XXXIV—GEOMETRICAL DEMONSTRATION

This exercise is regarded by logicians as one of the purest forms of argument. It is nothing more than an aid to a certain kind of perception.

Take, for instance, the fifth proposition of the first book of Euclid—'The angles at the base of an isosceles triangle are equal, and if the equal sides be produced the angles on the other side shall also be equal.' The proposition is accompanied by a diagram of an isosceles triangle with the equal sides already produced, so that the conditional phrasing of the proposition does not mean that the production of the sides, and what results therefrom, are future or possible events which neither Euclid nor anybody else has yet experienced, and the probability of which is an argumentative conclusion.

What the proposition means is this: an isosceles triangle of which the equal sides have been produced, has equal angles on the same side of the base both within and without the triangle. It is an affirmation of what is, not of what we must believe to be for reasons to be given.

The truth of the proposition is seen at once from simple inspection of the diagram. It is an association of properties related in a certain manner. It has many relations which the geometer does not mention in this proposition, but those which he mentions are seen to be correctly described as soon as we direct attention to them. If we have any doubt on the subject we remove it by measuring the angles.

Euclid however does not appeal to the powers of inspection we can exercise in this case, and he ignores our facilities for measurement. He appeals to simpler and easier kinds of perception expressed in his axioms, which he began by assuming we were capable of exercising without demonstration. They constitute what he considers the minimum power of relational perception, which if a man have not he cannot be taught geometry. Euclid also in this proposition refers to the result of a prior demonstration, the relation in which he supposes we have seized. By means of these antecedents he prompts our perceptive faculty up to the point of seeing the relations expressed in this proposition. If we saw them without the prompting, the latter is superfluous; if the relations do not stand the test of measurement, the prompting goes for nothing.

All Euclid's demonstrations are of this sort. They are pointings-out of what can be seen by inspection and sufficient attention. He is not bringing a case under a precedent—he is describing relations in things, that may serve as precedents in concrete or applied geometry. The service he performs is that of a connoisseur who points out the beauties of a picture or landscape to a careless or uninterested spectator. Relations are sometimes difficult to see—much more difficult than colours or masses—and there is a legitimate sphere of usefulness for people who point out what others are apt to overlook. There is no prediction in this. We are not asked to conceive anything that is not before us. Geometrical demonstration thus assists perception, but does not imply reasoning. Euclid does not argue—he prompts.

Those who maintain that Euclid is syllogistic do so on the ground that the axioms are generalisations, and that as often as one is cited there occurs the subsumption of an object under a class-notion. That would not be argument; but let us suppose it means bringing a case under a precedent. Then if the axioms be precedents and the demonstration an application of them to new cases, the theorem is a fallacy—a useless argument written to prove a foregone certainty, for the conclusion can be and generally is perfectly known without reference to the demonstration.

It appears to me more true to regard the axioms as the simplest relations, which everybody may be supposed capable of perceiving, and that geometrical demonstration consists in showing that other relations not so apparent are really varieties or combinations of the simpler relations. By using in concert with the axioms the relations already demonstrated, we are enabled to grasp relations that would not have been at all obvious on first beginning the geometrical praxis. Euclid's geometry is thus a series of graduated lessons in a special sort of observation, not a system of deductive arguments.

The educational theory that geometry is exceptionally good training for the reason—apart from its practical utility in mechanics—is thus evidently a mistake. Abstract geometry may induce habits of minute observation and exact definition, but reason nowhere enters into the study. As a rational gymnastic there is nothing better than the game of chess.

XXXV—INDUCTION

Those who contend that there is a kind of argument called Inductive different from the Deductive, illustrate their view by some such example as the following:—'This, that, and the other magnet' [that is, all the magnets we know] 'attract iron; therefore all possible magnets attract iron.' They say there is an irresistible compulsion in the mind to draw such a conclusion from information of the kind exemplified, and they contrast that type of thought with a deductive argument like—'All magnets attract iron; this object is a magnet; therefore it attracts iron.' They figure the former as a progress upwards, the latter as a regress downwards.

That is Induction as understood by J. S. Mill and Sir William Hamilton; on this point these philosophers happen to agree.

The first of those arguments is a deduction with the precedent omitted. Expressed in full it amounts to this—'Any relation observed several times to subsist between two classes of objects, and concerning which no exception has ever been observed, may be taken as universal; there is such a relation between known magnets and known iron; therefore it may be regarded as universal.' The precedent is not a mental compulsion, but a result of experience. Induction as above defined is therefore only a species of deductive conclusions.

