Russell’s Postulates of Inference
Part VI. Postulates of Scientific Inference
Russell builds on his review of probability and induction, giving five postulates (Ch. XI) under which inference may be justified. These amount to various kinds of regularity, which are not found in many systems of interest.
While the conclusions are of interest, so too are he arguments leading up to them, particularly for those who doubt some of Russell’s assumptions.
Ch. I Kinds of Knowledge
Russell discusses classification and expectation as heuristics rather than logical constructs.
Ch. II The Role of Induction
… I hold that the work of Keynes … suggests a change of emphasis, making induction n o longer a premiss, but an application of mathematical probability to premisses arrived at independently of induction. …
… To justify induction as such is impossible, since it can be shown to lead to falsehood as to truth. Nevertheless it remains important as a means of increasing the probability of generalization in suitable cases. …
As regards, the scientific use of induction, I accept the results reached by Keynes … .
Our problem … is to find principles which will make suitable generalizations probable in advance of evidence.
Russell discusses Keynes’ condition that the likelihood of a false hypothesis will tend to zero as the evidence increases. Russell notes:
It is difficult to see how this condition can fail in empirical material. …
Jack Good has shown how difficulties can arise for composite hypotheses, and proposes a work-around using generalized likelihoods.
Russell notes that whether one is sure that a proportion m/n have property B, or one thinks that, with probability m/n, all have property B, the probability of the next sample being a B is the same.
From the practical point of view … So far as expectations about the next A are concerned, it therefore comes to the same thing to be sure that most A’s are B’s, or to think it probable that all A’s are B’s. The case most likely to occur in practice is that in which it is probable that most A’s are B’s. This often suffices for rational expectation, and therefore for guidance in practice.
Thus Russell seems to side with those who interpret the Ellsberg paradox as showing that people are irrational. But Russell’s practical person has no consideration for the long term. He extracts as much value as he can out of a situation, moving on if necessary. In the long run the two types of probability are very different.
Ch. III The Postulate of Natural Kinds, or of Limited Variety
Russell discusses Keynes’ notion of limited variety (as corrected by Nicod). It excludes any innovation or surprise.
In seeking the postulate of postulates required to make inductive probabilities approach certainty as a limit … There must, of course, be no positive grounds for regarding a suggested postulate as false. In particular, it should be self-confirmatory, not self-refuting, i.e. inductions which assume it have conclusions consistent with it.
Russell’s view of classes as intensional, not extensional, means that in evolving systems ‘natural kinds’ correspond to equilibria in which all members have common characteristics, distinct from other classes, together with characteristics that may vary between members. Thus evolution typically involves members of one class evolving into one or more different classes, but while the evolution is in progress members may not belong to any class.
Such considerations suggest … a transformation of Keynes’ postulate into something more flexible and less reminiscent of a logical text-book than the principle that he enunciates. It would seem that there must be laws making certain kinds of combination more stable than other kinds, and demanding that when one character is slightly altered, another shall undergo some correlated slight alteration. This process leads to functional laws of correlation as probably more fundamental than natural kinds.
This is in line with Whitehead.
I conclude that the doctrine of natural kinds, though useful in establishing such pre-scientific inductions as “dogs bark” and “cats mew”, is only an approximate and transitional assumption on the road towards fundamental laws of a different kind. Both on this ground and because of its arbitrary character, I cannot accept is as one of the postulates of scientific inference.
Ch. IV Knowledge Transcending Experience
Some modern [1940s] empiricists – in particular, the majority of logical positivists – have, in my opinion, misconceived the relation of knowledge to experience. … It is maintained,
- on the one hand, that a statement is not “significant” unless there is some known method of verifying it;
- on the other hand, that we cannot know “something has this property” unless we can mention a specific subject that has the property.
… I wish to give reasons for rejecting both these opinions.
… The view which I wish to combat [2, above] … has been maintained by Brouwer in mathematics, and is maintained by some other philosophers in regard to empirical objects.
The paradoxes resulting from this opinion are very similar to those resulting from the above doctrine  as to verifiability. …
A. Meaning and Verification
There is a theory that the meaning of a proposition consists in its method of verification. It follows (a) that what cannot be verified or falsified is meaningless (b) that two propositions verified by the same occurrences have the same meaning.
I reject both, and I do not think that those who advocate them have fully realized their implications.
