an adjective, and consequently "is" as identical with the verb of incomplete predication in common speech.

It should be recognised that these syllogistic forms are purely artificial, invented for a purpose, namely, the simplification of syllogising. Aristotle indicated the precise usage on which his syllogism is based (Prior Analytics, i. 1 and 4). His form[²] for All S is P, is S is wholly in P; for No S is P, S is wholly not in P. His copula is not "is," but "is in," and it is a pity that this usage was not kept. "All S is in P" would have saved much confusion. But, doubtless for the sake of simplicity, the besetting sin of tutorial handbooks, All S is P crept in instead, illustrated by such examples as "All men are mortal".

Thus the "is" of the syllogistic form became confused with the "is" of common speech, and the syllogistic view of predication as being equivalent to inclusion in, or exclusion from a class, was misunderstood. The true Aristotelian doctrine is not that predication consists in referring subjects to classes, but only that for certain logical purposes it may be so regarded. The syllogistic forms are artificial forms. They were not originally intended to represent the actual processes of thought expressed in common speech. To argue that when I say "All crows are black," I do not form a class of black things, and contemplate crows within it as one circle is within another, is to contradict no intelligent logical doctrine.

The root of the confusion lies in quoting sentences from common speech as examples of the logical forms, forgetting that those forms are purely artificial. "Omnis homo est mortalis," "All men are mortal," is not an example formally of All S is P. P is a symbol for a substantive word or combination of words, and mortal is an adjective. Strictly speaking, there is no formal equivalent in common speech, that is, in the forms of ordinary use—no strict grammatical formal equivalent—for the syllogistic propositional symbols. We can make an equivalent, but it is not a form that men would use in ordinary intercourse. "All man is in mortal being" would be a strict equivalent, but it is not English grammar.

Instead of disputing confusedly whether All S is P should be interpreted in extension or in comprehension, it would be better to recognise the original and traditional use of the symbols S and P as class names, and employ other symbols for the expression in comprehension or connotation. Thus, let s and p stand for the connotation. Then the equivalent for All S is P would be All S has p, or p always accompanies s, or p belongs to all S.

It may be said that if predication is treated in this way, Logic is simplified to the extent of childishness. And indeed, the manipulation of the bare forms with the help of diagrams and mnemonics is a very humble exercise. The real discipline of Syllogistic Logic lies in the reduction of common speech to these forms.

This exercise is valuable because it promotes clear ideas about the use of general names in predication, their ground in thought and reality, and the liabilities to error that lurk in this fundamental instrument of speech.

[Footnote 1:] For perfect symmetry, the form of E should be All S is not P. "No S is P" is adopted for E to avoid conflict with a form of common speech, in which All S is not P conveys the meaning of the Particular Negative. "All advices are not safe" does not mean that safeness is denied of all advices, but that safeness cannot be affirmed of all, i.e., Not all advices are safe, i.e., some are not.

[Footnote 2:] His most precise form, I should say, for in "P is predicated of every S" he virtually follows common speech.

II.—The Practice of Syllogistic Analysis.

The basis of the analysis is the use of general names in predication. To say that in predication a subject is referred to a class, is only another way of saying that in every categorical sentence the predicate is a general name express or implied: that it is by means of general names that we convey our thoughts about things to others.

"Milton is a great poet." "Quoth Hudibras, I smell a rat." Great poet is a general name: it means certain qualities, and applies to anybody possessing them. Quoth implies a general name, a name for persons speaking, connoting or meaning a certain act and applicable to anybody in the performance of it. Quoth expresses also past time: thus it implies another general name, a name for persons in past time, connoting a quality which differentiates a species in the genus persons speaking, and making the predicate term "persons speaking in past time". Thus the proposition Quoth Hudibras, analysed into the syllogistic form S is in P, becomes S (Hudibras) is in P (persons speaking in past time). The Predicate term P is a class constituted on those properties. Smell a rat also implies a general name, meaning an act or state predicable of many individuals.

Even if we add the grammatical object of Quoth to the analysis, the Predicate term is still a general name. Hudibras is only one of the persons speaking in past time who have spoken of themselves as being in a certain mood of suspicion.[¹]

The learner may well ask what is the use of twisting plain speech into these uncouth forms. The use is certainly not obvious. The analysis may be directly useful, as Aristotle claimed for it, when we wish to ascertain exactly whether one proposition contradicts another, or forms with another or others a valid link in an argument. This is to admit that it is only in perplexing cases that the analysis is of direct use. The indirect use is to familiarise us with what the forms of common speech imply, and thus strengthen the intellect for interpreting the condensed and elliptical expression in which common speech abounds.

There are certain technical names applied to the components of many-worded general names, Categorematic and Syncategorematic, Subject and Attributive. The distinctions are really grammatical rather than logical, and of little practical value.

A word that can stand by itself as a term is said to be Categorematic. Man, poet, or any other common noun.

A word that can only form part of a term is Syncategorematic. Under this definition come all adjectives and adverbs.

The student's ingenuity may be exercised in applying the distinction to the various parts of speech. A verb may be said to be Hypercategorematic, implying, as it does, not only a term, but also a copula.

A nice point is whether the Adjective is categorematic or syncategorematic. The question depends on the definition of "term" in Logic. In common speech an adjective may stand by itself as a predicate, and so might be said to be Categorematic. "This heart is merry." But if a term is a class, or the name of a class, it is not Categorematic in the above definition. It can only help to specify a class when attached to the name of a higher genus.

Mr. Fowler's words Subject and Attributive express practically the same distinction, except that Attributive is of narrower extent than syncategorematic. An Attributive is a word that connotes an attribute or property, as hot, valorous, and is always grammatically an adjective.

The expression of Quantity, that is, of Universality or non-universality, is all-important in syllogistic formulæ. In them universality is expressed by All or None. In ordinary speech universality is expressed in various forms, concrete and abstract, plain and figurative, without the use of "all" or "none".

Uneasy lies the head that wears a crown.

He can't be wrong whose life is in the right.

What cat's averse to fish?

Can the leopard change his spots?

The longest road has an end.

Suspicion ever haunts the guilty mind.

Irresolution is always a sign of weakness.

Treason never prospers.

A proposition in which the quantity is not expressed is called by Aristotle Indefinite (ἀδιόριστος). For "indefinite"[²] Hamilton suggests Preindesignate, undesignated, that is, before being received from common speech for the syllogistic mill. A proposition is Predesignate when the quantity is definitely indicated. All the above propositions are "Predesignate" universals, and reducible to the form All S is P, or No S is P.

The following propositions are no less definitely particular, reducible to the form I or O. In them as in the preceding quantity is formally expressed, though the forms used are not the artificial syllogistic forms:—

Afflictions are often salutary.

Not every advice is a safe one.

All that glitters is not gold.

Rivers generally[³] run into the sea.

Often, however, it is really uncertain from the form of common speech whether it is intended to express a universal or a particular. The quantity is not formally expressed. This is especially the case with proverbs and loose floating sayings of a general tendency. For example:—

Haste makes waste.

Knowledge is power.

Light come, light go.

Left-handed men are awkward antagonists.

Veteran soldiers are the steadiest in fight.

Such sayings are in actual speech for the most part delivered as universals.[⁴] It is a useful exercise of the Socratic kind to decide whether they are really so. This can only be determined by a survey of facts. The best method of conducting such a survey is probably (1) to pick out the concrete subject, "hasty actions," "men possessed of knowledge," "things lightly acquired"; (2) to fix the attribute or attributes predicated; (3) to run over the individuals of the subject class and settle whether the attribute is as a matter of fact meant to be predicated of each and every one.

This is the operation of Induction. If one individual can be found of whom the attribute is not meant to be predicated, the proposition is not intended as Universal.

Mark the difference between settling what is intended and settling what is true. The conditions of truth and the errors incident to the attempt to determine it, are the business of the Logic of Rational Belief, commonly entitled Inductive Logic. The kind of "induction" here contemplated has for its aim merely to determine the quantity of a proposition in common acceptation, which can be done by considering in what cases the proposition would generally be alleged. This corresponds nearly as we shall see to Aristotelian Induction, the acceptance of a universal statement when no instance to the contrary is alleged.

It is to be observed that for this operation we do not practically use the syllogistic form All S is P. We do not raise the question Is All S, P? That is, we do not constitute in thought a class P: the class in our minds is S, and what we ask is whether an attribute predicated of this class is truly predicated of every individual of it.

Suppose we indicate by p the attribute, knot of attributes, or concept on which the class P is constituted, then All S is P is equivalent to "All S has p": and Has All S p? is the form of a question that we have in our minds when we make an inductive survey on the above method. I point this out to emphasise the fact that there is no prerogative in the form All S is P except for syllogistic purposes.

This inductive survey may be made a useful Collateral Discipline. The bare forms of Syllogistic are a useless item of knowledge unless they are applied to concrete thought. And determining the quantity of a common aphorism or saw, the limits within which it is meant to hold good, is a valuable discipline in exactness of understanding. In trying to penetrate to the inner intention of a loose general maxim, we discover that what it is really intended to assert is a general connexion of attributes, and a survey of concrete cases leads to a more exact apprehension of those attributes. Thus in considering whether Knowledge is power is meant to be asserted of all knowledge, we encounter along with such examples as the sailor's knowledge that wetting a rope shortens it, which enabled some masons to raise a stone to its desired position, or the knowledge of French roads possessed by the German invaders,—along with such examples as these we encounter cases where a knowledge of difficulties without a knowledge of the means of overcoming them is paralysing to action. Samuel Daniel says:—

Where timid knowledge stands considering

Audacious ignorance has done the deed.

Studying numerous cases where "Knowledge is power" is alleged or denied, we find that what is meant is that a knowledge of the right means of doing anything is power—in short, that the predicate is not made of all knowledge, but only of a species of knowledge.

Take, again, Custom blunts sensibility. Putting this in the concrete, and inquiring what predicate is made about "men accustomed to anything" (S), we have no difficulty in finding examples where such men are said to become indifferent to it. We find such illustrations as Lovelace's famous "Paradox":—

Through foul we follow fair

For had the world one face

And earth been bright as air

We had known neither place.

Indians smell not their nest

The Swiss and Finn taste best

The spices of the East.

So men accustomed to riches are not acutely sensible of their advantages: dwellers in noisy streets cease to be distracted by the din: the watchmaker ceases to hear the multitudinous ticking in his shop: the neighbours of chemical works are not annoyed by the smells like the casual passenger. But we find also that wine-tasters acquire by practice an unusual delicacy of sense; that the eyes once accustomed to a dim light begin to distinguish objects that were at first indistinguishable; and so on. What meanings of "custom" and of "sensibility" will reconcile these apparently conflicting examples? What are the exact attributes signified by the names? We should probably find that by sensibility is meant emotional sensibility as distinguished from intellectual discrimination, and that by custom is meant familiarity with impressions whose variations are not attended to, or subjection to one unvarying impression.

To verify the meaning of abstract proverbs in this way is to travel over the road by which the Greek dialecticians were led to feel the importance of definition. Of this more will be said presently. If it is contended that such excursions are beyond the bounds of Formal Logic, the answer is that the exercise is a useful one and that it starts naturally and conveniently from the formulæ of Logic. It is the practice and discipline that historically preceded the Aristotelian Logic, and in the absence of which the Aristotelian formulæ would have a narrowing and cramping effect.

Can all propositions be reduced to the syllogistic form? Probably: but this is a purely scientific inquiry, collateral to Practical Logic. The concern of Practical Logic is chiefly with forms of proposition that favour inaccuracy or inexactness of thought. When there is no room for ambiguity or other error, there is no virtue in artificial syllogistic form. The attempt so to reduce any and every proposition may lead, however, to the study of what Mr. Bosanquet happily calls the "Morphology" of Judgment, Judgment being the technical name for the mental act that accompanies the utterance of a proposition. Even in such sentences as "How hot it is!" or "It rains," the rudiment of subject and predicate may be detected. When a man says "How hot it is," he conveys the meaning, though there is no definitely formed subject in his mind, that the outer world at the moment of his speaking has a certain quality or attribute. So with "It rains". The study of such examples in their context, however, reveals the fact that the same form of Common speech may cover different subjects and predicates in different connexions. Thus in the argument:—

"Whatever is, is best.

It rains!"—

the Subject is Rain and the Predicate is now, "is at the present time," "is in the class of present events".

[Footnote 1:] Remember that when we speak of a general name, we do not necessarily mean a single word. A general name, logically viewed, is simply the name of a genus, kind, or class: and whether this is single-worded or many-worded is, strictly speaking, a grammatical question. "Man," "man-of-ability," "man-of-ability-and-courage," "man-of-ability-and-courage-and-gigantic-stature," "man-who-fought-at-Marathon"—these are all general names in their logical function. No matter how the constitutive properties of the class are indicated, by one word or in combination, that word or combination is a general name. In actual speech we can seldom indicate by a single word the meaning predicated.

[Footnote 2:] The objection taken to the word "indefinite," that the quantity of particular propositions is indefinite, some meaning any quantity less than all, is an example of the misplaced and frivolous subtlety that has done so much to disorder the tradition of Logic. By "indefinite" is simply meant not definitely expressed as either Universal or Particular, Total or Partial. The same objection might be taken to any word used to express the distinction: the degree of quantity in Some S is not "designate" any more than it is "definite" or "dioristic".

[Footnote 3:] Generally. In this word we have an instance of the frequent conflict between the words of common speech and logical terminology. How it arises shall be explained in next chapter. A General proposition is a synonym for a Universal proposition (if the forms A and E are so termed): but "generally" in common speech means "for the most part," and is represented by the symbol of particular quantity, Some.

[Footnote 4:] With some logicians it is a mechanical rule in reducing to syllogistic form to treat as I or O all sentences in which there is no formal expression of quantity. This is to err on the safe side, but common speakers are not so guarded, and it is to be presumed rather that they have a universal application in their minds when they do not expressly qualify.

III.—Some Technical Difficulties.

The formula for Exclusive Propositions. "None but the brave deserve the fair": "No admittance except on business": "Only Protestants can sit on the throne of England".

These propositions exemplify different ways in common speech of naming a subject exclusively, the predication being made of all outside a certain term. "None that are not brave, etc.;" "none that are not on business, etc.;" "none that are not Protestants, etc.". No not-S is P. It is only about all outside the given term that the universal assertion is made: we say nothing universally about the individuals within the term: we do not say that all Protestants are eligible, nor that all persons on business are admitted, nor that every one of the brave deserves the fair. All that we say is that the possession of the attribute named is an indispensable condition: a person may possess the attribute, and yet on other grounds may not be entitled to the predicate.

The justification for taking special note of this form in Logic is that we are apt by inadvertence to make an inclusive inference from it. Let it be said that None but those who work hard can reasonably expect to pass, and we are apt to take this as meaning that all who work hard may reasonably expect to pass. But what is denied of every Not-S is not necessarily affirmed of every S.

The expression of Tense or Time in the Syllogistic Forms. Seeing that the Copula in S is P or S is in P does not express time, but only a certain relation between S and P, the question arises Where are we to put time in the analytic formula? "Wheat is dear;" "All had fled;" time is expressed in these propositions, and our formula should render the whole content of what is given. Are we to include it in the Predicate term or in the Subject term? If it must not be left out altogether, and we cannot put it with the copula, we have a choice between the two terms.

It is a purely scholastic question. The common technical treatment is to view the tense as part of the predicate. "All had fled," All S is P, i.e., the whole subject is included in a class constituted on the attributes of flight at a given time. It may be that the Predicate is solely a predicate of time. "The Board met yesterday at noon." S is P, i.e., the meeting of the Board is one of the events characterised by having happened at a certain time, agreeing with other events in that respect.

But in some cases the time is more properly regarded as part of the subject. E.g., "Wheat is dear". S does not here stand for wheat collectively, but for the wheat now in the market, the wheat of the present time: it is concerning this that the attribute of dearness is predicated; it is this that is in the class of dear things.

The expression of Modality in the Syllogistic Forms. Propositions in which the predicate is qualified by an expression of necessity, contingency, possibility or impossibility [i.e., in English by must, may, can, or cannot], were called in Mediæval Logic Modal Propositions. "Two and two must make four." "Grubs may become butterflies." "Z can paint." "Y cannot fly."

There are two recognised ways of reducing such propositions to the form S is P. One is to distinguish between the Dictum and the Mode, the proposition and the qualification of its certainty, and to treat the Dictum as the Subject and the Mode as the Predicate. Thus: "That two and two make four is necessary"; "That Y can fly is impossible".

The other way is to treat the Mode as part of the predicate. The propriety of this is not obvious in the case of Necessary propositions, but it is unobjectionable in the case of the other three modes. Thus: "Grubs are things that have the potentiality of becoming butterflies"; "Z has the faculty of painting"; "Y has not the faculty of flying".

The chief risk of error is in determining the quantity of the subject about which the Contingent or Possible predicate is made. When it is said that "Victories may be gained by accident," is the predicate made concerning All victories or Some only? Here we are apt to confuse the meaning of the contingent assertion with the matter of fact on which in common belief it rests. It is true only that some victories have been gained by accident, and it is on this ground that we assert in the absence of certain knowledge concerning any victory that it may have been so gained. The latter is the effect of the contingent assertion: it is made about any victory in the absence of certain knowledge, that is to say, formally about all.