Most logicians take the word Induction in its etymological sense, as meaning systematic observation carried on with a view to obtaining a general idea of some class of objects; or of establishing a categorical relation between one object or class and another, by eliminating all the alternative correlatives. In neither operation would Induction be argument.

In science a 'perfect induction' is one in which all existing objects of a class, or all objects related in a certain manner, have been perceived, so that there is no other object concerning which a conclusion can be drawn. In such cases, says Mill, there is no induction—only a summary of experience. He evidently regarded the conclusion with respect to unknown cases as the essence of induction, whereas in the scientific sense the induction is the positive content of the idea, or the abstract relation—the unknown cases are ignored, or there may be none. In scientific writings induction sometimes means the method of observation rather than the result—the method of correcting inferences by perception, wherever possible.

XXXVI—ARISTOTLE'S DICTUM

This is usually put into English thus—'Whatever is affirmed or denied of a class, may be affirmed or denied of any part of that class,' and such an affirmation or denial is supposed to be an act of reason. Archbishop Whately expounds the Dictum in analysing the following theorem—Whatever exhibits marks of design had an intelligent author; the world exhibits marks of design; therefore the world had an intelligent author.

'In the first of these premises,' he says, 'we find it assumed universally of the class of "things which exhibit marks of design," that they had an intelligent author; and in the other premise, "the world" is referred to that class as comprehended in it: now it is evident that whatever is said of the whole of a class, may be said of anything comprehended in that class: so that we are thus authorised to say of the world, that "it had an intelligent author." Again, if we examine a syllogism with a negative conclusion, as, e.g. "nothing which exhibits marks of design could have been produced by chance; the world exhibits, &c.; therefore the world could not have been produced by chance:" the process of Reasoning will be found to be the same; since it is evident, that whatever is denied universally of any class may be denied of anything that is comprehended in that class. On further examination it will be found, that all valid arguments whatever may be easily reduced to such a form as that of the foregoing syllogisms; and that consequently the principle on which they are constructed is the Universal Principle of Reasoning.'[20]

The examples given by Whately are perfectly valid; the first is a constructive argument in the Sixth Category, the second a stigmatic in the Fifth. I have in several places admitted that the arguments adduced by syllogists are sometimes correct, the fault complained of being in the mode in which such correct arguments are interpreted. They are interpreted wrongly, and then other theorems are found or made agreeing with the interpretation, and the admitted soundness of the first theorems is used to procure acceptance for the second. Things brought under the same definition ought to be essentially alike, but they are not so when the utmost latitude is taken to 'assume' that predicates have properties which they obviously have not.

The objections we make to the Dictum as above interpreted are—(1) that in reasoning the precedent (major premise) need not be a class; (2) if it is a class, it consists of all known things of a similar kind, not of all possible things of a similar kind. When interpreted in the latter sense the Dictum becomes dialectically tautological, as has been often observed.

XXXVII—MEDIATE COMPARISON

A few pages further on Whately gives a totally different account of reasoning, without being aware of his inconsistency.

'Every syllogism has three, and only three terms: viz. the middle term and the two terms (or extremes, as they are commonly called) of the Conclusion or Question. Of these, first, the subject of the conclusion is called the minor term; second, its predicate, the major term; and third, the middle term, (called by the older logicians "Argumentum") is that with which each of them is separately compared, in order to judge of their agreement or disagreement with each other. If therefore there were two middle terms, the extremes or terms of conclusion not being both compared to the same, could not be conclusively compared to each other.'[21]

Here reasoning is made to consist in comparing two things by reference to a third which both resemble. There is not a word about classification, which is declared just before—in loud capitals—to be the universal principle of reasoning!

On this definition we remark—

(1) Comparison by mediation is untrustworthy, unless the qualities compared be rigidly defined or restricted, as in geometry and the use of standards (xxii). In geometry the only two qualities recognised are figure and magnitude. The axiom of mediate comparison means that things having the same magnitude as a third thing are to be considered equal, though they may have different outlines. But the axiom is liable to be untrue in things of three or more qualities. Add colour. Then a white sphere may resemble a white cube on the one side, and a black sphere on the other, but the white cube does not at all resemble the black sphere. This axiom is therefore inadmissible or at least extremely risky in logic, which treats of things having many qualities.

(2) Comparison, however correctly performed, is never the end, but only a means, of reasoning.

