A rational man is one who makes a proper use of reason: and this implies, among other things, that he correctly estimates the strength of evidence. In many instances, the result will be that he is able to vindicate his assertions by adducing other propositions which support them. But what is it for one proposition to support another? In the most favourable case, the premisses of an argument entail its conclusion, so that if they are true the conclusion also must be true. It would seem, however, that not all our reasoning takes the form of deductive inference. In many cases, and most conspicuously when we base an unrestricted generalisation on a limited set of data, we appear to run beyond our evidence: that is, we appear not to have a logical guarantee that even if our premisses are true, they convey their truth to the conclusion. But then what sort of inference are we making, and how can it be justified? These questions have not proved easy to answer, and their difficulty creates what philosophers call the problem of induction.

The attention which has been paid to this problem is primarily due to the work of David Hume; and it is worth taking some trouble to restate Hume's argument since, for all its essential simplicity, it has often been misunderstood. Hume starts from the assumption that we can have reason to believe in the truth of any proposition concerning an empirical matter of fact only in so far as we are able to connect the state of affairs which it describes with something that we now perceive or remember. Let us assign the neutral term "data'' to what one perceives at a given moment or what one then remembers having previously perceived, leaving aside the question what these data are. On any theory of perception, their range will be very limited. Then Hume maintains that one will have reason to believe in the existence of anything which is not a datum, at the time in question, only if one has reason to believe that it is connected with one's data in a law-like fashion. He puts this rather misleadingly by saying that "all reasonings concerning matters of fact seem to be founded on the relation of cause and effect.'' He then raises the question whether our belief in the existence of these law-like connections can ever be rationally justified, and he offers a proof that it cannot.

This proof may be set out in nine stages, as follows:

(i) An inference from one matter of fact to another is never demonstrative. This is not to say that when the inference is fully set out the conclusion does not follow validly from the premisses - there is no question but that "q'' does follow from "p'' and "if p then q''but rather that what one may call the guiding principle of the inference, the proposition "if p then q,'' when based on a supposed factual connection between the events referred to by "p'' and "q,'' is always an empirical proposition, and as such can be denied without contradiction. Hume's way of putting this, or one of his many ways of putting this, is to say that "knowledge of the relation of cause and effect is not, in any instance, attained by reasonings a priori, but arises entirely from experience.''

(ii) There is no such thing as a synthetic necessary connection between events. These are not, of course, the terms in which Hume puts it, but this is what it comes to. No matter what events A and B are, if A is presented to us in some spatio-temporal relation to B, there is nothing in this situation from which we could validly infer, without the help of other premisses, that events of the same type as A and B are connected in the same way on any other occasion. There is no such thing as seeing that A must be attended by B, and this not just because we lack the requisite power of vision but because there is nothing of this sort to be seen. No sense can be given to a "must'' of this type.

(iii) So the only ground that we can have for believing, in a case where A is observed by us and B not yet observed, that B does exist in such and such a spatio-temporal relation to A is our past experience of the constant conjunction of As and Bs.

(iv) But clearly the inference from the premiss "Events of the type A and B have invariably been found in conjunction,'' or to put it more shortly, "All hitherto observed As bear the relation R to Bs,'' to the conclusion "All As bear the relation R to Bs,'' or even to the conclusion "This A will have the relation R to some B,'' is not formally valid. There is what we may call an inductive jump.

(v) To make it valid an extra premiss is needed assuring us that what has held good in the past will hold good in the future. Hume's formulation of this principle in the Treatise of Human Nature is "that instances of which we have had no experience, must resemble those of which we have had experience, and that the course of nature continues always uniformly the same.''

(vi) But if all our reasonings about matters of fact are founded on this principle, we have no justification for them unless the principle itself is justifiable. But what justification could it have? There can be no demonstrative argument for it. It is clearly not a logical truth. In Hume's own words "we can at least conceive a change in the course of nature; which sufficiently proves that such a change is not absolutely impossible.''

(vii) Even if the principle cannot be demonstrated, perhaps we can at least show it to be probable. But a judgement of probability must have some foundation. And this foundation can lie only in our past experience. The only ground we can have for saying that it is even probable that the course of nature continues uniformly the same is that we have hitherto found this to be the case. But then we are arguing in a circle. To quote Hume again, "probability is founded on the presumption of a resemblance betwixt those objects of which we have had experience, and those of which we have had none; and therefore it is impossible that this presumption can arise from probability.''

(viii) The same objection would apply to any attempt to by-pass the general principle of the uniformity of nature and argue that inferences from one matter of fact to another, though admittedly not demonstrative, can nevertheless be shown to be probable. Again, this judgement of probability must have some foundation. But this foundation can lie only in our past experience. And so we have to assume the very principle that we are trying to by-pass, and the same objections arise.

(ix) We must, therefore, admit that since the inferences on which we base our beliefs about matters of fact are not formally valid, and since the conclusions to which they lead cannot be shown without circularity even to be probable, there is no justification for them at all. We just have the habit of making such inferences, and that is all there is to it. Logically, we ought to be complete skeptics, but in practice we shall continue to be guided by our natural beliefs.

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