Part 11

[Footnote 18: Whately complains of the disinclination shown by logicians to put their rules into practice. 'Whenever they have to treat of anything that is beyond the mere elements of Logic, they totally lay aside all reference to the principles they have been occupied in establishing and explaining, and have recourse to a loose, vague, and popular kind of language; such as would be best suited indeed to an exoterical discourse but seems strangely incongruous in a professed logical treatise.... Surely it affords but too much plausibility to the cavils of those who scoff at Logic altogether, that the very writers who profess to teach it should never themselves make any application of, or reference to, its principles, when, and _when only_, such application and reference are to be expected.' _Logic_, Book III. Introd. The fact here admitted proves that even logicians do not find their method of any practical use. But what is the meaning of the emphatic 'when only'? Why should a logical method be unsuitable for every sort of subject except those matters of logic that are beyond the mere elements?]

[Footnote 19: _Logic_, 'Fallacies,' c. 6.]

ACADEMICAL DIALECTIC

XXXI--ANALOGY

Logicians of Greek inspiration apply the term reasoning or argument to at least eight different intellectual operations, some of them important indeed but only one of them argument. This is Analogy--which receives but little notice from logicians because it does not give certain conclusions. The operations mistaken for argument are:

Immediate Inference-- Arithmetical Calculation-- Geometrical Demonstration-- Induction-- Aristotle's Dictum-- Mediate Comparison-- Syllogism.

XXXII--IMMEDIATE INFERENCE

Some logicians maintain that it is possible to draw a kind of conclusions from one judgment alone. These pretended conclusions are of two species.

The first is a restatement in different words of the whole or part of the single idea, and it is preceded by 'therefore' to give it the appearance of an argument. 'All men suffer, therefore some men suffer.' 'John is a man, therefore he is a living creature.' 'This weighs that down, therefore it is heavier.' These are all obvious tautologisms. It is not an inference to deny the opposite of what we have asserted, as 'The weather is warm, therefore it is not cold.' The conditional and dilemmatic examples of logicians abound in such 'inferences.' We cannot entirely avoid these locutions, as they give point and clearness to speech, but they are not argument, even when introduced by 'therefore.'

The other species of spurious conclusions arises out of what is technically called Conversion. This is a process permitted in Syllogistic in order to render propositions more explicit. The subject may change places with the predicate, a 'some' may be inserted, an 'all' suppressed, or a 'not' may be made to qualify one word instead of another. In all this there must be no change in the meaning of the proposition, and therefore there can be no inference. If the second proposition means something more or different from the first, another premise is unconsciously taken for granted, or the supposed interpretation amounts to interpolation. The reasoner may have inadvertently or sophistically added something to the original datum. Here is an example of inference by conversion--'All cabbages are plants, therefore _some_ plants are cabbages.' If it is not understood from the terms of the first proposition that plants are limited to such as are cabbages, the 'some' of the converted proposition is an interpolation supplied from the reasoner's knowledge of the matter. In this case the 'quantification' of plants is not a valid inference from the original information.

XXXIII--ARITHMETICAL CALCULATION

Arithmetic is first a manipulation of symbols called 'figures.' There are ten of these, and they are capable of many species of combination, and an indefinite number of individual operations under each species. Certain rules govern each sort of operation, and when the rules are properly understood and recollected the operations can be performed with absolute certainty. Although the figures have names relating to number, and the problems given for exercise make mention of acres, pounds, tons, miles, and all sorts of concrete objects, the symbolic calculations of books have no necessary relation to real things, numbers, or quantities. They are a purely conventional treatment of arbitrary marks that may mean anything or nothing. That is the arithmetic of the 'schools.' There is no trace of reasoning or argument in it--it is mere rule and recollection.

There is however real Number and there is real Quantity. Number is that quality in which a group of three things (for instance) is seen to differ from a group of four or seven, even when the things are otherwise quite similar. We begin by distinguishing ten primary degrees of this difference, and then consider other degrees as multiples or parts of these primary degrees.

Quantity is degree in size, and is a property quite different from number. But, for convenience, we assume that quantities are all units or fractions of certain standard quantities, and we are thus enabled to use the same terms for both number and quantity.

The names which written language provides for the numerical degrees and their combinations are inconvenient to use, and so a set of symbols was devised exclusively for numerical designation. These are the figures of arithmetic. They are the technical vocabulary of number, and of quantity considered as number.

Number and quantity admit of but two kinds of variation--increase and diminution. These variations can be denoted so correctly by figures, that any combination we first make in figures according to rule can be reproduced in real objects, provided the objects are in other respects possible. The result of this perfection of technical nomenclature is that our study of number and quantity has been transferred from real objects to figures. It has become symbolic and indirect, and most of us never go beyond the symbols; that is, what we call arithmetic is an affair of figures, not of true quantities and numbers. We talk of miles, tons, and pounds sterling, but we do not _think_ of miles, tons, and pounds sterling--we think of _figures_. A thousand shillings is to us, when arithmetically stated, '1000_s._,' just as it is here represented on paper; we do not think of silver coins, and we could not if we tried imagine a thousand things of any sort. There is in reality an enormous difference between '0001_s._' and '1000_s._,' but to the arithmetician the only objective difference is one of arrangement in figures.

From these considerations it follows that there are two sciences of number. There is the true science which deals with quantities really seen in objects and imagined in the mind, and an artificial science dealing with figures which have only a historical connection with real quantity. Of the latter, unfortunately, our arithmetical education chiefly consists. We are never taught to distinguish number and size in things by the 'eye,' that is, by reason. The symbolism that was originally intended to assist real arithmetical thought has ended by supplanting it. An ignorant shepherd, bricklayer, or carpenter, who is accustomed to make a rapid estimate of the number of things in a mass, or the area of planking in a log, has a better training in real arithmetical science than some mathematicians. If we are obliged to practise genuine arithmetical thought in engineering, astronomy, and other professions, our scholastic symbolism gets realised to some extent, and is a great assistance in arithmetical estimation. But without this it has no more reference to number and quantity than a musical education, based entirely on the printed or written notation, has to the appreciation of musical sounds. A book arithmetician is in the position of a person thoroughly acquainted with theoretical music, and who can even compose music _according to rule_, but who is unable to distinguish a high note from a low one or harmony from discord in actual sound.

It will thus be seen that it is only in the real arithmetic that reasoning can enter. The judgment in free arithmetical observation is the counting of actual groups and the measurement of actual surfaces, and the argument consists in estimating the number of individuals in other groups, and the size of other surfaces, without counting or measurement. But this exercise never enters into symbolic arithmetic. All the apparent conclusions of book arithmetic are tautological; they consist in repeating in one combination of symbols the whole or part of what has been already given in another combination. It is an exercise in expression--nothing more.

Arithmetical ratio has a resemblance to the rational parallel. 3:5::9:15 might be arranged thus--

5 | 15 --+--- 3 | 9

This is not argument, for two reasons. (1) The apparent conclusion is not an effort of rational imagination; it is a figure that can be obtained with infallible certainty by treating the other figures according to a rule, which has only to be recollected and applied. (2) The relation between the left-hand figures and the right-hand figures is not a categorical judgment; it is a form of resemblance, and so it cannot yield a valid conclusion.

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.