Induction: Keynes’ Treatise
In his Treatise on Probability, Keynes discusses induction at length. Part III ‘Induction and Analogy’ provides the introduction and overview, and covers ‘universal induction’, which purports to find a universal law, such as ‘all swans are white’. ‘Inductive correlation’, which purports to find a probabilistic statement, such as ‘most swans are white’, is covered in part V ‘The Foundations of Statistical Inference’.
Part III Induction and Analogy
Ch. XVIII Introduction
1. I [JMK] have described Probability as comprising that part of logic which deals with arguments which are rational but not conclusive.- By far the most important types of such arguments are those which are based on the methods of Induction and Analogy. Almost all empirical science rests on these. And the decisions dictated by experience in the ordinary conduct of life generally depend on them. To the analysis and logical justification of these methods the following chapters are directed.
… Induction has been admitted into the organon of scientific proof, without much help from the logicians, no one quite knows when.
5. In an inductive argument … we start with a number of instances similar in some respects AB, dissimilar in others C. We pick out one or more respects A in which the instances are similar, and argue that some of the other respects B in which they are also similar are likely to be associated with the characteristics A in other unexamined cases. The more comprehensive the essential characteristics A, the greater the variety amongst tlie non-essential characteristics C, and the less comprehensive the characteristics B which we seek to associate with A, the stronger is the likelihood or probability of the generalisation we seek to establish.
These are the three ultimate logical elements on which the probability of an empirical argument depends, the Positive and the Negative Analogies and the scope of the generalisation.
6. Amongst the generalisations arising out of empirical argument we can distinguish two separate types. The first of these may be termed universal induction. … The generalisations which they assert … claim universality, and are upset if a single exception to them can be discovered. Only in the more exact sciences, however, do we aim at establishing universal inductions. In the majority of cases we are content with that other kind of induction which leads up to laws upon which we can generally depend, but which does not claim, however adequately established, to assert a law of more than probable connection. 1 This second type may be termed Inductive Correlation. …
An inductive argument affirms, not that a certain matter of fact is so, but that relative to certain evidence there is a probability in its favour. The validity of the induction, relative to the original evidence, is not upset, therefore, if, as a fact, the truth turns out to be otherwise.
The clear apprehension of this truth profoundly modifies our attitude towards the solution of the inductive problem. The validity of the inductive method does not depend on the success of its predictions. Its repeated failure in the past may, of course, supply us with new evidence, the inclusion of which will modify the force of subsequent inductions. But the force of the old induction relative to the old evidence is untouched. The evidence with which our experience has supplied us in the past may have proved misleading, but this is entirely irrelevant to the question of what conclusion we ought reasonably to have drawn from the evidence then before us. The validity and reasonable nature of inductive generalisation is, therefore, a question of logic and not of experience, of formal and not of material laws. The actual constitution of the phenomenal universe determines the character of our evidence ; but it cannot determine what conclusions given evidence rationally supports.
It is sometimes supposed that the strength of induction rest only on the number of cases consider. Keynhes regards ‘negative analogy’ as also important. Thus, if we think that ‘all swans are white’ but have only ever considered British swans, we have very weak inductive grounds for supposing that New Zealand swans are all white. One way to appreciate this is by considering the induction ‘In all countries, all swans are white’. We may have seen many swans, but if they were all in one country, the induction is clearly weak. More generally, an inductive conclusion is a conclusion about the system from which the evidence is drawn. One may predict that the future will be like the past: but it may not, and such a change does not invalidate the proper conclusion of the induction. In terms of Whitehead, induction draws on evidence from ‘the current epoch’ and can only draw conclusions about the same epoch.
Ch. XIX The Nature of Argument by Analogy
12. … It has been my object … to inquire whether ultimate uniformities of method can be found beneath the innumerable modes, superficially differing from another, in which we do in fact argue.
Ch. XX The Value of … Pure Induction
Here Keynes is considering whetehwr induction can justify deterministic rules.
6. The conditions, which we have now established in order that the probability of a pure induction may tend towards certainty as the number of instances is increased, are:
- That xr/x1x2…xr-1¬gh falls short of certainty by a finite amount for all values of r beyond a specified value s.
[Where h is the a priori data and ¬g is the falsity of the candidate inductive generalisation.]
- That ps, the probability of the generalisation relative to a knowledge of these first s instances, exceeds impossibility by a finite amount.