The history of Modals in Logic is a good illustration of intricate confusion arising from disregard of a clear traditional definition. The treatment of them by Aristotle was simple, and had direct reference to tricks of disputation practised in his time. He specified four "modes," the four that descended to mediæval logic, and he concerned himself chiefly with the import of contradicting these modals. What is the true contradictory of such propositions as, "It is possible to be" (δυνατὸν εἶναι), "It admits of being" (ἐνδέχεται εἶναι), "It must be" (ἀναγκαῖον εἶναι), "It is impossible to be" (ἀδύνατον εἶναι)? What is implied in saying "No" to such propositions put interrogatively? "Is it possible for Socrates to fly?" "No." Does this mean that it is not possible for Socrates to fly, or that it is possible for Socrates not to fly?

A disputant who had trapped a respondent into admitting that it is possible for Socrates not to fly, might have pushed the concession farther in some such way as this: "Is it possible for Socrates not to walk?" "Certainly." "Is it possible for him to walk?" "Yes." "When you say that it is possible for a man to do anything do you not believe that it is possible for him to do it?" "Yes." "But you have admitted that it is possible for Socrates not to fly?"

It was in view of such perplexities as these that Aristotle set forth the true contradictories of his four Modals. We may laugh at such quibbles now and wonder that a grave logician should have thought them worth guarding against. But historically this is the origin of the Modals of Formal Logic, and to divert the names of them to signify other distinctions than those between modes of qualifying the certainty of a statement is to introduce confusion.

Thus we find "Alexander was a great general," given as an example of a Contingent Modal, on the ground that though as a matter of fact Alexander was so he might have been otherwise. It was not necessary that Alexander should be a great general: therefore the proposition is contingent. Now the distinction between Necessary truth and Contingent truth may be important philosophically: but it is merely confusing to call the character of propositions as one or the other by the name of Modality. The original Modality is a mode of expression: to apply the name to this character is to shift its meaning.

A more simple and obviously unwarrantable departure from tradition is to extend the name Modality to any grammatical qualification of a single verb in common speech. On this understanding "Alexander conquered Darius" is given by Hamilton as a Pure proposition, and "Alexander conquered Darius honourably" as a Modal. This is a merely grammatical distinction, a distinction in the mode of composing the predicate term in common speech. In logical tradition Modality is a mode of qualifying the certainty of an affirmation. "The conquest of Darius by Alexander was honourable," or "Alexander in conquering Darius was an honourable conqueror," is the syllogistic form of the proposition: it is simply assertory, not qualified in any "mode".

There is a similar misunderstanding in Mr. Shedden's treatment of "generally" as constituting a Modal in such sentences, as "Rivers generally flow into the sea". He argues that as generally is not part either of the Subject term or of the Predicate term, it must belong to the Copula, and is therefore a modal qualification of the whole assertion. He overlooked the fact that the word "generally" is an expression of Quantity: it determines the quantity of the Subject term.

Finally it is sometimes held (e.g., by Mr. Venn) that the question of Modality belongs properly to Scientific or Inductive Logic, and is out of place in Formal Logic. This is so far accurate that it is for Inductive Logic to expound the conditions of various degrees of certainty. The consideration of Modality is pertinent to Formal Logic only in so far as concerns special perplexities in the expression of it. The treatment of it by Logicians has been rendered intricate by torturing the old tradition to suit different conceptions of the end and aim of Logic.

PART II.

DEFINITION.

Chapter I.

IMPERFECT UNDERSTANDING OF WORDS AND THE REMEDIES THEREFOR.—DIALECTIC.—DEFINITION.

We cannot inquire far into the meaning of proverbs or traditional sayings without discovering that the common understanding of general and abstract names is loose and uncertain. Common speech is a quicksand.

Consider how we acquire our vocabulary, how we pick up the words that we use from our neighbours and from books, and why this is so soon becomes apparent. Theoretically we know the full meaning of a name when we know all the attributes that it connotes, and we are not justified in extending it except to objects that possess all the attributes. This is the logical ideal, but between the ought to be of Logic and the is of practical life, there is a vast difference. How seldom do we conceive words in their full meaning! And who is to instruct us in the full meaning? It is not as in the exact sciences, where we start with a knowledge of the full meaning. In Geometry, for example, we learn the definitions of the words used, point, line, parallel, etc., before we proceed to use them. But in common speech, we learn words first in their application to individual cases. Nobody ever defined good to us, or fair, or kind, or highly educated. We hear the words applied to individual objects: we utter them in the same connexion: we extend them to other objects that strike us as like without knowing the precise points of likeness that the convention of common speech includes. The more exact meaning we learn by gradual induction from individual cases. Ugly, beautiful, good, bad—we learn the words first as applicable to things and persons: gradually there arises a more or less definite sense of what the objects so designated have in common. The individual's extension of the name proceeds upon what in the objects has most impressed him when he caught the word: this may differ in different individuals; the usage of neighbours corrects individual eccentricities. The child in arms shouts Da at the passing stranger who reminds him of his father: for him at first it is a general name applicable to every man: by degrees he learns that for him it is a singular name.

The mode in which words are learnt and extended may be studied most simply in the nursery. A child, say, has learnt to say mambro when it sees its nurse. The nurse works a hand-turned sewing machine, and sings to it as she works. In the street the child sees an organ-grinder singing as he turns his handle: it calls mambro: the nurse catches the meaning and the child is overjoyed. The organ-grinder has a monkey: the child has an india-rubber monkey toy: it calls this also mambro. The name is extended to a monkey in a picture-book. It has a toy musical box with a handle: this also becomes mambro, the word being extended along another line of resemblance. A stroller with a French fiddle comes within the denotation of the word: a towel-rail is also called mambro from some fancied resemblance to the fiddle. A very swarthy hunch-back mambro frightens the child: this leads to the transference of the word to a terrific coalman with a bag of coals on his back. In a short time the word has become a name for a great variety of objects that have nothing whatever common to all of them, though each is strikingly like in some point to a predecessor in the series. When the application becomes too heterogeneous, the word ceases to be of use as a sign and is gradually abandoned, the most impressive meaning being the last to go. In a child's vocabulary where the word mambro had a run of nearly two years, its last use was as an adjective signifying ugly or horrible.

The history of such a word in a child's language is a type of what goes on in the language of men. In the larger history we see similar extensions under similar motives, checked and controlled in the same way by surrounding usage.

It is obvious that to avoid error and confusion, the meaning or connotation of names, the concepts, should somehow be fixed: names cannot otherwise have an identical reference in human intercourse. We may call this ideal fixed concept the Logical Concept: or we may call it the Scientific Concept, inasmuch as one of the main objects of the sciences is to attain such ideals in different departments of study. But in actual speech we have also the Personal Concept, which varies more or less with the individual user, and the Popular or Vernacular Concept, which, though roughly fixed, varies from social sect to social sect and from generation to generation.

The variations in Popular Concepts may be traced in linguistic history. Words change with things and with the aspects of things, as these change in public interest and importance. As long as the attributes that govern the application of words are simple, sensible attributes, little confusion need arise: the variations are matters of curious research for the philologist, but are logically insignificant. Murray's Dictionary, or such books as Trench's English Past and Present, supply endless examples, as many, indeed as there are words in the language. Clerk has almost as many connotations as our typical mambro: clerk in holy orders, church clerk, town clerk, clerk of assize, grocer's clerk. In Early English, the word meant "man in a religious order, cleric, clergyman"; ability to read, write, and keep accounts being a prominent attribute of the class, the word was extended on this simple ground till it has ceased altogether to cover its original field except as a formal designation. But no confusion is caused by the variation, because the property connoted is simple.[¹] So with any common noun: street, carriage, ship, house, merchant, lawyer, professor. We might be puzzled to give an exact definition of such words, to say precisely what they connote in common usage; but the risk of error in the use of them is small.

When we come to words of which the logical concept is a complex relation, an obscure or intangible attribute, the defects of the popular conception and its tendencies to change and confusion, are of the greatest practical importance. Take such words as Monarchy, tyranny, civil freedom, freedom of contract, landlord, gentleman, prig, culture, education, temperance, generosity. Not merely should we find it difficult to give an analytic definition of such words: we might be unable to do so, and yet flatter ourselves that we had a clear understanding of their meaning. But let two men begin to discuss any proposition in which any such word is involved, and it will often be found that they take the word in different senses. If the relation expressed is complex, they have different sides or lines of it in their minds; if the meaning is an obscure quality, they are guided in their application of it by different outward signs.

Monarchy, in its original meaning, is applied to a form of government in which the will of one man is supreme, to make laws or break them, to appoint or dismiss officers of state and justice, to determine peace or war, without control of statute or custom. But supreme power is never thus uncontrolled in reality; and the word has been extended to cover governments in which the power of the titular head is controlled in many different modes and degrees. The existence of a head, with the title of King or Emperor, is the simplest and most salient fact: and wherever this exists, the popular concept of a monarchy is realised. The President of the United States has more real power than the Sovereign of Great Britain; but the one government is called a Republic and the other a Monarchy. People discuss the advantages and disadvantages of monarchy without first deciding whether they take the word in its etymological sense of unlimited power, or its popular sense of titular kingship, or its logical sense of power definitely limited in certain ways. And often in debate, monarchy is really a singular term for the government of Great Britain.

Culture, religious, generous, are names for inward states or qualities: with most individuals some simple outward sign directs the application of the word—it may be manner, or bearing, or routine observances, or even nothing more significant than the cut of the clothes or of the hair. Small things undoubtedly are significant, and we must judge by small things when we have nothing else to go by: but instead of trying to get definite conceptions for our moral epithets, and suspending judgment till we know that the use of the epithet is justified, the trifling superficial sign becomes for us practically the whole meaning of the word. We feel that we must have a judgment of some sort at once: only simple signs are suited to our impatience.

It was with reference to this state of things that Hegel formulated his paradox that the true abstract thinker is the plain man who laughs at philosophy as what he calls abstract and unpractical. He holds decided opinions for or against this or the other abstraction, freedom, tyranny, revolution, reform, socialism, but what these words mean and within what limits the things signified are desirable or undesirable, he is in too great a hurry to pause and consider.

The disadvantages of this kind of "abstract" thinking are obvious. The accumulated wisdom of mankind is stored in language. Until we have cleared our conceptions, and penetrated to the full meaning of words, that wisdom is a sealed book to us. Wise maxims are interpreted by us hastily in accordance with our own narrow conceptions. All the vocables of a language may be more or less familiar to us, and yet we may not have learnt it as an instrument of thought. Outside the very limited range of names for what we see and use in the daily routine of life, food and clothes and the common occupations of men, words have little meaning for us, and are the vehicles merely of thin preconceptions and raw prejudices.

The remedy for "abstract" thinking is more thinking, and in pursuing this two aims may be specified for the sake of clearness, though they are closely allied, and progress towards both may often be made by one and the same operation. (1) We want to reach a clear and full conception of the meaning of names as used now or at a given time. Let us call this the Verification of the Meaning. (2) We want to fix such conceptions, and if necessary readjust their boundaries. This is the province of Definition, which cannot be effectually performed without Scientific Classification or Division.

I.—Verification of the Meaning—Dialectic.

This can only be done by assembling the objects to which the words are applied, and considering what they have in common. To ascertain the actual connotation we must run over the actual denotation. And since in such an operation two or more minds are better than one, discussion or dialectic is both more fruitful and more stimulating than solitary reflection or reading.

The first to practise this process on a memorable scale, and with a distinct method and purpose, was Socrates. To insist upon the necessity of clear conceptions, and to assist by his dialectic procedure in forming them, was his contribution to philosophy.

His plan was to take a common name, profess ignorance of its meaning, and ask his interlocutor whether he would apply it in such and such an instance, producing one after another. According to Xenophon's Memorabilia he habitually chose the commonest names, good, unjust, fitting, and so forth, and tried to set men thinking about them, and helped them by his questions to form an intelligent conception of the meaning.

For example, what is the meaning of injustice? Would you say that the man who cheats or deceives is unjust? Suppose a man deceives his enemies, is there any injustice in that? Can the definition be that a man who deceives his friends is unjust? But there are cases where friends are deceived for their own good: are these cases of injustice? A general may inspirit his soldiers by a falsehood. A man may cajole a weapon out of his friend's hand when he sees him about to commit suicide. A father may deceive his son into taking medicine. Would you call these men unjust? By some such process of interrogation we are brought to the definition that a man is unjust who deceives his friends to their hurt.

Observe that in much of his dialectic the aim of Socrates was merely to bring out the meaning lying vague and latent, as it were, in the common mind. His object was simply what we have called the verification of the meaning. And a dialectic that confines itself to the consideration of what is ordinarily meant as distinct from what ought to be meant may often serve a useful purpose. Disputes about words are not always as idle as is sometimes supposed. Mr. H. Sidgwick truly remarks (à propos of the terms of Political Economy) that there is often more profit in seeking a definition than in finding it. Conceptions are not merely cleared but deepened by the process. Mr. Sidgwick's remarks are so happy that I must take leave to quote them: they apply not merely to the verification of ordinary meaning but also to the study of special uses by authorities, and the reasons for those special uses.

"The truth is—as most readers of Plato know, only it is a truth difficult to retain and apply—that what we gain by discussing a definition is often but slightly represented in the superior fitness of the formula that we ultimately adopt; it consists chiefly in the greater clearness and fulness in which the characteristics of the matter to which the formula refers have been brought before the mind in the process of seeking for it. While we are apparently aiming at definitions of terms, our attention should be really fixed on distinctions and relations of fact. These latter are what we are concerned to know, contemplate, and as far as possible arrange and systematise; and in subjects where we cannot present them to the mind in ordinary fulness by the exercise of the organs of sense, there is no way of surveying them so convenient as that of reflecting on our use of common terms.... In comparing different definitions our aim should be far less to decide which we ought to adopt, than to apprehend and duly consider the grounds on which each has commended itself to reflective minds. We shall generally find that each writer has noted some relation, some resemblance or difference, which others have overlooked; and we shall gain in completeness, and often in precision, of view by following him in his observations, whether or not we follow him in his conclusions."[²]

Mr. Sidgwick's own discussions of Wealth, Value, and Money are models. A clue is often found to the meaning in examining startlingly discrepant statements connected with the same leading word. Thus we find some authorities declaring that "style" cannot be taught or learnt, while others declare that it can. But on trying to ascertain what they mean by "style," we find that those who say it cannot be taught mean either a certain marked individual character or manner of writing—as in Buffon's saying, Le style c'est l'homme même—or a certain felicity and dignity of expression, while those who say style can be taught mean lucid method in the structure of sentences or in the arrangement of a discourse. Again in discussions on the rank of poets, we find different conceptions of what constitutes greatness in poetry lying at the root of the inclusion of this or the other poet among great poets. We find one poet excluded from the first rank of greatness because his poetry was not serious; another because his poetry was not widely popular; another because he wrote comparatively little; another because he wrote only songs or odes and never attempted drama or epic. These various opinions point to different conceptions of what constitutes greatness in poets, different connotations of "great poet". Comparing different opinions concerning "education" we may be led to ask whether it means more than instruction in the details of certain subjects, whether it does not also import the formation of a disposition to learn or an interest in learning or instruction in a certain method of learning.

Historically, dialectic turning on the use of words preceded the attempt to formulate principles of Definition, and attempts at precise definition led to Division and Classification, that is to systematic arrangement of the objects to be defined. Attempt to define any such word as "education," and you gradually become sensible of the needs in respect of method that forced themselves upon mankind in the history of thought. You soon become aware that you cannot define it by itself alone; that you are beset by a swarm of more or less synonymous words, instruction, discipline, culture, training, and so on; that these various words represent distinctions and relations among things more or less allied; and that, if each must be fixed to a definite meaning, this must be done with reference to one another and to the whole department of things that they cover.

The first memorable attempts at scientific arrangement were Aristotle's treatises on Ethics and Politics, which had been the subjects of active dialectic for at least a century before. That these the most difficult of all departments to subject to scientific treatment should have been the first chosen was due simply to their preponderating interest: "The proper study of mankind is man". The systems of what are known as the Natural Sciences are of modern origin: the first, that of Botany, dates from Cesalpinus in the sixteenth century. But the principles on which Aristotle proceeded in dividing and defining, principles which have gradually themselves been more precisely formulated, are principles applicable to all systematic arrangements for purposes of orderly study. I give them in the precise formulæ which they have gradually assumed in the tradition of Logic. The principles of Division are often given in Formal Logic, and the principles of Classification in Inductive Logic, but there is no valid reason for the separation. The classification of objects in the Natural Sciences, of animals, plants, and stones, with a view to the thorough study of them in form, structure, and function, is more complex than classifications for more limited purposes, and the tendency is to restrict the word classification to these elaborate systems. But really they are only a series of divisions and subdivisions, and the same principles apply to each of the subordinate divisions as well as to the division of the whole department of study.

II.—Principles of Division or Classification and Definition.

Confusion in the boundaries of names arises from confused ideas regarding the resemblances and differences of things. As a protective against this confusion, things must be clearly distinguished in their points of likeness and difference, and this leads to their arrangement in systems, that is, to division and classification. A name is not secure against variation until it has a distinct place in such a system as a symbol for clearly distinguished attributes. Nor must we forget, further, that systems have their day, that the best system attainable is only temporary, and may have to be recast to correspond with changes of things and of man's way of looking at them.

The leading principles of Division may be stated as follows:—

I. Every division is made on the ground of differences in some attribute common to all the members of the whole to be divided.

This is merely a way of stating what a logical division is. It is a division of a generic whole or genus, an indefinite number of objects thought of together as possessing some common character or attribute. All have this attribute, which is technically called the fundamentum divisionis, or generic attribute. But the whole is divisible into smaller groups (species), each of which possesses the common character with a difference (differentia). Thus, mankind may be divided into White men, Black men, Yellow men, on ground of the differences in the colour of their skins: all have skins of some colour: this is the fundamentum divisionis: but each subdivision or species has a different colour: this is the differentia. Rectilineal figures are divided into triangles, quadrangles, pentagons, etc., on the ground of differences in the number of angles.