XXXVIII—SYLLOGISM

We have already had two distinct definitions of syllogism. According to the first it is the application of class-attributes to individuals known to belong to the class; according to the second it is the comparison of two things or terms by reference to a third which both resemble. When we arrive at the chapters in logic books devoted to the exposition of the syllogism in detail, we find that the theorems there discussed do not conform to either of those definitions. The only sort of syllogism that can be 'converted' is one consisting of two classifications, and a conclusion which predicates a classification, as thus—

All Englishmen are Europeans;
John Smith is an Englishman;
therefore John Smith is a European.

Observe the difference between this theorem and that adduced in illustration of the Dictum (xxxvi). In the latter the first premise is a categorical judgment and so therefore is the conclusion; in the theorem just given the first premise is a classification, and the conclusion is necessarily a classification.

We first remark that such an 'argument' is never met with in real spontaneous thinking—it occurs only in logic books. It is manufactured exclusively for Peripatetic consumption. The reason it is not to be found is simple—the conclusion it yields is a classification, and that is not enough for valid argument. In reasoning we may introduce a classification as the minor premise—that is, the proposition which brings the case under the precedent—but the applicate is never a general or class idea. It is one or more properties abstracted from the subject (whether the latter be a single object or general idea), and applied to the case. Merely to classify a case and so leave it would answer no rational purpose.

Logicians urge in recommendation of this syllogism that it gives a certain conclusion. The premises being correct, the conclusion is infallibly true.

No doubt it is, for in contemplating a thing we can mentally enter it into all the classes to which it appears to belong, whatever be their generality. Knowing the class European and the individual John Smith, we see at once that the latter is contained in the former, and we can do this without putting him first in the minor class English. It is like saying, 'The pavilion is in the garden, John Smith is in the pavilion, therefore he is in the garden.' Of course he is! The minor premise of a double classification is superfluous. The fact that such conclusions are certain, shows how nugatory they are. We are not certain of anything till it has been experienced. In legitimate reasoning the conclusion is never more than probable. The certainty of these double classifications shows that we are stating what we already know—not imagining an ideal addition to our positive knowledge.

Doctrine of the Predicate. So long as logicians are permitted to fabricate their own examples, all is plain sailing with the syllogism. But they are sometimes obliged to deal with genuine arguments. In this case what they do is to assume that for logical purposes every predicate of the precedent—that is, the applicate—is a general or class term. Even when an argument is good they spoil it with a bad theory.

Sir William Hamilton states that up to his time logicians recognised but one type of proposition—that called by him the proposition 'in extension,' which means the classifying of the subject. He announced that he intended to introduce a proposition 'in comprehension,' meaning a judgment in the category of inherence—as for instance, 'man is responsible.' He further said that he recognised a third type of proposition, that concerning 'cause and effect.'

But in the course of working out these logical novelties he seems to have discovered that they were irreconcilable with conversion, and so he dropped them. The judgment in comprehension, he then declared, was to all intents and purposes the same as one in extension, and as to causation—why, a cause is a class, and an effect is an individual belonging to that class![22]

Let us see what is the result of treating applicates as general ideas. Take an example in each of the categories.

'The paper is white.' This means that the paper has the property or attribute of whiteness. In logic it is interpreted to mean that paper is an individual of the class white. This is wrong, for there is no such class. No sane person would form a class out of salt, snow, milk, china, silver, the moon, and other white things; for though they have a common property it is not the sign of a common human utility.

The confusing a single property with a class is not always owing to exigencies of syllogism. It pervades the writings of most Western metaphysicians, and may be accounted for in this manner.

General ideas and abstract properties or ideas have in common that they are partial recognitions of what we perceive (xiv). The partition in each is however made in a different way, and for a different purpose. In generalisation the selection is done almost mechanically. We see many things that have some common relation, function, or utility for us, and we remember only so much of them as appears to be necessary for the recognition of that relation or utility—just so much of the Intellectual experience as has always accompanied the Sentimental experience. The process is very like that of putting a piece of wood or ivory in a turning-lathe, and whittling off all that we do not want. A general idea is the useful core of a multitude of superposed observations, each of which had something irrelevant—something which it is better to forget. We whittle this off and remember only the core.

Abstraction, on the other hand, is a conscious and deliberate operation from beginning to end. It consists in distinguishing one by one the properties of a thing, and even treating each property as if it had an independent existence. For this exercise it is not necessary to observe many things: we can analyse one alone, though an acquaintance with other cognate objects is sometimes necessary to call our attention to single properties. We need the shock of difference to be able to distinguish well a fine abstraction—the difference between shades of colours, for example. Abstraction is thus a minute attention to individuals, and need not for a moment be confounded with generalisation.

