Philosophy

It is customary in science to regard certain facts as “data”, from which laws and also other facts are “inferred”. We saw in Chapter VII that the practice of inference is much wider than the theories of any logician would justify, and that it is nothing other than the law of association or of “learned reactions”. In the present chapter, I wish to consider what the logicians have evolved from this primitive form of inference, and what grounds we have, as rational beings, for continuing to infer. But let us first get as clear a notion as we can of what should be meant by a “datum”.

The conception of a “datum” cannot be made absolute. Theoretically, it should mean something that we know without inference. But before this has any definite meaning, we must define both “knowledge” and “inference”. Both these terms have been considered in earlier chapters. For our present purpose it will simplify matters to take account only of such knowledge as is expressed in words. We considered in Chapter XXIV the conditions required in order that a form of words may be “true”; for present purposes, therefore, we may say that “knowledge” means “the assertion of a true form of words”. This definition is not quite adequate, since a man may be right by chance; but we may ignore this complication. We may then define a “datum” as follows: A “datum” is a form of words which a man utters as the result of a stimulus, with no intermediary of any learned reaction beyond what is involved in knowing how to speak. We must, however, permit such learned reactions as consist in adjustments of the sense-organs or in mere increase of sensitivity. These merely improve the receptivity to data, and do not involve anything that can be called inference.

If the above definition is accepted, all our data for knowledge of the external world must be of the nature of percepts. The belief in external objects is a learned reaction acquired in the first months of life, and it is the duty of the philosopher to treat it as an inference whose validity must be tested. A very little consideration shows that, logically, the inference cannot be demonstrative, but must be at best probable. It is not logically impossible that my life may be one long dream, in which I merely imagine all the objects that I believe to be external to me. If we are to reject this view, we must do so on the basis of an inductive or analogical argument, which cannot give complete certainty. We perceive other people behaving in a manner analogous to that in which we behave, and we assume that they have had similar stimuli. We may hear a whole crowd say “oh” at the moment when we see a rocket burst, and it is natural to suppose that the crowd saw it too. Nor are such arguments confined to living organisms. We can talk to a dictaphone and have it afterwards repeat what we said; this is most easily explained by the hypothesis that at the surface of the dictaphone events happened, while I was speaking, which were closely analogous to those that were happening just outside my ears. It remains possible that there is no dictaphone and I have no ears and there is no crowd watching the rocket; my percepts may be all that is happening in such cases. But, if so, it is difficult to arrive at any causal laws, and arguments from analogy are more misleading than we are inclined to think them. As a matter of fact, the whole structure of science, as well as the world of common sense, demands the use of induction and analogy if it is to be believed. These forms of inference, therefore, rather than deduction, are those that must be examined if we are to accept the world of science or any world outside of our own dreams.

Let us take a simple example of an induction which we have all performed in practice. If we are hungry, we eat certain things we see and not others—it may be said that we infer edibility inductively from a certain visual and olfactory appearance. The history of this process is that children a few months old put everything into their mouths unless they are stopped; sometimes the result is pleasant, sometimes unpleasant; they repeat the former rather than the latter. That is to say: given that an object having a certain visual and olfactory appearance has been found pleasant to eat, an object having a very similar appearance will be eaten; but when a certain appearance has been found connected with unpleasant consequences when eaten, a similar appearance does not lead to eating next time. The question is: what logical justification is there for our behaviour? Given all our past experience, are we more likely to be nourished by bread than by a stone? It is easy to see why we think so, but can we, as philosophers justify this way of thinking?

It is, of course, obvious that unless one thing can be a sign of another both science and daily life would be impossible. More particularly, reading involves this principle. One accepts printed words as signs, but this is only justifiable by means of induction. I do not mean that induction is necessary to establish the existence of other people, though that also, as we have seen, is true. I mean something simpler. Suppose you want your hair cut, and as you walk along the street you see a notice “hair-cutting, first floor”. It is only by means of induction that you can establish that this notice makes it in some degree probable that there is a hair-cutter’s establishment on the first floor. I do not mean that you employ the principle of induction; I mean that you act in accordance with it, and that you would have to appeal to it if you were accompanied by a long-haired sceptical philosopher who refused to go upstairs till he was persuaded there was some point in doing so.

The principle of induction, prima facie, is as follows: Let there be two kinds of events, A and B (e.g. lightning and thunder), and let many instances be known in which an event of the kind A has been quickly followed by one of the kind B, and no instances of the contrary. Then either a sufficient number of instances of this sequence, or instances of suitable kinds will make it increasingly probable that A is always followed by B, and in time the probability can be made to approach certainty without limit provided the right kind and number of instances can be found. This is the principle we have to examine. Scientific theories of induction generally try to substitute well-chosen instances for numerous instances, and represent number of instances as belonging to crude popular induction. But in fact popular induction depends upon the emotional interest of the instances, not upon their number. A child who has burnt its hand once in a candle-flame establishes an induction, but words take longer, because at first they are not emotionally interesting. The principle used in primitive practice is: Whatever, on a given occasion, immediately precedes something very painful or pleasant, is a sign of that interesting event. Number plays a secondary part as compared with emotional interest. That is one reason why rational thought is so difficult.