B. Inferential Existence-Propositions
… We must … find grounds for trusting inference before it is verified. And I defy the world to find any such grounds for trusting which will be verified, which are not equally grounds for trusting certain inferences which will be neither verified nor falsified, such as the inference to Napoleon III’s [unknown] father.
I incline to think that valid inductions, and generally, inferences going beyond my personal past and present experience, always depend on causation, sometimes supplemented by analogy. …
Ch. V Causal Lines
The concept of “cause”, as it occurs in the works of most philosophers, is one which is apparently not used in any advanced science. But … the primitive concept, as I try to show, still has importance as the source of approximate generalizations and pre-scientific inductions, and as a concept which is valid when suitably limited.
… [W]e have seen that induction cannot prove causation unless causation is antecedently probable. …
Whether from pure prejudice, or from the influence of tradition, or for some other reason, it is easier to believe that there is a law of nature to the effect that causes are always followed by their effects than to imagine that this usually happens. …
Belief in causation, whether valid or not, is deeply embedded in language. …
The concept of “cause”, as we have been considering it, is primitive and unscientific. In science it is replaced by “causal laws”. … Our law … becomes “A will cause B if nothing happens to prevent B”. Or, more simply: “A will cause B unless it doesn’t”. This is a poor sort of law, and no very useful as a basis for scientific knowledge.
There are three ways in which science overcomes this difficulty ; they are those of
(1) differential equations,
(3) statistical regularity.
[On 2:] “Given an event at a certain time, then at any slightly earlier or slightly later time there is, at some neighbouring place, a closely similar event”. I … assert .. that it happens … sufficiently often to give a high probability to an induction confirming it in a particular case.
That there are such more or less self-determined causal processes is in no degree logically necessary but is, I think, one of the fundamental postulates of science.
This is reminiscent of Whitehead.
(3) On statistical regularity it is not necessary to say much, since it appears to be an inference, not a postulate. …
Moreover the substitution of statistical for individual regularities has only been necessary in regard to atomic phenomena, all of which are inferred. All the phenomena that can be observed are macroscopic, and the problem of making such phenomena amenable to science remains what it was.
Thus science is quite different from economics, where in ‘the Great Moderation’ regularity was a postulate about compound macro phenomena.
Ch. VI Structure and Causal Laws
We will distinguish … “event structures” and “material structures”. …
It is to be observed that what we need in addition to actual experience is only a principle giving probability to certain sorts of inductions. What I am suggesting is that we are not merely to seek simple laws such as A causes B, but are to enumerate a principle of the following sort: given two identical structures, it is probable that that have a causal connection of one of two kinds.
The first kind consists of having a common causal ancestor – this is illustrated by the different visual sensations of a number of people looking at a given object, and by the different auditory sensations of a number of people hearing a given speech.
The second kind arises where two structures are composed of similar ingredients and there exists a causal law leading such ingredients to arrange themselves in a certain pattern. …
On the whole it may be said that the similarity of structure is taken as showing causal ancestry whenever the structure is very complex. …
Sometimes the change of structure is much more complete … . To such changes our principle of constancy of structure is inapplicable.
The principle considered in this chapter has only to do with persistence.
Structures generally persist, but can change.
We hve been chiefly concerned, in recent chapters, with a kind of causation that may be called “intrinsic”. … This principle affords a basis for a great many inductions, but it does not, prima facie, enable us to deal with what commonly count as interactions … .
It will be observed that I have not introduced a postulate to the effect that there are natural laws. My reason for not doing so is that, in any verifiable form, such a postulate would be would be either false or a tautology. …
Russell considers typical inferences, such as to the motion of the planets.
Mathematical probability does not play any part in the above inferences.
The conclusion … is that the fundamental postulate is that of “causal lines”. This postulate enables us to infer, from any given event, something (thought not much) as to what is probable at all neighbouring times and some neighbouring places. So long as a causal line is not entangled with another, a good deal can be inferred, but where there is entanglement (i.e. interaction) the postulate alone allows a much more restrictive inference. However, when quantitative measurement is possible, the measurably different possibilities after an interaction are finite in number, and therefore observation plus induction can make a general law highly probable. In this kind of way, step by step, it would seem that scientific generalizations can be justified.
If what comes after an interaction must be one of a finite number of possible self-sustaining, stable, systems – perhaps modelled as eigen-states of a system – then one may also have scientific generalizations about such stable possibilities.
Ch. VIII Analogy
This chapter is more tentative – and questionable. It develops a postulate which would justify us in supposing that people who seem to be human, are.