Another cause of the confusion in question can be traced to the use of the verb 'is' to represent both the relation of a thing to the general idea it has contributed to form, and the relation of a single property to the thing in which it inheres. We say 'The man is a British subject'—classifying him; we say also 'The man is cold'—mentioning one of his attributes. There is no class of cold men, and the two relations have nothing in common. A class does not inhere in a man as cold inheres in him. There is no object corresponding to class—it is a conceptual creation.

The ambiguity of 'is' favours the syllogistic doctrine of predication, and there is a rule to the effect that in syllogising propositions, all verbs are to be converted into 'is' (or its conjugates) with a participle or noun, so that if they were not before statements of classification they now become such. 'He walks' is clearly no classification; but 'he is walking' is assimilated by false analogy to such a classification as 'he is human,' and so is treated as a classification by those who reason according to the Letter.

The substantive verb has no positive and uniform meaning. As an auxiliary it is a mere sign of tense, and in other positions it is an indefinite mark of relationship, the precise meaning of which must be determined by the subject and the context. It may sometimes be dispensed with in classification, as 'Victoria Regina'—'Phillips, Dentist.'

In the second category we have such propositions as 'the book lies on the table.' In syllogistic this is first altered to 'the book is lying on the table,' and it is feigned that 'lying on the table' is a class or general idea, and 'book' an individual of that class. To interpret 'the groom stands by the horse' a class has to be created, composed of the persons who happen to be standing by horses.

'The mountain is ten miles off' is a judgment in perspection. Syllogistically we are asked to believe that a class of things exists having the common property of being ten miles off, and that the mountain is entered in that class. The absurdity of this doctrine is self-evident.

In the remaining categories the reduction to 'is' has, if possible, a worse effect. In changing 'Canada lies west of Ireland' into 'Canada is a country lying west of Ireland,' we lose the relation in concretion, and express instead a verbal definition. Instead of affirming a position we explain a name. In such a proposition as 'the town of A lies 100 miles due north of B,' it is plain the predicate cannot be a class, for only one place has the quality expressed.

In the fifth category we have such a proposition as 'water freezes when the temperature falls to zero Centigrade.' This is turned into a substantive sentence by saying 'water is that liquid which freezes,' &c., which is a verbal or identical proposition.

'Cecrops founded Athens' is a judgment in causation. In turning it into 'Cecrops was (or is) the founder of Athens,' we emphasise the man's name, but the relation signified by 'founded' is slurred over or lost sight of. Boole converts 'Caesar conquered the Gauls' into 'Caesar is he who conquered the Gauls,'[23] and this he interpreted as classification. We need not be surprised that he should suppose a class could be formed by one individual, for he elsewhere tells us that Nothing is a class.[24]

Classification is not judgment of any sort—it is a variety of recollection. Logicians imagine it is the only judgment, and so far as they can they degrade true judgments to that spurious form.

Moods of the Syllogism. Having persuaded themselves that classification is the beginning, middle, and end of reasoning, logicians next proceed to divide the matter of their science.

Modern logicians who have some acquaintance with real thinking as exemplified in works of physical science, can, if acting according to their natural intelligence, lay down correct rules for dividing a subject. These are simple and obvious: divide according to fundamental resemblance—let each division correspond to some definite human utility—let the more important properties take precedence of the less important, and so forth: the merest common sense.

But in the division of their own subject they follow Aristotle, and so lose their way.

It is plain that an act of reasoning is a mental thing in the first place, and only when uttered, and thus in a secondary sense, is it a material object. The classification of arguments should therefore follow mental characteristics. Logicians make it follow the material characteristics of the terms in which the arguments are uttered. Their moods of the syllogism are mere varieties of expression, not varieties of reason.

The number of these moods is accidental, depending on flexibility of language and ingenuity in inventing varieties of syntax. Mere transposition of premises constitutes a difference of mood. Logicians however pretend to base their numeration on a more general necessity. They calculate from the distinctive parts of the three propositions forming a syllogism, varied by negation, &c., that there ought to be sixty-four moods. Experience proves that in spite of their free and easy method of multiplying syllogistic varieties they cannot produce anything like that number. One logician has thirty-six moods, another thirty-two, a third eleven; the more orthodox fix the number at nineteen. But they all admit that every argument can be reduced to one of four fundamental types—the moods of the First Figure. Why then have more classes than these four? Because, says Whately, it would be 'occasionally tedious' to reduce every argument to the first figure.

