Harold Jeffreys on Logic and Scientific Inference

Between 1919 and 1923 Harold Jeffreys and Dorothy Wrinch wrote three papers on probability and scientific inference. These are:

D Wrinch and H Jeffreys, On Some Aspects of the Theory of Probability, Philosophical Magazine 38 (1919), 715-731.

D Wrinch and H Jeffreys, On Certain Fundamental Principles of Scientific Inquiry, Philosophical Magazine 42 (1921), 369-390,

D Wrinch and H Jeffreys, On Certain Fundamental Principles of Scientific Inquiry, Philosophical Magazine 45 (1923), 368-374.

In their views of probability they were influenced by William Ernest Johnson and John Maynard Keynes. Dorothy Wrinch had, in fact, attended lectures by Johnson. Jeffreys used the ideas from these papers in his book Scientific Inference published by Cambridge University Press in 1931. We give below an extract from Jeffreys' book on Logic and Scientific Inference

Logic and Scientific Inference

by
Harold Jeffreys

1.

The fundamental problem of this work is the question of the nature of scientific inference. The data available to the scientific worker, as well as to the man in the street, are composed of two classes. The first class consists of the crude data provided by the senses. These will be called sensations. The second class consists of general principles, which determine how the information provided by the senses is to be treated. It is actually treated in two different ways, which may be called description and inference. Description, in the strict sense, would involve only the cataloguing and classification of sensations already experienced. Inference is the use of sensations already experienced to derive information about sensations not yet experienced, to construct physical objects, and to describe the past and future of these physical objects. For pure description only an application of the principles of classification and the properties of classes is required; these are purely logical ideas.

Inference requires much more. However fully one's past experience has been described and indexed, nothing not included in it can be inferred without some principles not purely logical in character. As a matter of logic this is a commonplace. Actually one proceeds, in the simplest type of inference, on the supposition that what has been found to be true in previous instances will be repeated in new instances. The distinction between deductive logic and scientific inference may be illustrated by means of one of the classical instances of the former.

All men are mortal.
Socrates is a man.
Therefore Socrates is mortal.

This type of argument, the syllogism, is one of those chiefly used in pure logic; indeed it was believed for ages that there was no other. The first, or general, statement about all men is called the major premiss, the particular statement that Socrates is a man is called the minor premiss, and from the two together we draw the conclusion that Socrates is mortal. But as a scientific argument it is unsatisfactory. We question immediately whether the major premiss is true. It is not known by experience. We cannot state as a result of experience that a man is mortal until he is dead. At any instant men are living, and they all constitute unverified instances of the premiss; it is simply unknown by experience whether the general statement is true or not. Gulliver, arriving in Luggnagg, might have said equally well:

All men are mortal.
A Struldbrug is a man.
Therefore a Struldbrug is mortal.

But Gulliver knew better and did not argue with his informers.

There are several ways of treating the classical syllogism so as to make it somewhat more acceptable to scientific thought. One is to say that the general proposition is not asserted from experience at all, but is known to be true in all possible cases from previous knowledge. In such a case the syllogism becomes valid. But we avoid the difficulty only by admitting that there may be knowledge applicable to the study of experience and not itself derived from experience. This type of knowledge we call a priori. We do not say that it is the solution of the present difficulty, but a priori knowledge exists, and we shall have occasion later to consider instances of it at length.

The word mortal itself introduces difficulties of a type that will concern us later. Suppose that we accepted the syllogism and that Socrates had nevertheless survived till the present day. We should still not be compelled to reject the conclusion of the syllogism. If a doubter pointed out that Socrates had reached the advanced age of 2000 odd, that would not in the least prevent us from continuing to assert his mortality. Our reply would be that he might die to-morrow as far as the doubter knew; and that would close the matter unless the doubter thought of a new line of attack. But suppose he went on: "You are saying that Socrates will not live for ever. May I point out that even if he lives to be a million years old it could still be said that he would die some day? Your statement has the quality that no evidence could possibly be produced that would contradict it. Even if it is true it still gives no reason to suppose that a man cannot live till he is a million years old. In fact it is vague and unverifiable, and therefore uninteresting". The doubter has at this stage abandoned the attempt to show that the deduction has been falsified by experience; he says instead that it is futile because it is not capable of being compared with experience. This is the scientific attitude.

Both these criticisms of the classical syllogism have analogues in relation to certain modem theories of scientific knowledge, as we shall see later.

2.

An essential object of scientific inference is to increase knowledge. The syllogism has a place in it just so far as it assists this object. Some syllogisms do; others do not. Consider the following example:

All English policemen are over five feet nine inches in height.
Brown is an English policeman.
Therefore Brown is over five feet nine inches in height.