The logical problem of induction is to show that the proposition “A is always accompanied (or followed) by B” can be rendered probable by knowledge of instances in which this happens, provided the instances are suitably chosen or very numerous. Far the best examination of induction is contained in Mr. J. M. Keynes’s Treatise on Probability. There is a valuable doctor’s thesis by the late Jean Nicod, Le Problème logique de l’induction, which is very ably reviewed by R. B. Braithwaite in Mind, October 1925. A man who reads these three will know most of what is known about induction. The subject is technical and difficult, involving a good deal of mathematics, but I will attempt to give the gist of the results.

We will begin with the condition in which the problem had been left by J. S. Mill. He had four canons of induction, by means of which, given suitable examples, it could be demonstrated that A and B were causally connected, if the law of causation could be assumed. That is to say, given the law of causation, the scientific use of induction could be reduced to deduction. Roughly the method is this: We know that B must have a cause; the cause cannot be C or D or E, etc., because we find by experiment or observation that these may be present without producing B. On the other hand, we never succeed in finding A without its being accompanied (or followed) by B. If A and B are both capable of quantity, we may find further that the more there is of A the more there is of B. By such methods we eliminate all possible causes except A; therefore, since B must have a cause, that cause must be A. All this is not really induction at all; true induction only comes in in proving the law of causation. This law Mill regards as proved by mere enumeration of instances: we know vast numbers of events which have causes, and no events which can be shown to be uncaused; therefore, it is highly probable that all events have causes. Leaving out of account the fact that the law of causality cannot have quite the form that Mill supposed, we are left with the problem: Does mere number of instances afford a basis for induction? If not, is there any other basis? This is the problem to which Mr. Keynes addresses himself.

Mr. Keynes holds that an induction may be rendered more probable by number of instances, not because of their mere number, but because of the probability, if the instances are very numerous, that they will have nothing in common except the characteristics in question. We want, let us suppose, to find out whether some quality A is always associated with some quality B. We find instances in which this is the case; but it may happen that in all our instances some quality C is also present, and that it is C that is associated with B. If we can so choose our instances that they have nothing in common except the qualities A and B, then we have better grounds for holding that A is always associated with B. If our instances are very numerous, then, even if we do not know that they have no other common quality, it may become quite likely that this is the case. This, according to Mr. Keynes, is the sole value of many instances.

A few technical terms are useful. Suppose we want to establish inductively that there is some probability in favour of the generalisation: “Everything that has the property F also has the property f”. We will call this generalisation g. Suppose we have observed a number of instances in which F and f go together, and no instances to the contrary. These instances may have other common properties as well; the sum-total of their common properties is called the total positive analogy, and the sum-total of their known common qualities is called the known positive analogy. The properties belonging to some but not to all of the instances in question are called the negative analogy: all of them constitute the total negative analogy, all those that are known constitute the known negative analogy. To strengthen an induction, we want to diminish the positive analogy to the utmost possible extent; this, according to Mr. Keynes, is why numerous instances are useful.

On “pure” induction, where we rely solely upon number of instances, without knowing how they affect the analogy, Mr. Keynes concludes (Treatise in Probability, p. 236):

“We have shown that if each of the instances necessarily follows from the generalisation, then each additional instance increases the probability of the generalisation, so long as the new instance could not have been predicted with certainty from a knowledge of the former instances.... The common notion, that each successive verification of a doubtful principle strengthens it, is formally proved, therefore without any appeal to conceptions of law or of causality. But we have not proved that this probability approaches certainty as a limit, or even that our conclusion becomes more likely than not, as the number of verifications or instances is indefinitely increased.”

It is obvious that induction is not much use unless, with suitable care, its conclusions can be rendered more likely to be true than false. This problem therefore necessarily occupies Mr. Keynes.

It is found that an induction will approach certainty as a limit if two conditions are fulfilled:

(1) If the generalisation is false, the probability of its being true in a new instance when it has been found to be true in a certain number of instances, however great that number may be, falls short of certainty by a finite amount.

(2) There is a finite a priori probability in favour of our generalisation.

Mr. Keynes uses “finite” here in a special sense. He holds that not all probabilities are numerically measurable; a “finite” probability is one which exceeds some numerically measurable probability however small. E.g. our generalisation has a finite a priori probability if it is less unlikely than throwing heads a billion times running.

The difficulty is, however, that there is no easily discoverable way of estimating the a priori probability of a generalisation. In examining this question, Mr. Keynes is led to a very interesting postulate which, if true, will, he thinks, give the required finite a priori probability. His postulate as he gives it is not quite correct, but I shall give his form first, and then the necessary modification.