Unless there is a fund. div., i.e., unless the differences are differences in a common character, the division is not a logical division. To divide men into Europeans, opticians, tailors, blondes, brunettes, and dyspeptics is not to make a logical division. This is seen more clearly in connexion with the second condition of a perfect division.

II. In a perfect division, the subdivisions or species are mutually exclusive.

Every object possessing the common character should be in one or other of the groups, and none should be in more than one.

Confusion between classes, or overlapping, may arise from two causes. It may be due (1) to faulty division, to want of unity in the fundamentum divisionis; (2) to the indistinct character of the objects to be defined.

(1) Unless the division is based upon a single ground, unless each species is based upon some mode of the generic character, confusion is almost certain to arise. Suppose the field to be divided, the objects to be classified, are three-sided rectilineal plane figures, each group must be based upon some modification of the three sides. Divide them into equilateral, isosceles, and scalene according as the three sides are all of equal length, or two of equal length, or each of different length, and you have a perfect division. Similarly you can divide them perfectly according to the character of the angles into acute-angled, right-angled and obtuse-angled. But if you do not keep to a single basis, as in dividing them into equilateral, isosceles, scalene, and right-angled, you have a cross-division. The same triangle might be both right-angled and isosceles.

(2) Overlapping, however, may be unavoidable in practice owing to the nature of the objects. There may be objects in which the dividing characters are not distinctly marked, objects that possess the differentiæ of more than one group in a greater or less degree. Things are not always marked off from one another by hard and fast lines. They shade into one another by imperceptible gradations. A clear separation of them may be impossible. In that case you must allow a certain indeterminate margin between your classes, and sometimes it may be necessary to put an object into more than one class.

To insist that there is no essential difference unless a clear demarcation can be made is a fallacy. A sophistical trick called the Sorites or Heap from the classical example of it was based upon this difficulty of drawing sharp lines of definition. Assuming that it is possible to say how many stones constitute a heap, you begin by asking whether three stones form a heap. If your respondent says No, you ask whether four stones form a heap, then five, and so on and he is puzzled to say when the addition of a single stone makes that a heap which was not a heap before. Or you may begin by asking whether twenty stones form a heap, then nineteen, then eighteen, and so on, the difficulty being to say when what was a heap ceases to be so.

Where the objects classified are mixed states or affections, the products of interacting factors, or differently interlaced or interfused growths from common roots, as in the case of virtues, or emotions, or literary qualities, sharp demarcations are impossible. To distinguish between wit and humour, or humour and pathos, or pathos and sublimity is difficult because the same composition may partake of more than one character. The specific characters cannot be made rigidly exclusive one of another.

Even in the natural sciences, where the individuals are concrete objects of perception, it may be difficult to decide in which of two opposed groups an object should be included. Sydney Smith has commemorated the perplexities of Naturalists over the newly discovered animals and plants of Botany Bay, in especial with the Ornithorynchus,—"a quadruped as big as a large cat, with the eyes, colour, and skin of a mole, and the bill and web-feet of a duck—puzzling Dr. Shaw, and rendering the latter half of his life miserable, from his utter inability to determine whether it was a bird or a beast".

III. The classes in any scheme of division should be of co-ordinate rank.

The classes may be mutually exclusive, and yet the division imperfect, owing to their not being of equal rank. Thus in the ordinary division of the Parts of Speech, parts, that is, of a sentence, Prepositions and Conjunctions are not co-ordinate in respect of function, which is the basis of the division, with Nouns, Adjectives, Verbs, and Adverbs. The preposition is a part of a phrase which serves the same function as an adjective, e.g., royal army, army of the king; it is thus functionally part of a part, or a particle. So with the conjunction: it also is a part of a part, i.e., part of a clause serving the function of adjective or adverb.

IV. The basis of division (fundamentum divisionis) should be an attribute admitting of important differences.

The importance of the attribute chosen as basis may vary with the purpose of the division. An attribute that is of no importance in one division, may be important enough to be the basis of another division. Thus in a division of houses according to their architectural attributes, the number of windows or the rent is of little importance; but if houses are taxed or rated according to the number of windows or the rent, these attributes become important enough to be a basis of division for purposes of taxation or rating. They then admit of important differences.

That the importance is relative to the purpose of the division should be borne in mind because there is a tendency to regard attributes that are of importance in any familiar or pre-eminent division as if they had an absolute importance. In short, disregard of this relativity is a fallacy to be guarded against.

In the sciences, the purpose being the attainment and preservation of knowledge, the objects of study are divided so as to serve that purpose. Groups must be formed so as to bring together the objects that have most in common. The question, Who are to be placed together? in any arrangement for purposes of study, receives the same answer for individuals and for classes that have to be grouped into higher classes, namely, Those that have most in common. This is what Dr. Bain happily calls "the golden rule" of scientific classification: "Of the various groupings of resembling things, preference is given to such as have the greatest number of attributes in common". I slightly modify Dr. Bain's statement: he says "the most numerous and the most important attributes in common". But for scientific purposes number of attributes constitutes importance, as is well recognised by Dr. Fowler when he says that the test of importance in an attribute proposed as a basis of classification is the number of other attributes of which it is an index or invariable accompaniment. Thus in Zoology the squirrel, the rat, and the beaver are classed together as Rodents, the difference between their teeth and the teeth of other Mammalia being the basis of division, because the difference in teeth is accompanied by differences in many other properties. So the hedge-hog, the shrew-mouse, and the mole, though very unlike in outward appearance and habits, are classed together as Insectivora, the difference in what they feed on being accompanied by a number of other differences.

The Principles of Definition. The word "definition" as used in Logic shows the usual tendency of words to wander from a strict meaning and become ambiguous. Throughout most of its uses it retains this much of a common signification, the fixing or determining of the boundaries of a class[³] by making clear its constituent attributes. Now in this making clear two processes may be distinguished, a material process and a verbal process. We have (1) the clearing up of the common attributes by a careful examination of the objects included in the class: and we have (2) the statement of these common attributes in language. The rules of definition given by Dr. Bain, who devotes a separate Book in his Logic to the subject of Definition, concern the first of these processes: the rules more commonly given concern mainly the second.

One eminent merit in Dr. Bain's treatment is that it recognises the close connexion between Definition and Classification. His cardinal rules are reduced to two.

I. Assemble for comparison representative individuals of the class.

II. Assemble for comparison representative individuals of the contrasted class or classes.

Seeing that the contrasted classes are contrasted on some basis of division, this is in effect to recognise that you cannot clearly define any class except in a scheme of classification. You must have a wide genus with its fundamentum divisionis, and, within this, species distinguished by their several differentiæ.

Next, as to the verbal process, rules are commonly laid down mostly of a trifling and obvious character. That "a definition should state neither more nor less than the common attributes of the class," or than the attributes signified by the class-name, is sometimes given as a rule of definition. This is really an explanation of what a definition is, a definition of a definition. And as far as mere statement goes it is not strictly accurate, for when the attributes of a genus are known it is not necessary to give all the attributes of the species, which include the generic attributes as well, but it is sufficient to give the generic name and the differentia. Thus Poetry may be defined as "a Fine Art having metrical language as its instrument". This is technically known as definition per genus et differentiam. This mode of statement is a recognition of the connexion between Definition and Division.

The rule that "a definition should not be a synonymous repetition of the name of the class to be defined," is too obvious to require formal statement. To describe a Viceroy as a man who exercises viceregal functions, may have point as an epigram in the case of a faineant viceroy, but it is not a definition.

So with the rule that "a definition should not be couched in ambiguous unfamiliar, or figurative language". To call the camel "the ship of the desert" is a suggestive and luminous description of a property, but it is not a definition. So with the noble description of Faith as "the substance of things hoped for, the evidence of things not seen". But if one wonders why so obvious a "rule" should be laid down, the answer is that it has its historical origin in the caprices of two classes of offenders, mystical philosophers and pompous lexicographers.[⁴]

That "the definition should be simply convertible with the term for the class defined," so that we may say, for example, either: "Wine is the juice of the grape," or, "The juice of the grape is wine," is an obvious corollary from the nature of definition, but should hardly be dignified with the name of a "rule".

The Principles of Naming. Rules have been formulated for the choice of names in scientific definition and classification, but it may be doubted whether such choice can be reduced to precise rule. It is enough to draw attention to certain considerations obvious enough on reflection.

We may take for granted that there should be distinct names for every defining attribute (a Terminology) and for every group or class (a Nomenclature). What about the selection of the names? Suppose an investigator is struck with likenesses and differences that seem to him important enough to be the basis of a new division, how should he be guided in his choice of names for the new groups that he proposes? Should he coin new names, or should he take old names and try to fit them with new definitions?

The balance of advantages is probably in favour of Dr. Whewell's dictum that "in framing scientific terms, the appropriation of old words is preferable to the invention of new ones". Only care must be taken to keep as close as possible to the current meaning of the old word, and not to run counter to strong associations. This is an obvious precept with a view to avoiding confusion. Suppose, for example, that in dividing Governments you take the distribution of political power as your basis of division and come to the conclusion that the most important differences are whether this power is vested in a few or in the majority of the community. You want names to fix this broad division. You decide instead of coining the new word Pollarchy to express the opposite of Oligarchy to use the old words Republic and Oligarchy. You would find, as Sir George Cornewall Lewis found, that however carefully you defined the word Republic, a division under which the British Government had to be ranked among Republics would not be generally understood and accepted. Using the word in the sense explained above, Mr. Bagehot maintained that the constitution of Great Britain was more Republican than that of the United States, but his meaning was not taken except by a few.

The difficulty in choosing between new words and old words to express new meanings is hardly felt in the exact sciences. It is at least at a minimum. The innovator may encounter violent prejudice, but, arguing with experts, he can at least make sure of being understood, if his new division is based upon real and important differences. But in other subjects the difficulty of transmitting truth or of expressing it in language suited for precise transmission, is almost greater than the difficulty of arriving at truth. Between new names and old names redefined, the possessor of fresh knowledge, assuming it to be perfectly verified, is in a quandary. The objects with which he deals are already named in accordance with loose divisions resting on strong prejudices. The names in current use are absolutely incapable of conveying his meaning. He must redefine them if he is to use them. But in that case he runs the risk of being misunderstood from people being too impatient to master his redefinition. His right to redefine may even be challenged without any reference to the facts to be expressed: he may simply be accused of circulating false linguistic coin, of debasing the verbal currency. The other alternative open to him is to coin new words. In that case he runs the risk of not being read at all. His contribution to verified knowledge is passed by as pedantic and unintelligible. There is no simple rule of safety: between Scylla and Charybdis the mariner must steer as best he may. Practically the advantage lies with old words redefined, because thereby discussion is provoked and discussion clears the air.

Whether it is best to attempt a formal definition or to use words in a private, peculiar, or esoteric sense, and leave this to be gathered by the reader from the general tenor of your utterances, is a question of policy outside the limits of Logic. It is for the logician to expound the method of Definition and the conditions of its application: how far there are subjects that do not admit of its application profitably must be decided on other grounds. But it is probably true that no man who declines to be bound by a formal definition of his terms is capable of carrying them in a clear unambiguous sense through a heated controversy.

[Footnote 1:] Except, perhaps, in new offices to which the name is extended, such as Clerk of School Board. The name, bearing its most simple and common meaning, may cause popular misapprehension of the nature of the duties. Any uncertainty in meaning may be dangerous in practice: elections have been affected by the ambiguity of this word.

[Footnote 2:] Sidgwick's Political Economy, pp. 52-3. Ed. 1883.

[Footnote 3:] Some logicians, however, speak of defining a thing, and illustrate this as if by a thing they meant a concrete individual, the realistic treatment of Universals lending itself to such expressions. But though the authority of Aristotle might be claimed for this, it is better to confine the name in strictness to the main process of defining a class. Since, however, the method is the same whether it is an individual or a class that we want to make distinct, there is no harm in the extension of the word definition to both varieties. See Davidson's Logic of Definition, chap. ii.

[Footnote 4:] See Davidson's Logic of Definition, chap. iii.

Chapter II.

THE FIVE PREDICABLES.—VERBAL AND REAL PREDICATION.

We give a separate chapter to this topic out of respect for the space that it occupies in the history of Logic. But except as an exercise in subtle distinction for its own sake, all that falls to be said about the Predicables might be given as a simple appendix to the chapter on Definition.

Primarily, the Five Predicables or Heads of Predicables—Genus, Species, Differentia, Proprium, and Accidens—are not predicables at all, but merely a list or enumeration of terms used in dividing and defining on the basis of attributes. They have no meaning except in connexion with a fixed scheme of division. Given such a scheme, and we can distinguish in it the whole to be divided (the genus), the subordinate divisions (the species), the attribute or combination of attributes on which each species is constituted (the differentia), and other attributes, which belong to some or all of the individuals but are not reckoned for purposes of division and definition (Propria and Accidentia). The list is not itself a logical division: it is heterogeneous, not homogeneous; the two first are names of classes, the three last of attributes. But corresponding to it we might make a homogeneous division of attributes, as follows:—

The origin of the title Predicables as applied to these five terms is curious, and may be worth noting as an instance of the difficulty of keeping names precise, and of the confusion arising from forgetting the purpose of a name. Porphyry in his εἰσαγωγὴ or Introduction explains the five words (φωναὶ) simply as terms that it is useful for various purposes to know, expressly mentioning definition and division. But he casually remarks that Singular names, "This man," "Socrates," can be predicated only of one individual, whereas Genera, Species, Differentiæ, etc., are predicables of many. That is to say he describes them as Predicables simply by contradistinction from Singular names. A name, however, was wanted for the five terms taken all together, and since they are not a logical division, but merely a list of terms used in dividing and defining, there was no apt general designation for them such as would occur spontaneously. Thus it became the custom to refer to them as the Predicables, a means of reference to them collectively being desiderated, while the meaning of this descriptive title was forgotten.

To call the five divisional elements or Divisoria Predicables is to present them as a division of Predicate Terms on the basis of their relation to the Subject Term: to suggest that every Predicate Term must be either a Genus or a Species, or a Differentia, or a Proprium, or an Accidens of the Subject Term. They are sometimes criticised as such, and it is rightly pointed out that the Predicate is never a species of or with reference to the Subject. But, in truth, the five so-called Predicables were never meant as a division of predicates in relation to the subject: it is only the title that makes this misleading suggestion.

To complete the confusion it so happens that Aristotle used three of the Five terms in what was virtually a division of Predicates inasmuch as it was a division of Problems or Questions. In expounding the methods of Dialectic in the Topica he divided Problems into four classes according to the relation of the Predicate to the Subject. The Predicate must either be simply convertible with the subject or not. If simply convertible, the two must be coextensive, and the Predicate must be either a Proprium or the Definition. If not simply convertible, the Predicate must either be part of the Definition or not. If part of the Definition it must be either a Generic Property or a Differentia (both of which in this connexion Aristotle includes under Genus): if not part of the Definition, it is an Accident. Aristotle thus arrives at a fourfold division of Problems or Predicates:—γένος (Genus, including Differentia, διαφορὰ); ὄρος (Definition); τὸ ἴδιον (Proprium); and τὸ συμβεβηκὸς (Accidens). The object of it was to provide a basis for his systematic exposition; each of the four kinds admitted of differences in dialectic method. For us it is a matter of simple curiosity and ingenuity. It serves as a monument of how much Greek dialectic turned on Definition, and it corresponds exactly to the division of attributes into Defining and Non-defining given above. It is sometimes said that Aristotle showed a more scientific mind than Porphyry in making the Predicables four instead of five. This is true if Porphyry's list had been meant as a division of attributes: but it was not so meant.

The distinction between Verbal or Analytic and Real or Synthetic Predication corresponds to the distinction between Defining and Non-defining attributes, and also has no significance except with reference to some scheme of Division, scientific and precise or loose and popular.

When a proposition predicates of a subject something contained in the full notion, concept, or definition of the subject term, it is called Verbal, Analytic, or Explicative: verbal, inasmuch as it merely explains the meaning of a name; explicative for the same reason; analytic, inasmuch as it unties the bundle of attributes held together in the concept and pays out one, or all one by one.

When the attributes of the Predicate are not contained in the concept of the Subject, the proposition is called Real, Synthetic, or Ampliative, for parallel reasons.

Thus: "A triangle is a three-sided rectilinear figure" is Verbal or Analytic; "Triangles have three angles together equal to two right angles," or "Triangles are studied in schools," is Real or Synthetic.

According to this distinction, predications of the whole Definition or of a Generic attribute or of a Specific attribute are Verbal; predications of Accident are Real. A nice point is whether Propria are Verbal or Real. They can hardly be classed with Verbal, inasmuch as one may know the full meaning of the name without knowing them: but it might be argued that they are Analytic, inasmuch as they are implicitly contained in the defining attributes as being deducible from them.

Observe, however, that the whole distinction is really valid only in relation to some fixed or accepted scheme of classification or division. Otherwise, what is Verbal or Analytic to one man may be Real or Synthetic to another. It might even be argued that every proposition is Analytic to the man who utters it and Synthetic to the man who receives it. We must make some analysis of a whole of thought before paying it out in words: and in the process of apprehending the meaning of what we hear or read we must add the other members of the sentence on to the subject. Whether or not this is super-subtle, it clearly holds good that what is Verbal (in the sense defined) to the learned man of science may be Real to the learner. That the horse has six incisors in each jaw or that the domestic dog has a curly tail, is a Verbal Proposition to the Natural Historian, a mere exposition of defining marks; but the plain man has a notion of horse or dog into which this defining attribute does not enter, and to him accordingly the proposition is Real.