The syllogism as it stands is perfect. The conclusion is free from the difficulties of that of the classical syllogism; it is perfectly possible to measure Brown's height. But how do we know the general proposition? If it is known by experience, we have already measured the heights of all English policemen, and therefore we have measured P.C. Brown in particular, and we know directly that his height is over five feet nine inches. The major premiss, in fact, contains the conclusion, and the syllogism tells us nothing that we did not know already. But suppose that we have not made extensive measurements of the heights of policemen, but that we know of the official regulation that no man is appointed to be a policeman unless his height is at least five feet nine. The general proposition is now part of our knowledge without having been verified in all its instances; it is previous knowledge. The inference concerning P.C. Brown is now new knowledge; the syllogism tells us something to expect about his appearance when we meet him that we should not have known without it. Thus the same syllogism may or may not provide new knowledge, according to the means of knowing its premisses.

The syllogism about Socrates raises the same question in a more complicated form. Its author may have had previous intuitive or divinely revealed knowledge, independent of experience, that all men were mortal. If so, he could construct his syllogism and derive new knowledge about the particular man Socrates. But this is not the practical case; belief in human mortality is based on experience, A contemporary of Socrates might proceed in the following way. He would summarize what he knew of the duration of human life. No case was known of a man's having lived for 200 years, and few for 100. This suggests a general rule: all men die before reaching 200 years of age, most before reaching 100. He might look for exceptions among living persons, of whom few were over 100 and none over 200. The general rule was verified with regard to all dead persons, and not contradicted by living ones. It is then stated as a result of experience. The inference concerning the life of a living person could then be drawn from the rule. It could be said that "Socrates will not live to be 200; he will probably not live to be 100". The fundamental difference between the two methods of approach is that in the former, where the major premiss is known a priori, we always proceed from the general to the particular; in the latter we get the major premiss itself by asserting as a general proposition what was previously known only in particular instances. The former method is deduction, the latter induction. In both cases we proceed from the premisses to the conclusion by means of an apparent syllogism; but there is a significant difference, due to the difference in the nature of the available knowledge about the major premiss. Suppose that two people, while Socrates was alive, both drew the inference that he would not live to be 200, one basing his beliefs on human life in general on intuitive knowledge, the other on previous experience; and suppose that Socrates nevertheless lived to be 2000. Suppose further that our doubter paid a visit to Elysium and interviewed their shades. The former would have to admit that his intuitive knowledge, which he had held with certainty, was wrong, or to say that Socrates was not a man but an immortal god, or perhaps to resort to abuse. The latter would explain that the major premiss in his inference was not known with certainty, but that it was extremely probable on the evidence before him. The inference had been correct for some thousands of millions of people that lived when or after it was drawn, and in the circumstances it was not so bad that there had been one exception to the general rule. If he chose to be aggressive he might ask whether Socrates had been medically examined recently with a view to finding out the causes of his anomalous behaviour; for one of the chief functions of exceptions is to improve the rule.

The inference with regard to Socrates has actually been verified, but the situation has arisen with respect to many. other scientific laws. At present we are faced with the inaccuracy of Euclid's parallel axiom, which for millennia was considered intuitively obvious; with the inaccuracy of Newton's law of gravitation, which had been well established by experience and had been believed for centuries to be exact; with the failure in stars of the law of the indestructibility of matter; and with the discordance of the classical undulatory theory of light with the group of facts known as quantum phenomena. For twenty years physical science has been modifying and reconstructing its most fundamental laws as a result of new knowledge. The reconstruction has followed, and will continue to follow, the old method, but the results will be different because new facts have to be fitted in. Will modem physics suffer in turn the fate of the old? Perhaps; nobody knows. But in the circumstances we must raise a group of questions more fundamental and general than any physical law. Have recent developments shown that scientific method itself is open to suspicion, and if so, is there a better one? Just how much do we mean when we assert the truth of a scientific generalization? Men we have made such a generalization, what reason have we for supposing that further instances of it will be true?

3.

The answers to these questions may be stated at once. There is no more ground now than thirty years ago for doubting the general validity of scientific method, and there is no adequate substitute for it. Men we make a scientific generalization we do not assert the generalization or its consequences with certainty; we assert that they have a high degree of probability on the knowledge available to us at the time, but that this probability may be modified by additional knowledge. Our answer is that returned to the doubter by the second shade. The more facts are shown to be co-ordinated by a law, the higher the probability of that law and of further inferences from it. But we can never be entirely sure that additional knowledge will not some day show that the law is in need of modification. The law is provisional, not final; but scientific method provides its own means of assimilating new knowledge and improving its results. The notion of probability, which plays no part in logic, is fundamental in scientific inference. But the mere notion does not take us far. We must consider what general rules it satisfies, what probabilities are attached to propositions in particular cases, and how the theory of probability can be developed so as to derive estimates of the probabilities of propositions inferred from others and not directly known by experience.

At the same time. a remarkable thing happens. It is found that general propositions with high probabilities must have the property of mathematical or logical simplicity. This leads to a reaction upon. the descriptive part of science itself. The number of possible methods of classifying sensations is colossal, perhaps infinite. But the importance of simple laws in inference leads us to concentrate on those properties of sensations that actually satisfy simple laws as far as they have been tested. Thus the classifications of sensations actually adopted in practical description are determined by considerations derived from the theory of inference; and probability, from being a despised and generally avoided subject, becomes the most fundamental and general guiding principle of the whole of science.

See also Jeffreys Probability


JOC/EFR August 2007

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