Ch. IX Summary of postulates
… The five postulates to which previous analyses have led may be called:
I. The postulate of quasi-permance.
II. The postulate of separable causal lines.
III. The postulate of spatio-temporal continuity in casual lines.
IV. The … structural postulate.
V. The postulate of analogy.
Each of these postulates asserts that something happens often, but not necessarily always; each therefore justifies, in a particular case, a rational expectation which falls short of certainty. Each has an objective and a subjective aspect: objectively, it asserts that something happens in most cases of a certain sort; subjectively, it asserts that, in certain circumstances, an expectation falling short of certainty in a greater or less degree has rational credibility. The postulates collectively are intended to justify provide the antecedent probabilities required to justify inductions.
These postulates are not axioms in the sense that we have licence to assume that they are always true. Rather they are defaults that we are advised to consider, and only rely on probabilistic reasoning and induction in so far as they do seem to hold. Moreover, even if the postulates seem to be true our induction may still be in error: but if so we should reconsider the postulates. It is not ‘If postulates then induction’ but ‘If not induction then not some postulate’.
It should also be born in mind that Russell is considering the short-run. In the long-run the postulates fail, so one can only use induction in so far as the postulates do hold.
Ch. X The Limits of Empiricism
Empiricism may be defined as the assertion “all synthetic knowledge is based on experience”. I wish to consider what, exactly, this statement can signify, and whether it is wholly true, or only true, or only true with certain limitations.
It is agreed that everything inferred from a piece of knowledge by a demonstrative argument is itself knowledge. But … since most inferences are non-demonstrative, we have to consider when such an inference makes its conclusion a piece of “knowledge”, granted that we know the premises.
This … question has sometimes a precise answer. …
But such precision is seldom possible. We do not usually know any mathematical measure of the probability conferred by a non-demonstrative inference, and we hardly ever know the degree of doubtfulness of our premisses. …
[T]o the principle that words which I can understand derive their meaning from my experience there is no need to admit any exceptions whatever. This part of empiricist theory appears to be true without any qualification.
This has implications for the relationship between decision-makers and expert advisers, below.
Universal propositions based on perception alone apply only to a definite period of time, during which there has been continuous observation; they cannot tell us what happens at other times. In particular, they tell us nothing about the future. The whole practical utility of knowledge depends on its power of foretelling the future, and if this is to be possible we must have universal knowledge of the above sort.
But it is also useful to know when a prediction that one might otherwise rely on is invalid and probably wrong.
Induction, we have seen, is not quite the universal proposition that we need to justify scientific inference. But most certainly we do need some universal proposition or propositions, whether the five canons suggested [above] or something different. And whatever these principles of inference may be, they certainly cannot be logically deduced from facts of experience. Either, therefore, we know something independently of experience, or science is moonshine.
We can now sum up our conclusions as to the degree of truth in the doctrine that all our synthetic knowledge is based on experience.
In the first place, this doctrine, if true, cannot be known, since it is a universal proposition of just the sort that experience that experience alone cannot prove. This does not prove that the doctrine is not true ; it proves only that it is either false or unknowable. …
[I]nferences from facts to other facts can only be valid if the world has certain characteristics which are not necessary. Are these characteristics known to us by experience? It would seem not.
Our knowledge of these principles – if it can be called “knowledge” – exists at first solely in the form of propensity to inferences of the kind that they justify. It is by reflecting upon such inferences that we make the principles explicit. And when they have been explicit, we can use logical technique to improve the form in which they are stated, and remove unnecessary accretions.
The approach of experience, reflect, refine could, perhaps, be applied more widely.
… As mankind have advanced in intelligence, their inferential habits have come gradually nearer to agreement with the laws of nature which have made these habits, throughout, more often a source of true habits than of false ones. The forming of inferential habits which lead to true expectations is part of the adaptation to the environment upon which biological survival depends.
… [S]uch inadequacies as we have seemed to find in empiricism have been found by strict adherence to a doctrine by which empiricist philosophy has been inspired: that all human knowledge is uncertain, inexact, and partial. To this doctrine we have not found any limitation whatever.
This postulates may perhaps be taken as standards, to be interpreted, assessed and perhaps developed or gone beyond in particular cases. Thus the result of inference will merit more confidence and be more suitable to take as a prediction the more the postulates, or stronger, are credible.
The main review of Human Knowledge has some broader comments.