Philosophy

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.