But what of propositions that the plain man would at once recognise as Verbal? Charles Lamb, for example, remarks that the statement that "a good name shows the estimation in which a man is held in the world" is a verbal proposition. Where is the fixed scheme of division there? The answer is that by a fixed scheme of division we do not necessarily mean a scheme that is rigidly, definitely and precisely fixed. To make such schemes is the business of Science. But the ordinary vocabulary of common intercourse as a matter of fact proceeds upon schemes of division, though the names used in common speech are not always scientifically accurate, not always the best that could be devised for the easy acquisition and sure transmission of thorough knowledge. The plain man's vocabulary, though often twisted aside by such causes as we have specified, is roughly moulded on the most marked distinguishing attributes of things. This was practically recognised by Aristotle when he made one of his modes of definition consist in something like what we have called verifying the meaning of a name, ascertaining the attributes that it signifies in common speech or in the speech of sensible men. This is to ascertain the essence, οὐσία, or Substantia, of things, the most salient attributes that strike the common eye either at once or after the closer inspection that comes of long companionship, and form the basis of the ordinary vocabulary. "Properly speaking," Mansel says,[¹] "All Definition is an inquiry into Attributes. Our complex notions of Substances can only be resolved into various Attributes, with the addition of an unknown substratum: a something to which we are compelled to regard these attributes as belonging. Man, for example, is analysed into Animality, Rationality, and the something which exhibits these phenomena. Pursue the analysis and the result is the same. We have a something corporeal, animated, sensible, rational. An unknown constant must always be added to complete the integration." This "unknown constant" was what Locke called the Real Essence, as distinguished from the Nominal Essence, or complex of attributes. It is upon this nominal essence, upon divisions of things according to attributes, that common speech rests, and if it involves many cross-divisions, this is because the divisions have been made for limited and conflicting purposes.

[Footnote 1:] Aldrich's Compendium, Appendix, Note C. The reader may be referred to Mansel's Notes A and C for valuable historical notices of the Predicables and Definition.

Chapter III.

ARISTOTLE'S CATEGORIES.

In deference to tradition a place must be found in every logical treatise for Aristotle's Categories. No writing of the same length has exercised a tithe of its influence on human thought. It governed scholastic thought and expression for many centuries, being from its shortness and consequent easiness of transcription one of the few books in every educated man's library. It still regulates the subdivisions of Parts of Speech in our grammars. Its universality of acceptance is shown in the fact that the words category (κατηγορία) and predicament, its Latin translation, have passed into common speech.

The Categories have been much criticised and often condemned as a division, but, strange to say, few have inquired what they originally professed to be a division of, or what was the original author's basis of division. Whether the basis is itself important, is another question: but to call the division imperfect, without reference to the author's intention, is merely confusing, and serves only to illustrate the fact that the same objects may be differently divided on different principles of division. Ramus was right in saying that the Categories had no logical significance, inasmuch as they could not be made a basis for departments of logical method; and Kant and Mill in saying that they had no philosophical significance, inasmuch as they are founded upon no theory of Knowing and Being: but this is to condemn them for not being what they were never intended to be.

The sentence in which Aristotle states the objects to be divided, and his division of them is so brief and bold that bearing in mind the subsequent history of the Categories, one first comes upon it with a certain surprise. He says simply:—

"Of things expressed without syntax (i.e., single words), each signifies either substance, or quantity, or quality, or relation, or place, or time, or disposition (i.e., attitude or internal arrangement), or appurtenance, or action (doing), or suffering (being done to)."[¹]

The objects, then, that Aristotle proposed to classify were single words (the themata simplicia of the Schoolmen). He explains that by "out of syntax" (ἄνευ συμπλοκῆς)[*] he means without reference to truth or falsehood: there can be no declaration of truth or falsehood without a sentence, a combination, or syntax: "man runs" is either true or false, "man" by itself, "runs" by itself, is neither. His division, therefore, was a division of single words according to their differences of signification, and without reference to the truth or falsehood of their predication.[²]

Signification was thus the basis of division. But according to what differences? The Categories themselves are so abstract that this question might be discussed on their bare titles interminably. But often when abstract terms are doubtful, an author's intention may be gathered from his examples. And when Aristotle's examples are ranged in a table, certain principles of subdivision leap to the eyes. Thus:—

Substance (οὐσία) (Substantia)	Man (ἄνθρωπος)		Common Noun		Substance
Quantity (ποσὸν) (Quantitas) Quality (ποιὸν) (Qualitas) Relation (πρός τι) (Relatio)	Five-feet-five (τρίπηχυ) Scholarly (γραμματικὸν) Bigger (μεῖζον)		Adjective		Permanent Attribute
Place (ποῦ) (Ubi) Time (ποτὲ) (Quando)	In-the-Lyceum (ἐν Λυκείῳ) Yesterday (χθὲς)		Adverb		Temporary Attribute
Disposition (κεῖσθαι) (Positio) Appurtenance (ἔχειν) (Habitus) Action (ποιεῖν) (Actio) Passion (πάσχειν) (Passio)	Reclines (ἀνάκειται) Has-shoes-on (ὑποδέδεται) Cuts (τέμνει) Is cut (τέμνεται)		Verb

In looking at the examples, our first impression is that Aristotle has fallen into a confusion. He professes to classify words out of syntax, yet he gives words with the marks of syntax on them. Thus his division is accidentally grammatical, a division of parts of speech, parts of a sentence, into Nouns, Adjectives, Adverbs, and Verbs. And his subdivisions of these parts are still followed in our grammars. But really it is not the grammatical function that he attends to, but the signification: and looking further at the examples, we see what differences of signification he had in his mind. It is differences relative to a concrete individual, differences in the words applied to him according as they signify the substance of him or his attributes, permanent or temporary.

Take any concrete thing, Socrates, this book, this table. It must be some kind of a thing, a man, a book. It must have some size or quantity, six feet high, three inches broad. It must have some quality, white, learned, hard. It must have relations with other things, half this, double that, the son of a father. It must be somewhere, at some time, in some attitude, with some "havings," appendages, appurtenances, or belongings, doing something, or having something done to it. Can you conceive any name (simple or composite) applicable to any object of perception, whose signification does not fall into one or other of these classes? If you cannot, the categories are justified as an exhaustive division of significations. They are a complete list of the most general resemblances among individual things, in other words, of the summa genera, the genera generalissima of predicates concerning this, that or the other concrete individual. No individual thing is sui generis: everything is like other things: the categories are the most general likenesses.

The categories are exhaustive, but do they fulfil another requisite of a good division—are they mutually exclusive? Aristotle himself raised this question, and some of his answers to difficulties are instructive. Particularly his discussion of the distinction between Second Substances or Essences and Qualities. Here he approximates to the modern doctrine of the distinction between Substance and Attribute as set forth in our quotation from Mansel at p. 110. Aristotle's Second Essences (δεύτεραι οὐσίαι) are common nouns or general names, Species and Genera, man, horse, animal, as distinguished from Singular names, this man, this horse, which he calls First Substances (πρῶται οὐσίαι), essences par excellence, to which real existence in the highest sense is attributed. Common nouns are put in the First Category because they are predicated in answer to the question, What is this? But he raises the difficulty whether they may not rather be regarded as being in the Third Category, that of Quality (τὸ ποιὸν). When we say, "This is a man," do we not declare what sort of a thing he is? do we not declare his Quality? If Aristotle had gone farther along this line, he would have arrived at the modern point of view that a man is a man in virtue of his possessing certain attributes, that general names are applied in virtue of their connotation. This would have been to make the line of distinction between the First Category and the Third pass between First Essence and Second, ranking the Second Essences with Qualities. But Aristotle did not get out of the difficulty in this way. He solved it by falling back on the differences in common speech. "Man" does not signify the quality simply, as "whiteness" does. "Whiteness" signifies nothing but the quality. That is to say, there is no separate name in common speech for the common attributes of man. His further obscure remark that general names "define quality round essence" περὶ οὐσίαν, inasmuch as they signify what sort a certain essence is, and that genera make this definition more widely than species, bore fruit in the mediæval discussions between Realists and Nominalists by which the signification of general names was cleared up.

Another difficulty about the mutual exclusiveness of the Categories was started by Aristotle in connexion with the Fourth Category, Relation (πρός τι Ad aliquid, To something). Mill remarks that "that could not be a very comprehensive view of the nature of Relation which would exclude action, passivity, and local situation from that Category," and many commentators, from Simplicius down to Hamilton, have remarked that all the last six Categories might be included under Relation. This is so far correct that the word Relation is one of the vaguest and most extensive of words; but the criticism ignores the strictness with which Aristotle confined himself in his Categories to the forms of common speech. It is clear from his examples that in his Fourth Category he was thinking only of "relation" as definitely expressed in common speech. In his meaning, any word is a relative which is joined with another in a sentence by means of a preposition or a case-inflection. Thus "disposition" is a relative: it is the disposition of something. This kind of relation is perfect when the related terms reciprocate grammatically; thus "master," "servant," since we can say either "the master of the servant," or "the servant of the master". In mediæval logic the term Relata was confined to these perfect cases, but the Category had a wider scope with Aristotle. And he expressly raised the question whether a word might not have as much right to be put in another Category as in this. Indeed, he went further than his critics in his suggestions of what Relation might be made to include. Thus: "big" signifies Quality; yet a thing is big with reference to something else, and is so far a Relative. Knowledge must be knowledge of something, and is a relative: why then should we put "knowing" (i.e., learned) in the Category of Quality. "Hope" is a relative, as being the hope of a man and the hope of something. Yet we say, "I have hope," and there hope would be in the category of Having, Appurtenance. For the solution of all such difficulties, Aristotle falls back upon the forms of common speech, and decides the place of words in his categories according to them. This was hardly consistent with his proposal to deal with separate words out of syntax, if by this was meant anything more than dealing with them without reference to truth or falsehood. He did not and could not succeed in dealing with separate words otherwise than as parts of sentences, owing their signification to their position as parts of a transient plexus of thought. In so far as words have their being in common speech, and it is their being in this sense that Aristotle considers in the Categories, it is a transient being. What being they represent besides is, in the words of Porphyry, a very deep affair, and one that needs other and greater investigation.

[Footnote 1:] τῶν κατὰ μηδεμίαν συμπλοκὴν λεγομένων ἕκαστον ἢτοι οὐσίαν σημαίνει, ἢ ποσὸν, ἢ ποιὸν, ἢ πρός τι, ἢ ποῦ, ἢ ποτὲ, ἢ κεῖσθαι, ἢ ἔχειν, ἢ ποιεῖν, ἢ πάσχειν (Categ. ii. 5.)

[Footnote 2:] To describe the Categories as a grammatical division, as Mansel does in his instructive Appendix C to Aldrich, is a little misleading without a qualification. They are non-logical inasmuch as they have no bearing on any logical purpose. But they are grammatical only in so far as they are concerned with words. They are not grammatical in the sense of being concerned with the function of words in predication. The unit of grammar in this sense is the sentence, a combination of words in syntax; and it is expressly with words out of syntax that Aristotle deals, with single words not in relation to the other parts of a sentence, but in relation to the things signified. In any strict definition of the provinces of Grammar and Logic, the Categories are neither grammatical nor logical: the grammarians have appropriated them for the subdivision of certain parts of the sentence, but with no more right than the logicians. They really form a treatise by themselves, which is in the main ontological, a discussion of substances and attributes as underlying the forms of common speech. In saying this I use the word substance in the modern sense: but it must be remembered that Aristotle's οὐσία, translated substantia, covered the word as well as the thing signified, and that his Categories are primarily classes of words. The union between names and things would seem to have been closer in the Greek mind than we can now realise. To get at it we must note that every separate word τὸ λεγόμενον is conceived as having a being or thing τὸ ὄν corresponding to it, so that beings or things τὰ ὄντα are coextensive with single words: a being or thing is whatever receives a separate name. This is clear and simple enough, but perplexity begins when we try to distinguish between this nameable being and concrete being, which last is Aristotle's category of οὐσία, the being signified by a Proper or a Common as distinguished from an Abstract Noun. As we shall see, it is relatively to the highest sense of this last kind of being, namely, the being signified by a Proper name, that he considers the other kinds of being.

Chapter IV.

THE CONTROVERSY ABOUT UNIVERSALS. —DIFFICULTIES CONCERNING THE RELATION OF GENERAL NAMES TO THOUGHT AND TO REALITY.

In the opening sentences of his Isagoge, before giving his simple explanation of the Five Predicables, Porphyry mentions certain questions concerning Genera and Species, which he passes over as being too difficult for the beginner. "Concerning genera and species," he says, "the question whether they subsist (i.e., have real substance), or whether they lie in the mere thoughts only, or whether, granting them to subsist, they are corporeal or incorporeal, or whether they subsist apart, or in sensible things and cohering round them—this I shall pass over, such a question being a very deep affair and one that needs other and greater investigation."

This passage, written about the end of the third century, A.D., is a kind of isthmus between Greek Philosophy and Mediæval: it summarises questions which had been turned over on every side and most intricately discussed by Plato and Aristotle and their successors, and the bald summary became a starting-point for equally intricate discussions among the Schoolmen, among whom every conceivable variety of doctrine found champions. The dispute became known as the dispute about Universals, and three ultra-typical forms of doctrine were developed, known respectively as Realism, Nominalism, and Conceptualism. Undoubtedly the dispute, with all its waste of ingenuity, had a clearing effect, and we may fairly try now what Porphyry shrank from, to gather some simple results for the better understanding of general names and their relations to thoughts and to things. The rival schools had each some aspect of the general name in view, which their exaggeration served to render more distinct.

What does a general name signify? For logical purposes it is sufficient to answer—the points of resemblance as grasped in the mind, fixed by a name applicable to each of the resembling individuals. This is the signification of the general name logically, its connotation or concept, the identical element of objective reference in all uses of a general name.

But other questions may be asked that cannot be so simply answered. What is this concept in thought? What is there in our minds corresponding to the general name when we utter it? How is its signification conceived? What is the signification psychologically?

We may ask, further, What is there in nature that the general name signifies? What is its relation to reality? What corresponds to it in the real world? Has the unity that it represents among individuals no existence except in the mind? Calling this unity, this one in the many, the Universal (Universale, τὸ πᾶν), what is the Universal ontologically?

It was this ontological question that was so hotly and bewilderingly debated among the Schoolmen. Before giving the ultra-typical answers to it, it may be well to note how this question was mixed up with still other questions of Theology and Cosmogony. Recognising that there is a unity signified by the general name, we may go on to inquire into the ground of the unity. Why are things essentially like one another? How is the unity maintained? How is it continued? Where does the common pattern come from? The question of the nature of the Universal thus links itself with metaphysical theories of the construction of the world, or even with the Darwinian theory of the origin of species.

Passing by these remoter questions, we may give the answers of the three extreme schools to the ontological question, What is a Universal?

The answer of the Ultra-Realists, broadly put, was that a Universal is a substance having an independent existence in nature.

Of the Ultra-Nominalists, that the Universal is a name and nothing else, vox et præterea nihil; that this name is the only unity among the individuals of a species, all that they have in common.

Of the Ultra-Conceptualists, that the individuals have more in common than the name, that they have the name plus the meaning, vox + significatio, but that the Universals, the genera and species, exist only in the mind.

Now these extreme doctrines, as literally interpreted by opponents, are so easily refuted and so manifestly untenable, that it may be doubted whether they were ever held by any thinker, and therefore I call them Ultra-Realism, Ultra-Nominalism, and Ultra-Conceptualism. They are mere exaggerations or caricatures, set up by opponents because they can be easily knocked down.

To the Ultra-Realists, it is sufficient to say that if there existed anywhere a substance having all the common attributes of a species and only these, having none of the attributes peculiar to any of the individuals of that species, corresponding to the general name as an individual corresponds to a Proper or Singular name, it would not be the Universal, the unity pervading the individuals, but only another individual.

To the Ultra-Nominalists, it is sufficient to say that the individuals must have more in common than the name, because the name is not applied arbitrarily, but on some ground. The individuals must have in common that on account of which they receive the common name: to call them by the same name is not to make them of the same species.

To the Ultra-Conceptualists, it is sufficient to say that when we employ a general name, as when we say "Socrates is a man," we do not refer to any passing thought or state of mind, but to certain attributes independent of what is passing in our minds. We cannot make a thing of this or that species by merely thinking of it as such.

The ultra-forms of these doctrines are thus easily shown to be inadequate, yet each of the three, Realism, Nominalism, and Conceptualism, represents a phase of the whole truth.

Thus, take Realism. Although it is not true that there is anything in reality corresponding to the general name such as there is corresponding to the singular name, the general name merely signifying attributes of what the singular name signifies, it does not follow, as the opponents of Ultra-Realism hastily assume, that there is nothing in the real world corresponding to the general name. Three senses may be particularised in which Realism is justified.

(1) The points of resemblance from which the concept is formed are as real as the individuals themselves. It is true in a sense that it is our thought that gives unity to the individuals of a class, that gathers the many into one, and so far the Conceptualists are right. Still we should not gather them into one if they did not resemble one another: that is the reason why we think of them together: and the respects in which they resemble one another are as much independent of us and our thinking as the individuals themselves, as much beyond the power of our thought to change. We must go behind the activity of the mind in unifying to the reason for the unification: and the ground of unity is found in what really exists. We do not confer the unity: we do not make all men or all dogs alike: we find them so. The curly tails in a thousand domestic dogs, which serve to distinguish them from wolves and foxes, are as real as the thousand individual domestic dogs. In this sense the Aristotelian doctrine, Universalia in re, expresses a plain truth.

