Domestic animals which are habitually fed by a certain person will run towards that person as soon as they see him. We say that they expect food, and in fact their behaviour is very like what it would be if they saw food. But really we have only an example of “conditioning”: they have often seen first the farmer and then the food, so that in time they react to the farmer as they originally reacted to the food. Infants soon learn to react to the sight of the bottle, although at first they only react to the touch of it. When they can speak, the same law makes them say “dinner” when they hear the dinner-bell. It is quite unnecessary to suppose that they first think “that bell means dinner”, and then say “dinner”. The sight of dinner (by previous “learned reaction”) causes the word “dinner”: the bell frequently precedes the sight of dinner; therefore in time the bell produces the word “dinner”. It is only subsequent reflection, probably at a much later age, that makes the child say “I knew dinner was ready because I heard the bell”. Long before he can say this, he is acting as if he knew it. And there is no good reason for denying that he knows it, when he acts as if he did. If knowledge is to be displayed by behaviour, there is no reason to confine ourselves to verbal behaviour as the sole kind by which knowledge can manifest itself.
The situation, stated abstractly, is as follows. Originally, stimulus A produced reaction C; now stimulus B produces it, as a result of association. Thus B has become a “sign” of A, in the sense that it causes the behaviour appropriate to A. All sorts of things may be signs of other things, but with human beings words are the supreme example of signs. All signs depend upon some practical induction. Whenever we read or hear a statement, its effect upon us depends upon induction in this sense, since the words are signs of what they mean, in the sense that we react to them, in certain respects, as we should to what they stand for. If some one says to you “your house is on fire”, the effect upon you is practically the same as if you saw the conflagration. You may, of course, be the victim of a hoax, and in that case your behaviour will not be such as achieves any purpose you have in view. This risk of error exists always, since the fact that two things have occurred together in the past cannot prove conclusively that they will occur together in the future.
Scientific induction is an attempt to regularise the above process, which we may call “physiological induction”. It is obvious that, as practised by animals, infants, and savages, physiological induction is a frequent source of error. There is Dr. Watson’s infant who induced, from two examples, that whenever he saw a certain rat there would be a loud noise. There is Edmund Burke, who induced from one example (Cromwell) that revolutions lead to military tyrannies. There are savages who argue, from one bad season, that the arrival of a white man causes bad crops. The inhabitants of Siena, in 1348, thought that the Black Death was a punishment for their pride in starting to build too large a cathedral. Of such examples there is no end. It is very necessary, therefore, if possible, to find some method by which induction can be practised so as to lead, in general, to correct results. But this is a problem of scientific method, with which we will not yet concern ourselves.
What does concern us at present is the fact that all inference, of the sort that really occurs, is a development of this one principle of conditioning. In practice, inference is of two kinds, one typified by induction, the other by mathematical reasoning. The former is by far the more important, since, as we have seen, it covers all use of signs and all empirical generalisations as well as the habits of which they are the verbal expression. I know that, from the traditional standpoint, it seems absurd to talk of inference in most cases of this sort. For example, you find it stated in the paper that such and such a horse has won the Derby. According to my own use of words, you practise an induction when you arrive thence at the belief that that horse has won. The stimulus consists of certain black marks on white paper—or perhaps on pink paper. This stimulus is only connected with horses and the Derby by association, yet your reaction is one appropriate to the Derby. Traditionally, there was only inference where there was a “mental process”, which, after dwelling upon the “premisses”, was led to assert the “conclusion” by means of insight into their logical connection. I am not saying that the process which such words as the above are intended to describe never takes place; it certainly does. What I am saying is that, genetically and causally, there is no important difference between the most elaborate induction and the most elementary “learned reaction”. The one is merely a developed form of the other, not something radically different. And our determination to believe in the results of inductions, even if, as logicians, we see no reason to do so, is really due to the potency of the principle of association; it is an example—perhaps the most important example—of what Dr. Santayana calls “animal faith”.