In other words Pure Induction can be usefully employed to strengthen an argument if, after a certain number of instances have been examined, we have, from some other source, a finite probability in favour of the generalisation, and, assuming the generalisation is false, a finite uncertainty as to its conclusion being satisfied by the next hitherto unexamined instance which satisfies its premiss. To take an example, Pure Induction can be used to support the generalisation that the sun will rise every morning for the next million years, provided that with the experience we have actually had there are finite probabilities, however small, derived from some other source, first, in favour of the generalisation, and, second, in favour of the sun’s not rising to-morrow assuming the generalisation to be false. Given these finite probabilities, obtained otherwise, however small, then the probability can be strengthened and can tend to increase towards certainty by the mere multiplication of instances provided that these instances are so far distinct that they are not inferrible one from another.
9. Apart from analysis, careful reflection would hardly lead us to expect that a conclusion which is based on no other than grounds of pure induction, defined as I have defined them as consisting of repetition of instances merely, could attain in this way to a high degree of probability. To this extent we ought all of us to agree with Hume. We have found that the suggestions of common sense are supported by more precise methods. Moreover, we constantly distinguish between arguments, which we call inductive, upon other grounds than the number of instances upon which they are based; and under certain conditions we regard as crucial an insignificant number of experiments. The method of pure induction may be a useful means of strengthening a probability based on some other ground. In the case, however, of most scientific arguments, which would commonly be called inductive, the probability that we are right, when we make predictions on the basis of past experience, depends not so much on the number of past experiences upon which we rely, as on the degree in which the circumstances of these experiences resemble the known circumstances in winch the prediction is to take effect. Scientific method, indeed, is mainly devoted to discovering means of so heightening the known analogy that we may dispense as far as possible with the methods of pure induction.
This form of induction depends on our having certain a priori knowledge, h, such that the generalisation, g, being considered is the only theory that predicts the data. In other words, it depends on our knowing for sure that the data is diagnostic. Keynes does not give an example, so here is one:
Suppose we had a coin which we knew (h) was either normal or double-sided. Suppose that, on tossing, we get a run of heads. Let g be the generalisation that it will always give heads (i.e., is double-headed). The by pure induction, the longer the run of heads the more probable g.
This is rather artificial, since h severly limits ¬g in a very rare way. More typically, the sun rising is not diagnostic for the sun rising every day for the next million years: it may rise every day for a shorter period.
Ch. XXI The Nature of Inductive Argument continued
3. … Induction tells us that, on the basis of certain evidence, a certain conclusion is reasonable, not that it is true. If the sun does not rise to-morrow, if Queen Anne still lives, this will not prove that it was foolish or unreasonable of us to have believed the contrary.
4. It will be worth while to say a little more in this connection about the not infrequent failure to distinguish the rational from the true. The excessive ridicule, which this mistake has visited on the supposed irrationality of barbarous and primitive peoples, affords some good examples. … The first introduction of iron ploughshares into Poland, … having been followed by a succession of bad harvests, the farmers attributed the badness of the crops to the iron ploughshares, and discarded them for the old wooden ones. The method of reasoning of the farmers is not different from that of science, and may, surely, have had for them some appreciable probability in its favour.
Ch. XXII The Justification of these Methods
9. … If … our generalisation is to be universal, so that it breaks down if there is a single exception to it, we must obtain, by some means or other, a finite probability that the set of characters, which condition the generalisation, are not the possible effect of more than one distinct set of fundamental properties. I do not know upon what ground we could establish a finite probability to this effect. The necessity for this seemingly arbitrary hypothesis strongly suggests that our conclusions should be in the form of inductive correlations, rather than of universal generalisations. Perhaps our generalisations should always run: ‘It is probable that any given φ is f rather than, ‘It is probable that all φ are f’. Certainly, what we commonly seem to hold with conviction is the belief that the sun will rise to-morrow, rather than the belief that the sun will always rise so long as the conditions explicitly known to us are fulfilled.
11. … Our assumption, in its most limited form, then, amounts to this, that we have a finite a priori probability in favour of the Inductive Hypothesis as to there being some limitation of independent variety … in the objects of our generalisation. Our experience might have been such as to diminish this probability a posteriori. It has, in fact, been such as to increase it. It is because there has been so much repetition and uniformity in our experience that we place great confidence in it. To this extent the popular opinion that Induction depends upon experience for its validity is justified and does not involve a circular argument.
It seems reasonable to rely on induction under circumstances where induction has previously been found to be reliable, but perhaps not where nature (or man) has shown creativity or other unprojected variety.
Ch. XXIII Some Historical Notes …
In distinguishing, therefore, analogy from pure induction, and in justifying it by the assumption of a limited complexity in the problems which we investigate, I am, I think, pursuing, with numerous differences, the line of thought which Bacon first pursued and which Mill popularised. The method of treatment is dissimilar, but the subject-matter and the underlying beliefs are, in each case, the same.
We are rarely in a position to apply pure induction, but the caveats are often relevant. In particular, scientific methods, in so far as they seem to establish ‘laws of nature’, rest on assumptions that some such laws do in fact hold for the things being observed.