(2) The Platonic doctrine, formulated by the Schoolmen as Universalia ante rem, has also a plain validity. Individuals come and go, but the type, the Universal, is more abiding. Men are born and die: man remains throughout. The snows of last year have vanished, but snow is still a reality to be faced. Wisdom does not perish with the wise men of any generation. In this plain sense, at least, it is true that Universals exist before Individuals, have a greater permanence, or, if we like to say so, a higher, as it is a more enduring, reality.

(3) Further, the "idea," concept, or universal, though it cannot be separated from the individual, and whether or not we ascribe to it the separate suprasensual existence of the archetypal forms of Plato's poetical fancy, is a very potent factor in the real world. Ideals of conduct, of manners, of art, of policy, have a traditional life: they do not pass away with the individuals in whom they have existed, in whom they are temporarily materialised: they survive as potent influences from age to age. The "idea" of Chaucer's Man of Law, who always "seemed busier than he was," is still with us. Mediæval conceptions of chivalry still govern conduct. The Universal enters into the Individual, takes possession of him, makes of him its temporary manifestation.

Nevertheless, the Nominalists are right in insisting on the importance of names. What we call the real world is a common object of perception and knowledge to you and me: we cannot arrive at a knowledge of it without some means of communication with one another: our means of communication is language. It may be doubted whether even thinking could go far without symbols with the help of which conceptions may be made definite. A concept cannot be explained without reference to a symbol. There is even a sense in which the Ultra-Nominalist doctrine that the individuals in a class have nothing in common but the name is tenable. Denotability by the same name is the only respect in which those individuals are absolutely identical: in this sense the name alone is common to them, though it is applied in virtue of their resemblance to one another.

Finally, the Conceptualists are right in insisting on the mind's activity in connexion with general names. Genera and species are not mere arbitrary subjective collections: the union is determined by the characters of the things collected. Still it is with the concept in each man's mind that the name is connected: it is by the activity of thought in recognising likenesses and forming concepts that we are able to master the diversity of our impressions, to introduce unity into the manifold of sense, to reduce our various recollections to order and coherence.

So much for the Ontological question. Now for the Psychological. What is in the mind when we employ a general name? What is the Universal psychologically? How is it conceived?

What breeds confusion in these subtle inquiries is the want of fixed unambiguous names for the things to be distinguished. It is only by means of such names that we can hold on to the distinctions, and keep from puzzling ourselves. Now there are three things to be distinguished in this inquiry, which we may call the Concept, the Conception, and the Conceptual or Generic Image. Let us call them by these names, and proceed to explain them.

By the Concept, I understand the meaning of the general name, what the general name signifies: by the Conception, the mental act or state of him who conceives this meaning. The concept of "triangle," i.e., what you and I mean by the word, is not my act of mind or your act of mind when we think or speak of a triangle. The Conception, which is this act, is an event or incident in our mental history, a psychical act or state, a distinct occurrence, a particular fact in time as much as the battle of Waterloo. The concept is the objective reference of the name, which is the same, or at least is understood to be the same, every time we use it. I make a figure on paper with ink or on a blackboard with chalk, and recognise or conceive it as a triangle: you also conceive it as such: we do the same to-morrow: we did the same yesterday: each act of conception is a different event, but the concept is the same throughout.

Now the psychological question about the Universal is, What is this conception? We cannot define it positively further than by saying that it consists in realising the meaning of a general name: the act being unique, we can only make it intelligible by producing an example of it. But we may define it negatively by distinguishing it from the conceptual image. Whenever we conceive anything, "man," "horse," there is generally present to our minds an image of a man or horse, with accidents of size, colour, position or other categories. But this conceptual image is not the concept, and the mental act of forming it is not conception.

This distinction between mental picturing or imaging and the conception of common attributes is variously expressed. The correlative terms Intuitive and Symbolical Thinking, Presentative and Representative Knowledge have been employed.[¹] But whatever terms we use, the distinction itself is vital, and the want of it leads to confusion.

Thus the fact that we cannot form a conceptual image composed solely of common attributes has been used to support the argument of Ultra-Nominalism, that the individuals classed under a common name have nothing in common but the name. What the word "dog" signifies, i.e., the "concept" of dog, is neither big nor little, neither black nor tan, neither here nor there, neither Newfoundland, nor Retriever, nor Terrier, nor Greyhound, nor Pug, nor Bulldog. The concept consists only of the attributes common to all dogs apart from any that are peculiar to any variety or any individual. Now we cannot form any such conceptual image. Our conceptual image is always of some definite size and shape. Therefore, it is argued, we cannot conceive what a dog means, and dogs have nothing in common but the name. This, however, does not follow. The concept is not the conceptual image, and forming the image is not conception. We may even, as in the case of a chiliagon, or thousand-sided figure, conceive the meaning without being able to form any definite image.

How then, do we ordinarily proceed in conceiving, if we cannot picture the common attributes alone and apart from particulars? We attend, or strive to attend, only to those aspects of an image which it has in common with the individual things denoted. And if we want to make our conception definite, we pass in review an indefinite number of the individuals, case after case.

A minor psychological question concerns the nature of the conceptual image. Is it a copy of some particular impression, or a confused blur or blend of many? Possibly neither: possibly it is something like one of Mr. Galton's composite photographs, photographs produced by exposing the same surface to the impressions of a number of different photographs in succession. If the individuals are nearly alike, the result is an image that is not an exact copy of any one of the components and yet is perfectly distinct. Possibly the image that comes into our mind's eye when we hear such a word as "horse" or "man" is of this character, the result of the impressions of a number of similar things, but not identical with any one. As, however, different persons have different conceptual images of the same concept, so we may have different conceptual images at different times. It is only the concept that remains the same.

But how, it may be asked, can the concept remain the same? If the universal or concept psychologically is an intellectual act, repeated every time we conceive, what guarantee have we for the permanence of the concept? Does this theory not do away with all possibility of defining and fixing concepts?

This brings us back to the doctrine already laid down about the truth of Realism. The theory of the concept is not exhausted when it is viewed only psychologically, as a psychic act. If we would understand it fully, we must consider the act in its relations to the real experience of ourselves and others. To fix this act, we give it a separate name, calling it the conception: and then we must go behind the activity of the mind to the objects on which it is exercised. The element of fixity is found in them. And here also the truth of Nominalism comes in. By means of words we enter into communication with other minds. It is thus that we discover what is real, and what is merely personal to ourselves.

[Footnote 1:] The only objection to these terms is that they have slipped from their moorings in philosophical usage. Thus instead of Leibnitz's use of Intuitive and Symbolical, which corresponds to the above distinction between Imaging and Conception, Mr. Jevons employs the terms to express a distinction among conceptions proper. We can understand what a chiliagon means, but we cannot form an image of it in our minds, except in a very confused and imperfect way; whereas we can form a distinct image of a triangle. Mr. Jevons would call the conception of the triangle Intuitive, of the chiliagon Symbolical.

Again, while Mansel uses the words Presentative and Representative to express our distinction, a more common usage is to call actual Perception Presentative Knowledge, and ideation or recollection in idea Representative.

PART III.

THE INTERPRETATION OF PROPOSITIONS. —OPPOSITION AND IMMEDIATE INFERENCE.

Chapter I.

THEORIES OF PREDICATION.—THEORIES OF JUDGMENT.

We may now return to the Syllogistic Forms, and the consideration of the compatibility or incompatibility, implication, and interdependence of propositions.

It was to make this consideration clear and simple that what we have called the Syllogistic Form of propositions was devised. When are propositions incompatible? When do they imply one another? When do two imply a third? We have seen in the Introduction how such questions were forced upon Aristotle by the disputative habits of his time. It was to facilitate the answer that he analysed propositions into Subject and Predicate, and viewed the Predicate as a reference to a class: in other words, analysed the Predicate further into a Copula and a Class Term.

But before showing how he exhibited the interconnexion of propositions on this plan, we may turn aside to consider various so-called Theories of Predication or of Judgment. Strictly speaking, they are not altogether relevant to Logic, that is to say, as a practical science: they are partly logical, partly psychological theories: some of them have no bearing whatever on practice, but are matters of pure scientific curiosity: but historically they are connected with the logical treatment of propositions as having been developed out of this.

The least confusing way of presenting these theories is to state them and examine them both logically and psychologically. The logical question is, Has the view any advantage for logical purposes? Does it help to prevent error, to clear up confusion? Does it lead to firmer conceptions of the truth? The psychological question is, Is this a correct theory of how men actually think when they make propositions? It is a question of what is in the one case, and of what ought to be for a certain purpose in the other.

Whether we speak of Proposition or of Judgment does not materially affect our answer. A Judgment is the mental act accompanying a Proposition, or that may be expressed in a proposition and cannot be expressed otherwise: we can give no other intelligible definition or description of a judgment. So a proposition can only be defined as the expression of a judgment: unless there is a judgment underneath them, a form of words is not a proposition.

Let us take, then, the different theories in turn. We shall find that they are not really antagonistic, but only different: that each is substantially right from its own point of view: and that they seem to contradict one another only when the point of view is misunderstood.

I. That the Predicate term may be regarded as a class in or from which the Subject is included or excluded. Known as the Class-Inclusion, Class-Reference, or Denotative view.

This way of analysing propositions is possible, as we have seen, because every statement implies a general name, and the extension or denotation of a general name is a class defined by the common attribute or attributes. It is useful for syllogistic purposes: certain relations among propositions can be most simply exhibited in this way.

But if this is called a Theory of Predication or Judgment, and taken psychologically as a theory of what is in men's minds whenever they utter a significant Sentence, it is manifestly wrong. When discussed as such, it is very properly rejected. When a man says "P struck Q," he has not necessarily a class of "strikers of Q" definitely in his mind. What he has in his mind is the logical equivalent of this, but it is not this directly. Similarly, Mr. Bradley would be quite justified in speaking of Two Terms and a Copula as a superstition, if it were meant that these analytic elements are present to the mind of an ordinary speaker.

II. That every Proposition may be regarded as affirming or denying an attribute of a subject. Known sometimes as the Connotative or the Denotative-Connotative view. This also follows from the implicit presence of a general name in every sentence. But it should not be taken as meaning that the man who says: "Tom came here yesterday," or "James generally sits there," has a clearly analysed Subject and Attribute in his mind. Otherwise it is as far wrong as the other view.

III. That every proposition may be regarded as an equation between two terms. Known as the Equational View.

This is obviously not true for common speech or ordinary thought. But it is a possible way of regarding the analytic components of a proposition, legitimate enough if it serves any purpose. It is a modification of the Class-Reference analysis, obtained by what is known as Quantification of the Predicate. In "All S is in P," P is undistributed, and has no symbol of Quantity. But since the proposition imports that All S is a part of P, i.e., Some P, we may, if we choose, prefix the symbol of Quantity, and then the proposition may be read "All S = Some P". And so with the other forms.

Is there any advantage in this? Yes: it enables us to subject the formulæ to algebraic manipulation. But any logical advantage—any help to thinking? None whatever. The elaborate syllogistic systems of Boole, De Morgan, and Jevons are not of the slightest use in helping men to reason correctly. The value ascribed to them is merely an illustration of the Bias of Happy Exercise. They are beautifully ingenious, but they leave every recorded instance of learned Scholastic trifling miles behind.

IV. That every proposition is the expression of a comparison between concepts. Sometimes called the Conceptualist View.

"To judge," Hamilton says, "is to recognise the relation of congruence or confliction in which two concepts, two individual things, or a concept and an individual compared together stand to each other."

This way of regarding propositions is permissible or not according to our interpretation of the words "congruence" and "confliction," and the word "concept". If by concept we mean a conceived attribute of a thing, and if by saying that two concepts are congruent or conflicting, we mean that they may or may not cohere in the same thing, and by saying that a concept is congruent or conflicting with an individual that it may or may not belong to that individual, then the theory is a corollary from Aristotle's analysis. Seeing that we must pass through that analysis to reach it, it is obviously not a theory of ordinary thought, but of the thought of a logician performing that analysis.

The precise point of Hamilton's theory was that the logician does not concern himself with the question whether two concepts are or are not as a matter of fact found in the same subject, but only with the question whether they are of such a character that they may be found, or cannot be found, in the same subject. In so far as his theory is sound, it is an abstruse and technical way of saying that we may consider the consistency of propositions without considering whether or not they are true, and that consistency is the peculiar business of syllogistic logic.

V. That the ultimate subject of every judgment is reality.

This is the form in which Mr. Bradley and Mr. Bosanquet deny the Ultra-Conceptualist position. The same view is expressed by Mill when he says that "propositions are concerned with things and not with our ideas of them".

The least consideration shows that there is justice in the view thus enounced. Take a number of propositions:—

The streets are wet.

George has blue eyes.

The Earth goes round the Sun.

Two and two make four.

Obviously, in any of these propositions, there is a reference beyond the conceptions in the speaker's mind, viewed merely as incidents in his mental history. They express beliefs about things and the relations among things in rerum natura: when any one understands them and gives his assent to them, he never stops to think of the speaker's state of mind, but of what the words represent. When states of mind are spoken of, as when we say that our ideas are confused, or that a man's conception of duty influences his conduct, those states of mind are viewed as objective facts in the world of realities. Even when we speak of things that have in a sense no reality, as when we say that a centaur is a combination of man and horse, or that centaurs were fabled to live in the vales of Thessaly, it is not the passing state of mind expressed by the speaker as such that we attend to or think of; we pass at once to the objective reference of the words.

Psychologically, then, the theory is sound: what is its logical value? It is sometimes put forward as if it were inconsistent with the Class-reference theory or the theory that judgment consists in a comparison of concepts. Historically the origin of its formal statement is its supposed opposition to those theories. But really it is only a misconception of them that it contradicts. It is inconsistent with the Class-reference view only if by a class we understand an arbitrary subjective collection, not a collection of things on the ground of common attributes. And it is inconsistent with the Conceptualist theory only if by a concept we understand not the objective reference of a general name, but what we have distinguished as a conception or a conceptual image. The theory that the ultimate subject is reality is assumed in both the other theories, rightly understood. If every proposition is the utterance of a judgment, and every proposition implies a general name, and every general name has a meaning or connotation, and every such meaning is an attribute of things and not a mental state, it is implied that the ultimate subject of every proposition is reality. But we may consider whether or not propositions are consistent without considering whether or not they are true, and it is only their mutual consistency that is considered in the syllogistic formulæ. Thus, while it is perfectly correct to say that every proposition expresses either truth or falsehood, or that the characteristic quality of a judgment is to be true or false, it is none the less correct to say that we may temporarily suspend consideration of truth or falsehood, and that this is done in what is commonly known as Formal Logic.

VI. That every proposition may be regarded as expressing relations between phenomena.

Bain follows Mill in treating this as the final import of Predication. But he indicates more accurately the logical value of this view in speaking of it as important for laying out the divisions of Inductive Logic. They differ slightly in their lists of Universal Predicates based upon Import in this sense—Mill's being Resemblance, Coexistence, Simple Sequence, and Causal Sequence, and Bain's being Coexistence, Succession, and Equality or Inequality. But both lay stress upon Coexistence and Succession, and we shall find that the distinctions between Simple Sequence and Causal Sequence, and between Repeated and Occasional Coexistence, are all-important in the Logic of Investigation. But for syllogistic purposes the distinctions have no relevance.

Chapter II.

THE "OPPOSITION" OF PROPOSITIONS.—THE INTERPRETATION OF "NO".

Propositions are technically said to be "opposed" when, having the same terms in Subject and Predicate, they differ in Quantity, or in Quality, or in both.[¹]

The practical question from which the technical doctrine has been developed was how to determine the significance of contradiction. What is meant by giving the answer "No" to a proposition put interrogatively? What is the interpretation of "No"? What is the respondent committed to thereby?

"Have all ratepayers a vote?" If you answer "No," you are bound to admit that some ratepayers have not. O is the Contradictory of A. If A is false, O must be true. So if you deny O, you are bound to admit A: one or other must be true: either Some ratepayers have not a vote or All have.

Is it the case that no man can live without sleep? Deny this, and you commit yourself to maintaining that Some man, one at least, can live without sleep. I is the Contradictory of E; and vice versâ.

Contradictory opposition is distinguished from Contrary, the opposition of one Universal to another, of A to E and E to A. There is a natural tendency to meet a strong assertion with the very reverse. Let it be maintained that women are essentially faithless or that "the poor in a lump is bad," and disputants are apt to meet this extreme with another, that constancy is to be found only in women or true virtue only among the poor. Both extremes, both A and E, may be false: the truth may lie between: Some are, Some not.

Logically, the denial of A or E implies only the admission of O or I. You are not committed to the full contrary. But the implication of the Contradictory is absolute; there is no half-way house where the truth may reside. Hence the name of Excluded Middle is applied to the principle that "Of two Contradictories one or other must be true: they cannot both be false".

While both Contraries may be false, they cannot both be true.

It is sometimes said that in the case of Singular propositions, the Contradictory and the Contrary coincide. A more correct doctrine is that in the case of Singular propositions, the distinction is not needed and does not apply. Put the question "Is Socrates wise?" or "Is this paper white?" and the answer "No" admits of only one interpretation, provided the terms remain the same. Socrates may become foolish, or this paper may hereafter be coloured differently, but in either case the subject term is not the same about which the question was asked. Contrary opposition belongs only to general terms taken universally as subjects. Concerning individual subjects an attribute must be either affirmed or denied simply: there is no middle course. Such a proposition as "Socrates is sometimes not wise," is not a true Singular proposition, though it has a Singular term as grammatical subject. Logically, it is a Particular proposition, of which the subject-term is the actions or judgments of Socrates.[²]

Opposition, in the ordinary sense, is the opposition of incompatible propositions, and it was with this only that Aristotle concerned himself. But from an early period in the history of Logic, the word was extended to cover mere differences in Quantity and Quality among the four forms A E I O, which differences have been named and exhibited symmetrically in a diagram known as: The Square of Opposition.