The question of mathematical reasoning is more difficult. I think we may lay it down that, in mathematics, the conclusion always asserts merely the whole or part of the premisses, though usually in new language. The difficulty of mathematics consists in seeing that this is so in particular cases. In practice, the mathematician has a set of rules according to which his symbols can be manipulated, and he acquires technical skill in working according to the rules in the same sort of way as a billiard-player does. But there is a difference between mathematics and billiards: the rules of billiards are arbitrary, whereas in mathematics some at least are in some sense “true”. A man cannot be said to understand mathematics unless he has “seen” that these rules are right. Now what does this consist of? I think it is only a more complicated example of the process of understanding that “Napoleon” and “Bonaparte” refer to the same person. To explain this, however, we must revert to what was said, in the [chapter on “Language”], about the understanding of form.
Human beings possess the power of reacting to form. No doubt some of the higher animals also possess it, though to nothing like the same extent as men do; and all animals, except a few of the most intelligent species, appear to be nearly devoid of it. Among human beings, it differs greatly from one individual to another, and increases, as a rule, up to adolescence. I should take it as what chiefly characterises “intellect”. But let us see, first, in what the power consists.
When a child is being taught to read, he learns to recognise a given letter, say H, whether it is large or small, black or white or red. However it may vary in these respects his reaction is the same: he says “H”. That is to say, the essential feature in the stimulus is its form. When my boy, at the age of just under three, was about to eat a three-cornered piece of bread and butter, I told him it was a triangle. (His slices were generally rectangular.) Next day, unprompted, he pointed to triangular bits in the pavement of the Albert Memorial, and called them “triangles”. Thus the form of the bread and butter, as opposed to its edibility, its softness, its colour, etc., was what had impressed him. This sort of thing constitutes the most elementary kind of reaction to form.
Now “matter” and “form” can be placed, as in the Aristotelian philosophy, in a hierarchy. From a triangle we can advance to a polygon, thence to a figure, thence to a manifold of points. Then we can go on and turn “point” into a formal concept, meaning “something that has relations which resemble spatial relations in certain formal respects”. Each of these is a step away from “matter” and further into the region of “form”. At each stage the difficulty increases. The difficulty consists in having a uniform reaction (other than boredom) to a stimulus of this kind. When we “understand” a mathematical expression, that means that we can react to it in an appropriate manner, in fact, that it has “meaning” for us. This is also what we mean by “understanding” the word “cat”. But it is easier to understand the word “cat”, because the resemblances between different cats are of a sort which causes even animals to have a uniform reaction to all cats. When we come to algebra, and have to operate with x and y, there is a natural desire to know what x and y really are. That, at least, was my feeling: I always thought the teacher knew what they really were, but would not tell me. To “understand” even the simplest formula in algebra, say (x + y)² = x² + 2xy + y², is to be able to react to two sets of symbols in virtue of the form which they express, and to perceive that the form is the same in both cases. This is a very elaborate business, and it is no wonder that boys and girls find algebra a bugbear. But there is no novelty in principle after the first elementary perceptions of form. And perception of form consists merely in reacting alike to two stimuli which are alike in form but very different in other respects. For, when we can do that, we can say, on the appropriate occasion, “that is a triangle”; and this is enough to satisfy the examiner that we know what a triangle is, unless he is so old-fashioned as to expect us to reproduce the verbal definition, which is of course a far easier matter, in which, with patience, we might teach even a parrot to succeed.
The meanings of complex mathematical symbols are always fixed by rules in relation to the meaning of simpler symbols; thus their meanings are analogous to those of sentences, not to those of single words. What was said earlier about the understanding of sentences applies, therefore, to any group of symbols which, in mathematics, will be declared to have the same meaning as another group, or part of that meaning.
We may sum up this discussion by saying that mathematical inference consists in attaching the same reactions to two different groups of signs, whose meanings are fixed by convention in relation to their constituent parts, whereas induction consists, first, in taking something as a sign of something else, and later, when we have learned to take A as a sign of B, in taking A as also a sign of C. Thus the usual cases of induction and deduction are distinguished by the fact that, in the former, the inference consists in taking one sign as a sign of two different things, while in the latter the inference consists in taking two different signs as signs of the same thing. This statement is a little too antithetical to be taken as an exact expression of the whole truth in the matter. What is true, however, is that both kinds of inferences are concerned with the relation of a sign to what it signifies, and therefore come within the scope of the law of association.