Mr. Keynes supposes that the qualities of objects cohere in groups, so that the number of independent qualities is much less than the total number of qualities. We may conceive this after the analogy of biological species: a cat has a number of distinctive qualities which are found in all cats, a dog has a number of other distinctive qualities which are found in all dogs. The method of induction can, he says, be justified if we assume “that the objects in the field, over which our generalisations extend, do not have an infinite number of independent qualities; that, in other words, their characteristics, however numerous, cohere together in groups of invariable connection, which are finite in number” (p. 256). Again (p. 258): “As a biological foundation for Analogy, therefore, we seem to need some such assumption as that the amount of variety in the universe is limited in such a way that there is no one object so complex that its qualities fall into an infinite number of independent groups ... or rather that none of the objects about which we generalise are as complex as this; or at least that, though some objects may be infinitely complex, we sometimes have a finite probability that an object about which we seek to generalise is not infinitely complex.”

This postulate is called the “principle of limitation of variety”. Mr. Keynes again finds that it is needed in attempts to establish laws by statistics; if he is right, it is needed for all our scientific knowledge outside pure mathematics. Jean Nicod pointed out that it is not quite sufficiently stringent. We need, according to Mr. Keynes, a finite probability that the object in question has only a finite number of independent qualities; but what we really need is a finite probability that the number of its independent qualities is less than some assigned finite number. This is a very different thing, as may be seen by the following illustration. Suppose there is some number of which we know only that it is finite; it is infinitely improbable that it will be less than a million, or a billion, or any other assigned finite number, because, whatever such number we take, the number of smaller numbers is finite and the number of greater numbers is infinite. Nicod requires us to assume that there is a finite number n such that there is a finite probability that the number of independent qualities of our object is less than n. This is a much stronger assumption than Mr. Keynes’s, which is merely that the number of independent qualities is finite. It is the stronger assumption which is needed to justify induction.

This result is very interesting and very important. It is remarkable that it is in line with the trend of modern science. Eddington has pointed out that there is a certain finite number which is fundamental in the universe, namely the number of electrons. According to the quantum theory, it would seem that the number of possible arrangements of electrons may well also be finite, since they cannot move in all possible orbits, but only in such as make the action in one complete revolution conform to the quantum principle. If all this is true, the principle of limitation of variety may well also be true. We cannot, however, arrive at a proof of our principle in this way, because physics uses induction, and is therefore presumably invalid unless the principle is true. What we can say, in a general way, is that the principle does not refute itself, but, on the contrary, leads to results which confirm it. To this extent, the trend of modern science may be regarded as increasing the plausibility of the principle.

It is important to realise the fundamental position of probability in science. At the very best, induction and analogy only give probability. Every inference worthy of the name is inductive, therefore all inferred knowledge is at best probable. As to what is meant by probability, opinions differ. Mr. Keynes takes it as a fundamental logical category: certain premisses may make a conclusion more or less probable, without making it certain. For him, probability is a relation between a premiss and a conclusion. A proposition does not have a definite probability on its own account; in itself, it is merely true or false. But it has probabilities of different amounts in regard to different premisses. When we speak, elliptically, of the probability of a proposition, we mean its probability in relation to all our relevant knowledge. A proposition in probability cannot be refuted by mere observation: improbable things may happen and probable things may fail to happen. Nor is an estimate of probability relevant to given evidence proved wrong when further evidence alters the probability.

For this reason the inductive principle cannot be proved or disproved by experience. We might prove validly that such and such a conclusion was enormously probable, and yet it might not happen. We might prove invalidly that it was probable, and yet it might happen. What happens affects the probability of a proposition, since it is relevant evidence; but it never alters the probability relative to the previously available evidence. The whole subject of probability, therefore, on Mr. Keynes’s theory, is strictly a priori and independent of experience.

There is however another theory, called the “frequency theory”, which would make probability not indefinable, and would allow empirical evidence to affect our estimates of probability relative to given premisses. According to this theory in its crude form, the probability that an object having the property F will have the property f is simply the proportion of the objects having both properties to all those having the property F. For example, in a monogamous country the probability of a married person being male is exactly a half. Mr. Keynes advances strong arguments against all forms of this theory that existed when his book was written. There is however an article by R. H. Nisbet on “The Foundations of Probability” in Mind for January 1926, which undertakes to rehabilitate the frequency theory. His arguments are interesting, and suffice to show that the controversy is still an open one, but they do not, in my opinion, amount to decisive proof. It is to be observed, however, that the frequency theory, if it could be maintained, would be preferable to Mr. Keynes’s, because it would get rid of the necessity for treating probability as indefinable, and would bring probability into much closer touch with what actually occurs. Mr. Keynes leaves an uncomfortable gap between probability and fact, so that it is far from clear why a rational man will act upon a probability. Nevertheless, the difficulties of the frequency theory are so considerable that I cannot venture to advocate it definitely. Meanwhile, the details of the discussion are unaffected by the view we may take on this fundamental philosophical question. And on either view the principle of limitation of variety will be equally necessary to give validity to the inferences by induction and analogy upon which science and daily life depend.