If the 11, 19, 32, or 36 classes were natural arguments taken down untouched from men's lips, and it was found to be useless and troublesome to reduce them to four artificial forms, the plea might be admitted. But the so-called valid syllogisms are themselves artificial, and just as tedious to make as the moods of the first figure. Not only so, but an elaborate system of mnemonic rules is provided for reducing the valid moods to the fundamental moods, thus admitting that the former are only intermediate halting places between the natural speech and the fundamental moods. It is expected that the intermediates should be reduced to the first figure.

Is there anything analogous to this sort of division in any science or branch of practical thought? Would logicians themselves sanction such a classification in a natural science? If a zoologist, for example, were to determine beforehand how many classes of animals there ought to be, would they not say he was acting improperly? If, after discovering that he had five times as many classes as he could find animals to put into them, he still retained his classification and required his pupils to write out the names or symbols of all the useless classes—would not logicians be apt to call him a pedant? Yet in a modern work on logic such a task is prescribed for students:—

'Write out the sixty-four moods of the syllogism, and strike out the fifty-three invalid ones.'

We might have excused the existence of a merely verbal classification in logic, if it were accompanied by and subordinated to a classification of theorems considered as mental facts. But in syllogistic the verbal is the dominant classification, and we have seen from the procedure of Sir William Hamilton—in dropping his categorical judgments—that when the two principles of division conflict, it is the mental which has to give way. The Letter is allowed to kill the Spirit.

All the Moods reducible to One. Syllogists appear not to know their own schematism very well. They say there are four ultimate moods, which it is impossible to reduce to any lower number. But since each of the four is, mentally, a double classification, it must be possible to reflect this common property in the mode of expression. The difference between them can only be verbal. Let us adopt another than the ordinary symbolism.

Cut a card into three triangular pieces of unequal size, and call them by the letters A, B, C, beginning with the largest. These are the terms of the syllogism.

Barbara. Celarent.

Darii. Ferio.

The first mood Barbara is formed by placing the cards on top of each other, so that B is within the margin of A, and C within the margin of B. This is the syllogism, 'All B is A, all C is B, therefore all C is A.'

Next let B and C be as above, but let A be wholly apart from both. This is Celarent: 'No B is A, all C is B, therefore no C is A.'

In Darii the whole of B is in A, but only a part of C coincides with B. The syllogism is: 'All B is A, some C is B, therefore some C is A.'

In Ferio A is again wholly separated from the others, and C is only partially in B. Argument: 'No B is A, some C is B, therefore some C is not A.'

It is to be remembered that all the other figures and moods are reducible to the above figure of four moods, so that the reduction applicable to the latter is equally applicable to the former.

To reduce Darii to Barbara all that is necessary is to ignore the dotted part of C. That is suggested by the use of the word 'some,' which has a correlative 'all' or 'others.' But the correlative quantity does not enter into the syllogism, and we know nothing about it. It may not even exist. We are therefore at liberty to substitute for 'some C' the name D, and consider it an integer instead of a fraction. Then we have the Barbara syllogism: 'All B is A, all D (= some C) is B, therefore all D is A.' The phrase 'all of some' is quite allowable: 'I met some firemen, all of whom wore brass helmets.'

Ferio in the same manner is reduced to Celarent. The dotted part of C is cut away, and the part really significant in the syllogism is called E. Then 'No B is A, all E is B, no E is A.'

Finally Celarent can be reduced to Barbara. B cannot indeed be enclosed in A, but we assume the existence of a whole having all the characters which A has not, or having none of the characters which A has. This is the whole F = Not-A. Then Celarent becomes Barbara thus: 'All B is F, all C is B, therefore all C is F.'

This demonstrates that there is only one fundamental operation where syllogists suppose there are at least four. The difference is wholly a matter of language, and disappears on changing the names of the terms and ignoring irrelevant suggestions. But the syllogism, I repeat, does not represent the act of reasoning, and its moods and figures are fit only to be a game for children.

[20:] Logic, Book I. § 3.

[21:] Logic, Book II. c. 3. § 2.

[22:] Lectures, iii. pp. 287 and 356. The impossibility of reconciling their definitions and rules to real thinking and argument is the despair of logicians. Most of them take to symbols, which are more accommodating than real experience, having just such properties as their makers choose to put in them. Sir William Hamilton had the courage to declare that a logician might use arguments of a concrete or real form, but that it is not necessary they should agree with real fact. 'The logician has a right to suppose any material impossibility, any material falsity; he takes no account of what is objectively impossible or false, he has a right to assume what premises he please, provided that they do not involve a contradiction in terms.'—Id. 322. That means in plain English that a logician may misrepresent matters of fact, if he cannot otherwise establish his theory!

[23:] Laws of Thought, p. 35.

[24:] Ibid. p. 47.