The four forms being placed at the four corners of the Square, and the sides and diagonals representing relations between them thus separated, a very pretty and symmetrical doctrine is the result.

Contradictories, A and O, E and I, differ both in Quantity and in Quality.

Contraries, A and E, differ in Quality but not in Quantity, and are both Universal.

Sub-contraries, I and O, differ in Quality but not in Quantity, and are both Particular.

Subalterns, A and I, E and O, differ in Quantity but not in Quality.

Again, in respect of concurrent truth and falsehood there is a certain symmetry.

Contradictories cannot both be true, nor can they both be false.

Contraries may both be false, but cannot both be true.

Sub-contraries may both be true, but cannot both be false.

Subalterns may both be false and both true. If the Universal is true, its subalternate Particular is true: but the truth of the Particular does not similarly imply the truth of its Subalternating Universal.

This last is another way of saying that the truth of the Contrary involves the truth of the Contradictory, but the truth of the Contradictory does not imply the truth of the Contrary.

There, however, the symmetry ends. The sides and the diagonals of the Square do not symmetrically represent degrees of incompatibility, or opposition in the ordinary sense.

There is no incompatibility between two Sub-contraries or a Subaltern and its Subalternant. Both may be true at the same time. Indeed, as Aristotle remarked of I and O, the truth of the one commonly implies the truth of the other: to say that some of the crew were drowned, implies that some were not, and vice versâ. Subaltern and Subalternant also are compatible, and something more. If a man has admitted A or E, he cannot refuse to admit I or O, the Particular of the same Quality. If All poets are irritable, it cannot be denied that some are so; if None is, that Some are not. The admission of the Contrary includes the admission of the Contradictory.

Consideration of Subalterns, however, brings to light a nice ambiguity in Some. It is only when I is regarded as the Contradictory of E, that it can properly be said to be Subalternate to A. In that case the meaning of Some is "not none," i.e., "Some at least". But when Some is taken as the sign of Particular quantity simply, i.e., as meaning "not all," or "some at most," I is not Subalternate to A, but opposed to it in the sense that the truth of the one is incompatible with the truth of the other.

Again, in the diagram Contrary opposition is represented by a side and Contradictory by the diagonal; that is to say, the stronger form of opposition by the shorter line. The Contrary is more than a denial: it is a counter-assertion of the very reverse, τὸ ἐνάντιον. "Are good administrators always good speakers?" "On the contrary, they never are." This is a much stronger opposition, in the ordinary sense, than a modest contradictory, which is warranted by the existence of a single exception. If the diagram were to represent incompatibility accurately, the Contrary ought to have a longer line than the Contradictory, and this it seems to have had in the diagram that Aristotle had in mind (De Interpret., c. 10).

It is only when Opposition is taken to mean merely difference in Quantity and Quality that there can be said to be greater opposition between Contradictories than between Contraries. Contradictories differ both in Quantity and in Quality: Contraries, in Quality only.

There is another sense in which the Particular Contradictory may be said to be a stronger opposite than the Contrary. It is a stronger position to take up argumentatively. It is easier to defend than a Contrary. But this is because it offers a narrower and more limited opposition.

We deal with what is called Immediate Inference in the next chapter. Pending an exact definition of the process, it is obvious that two immediate inferences are open under the above doctrines, (1) Granted the truth of any proposition, you may immediately infer the falsehood of its Contradictory. (2) Granted the truth of any Contrary, you may immediately infer the truth of its Subaltern.[³]

[Footnote 1:] This is the traditional definition of Opposition from an early period, though the tradition does not start from Aristotle. With him opposition (ἀντικεῖσθαι) meant, as it still means in ordinary speech, incompatibility. The technical meaning of Opposition is based on the diagram (given afterwards in the text) known as the Square of Opposition, and probably originated in a confused apprehension of the reason why it received that name. It was called the Square of Opposition, because it was intended to illustrate the doctrine of Opposition in Aristotle's sense and the ordinary sense of repugnance or incompatibility. What the Square brings out is this. If the four forms A E I O are arranged symmetrically according as they differ in quantity, or quality, or both, it is seen that these differences do not correspond symmetrically to compatibility and incompatibility: that propositions may differ in quantity or in quality without being incompatible, and that they may differ in both (as Contradictories) and be less violently incompatible than when they differ in one only (as Contraries). The original purpose of the diagram was to bring this out, as is done in every exposition of it. Hence it was called the Square of Opposition. But as a descriptive title this is a misnomer: it should have been the Square of Differences in Quantity or Quality. This misnomer has been perpetuated by appropriating Opposition as a common name for difference in Quantity or Quality when the terms are the same and in the same order, and distinguishing it in this sense from Repugnance or Incompatibility (Tataretus in Summulas, De Oppositionibus [1501], Keynes, The Opposition of Propositions [1887]). Seeing that there never is occasion to speak of Opposition in the limited sense except in connexion with the Square, there is no real risk of confusion. A common name is certainly wanted in that connexion, if only to say that Opposition (in the limited or diagrammatic sense) does not mean incompatibility.

[Footnote 2:] Cp. Keynes, pt. ii. ch. ii. s. 57. Aristotle laid down the distinction between Contrary and Contradictory to meet another quibble in contradiction, based on taking the Universal as a whole and indivisible subject like an Individual, of which a given predicate must be either affirmed or denied.

[Footnote 3:] I have said that there is little risk of confusion in using the word Opposition in its technical or limited sense. There is, however, a little. When it is said that these Inferences are based on Opposition, or that Opposition is a mode of Immediate Inference, there is confusion of ideas unless it is pointed out that when this is said, it is Opposition in the ordinary sense that is meant. The inferences are really based on the rules of Contrary and Contradictory Opposition; Contraries cannot both be true, and of Contradictories one or other must be.

Chapter III.

THE IMPLICATION OF PROPOSITIONS. —IMMEDIATE FORMAL INFERENCE.—EDUCATION.

The meaning of Inference generally is a subject of dispute, and to avoid entering upon debatable ground at this stage, instead of attempting to define Inference generally, I will confine myself to defining what is called Formal Inference, about which there is comparatively little difference of opinion.

Formal Inference then is the apprehension of what is implied in a certain datum or admission: the derivation of one proposition, called the Conclusion, from one or more given, admitted, or assumed propositions, called the Premiss or Premisses.

When the conclusion is drawn from one proposition, the inference is said to be immediate; when more than one proposition is necessary to the conclusion, the inference is said to be mediate.

Given the proposition, "All poets are irritable," we can immediately infer that "Nobody that is not irritable is a poet"; and the one admission implies the other. But we cannot infer immediately that "all poets make bad husbands". Before we can do this we must have a second proposition conceded, that "All irritable persons make bad husbands". The inference in the second case is called Mediate.[¹]

The modes and conditions of valid Mediate Inference constitute Syllogism, which is in effect the reasoning together of separate admissions. With this we shall deal presently. Meantime of Immediate Inference.

To state all the implications of a certain form of proposition, to make explicit all that it implies, is the same thing with showing what immediate inferences from it are legitimate. Formal inference, in short, is the eduction of all that a proposition implies.

Most of the modes of Immediate Inference formulated by logicians are preliminary to the Syllogistic process, and have no other practical application. The most important of them technically is the process known as Conversion, but others have been judged worthy of attention.

Æquipollent or Equivalent Forms—Obversion.

Æquipollence or Equivalence (Ισοδυναμία) is defined as the perfect agreement in sense of two propositions that differ somehow in expression.[²]

The history of Æquipollence in logical treatises illustrates two tendencies. There is a tendency on the one hand to narrow a theme down to definite and manageable forms. But when a useful exercise is discarded from one place it has a tendency to break out in another under another name. A third tendency may also be said to be specially well illustrated—the tendency to change the traditional application of logical terms.

In accordance with the above definition of Æquipollence or Equivalence, which corresponds with ordinary acceptation, the term would apply to all cases of "identical meaning under difference of expression". Most examples of the reduction of ordinary speech into syllogistic form would be examples of æquipollence; all, in fact, would be so were it not that ordinary speech loses somewhat in the process, owing to the indefiniteness of the syllogistic symbol for particular quality, Some. And in truth all such transmutations of expression are as much entitled to the dignity of being called Immediate Inferences as most of the processes so entitled.

Dr. Bain uses the word with an approach to this width of application in discussing all that is now most commonly called Immediate Inference under the title of Equivalent Forms. The chief objection to this usage is that the Converse per accidens is not strictly equivalent. A debater may want for his argument less than the strict equivalent, and content himself with educing this much from his opponent's admission. (Whether Dr. Bain is right in treating the Minor and Conclusion of a Hypothetical Syllogism as being equivalent to the Major, is not so much a question of naming.)

But in the history of the subject, the traditional usage has been to confine Æquipollence to cases of equivalence between positive and negative forms of expression. "Not all are," is equivalent to "Some are not": "Not none is," to "Some are". In Pre-Aldrichian text-books, Æquipollence corresponds mainly to what it is now customary to call (e.g., Fowler, pt. iii. c. ii., Keynes, pt. ii. c. vii.) Immediate Inference based on Opposition. The denial of any proposition involves the admission of its contradictory. Thus, if the negative particle "Not" is placed before the sign of Quantity, All or Some, in a proposition, the resulting proposition is equivalent to the Contradictory of the original. Not all S is P = Some S is not P. Not any S is P = No S is P. The mediæval logicians tabulated these equivalents, and also the forms resulting from placing the negative particle after, or both before and after, the sign of Quantity. Under the title of Æquipollence, in fact, they considered the interpretation of the negative particle generally. If the negative is placed after the universal sign, it results in the Contrary: if both before and after, in the Subaltern. The statement of these equivalents is a puzzling exercise which no doubt accounts for the prominence given it by Aristotle and the Schoolmen. The latter helped the student with the following Mnemonic line: Præ Contradic., post Contrar., præ postque Subaltern.[³]

To Æquipollence belonged also the manipulation of the forms known after the Summulæ as Exponibiles, notably Exclusive and Exceptive propositions, such as None but barristers are eligible, The virtuous alone are happy. The introduction of a negative particle into these already negative forms makes a very trying problem in interpretation. The æquipollence of the Exponibiles was dropped from text-books long before Aldrich, and it is the custom to laugh at them as extreme examples of frivolous scholastic subtlety: but most modern text-books deal with part of the doctrine of the Exponibiles in casual exercises.

Curiously enough, a form left unnamed by the scholastic logicians because too simple and useless, has the name Æquipollent appropriated to it, and to it alone, by Ueberweg, and has been adopted under various names into all recent treatises.

Bain calls it the Formal Obverse,[⁴] and the title of Obversion (which has the advantage of rhyming with Conversion) has been adopted by Keynes, Miss Johnson, and others.

Fowler (following Karslake) calls it Permutation. The title is not a happy one, having neither rhyme nor reason in its favour, but it is also extensively used.

This immediate inference is a very simple affair to have been honoured with such a choice of terminology. "This road is long: therefore, it is not short," is an easy inference: the second proposition is the Obverse, or Permutation, or Æquipollent, or (in Jevons's title) the Immediate Inference by Privative Conception, of the first.

The inference, such as it is, depends on the Law of Excluded Middle. Either a term P, or its contradictory, not-P, must be true of any given subject, S: hence to affirm P of all or some S, is equivalent to denying not-P of the same: and, similarly, to deny P, is to affirm not-P. Hence the rule of Obversion;—Substitute for the predicate term its Contrapositive,[⁵] and change the Quality of the proposition.

All S is P = No S is not-P.

No S is P = All S is not-P.

Some S is P = Some S is not not-P.

Some S is not P = Some S is not-P.

Conversion.

The process takes its name from the interchange of the terms. The Predicate-term becomes the Subject-term, and the Subject-term the Predicate-term.

When propositions are analysed into relations of inclusion or exclusion between terms, the assertion of any such relation between one term and another, implies a Converse relation between the second term and the first. The statement of this implied assertion is technically known as the Converse of the original proposition, which may be called the Convertend.

Three modes of Conversion are commonly recognised:—(a) Simple Conversion; (b) Conversion per accidens or by limitation; (c) Conversion by Contraposition.

(a) E and I can be simply converted, only the terms being interchanged, and Quantity and Quality remaining the same.

If S is wholly excluded from P, P must be wholly excluded from S. If Some S is contained in P, then Some P must be contained in S.

(b) A cannot be simply converted. To know that All S is contained in P, gives you no information about that portion of P which is outside S. It only enables you to assert that Some P is S; that portion of P, namely, which coincides with S.

O cannot be converted either simply or per accidens. Some S is not P does not enable you to make any converse assertion about P. All P may be S, or No P may be S, or Some P may be not S. All the three following diagrams are compatible with Some S being excluded from P.

(c) Another mode of Conversion, known by mediæval logicians following Boethius as Conversio per contra positionem terminorum, is useful in some syllogistic manipulations. This Converse is obtained by substituting for the predicate term its Contrapositive or Contradictory, not-P, making the consequent change of Quality, and simply converting. Thus All S is P is converted into the equivalent No not-P is S.[⁶]

Some have called it "Conversion by Negation," but "negation" is manifestly too wide and common a word to be thus arbitrarily restricted to the process of substituting for one term its opposite.

Others (and this has some mediæval usage in its favour, though not the most intelligent) would call the form All not-P is not-S (the Obverse or Permutation of No not-P is S), the Converse by Contraposition. This is to conform to an imaginary rule that in Conversion the Converse must be of the same Quality with the Convertend. But the essence of Conversion is the interchange of Subject and Predicate: the Quality is not in the definition except by a bungle: it is an accident. No not-P is S, and Some not-P is S are the forms used in Syllogism, and therefore specially named. Unless a form had a use, it was left unnamed, like the Subalternate forms of Syllogism: Nomen habent nullum: nec, si bene colligis, usum.

Table of Contrapositive Converses.

When not-P is substituted for P, Some S is P becomes Some S is not not-P, and this form is inconvertible.

Other Forms of Immediate Inference.

I have already spoken of the Immediate Inferences based on the rules of Contradictory and Contrary Opposition ([see p. 145])

Another process was observed by Thomson, and named Immediate Inference by Added Determinants. If it is granted that "A negro is a fellow-creature," it follows that "A negro in suffering is a fellow-creature in suffering". But that this does not follow for every attribute[⁷] is manifest if you take another case:—"A tortoise is an animal: therefore, a fast tortoise is a fast animal". The form, indeed, holds in cases not worth specifying: and is a mere handle for quibbling. It could not be erected into a general rule unless it were true that whatever distinguishes a species within a class, will equally distinguish it in every class in which the first is included.

Modal Consequence has also been named among the forms of Immediate Inference. By this is meant the inference of the lower degrees of certainty from the higher. Thus must be is said to imply may be; and None can be to imply None is.

Dr. Bain includes also Material Obversion, the analogue of Formal Obversion applied to a Subject. Thus Peace is beneficial to commerce, implies that War is injurious to commerce. Dr. Bain calls this Material Obversion because it cannot be practised safely without reference to the matter of the proposition. We shall recur to the subject in another chapter.

[Footnote 1:] I purposely chose disputable propositions to emphasise the fact that Formal Logic has no concern with the truth, but only with the interdependence of its propositions.

[Footnote 2:] Mark Duncan, Inst. Log., ii. 5, 1612.

[Footnote 3:] There can be no doubt that in their doctrine of Æquipollents, the Schoolmen were trying to make plain a real difficulty in interpretation, the interpretation of the force of negatives. Their results would have been more obviously useful if they had seen their way to generalising them. Perhaps too they wasted their strength in applying it to the artificial syllogistic forms, which men do not ordinarily encounter except in the manipulation of syllogisms. Their results might have been generalised as follows:—

(1) A "not" placed before the sign of Quantity contradicts the whole proposition. Not "All S is P," not "No S is P," not "Some S is P," not "Some S is not P," are equivalent respectively to contradictories of the propositions thus negatived.

(2) A "not" placed after the sign of Quantity affects the copula, and amounts to inverting its Quality, thus denying the predicate term of the same quantity of the subject term of which it was originally affirmed, and vice versâ.

(3) If a "not" is placed before as well as after, the resulting forms are obviously equivalent (under Rule 1) to the assertion of the contradictories of the forms on the right (in the illustration of Rule 2).

[Footnote 4:] Formal to distinguish it from what he called the Material Obverse, about which more presently.

[Footnote 5:] The mediæval word for the opposite of a term, the word Contradictory being confined to the propositional form.

[Footnote 6:] It is to be regretted that a practice has recently crept in of calling this form, for shortness, the Contrapositive simply. By long-established usage, dating from Boethius, the word Contrapositive is a technical name for a terminal form, not-A, and it is still wanted for this use. There is no reason why the propositional form should not be called the Converse by Contraposition, or the Contrapositive Converse, in accordance with traditional usage.

[Footnote 7:] Cf. Stock, part iii. c. vii.; Bain, Deduction, p. 109.

Chapter IV.

THE COUNTER-IMPLICATION OF PROPOSITIONS.

In discussing the Axioms of Dialectic, I indicated that the propositions of common speech have a certain negative implication, though this does not depend upon any of the so-called Laws of Thought, Identity, Contradiction, and Excluded Middle. Since, however, the counter-implicate is an important guide in the interpretation of propositions, it is desirable to recognise it among the modes of Immediate Inference.

I propose, then, first, to show that people do ordinarily infer at once to a counter-sense; second, to explain briefly the Law of Thought on which such an inference is justified; and, third, how this law may be applied in the interpretation of propositions, with a view to making subject and predicate more definite.

Every affirmation about anything is an implicit negation about something else. Every say is a gainsay. That people ordinarily act upon this as a rule of interpretation a little observation is sufficient to show: and we find also that those who object to having their utterances interpreted by this rule often shelter themselves under the name of Logic.

Suppose, for example, that a friend remarks, when the conversation turns on children, that John is a good boy, the natural inference is that the speaker has in his mind another child who is not a good boy. Such an inference would at once be drawn by any actual hearer, and the speaker would protest in vain that he said nothing about anybody but John. Suppose there are two candidates for a school appointment, A and B, and that stress is laid upon the fact that A is an excellent teacher. A's advocate would at once be understood to mean that B was not equally excellent as a teacher.

The fairness of such inferences is generally recognised. A reviewer, for example, of one of Mrs. Oliphant's historical works, after pointing out some small errors, went on to say that to confine himself to censure of small points, was to acknowledge by implication that there were no important points to find fault with.

Yet such negative implications are often repudiated as illogical. It would be more accurate to call them extra-logical. They are not condemned by any logical doctrine: they are simply ignored. They are extra-logical only because they are not legitimated by the Laws of Identity, Contradiction, and Excluded Middle: and the reason why Logic confines itself to those laws is that they are sufficient for Syllogism and its subsidiary processes.

But, though extra-logical, to infer a counter-implicate is not unreasonable: indeed, if Definition, clear vision of things in their exact relations, is our goal rather than Syllogism, a knowledge of the counter-implicate is of the utmost consequence. Such an implicate there must always be under an all-pervading Law of Thought which has not yet been named, but which may be called tentatively the law of Homogeneous Counter-relativity. The title, one hopes, is sufficiently technical-looking: though cumbrous, it is descriptive. The law itself is simple, and may be thus stated and explained.

The Law of Homogeneous Counter-relativity.

Every positive in thought has a contrapositive, and the positive and contrapositive are of the same kind.

The first clause of our law corresponds with Dr. Bain's law of Discrimination or Relativity: it is, indeed, an expansion and completion of that law. Nothing is known absolutely or in isolation; the various items of our knowledge are inter-relative; everything is known by distinction from other things. Light is known as the opposite of darkness, poverty of riches, freedom of slavery, in of out; each shade of colour by contrast to other shades. What Dr. Bain lays stress upon is the element of difference in this inter-relativity. He bases this law of our knowledge on the fundamental law of our sensibility that change of impression is necessary to consciousness. A long continuance of any unvaried impression results in insensibility to it. We have seen instances of this in illustrating the maxim that custom blunts sensibility (p. 74). Poets have been beforehand with philosophers in formulating this principle. It is expressed with the greatest precision by Barbour in his poem of "The Bruce," where he insists that men who have never known slavery do not know what freedom is.

Thus contrar thingis evermare

Discoverings of t' other are.

Since, then, everything that comes within our consciousness comes as a change or transition from something else, it results that our knowledge is counter-relative. It is in the clash or conflict of impressions that knowledge emerges: every item of knowledge has its illuminating foil, by which it is revealed, over against which it is defined. Every positive in thought has its contrapositive.

So much for the element of difference. But this is not the whole of the inter-relativity. The Hegelians rightly lay stress on the common likeness that connects the opposed items of knowledge.

"Thought is not only distinction; it is, at the same time, relation.[¹] If it marks off one thing from another, it, at the same time, connects one thing with another. Nor can either of these functions of thought be separated from the other: as Aristotle himself said, the knowledge of opposites is one. A thing which has nothing to distinguish it is unthinkable, but equally unthinkable is a thing which is so separated from all other things as to have no community with them. If then the law of contradiction be taken as asserting the self-identity of things or thoughts in a sense that excludes their community—in other words, if it be not taken as limited by another law which asserts the relativity of the things or thoughts distinguished—it involves a false abstraction.... If, then, the world, as an intelligible world, is a world of distinction, differentiation, individuality, it is equally true that in it as an intelligible world there are no absolute separations or oppositions, no antagonisms which cannot be reconciled."[²]

In the penultimate sentence of this quotation Dr. Caird differentiates his theory against a Logical counter-theory of the Law of Identity, and in the last sentence against an Ethical counter-theory: but the point here is that he insists on the relation of likeness among opposites. Every impression felt is felt as a change or transition from something else: but it is a variation of the same impression—the something else, the contrapositive, is not entirely different. Change itself is felt as the opposite of sameness, difference of likeness, and likeness of difference. We do not differentiate our impression against the whole world, as it were, but against something nearly akin to it—upon some common ground. The positive and the contrapositive are of the same kind.

Let us surprise ourselves in the act of thinking and we shall find that our thoughts obey this law. We take note, say, of the colour of the book before us: we differentiate it against some other colour actually before us in our field of vision or imagined in our minds. Let us think of the blackboard as black: the blackness is defined against the whiteness of the figures chalked or chalkable upon it, or against the colour of the adjacent wall. Let us think of a man as a soldier; the opposite in our minds is not the colour of his hair, or his height, or his birthplace, or his nationality, but some other profession—soldier, sailor, tinker, tailor. It is always by means of some contrapositive that we make the object of our thoughts definite; it is not necessarily always the same opposite, but against whatever opposite it is, they are always homogeneous. One colour is contradistinguished from another colour, one shade from another shade: colour may be contradistinguished from shape, but it is within the common genus of sensible qualities.

A curious confirmation of this law of our thinking has been pointed out by Mr. Carl Abel.[³] In Egyptian hieroglyphics, the oldest extant language, we find, he says, a large number of symbols with two meanings, the one the exact opposite of the other. Thus the same symbol represents strong and weak; above—below; with—without; for—against. This is what the Hegelians mean by the reconciliation of antagonisms in higher unities. They do not mean that black is white, but only that black and white have something in common—they are both colours.

I have said that this law of Homogeneous Counter-relativity has not been recognised by logicians. This, however, is only to say that it has not been explicitly formulated and named, as not being required for Syllogism; a law so all-pervading could not escape recognition, tacit or express. And accordingly we find that it is practically assumed in Definition: it is really the basis of definition per genus et differentiam. When we wish to have a definite conception of anything, to apprehend what it is, we place it in some genus and distinguish it from species of the same. In fact our law might be called the Law of Specification: in obeying the logical law of what we ought to do with a view to clear thinking, we are only doing with exactness and conscious method what we all do and cannot help doing with more or less definiteness in our ordinary thinking.

It is thus seen that logicians conform to this law when they are not occupied with the narrow considerations proper to Syllogism. And another unconscious recognition of it may be found in most logical text-books. Theoretically the not-A of the Law of Contradiction—(A is not not-A)—is an infinite term. It stands for everything but A. This is all that needs to be assumed for Conversion and Syllogism. But take the examples given of the Formal Obverse or Permutation, "All men are fallible". Most authorities would give as the Formal Obverse of this, "No men are infallible". But, strictly speaking, "infallible" is of more limited and definite signification than not-fallible. Not-fallible, other than fallible, is brown, black, chair, table, and every other nameable thing except fallible. Thus in Obversion and Conversion by Contraposition, the homogeneity of the negative term is tacitly assumed; it is assumed that A and not-A are of the same kind.

Now to apply this Law of our Thought to the interpretation of propositions. Whenever a proposition is uttered we are entitled to infer at once (or immediately) that the speaker has in his mind some counter-proposition, in which what is overtly asserted of the ostensible subject is covertly denied of another subject. And we must know what this counter-proposition, the counter-implicate is, before we can fully and clearly understand his meaning. But inasmuch as any positive may have more than one contrapositive, we cannot tell immediately or without some knowledge of the circumstances or context, what the precise counter-implicate is. The peculiar fallacy incident to this mode of interpretation is, knowing that there must be some counter-implicate, to jump rashly or unwarily to the conclusion that it is some definite one.

Dr. Bain applies the term Material Obverse to the form, Not-S is not P, as distinguished from the form S is not not-P, which he calls the Formal Obverse, on the ground that we can infer the Predicate-contrapositive at once from the form, whereas we cannot tell the Subject-contrapositive without an examination of the matter. But in truth we cannot tell either Predicate-contrapositive or Subject-contrapositive as it is in the mind of the speaker from the bare utterance. We can only tell that if he has in his mind a proposition definitely analysed into subject and predicate, he must have contrapositives in his mind of both, and that they must be homogeneous. Let a man say, "This book is a quarto". For all that we know he may mean that it is not a folio or that it is not an octavo: we only know for certain, under the law of Homogeneous Counter-relativity, that he means some definite other size. Under the same law, we know that he has a homogeneous contrapositive of the subject, a subject that admits of the same predicate, some other book in short. What the particular book is we do not know.

It would however be a waste of ingenuity to dwell upon the manipulation of formulæ founded on this law. The practical concern is to know that for the interpretation of a proposition, a knowledge of the counter-implicate, a knowledge of what it is meant to deny, is essential.

The manipulation of formulæ, indeed, has its own special snare. We are apt to look for the counterparts of them in the grammatical forms of common speech. Thus, it might seem to be a fair application of our law to infer from the sentence, "Wheat is dear," that the speaker had in his mind that Oats or Sugar or Shirting or some other commodity is cheap. But this would be a rash conclusion. The speaker may mean this, but he may also mean that wheat is dear now as compared with some other time: that is, the Positive subject in his mind may be "Wheat as now," and the Contrapositive "Wheat as then". So a man may say, "All men are mortal," meaning that the angels never taste death, "angels" being the contrapositive of his subject "men". Or he may mean merely that mortality is a sad thing, his positive subject being men as they are, and his contrapositive men as he desires them to be. Or his emphasis may be upon the all, and he may mean only to deny that some one man in his mind (Mr. Gladstone, for example) is immortal. It would be misleading, therefore, to prescribe propositions as exercises in Material Obversion, if we give that name to the explicit expression of the Contrapositive Subject: it is only from the context that we can tell what this is. The man who wishes to be clearly understood gives us this information, as when the epigrammatist said: "We are all fallible—even the youngest of us".

But the chief practical value of the law is as a guide in studying the development of opinions. Every doctrine ever put forward has been put forward in opposition to a previous doctrine on the same subject. Until we know what the opposed doctrine is, we cannot be certain of the meaning. We cannot gather it with precision from a mere study of the grammatical or even (in the narrow sense of the word) the logical content of the words used. This is because the framers of doctrines have not always been careful to put them in a clear form of subject and predicate, while their impugners have not moulded their denial exactly on the language of the original. No doubt it would have been more conducive to clearness if they had done so. But they have not, and we must take them as they are. Thus we have seen that the Hegelian doctrine of Relativity is directed against certain other doctrines in Logic and in Ethics; that Ultra-Nominalism is a contradiction of a certain form of Ultra-Realism; and that various theories of Predication each has a backward look at some predecessor.

I quote from Mr. A.B. Walkley a very happy application of this principle of interpretation:—

"It has always been a matter for speculation why so sagacious an observer as Diderot should have formulated the wild paradox that the greatest actor is he who feels his part the least. Mr. Archer's bibliographical research has solved this riddle. Diderot's paradox was a protest against a still wilder one. It seems that a previous eighteenth century writer on the stage, a certain Saint-Albine, had advanced the fantastic propositions that none but a magnanimous man can act magnanimity, that only lovers can do justice to a love scene, and kindred assertions that read like variations on the familiar 'Who drives fat oxen must himself be fat'. Diderot saw the absurdity of this; he saw also the essentially artificial nature of the French tragedy and comedy of his own day; and he hastily took up the position which Mr. Archer has now shown to be untenable."

This instance illustrates another principle that has to be borne in mind in the interpretation of doctrines from their historical context of counter-implication. This is the tendency that men have to put doctrines in too universal a form, and to oppose universal to universal, that is, to deny with the flat contrary, the very reverse, when the more humble contradictory is all that the truth admits of. If a name is wanted for this tendency, it might be called the tendency to Over-Contradiction. Between "All are" and "None are," the sober truth often is that "Some are" and "Some are not," and the process of evolution has often consisted in the substitution of these sober forms for their more violent predecessors.

[Footnote 1:] It is significant of the unsuitableness of the vague unqualified word Relativity to express a logical distinction that Dr. Bain calls his law the Law of Relativity simply, having regard to the relation of difference, i.e., to Counter-Relativity, while Dr. Caird applies the name Relativity simply to the relation of likeness, i.e., to Co-relativity. It is with a view to taking both forms of relation into account that I name our law the Law of Homogeneous Counter-relativity. The Protagorean Law of Relativity has regard to yet another relation, the relation of knowledge to the knowing mind: these other logical laws are of relations among the various items of knowledge. Aristotle's category of Relation is a fourth kind of relation not to be confused with the others. "Father—son," "uncle—nephew," "slave—master," are relata in Aristotle's sense: "father," "uncle" are homogeneous counter-relatives, varieties of kinship; so "slave," "freeman" are counter-relatives in social status.

[Footnote 2:] Dr. Caird's Hegel, p. 134.

[Footnote 3:] See article on Counter-Sense, Contemporary Review, April, 1884.

PART IV.

THE INTERDEPENDENCE OF PROPOSITIONS.—MEDIATE INFERENCE.—SYLLOGISM.

Chapter I.

THE SYLLOGISM.

We have already defined mediate inference as the derivation of a conclusion from more than one proposition. The type or form of a mediate inference fully expressed consists of three propositions so related that one of them is involved or implied in the other two.

Distraction is exhausting.

Modern life is full of distraction

.^.. Modern life is exhausting.

We say nothing of the truth of these propositions. I purposely choose questionable ones. But do they hang together? If you admit the first two, are you bound in consistency to admit the third? Is the truth of the conclusion a necessary consequence of the truth of the premisses? If so, it is a valid mediate inference from them.

When one of the two premisses is more general than the conclusion, the argument is said to be Deductive. You lead down from the more general to the less general. The general proposition is called the Major Premiss, or Grounding Proposition, or Sumption: the other premiss the Minor, or Applying Proposition, or Subsumption.

Undue haste makes waste.

This is a case of undue hasting.

.^.. It is a case of undue wasting.

We may, and constantly do, apply principles and draw conclusions in this way without making any formal analysis of the propositions. Indeed we reason mediately and deductively whenever we make any application of previous knowledge, although the process is not expressed in propositions at all and is performed so rapidly that we are not conscious of the steps.

For example, I enter a room, see a book, open it and begin to read. I want to make a note of something: I look round, see a paper case, open it, take a sheet of paper and a pen, dip the pen in the ink and proceed to write. In the course of all this, I act upon certain inferences which might be drawn out in the form of Syllogisms. First, in virtue of previous knowledge I recognise what lies before me as a book. The process by which I reach the conclusion, though it passes in a flash, might be analysed and expressed in propositions.

Whatever presents certain outward appearances, contains readable print.

This presents such appearances.

.^.. It contains readable print.

So with the paper case, and the pen, and the ink. I infer from peculiar appearances that what I see contains paper, that the liquid will make a black mark on the white sheet, and so forth.

We are constantly in daily life subsuming particulars under known universals in this way. "Whatever has certain visible properties, has certain other properties: this has the visible ones: therefore, it has the others" is a form of reasoning constantly latent in our minds.

The Syllogism may be regarded as the explicit expression of this type of deductive reasoning; that is, as the analysis and formal expression of this every-day process of applying known universals to particular cases. Thus viewed it is simply the analysis of a mental process, as a psychological fact; the analysis of the procedure of all men when they reason from signs; the analysis of the kind of assumptions they make when they apply knowledge to particular cases. The assumptions may be warranted, or they may not: but as a matter of fact the individual who makes the confident inference has such assumptions and subsumptions latent in his mind.

But practically viewed, that is logically viewed, if you regard Logic as a practical science, the Syllogism is a contrivance to assist the correct performance of reasoning together or syllogising in difficult cases. It applies not to mental processes but to results of such expressed in words, that is, to propositions. Where the Syllogism comes in as a useful form is when certain propositions are delivered to you ab extra as containing a certain conclusion; and the connexion is not apparent. These propositions are analysed and thrown into a form in which it is at once apparent whether the alleged connexion exists. This form is the Syllogism: it is, in effect, an analysis of given arguments.

It was as a practical engine or organon that it was invented by Aristotle, an organon for the syllogising of admissions in Dialectic. The germ of the invention was the analysis of propositions into terms. The syllogism was conceived by Aristotle as a reasoning together of terms. His prime discovery was that whenever two propositions necessarily contain or imply a conclusion, they have a common term, that is, only three terms between them: that the other two terms which differ in each are the terms of the conclusion; and that the relation asserted in the conclusion between its two terms is a necessary consequence of their relations with the third term as declared in the premisses.

Such was Aristotle's conception of the Syllogism and such it has remained in Logic. It is still, strictly speaking, a syllogism of terms: of propositions only secondarily and after they have been analysed. The conclusion is conceived analytically as a relation between two terms. In how many ways may this relation be established through a third term? The various moods and figures of the Syllogism give the answer to that question.

The use of the very abstract word "relation" makes the problem appear much more difficult than it really is. The great charm of Aristotle's Syllogism is its simplicity. The assertion of the conclusion is reduced to its simplest possible kind, a relation of inclusion or exclusion, contained or not contained. To show that the one term is or is not contained in the other we have only to find a third which contains the one and is contained or not contained in the other.

The practical difficulties, of course, consist in the reduction of the conclusions and arguments of common speech to definite terms thus simply related. Once they are so reduced, their independence or the opposite is obvious. Therein lies the virtue of the Syllogism.

Before proceeding to show in how many ways two terms may be Syllogised through a third, we must have technical names for the elements.

The third term is called the Middle (M) (τὸ μέσον): the other two the Extremes (ἄκρα).

The Extremes are the Subject (S) and the Predicate (P) of the conclusion.

In an affirmative proposition (the normal form) S is contained in P: hence P is called the Major[¹] term (τὸ μεῖζον), and S the Minor (τὸ ἔλαττον), being respectively larger and smaller in extension. All difficulty about the names disappears if we remember that in bestowing them we start from the conclusion. That was the problem (προβλῆμα) or thesis in dialectic, the question in dispute.

The two Premisses, or propositions giving the relations between the two Extremes and the Middle, are named on an equally simple ground.

One of them gives the relation between the Minor Term, S, and the Middle, M. S, All or Some, is or is not in M. This is called the Minor Premiss.

The other gives the relation between the Major Term and the Middle. M, All or Some, is or is not in P. This is called the Major Premiss.[²]

[Footnote 1:] Aristotle calls the Major the First (τὸ πρῶτον) and the Minor the last (τὸ ἔσχατον), probably because that was their order in the conclusion when stated in his most usual form, "P is predicated of S," or "P belongs to S".

[Footnote 2:] When we speak of the Minor or the Major simply, the reference is to the terms. To avoid a confusion into which beginners are apt to stumble, and at the same time to emphasise the origin of the names, the Premisses might be spoken of at first as the Minor's Premiss and the Major's Premiss. It was only in the Middle Ages when the origin of the Syllogism had been forgotten, that the idea arose that the terms were called Major and Minor because they occurred in the Major and the Minor Premiss respectively.

Chapter II.

FIGURES AND MOODS OF THE SYLLOGISM.

I.—The First Figure.

The forms (technically called Moods, i.e., modes) of the First Figure are founded on the simplest relations with the Middle that will yield or that necessarily involve the disputed relation between the Extremes.

The simplest type is stated by Aristotle as follows: "When three terms are so related that the last (the Minor) is wholly in the Middle, and the Middle wholly either in or not in the first (the Major) there must be a perfect syllogism of the Extremes".[¹]

When the Minor is partly in the Middle, the Syllogism holds equally good. Thus there are four possible ways in which two terms (ὅροι, plane enclosures) may be connected or disconnected through a third. They are usually represented by circles as being the neatest of figures, but any enclosing outline answers the purpose, and the rougher and more irregular it is the more truly will it represent the extension of a word.

These four forms constitute what are known as the moods of the First Figure of the Syllogism. Seeing that all propositions may be reduced to one or other of the four forms, A, E, I, or O, we have in these premisses abstract types of every possible valid argument from general principles. It is all the same whatever be the matter of the proposition. Whether the subject of debate is mathematical, physical, social or political, once premisses in these forms are conceded, the conclusion follows irresistibly, ex vi formæ, ex necessitate formæ. If an argument can be analysed into these forms, and you admit its propositions, you are bound in consistency to admit the conclusion—unless you are prepared to deny that if one thing is in another and that other in a third, the first is in the third, or if one thing is in another and that other wholly outside a third, the first is also outside the third.

This is called the Axiom of Syllogism. The most common form of it in Logic is that known as the Dictum, or Regula de Omni et Nullo: "Whatever is predicated of All or None of a term, is predicated of whatever is contained in that term". It has been expressed with many little variations, and there has been a good deal of discussion as to the best way of expressing it, the relativity of the word best being often left out of sight. Best for what purpose? Practically that form is the best which best commands general assent, and for this purpose there is little to choose between various ways of expressing it. To make it easy and obvious it is perhaps best to have two separate forms, one for affirmative conclusions and one for negative. Thus: "Whatever is affirmed of all M, is affirmed of whatever is contained in M: and whatever is denied of all M, is denied of whatever is contained in M". The only advantage of including the two forms in one expression, is compendious neatness. "A part of a part is a part of the whole," is a neat form, it being understood that an individual or a species is part of a genus. "What is said of a whole, is said of every one of its parts," is really a sufficient statement of the principle: the whole being the Middle Term, and the Minor being a part of it, the Major is predicable of the Minor affirmatively or negatively if it is predicable similarly of the Middle.

This Axiom, as the name imports, is indemonstrable. As Aristotle pointed out in the case of the Axiom of Contradiction, it can be vindicated, if challenged, only by reducing the challenger to a practical absurdity. You can no more deny it than you can deny that if a leaf is in a book and the book is in your pocket, the leaf is in your pocket. If you say that you have a sovereign in your purse and your purse is in your pocket, and yet that the sovereign is not in your pocket: will you give me what is in your pocket for the value of the purse?

II.—The Minor Figures Of the Syllogism, And Their Reduction To the First.

The word Figure (σχῆμα) applies to the form or figure of the premisses, that is, the order of the terms in the statement of the premisses, when the Major Premiss is put first, and the Minor second.

In the First Figure the order is

M P

S M

But there are three other possible orders or figures, namely:—

It results from the doctrines of Conversion that valid arguments may be stated in these forms, inasmuch as a proposition in one order of terms may be equivalent to a proposition in another. Thus No M is in P is convertible with No P is in M: consequently the argument

No P is in M

All S is in M,

in the Second Figure is as much valid as when it is stated in the First—

No M is in P

All S is in M.

Similarly, since All M is in S is convertible into Some S is in M, the following arguments are equally valid:—

Using both the above Converses in place of their Convertends, we have—

It can be demonstrated (we shall see presently how) that altogether there are possible four valid forms or moods of the Second Figure, six of the Third, and five of the Fourth. An ingenious Mnemonic of these various moods and their reduction to the First Figure by the transposition of terms and premisses has come down from the thirteenth century. The first line names the moods of the First, Normal, or Standard Figure.

BArbArA, CElArEnt, DArII, FErIOque prioris;

CEsArE, CAmEstrEs, FEstInO, BArOkO, secundæ;

Tertia DArAptI, DIsAmIs, DAtIsI, FElAptOn,

BOkArdO, FErIsOque, habet; quarta insuper addit,

BrAmAntIP, CAmEnEs, DImArIs, FEsApO, FrEsIsOn.

The vowels in the names of the Moods indicate the propositions of the Syllogism in the four forms, A E I O. To write out any Mood at length you have only to remember the Figure, and transcribe the propositions in the order of Major Premiss, Minor Premiss, and Conclusion. Thus, the Second Figure being

PM

SM

FEstInO is written—

No P is in M.

Some S is in M.

Some S is not in P.

The Fourth Figure being

PM

MS

DImArIs is

Some P is in M.

All M is in S.

Some S is in P.

The initial letter in a Minor Mood indicates that Mood of the First to which it may be reduced. Thus Festino is reduced to Ferio, and Dimaris to Darii. In the cases of Baroko and Bokardo, B indicates that you may employ Barbara to bring any impugner to confusion, as shall be afterwards explained.

The letters s, m, and p are also significant. Placed after a vowel, s indicates that the proposition has to be simply converted. Thus, FEstInO:—

No P is in M.

Some S is in M.

Some S is not in P.

Simply convert the Major Premiss, and you get FErIO, of the First.

No M is in P.

Some S is in M.

Some S is not in P.

m (muta, or move) indicates that the premisses have to be transposed. Thus, in CAmEstrEs, you have to transpose the premisses, as well as simply convert the Minor Premiss before reaching the figure of CElArEnt.

From this it follows in CElArEnt that No P is in S, and this simply converted yields No S is in P.

A simple transposition of the premisses in DImArIs of the Fourth

Some P is in M

All M is in S

yields the premisses of DArII

All M is in S

Some P is in M,

but the conclusion Some P is in S has to be simply converted.

Placed after a vowel, p indicates that the proposition has to be converted per accidens. Thus in FElAptOn of the Third (MP, MS)

No M is in P

All M is in S

Some S is not in P

you have to substitute for All M is in S its converse by limitation to get the premisses of FErIO.

Two of the Minor Moods, Baroko of the Second Figure, and Bokardo of the Third, cannot be reduced to the First Figure by the ordinary processes of Conversion and Transposition. It is for dealing with these intractable moods that Contraposition is required. Thus in BArOkO of the Second (PM, SM)

All P is in M.

Some S is not in M.

Substitute for the Major Premiss its Converse by Contraposition, and for the Minor its Formal Obverse or Permutation, and you have FErIO of the First, with not-M as the Middle.

No not-M is in P.

Some S is in not-M,

Some S is not in P.

The processes might be indicated by the Mnemonic FAcsOcO, with c indicating the contraposition of the predicate term or Formal Obversion.

The reduction of BOkArdO,

Some M is not in P

All M is in S

Some S is not in P,

is somewhat more intricate. It may be indicated by DOcsAmOsc. You substitute for the Major Premiss its Converse by Contraposition, transpose the Premisses and you have DArII.

All M is in S.

Some not-P is in M.

Some not-P is in S.

Convert now the conclusion by Contraposition, and you have Some S is not in P.

The author of the Mnemonic apparently did not recognise Contraposition, though it was admitted by Boethius; and, it being impossible without this to demonstrate the validity of Baroko and Bokardo by showing them to be equivalent with valid moods of the First Figure, he provided for their demonstration by the special process known as Reductio ad absurdum. B indicates that Barbara is the medium.

The rationale of the process is this. It is an imaginary opponent that you reduce to an absurdity or self-contradiction. You show that it is impossible with consistency to admit the premisses and at the same time deny the conclusion. For, let this be done; let it be admitted as in BArOkO that,

All P is in M

Some S is not in M,

but denied that Some S is not in P. The denial of a proposition implies the admission of its Contradictory. If it is not true that Some S is not in P, it must be true that All S is in P. Take this along with the admission that All P is in M, and you have a syllogism in BArbArA,

All P is in M

All S is in P,

yielding the conclusion All S is in M. If then the original conclusion is denied, it follows that All S is in M. But this contradicts the Minor Premiss, which has been admitted to be true. It is thus shown that an opponent cannot admit the premisses and deny the conclusion without contradicting himself.

The same process may be applied to Bokardo.

Some M is not in P.

All M is in S.

Some S is not in P.

Deny the conclusion, and you must admit that All S is in P. Syllogised in Barbara with All M is in S, this yields the conclusion that All M is in P, the contradictory of the Major Premiss.

The beginner may be reminded that the argument ad absurdum is not necessarily confined to Baroko and Bokardo. It is applied to them simply because they are not reducible by the ordinary processes to the First Figure. It might be applied with equal effect to other Moods, DImArIs, e.g., of the Third.

Some M is in P.

All M is in S.

Some S is in P.

Let Some S is in P be denied, and No S is in P must be admitted. But if No S is in P and All M is in S, it follows (in Celarent) that No M is in P, which an opponent cannot hold consistently with his admission that Some M is in P.

The beginner sometimes asks: What is the use of reducing the Minor Figures to the First? The reason is that it is only when the relations between the terms are stated in the First Figure that it is at once apparent whether or not the argument is valid under the Axiom or Dictum de Omni. It is then undeniably evident that if the Dictum holds the argument holds. And if the Moods of the First Figure hold, their equivalents in the other Figures must hold too.

Aristotle recognised only two of the Minor Figures, the Second and Third, and thus had in all only fourteen valid moods.

The recognition of the Fourth Figure is attributed by Averroes to Galen. Averroes himself rejects it on the ground that no arguments expressed naturally, that is, in accordance with common usage, fall into that form. This is a sufficient reason for not spending time upon it, if Logic is conceived as a science that has a bearing upon the actual practice of discussion or discursive thought. And this was probably the reason why Aristotle passed it over.

If however the Syllogism of Terms is to be completed as an abstract doctrine, the Fourth Figure must be noticed as one of the forms of premisses that contain the required relation between the extremes. There is a valid syllogism between the extremes when the relations of the three terms are as stated in certain premisses of the Fourth Figure.

III.—The Sorites.

A chain of Syllogisms is called a Sorites. Thus:—

All A is in B.

All B is in C.

All C is in D.

:

:

:

:

All X is in Z.

.^.. All A is in Z.

A Minor Premiss can thus be carried through a series of Universal Propositions each serving in turn as a Major to yield a conclusion which can be syllogised with the next. Obviously a Sorites may contain one particular premiss, provided it is the first; and one universal negative premiss, provided it is the last. A particular or a negative at any other point in the chain is an insuperable bar.

[Footnote 1:] Ὅταν οὒν ὅροι τρεῖς αὔτως ἔχωσι πρὸς ἀλλήλους ὥστε τὸν ἔσχατον ἐν ὅλῳ εἶναι τῷ μέσῳ, καὶ τὸν μέσον ἐν ὅλῳ τῷ κρώτῳ ἢ εἶναι ἢ μὴ εἶναι, ἀνάγκη τῶν ἀκρων εἶναι συλλογισμὸν τέλειον (Anal. Prior., i. 4.)

Chapter III.

THE DEMONSTRATION OF THE SYLLOGISTIC MOODS. —THE CANONS OF THE SYLLOGISM.

How do we know that the nineteen moods are the only possible forms of valid syllogism?

Aristotle treated this as being self-evident upon trial and simple inspection of all possible forms in each of his three Figures.

Granted the parity between predication and position in or out of a limited enclosure (term, ὄρος), it is a matter of the simplest possible reasoning. You have three such terms or enclosures, S, P and M; and you are given the relative positions of two of them to the third as a clue to their relative positions to one another. Is S in or out of P, and is it wholly in or wholly out or partly in or partly out? You know how each of them lies toward the third: when can you tell from this how S lies towards P?

We have seen that when M is wholly in or out of P, and S wholly or partly in M, S is wholly or partly in or out of P.

Try any other given positions in the First Figure, and you find that you cannot tell from them how S lies relatively to P. Unless the Major Premiss is Universal, that is, unless M lies wholly in or out of P, you can draw no conclusion, whatever the Minor Premiss may give. Given, e.g., All S is in M, it may be that All S is in P, or that No S is in P, or that Some S is in P, or that Some S is not in P.

Again, unless the Minor Premiss is affirmative, no matter what the Major Premiss may be, you can draw no conclusion. For if the Minor Premiss is negative, all that you know is that All S or Some S lies somewhere outside M; and however M may be situated relatively to P, that knowledge cannot help towards knowing how S lies relatively to P. All S may be P, or none of it, or part of it. Given all M is in P; the All S (or Some S) which we know to be outside of M may lie anywhere in P or out of it.

Similarly, in the Second Figure, trial and simple inspection of all possible conditions shows that there can be no conclusion unless the Major Premiss is universal, and one of the premisses negative.

Another and more common way of eliminating the invalid forms, elaborated in the Middle Ages, is to formulate principles applicable irrespective of Figure, and to rule out of each Figure the moods that do not conform to them. These regulative principles are known as The Canons of the Syllogism.

Canon I. In every syllogism there should be three, and not more than three, terms, and the terms must be used throughout in the same sense.

It sometimes happens, owing to the ambiguity of words, that there seem to be three terms when there are really four. An instance of this is seen in the sophism:—

He who is most hungry eats most.

He who eats least is most hungry.

.^.. He who eats least eats most.

This Canon, however, though it points to a real danger of error in the application of the syllogism to actual propositions, is superfluous in the consideration of purely formal implication, it being a primary assumption that terms are univocal, and remain constant through any process of inference.

Under this Canon, Mark Duncan says (Inst. Log., iv. 3, 2), is comprehended another commonly expressed in this form: There should be nothing in the conclusion that was not in the premisses: inasmuch as if there were anything in the conclusion that was in neither of the premisses, there would be four terms in the syllogism.

The rule that in every syllogism there must be three, and only three, propositions, sometimes given as a separate Canon, is only a corollary from Canon I.

Canon II. The Middle Term must be distributed once at least in the Premisses.

The Middle Term must either be wholly in, or wholly out of, one or other of the Extremes before it can be the means of establishing a connexion between them. If you know only that it is partly in both, you cannot know from that how they lie relatively to one another: and similarly if you know only that it is partly outside both.

The Canon of Distributed Middle is a sort of counter-relative supplement to the Dictum de Omni. Whatever is predicable of a whole distributively is predicable of all its several parts. If in neither premiss there is a predication about the whole, there is no case for the application of the axiom.

Canon III. No term should be distributed in the conclusion that was not distributed in the premisses.

If an assertion is not made about the whole of a term in the premisses, it cannot be made about the whole of that term in the conclusion without going beyond what has been given.

The breach of this rule in the case of the Major term is technically known as the Illicit Process of the Major: in the case of the Minor term, Illicit Process of the Minor.

Great use is made of this canon in cutting off invalid moods. It must be remembered that the Predicate term is "distributed" or taken universally in O (Some S is not in P) as well as in E (No S is in P); and that P is never distributed in affirmative propositions.

Canon IV. No conclusion can be drawn from two negative premisses.

Two negative premisses are really tantamount to a declaration that there is no connexion whatever between the Major and Minor (as quantified in the premisses) and the term common to both premisses; in short, that this is not a Middle term—that the condition of a valid Syllogism does not exist.

There is an apparent exception to this when the real Middle in an argument is a contrapositive term, not-M. Thus:—

Nobody who is not thirsty is suffering from fever.

This person is not thirsty.

.^.. He is not suffering from fever.

But in such cases it is really the absence of a quality or rather the presence of an opposite quality on which we reason; and the Minor Premiss is really Affirmative of the form S is in not-M.

Canon V. If one premiss is negative, the conclusion must be negative.

If one premiss is negative, one of the Extremes must be excluded in whole or in part from the Middle term. The other must therefore (under Canon IV.) declare some coincidence between the Middle term and the other extreme; and the conclusion can only affirm exclusion in whole or in part from the area of this coincidence.

Canon VI. No conclusion can be drawn from two particular premisses.