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What kind of proposition is Hume's sceptical conclusion (j), `All predictive-inductive inferences are unreasonable'? The same kind, at any rate, as his earlier conclusion (d), `All a priori inferences are unreasonable'. I have argued, in Chapter 2 section (iii), that these two propositions are not psychological generalizations, but rather propositions evaluative, in some sense, of the two classes of inferences which are their respective subjects. In the present section my first object is to prove a more general result, that they cannot be factual propositions of any kind whatever.
There are, indeed, it deserves to be noticed, two factual propositions about the a priori and the predictive-inductive inference respectively, which Hume does repeatedly assert and imply. These propositions concern the degrees of belief which a knowledge of the premisses of the two inferences actually produces in the minds of men. These degrees of belief are, of course, according to Hume, very different in the two cases. Knowledge of `This is a flame', the premiss of an a priori inference, is not sufficient, as a matter of fact, to produce in us any belief whatever in the conclusion `This is hot'. The first flame we observe is no more effective, in leading us to anticipate heat, than is a complete absence of experience. Experience of the conjunction, in all of many instances, of heat with flame is necessary to make us believe, when we observe a future flame, that it will be hot [1]. Such experience once acquired, however, is sufficient, as well as necessary, to produce such belief; and it even produces, according to Hume, `the last degree of assurance' [2] in the truth of the conclusion `This is hot'. (The predictive-inductive inference, it will be recalled, was distinguished by Hume from `probable arguments' in his narrowed sense, precisely and solely by its being, in practice and among men, `entirely free from doubt and uncertainty').
Clearly, these propositions are factual and indeed psychological ones. They are propositions about how `convincing' [3] the a priori and the predictive-inductive inferences respectively are in fact to men. Though they are not to be confused with (d) and (j), they will prove to be useful, by way of contrast, in determining the nature of those two conclusions.
What, then, is the nature of propositions (d) and (j)? Any answer to this question must be consistent with the following evident facts about those two propositions. (1) They are propositions about certain inferences. (2) They are universal propositions. Whatever it is that (j), for example, says about predictive-inductive inferences, it says about all inferences belonging to that class. (3) They are evaluations, of some kind, of the relevant classes of inferences. (4) They are extremely adverse evaluations, of the a priori and predictive-inductive inferences respectively.
These desiderata do not suffice to exclude a factual interpretation of (j) and (d). A proposition could perfectly well be an adverse evaluation of all predictive-inductive inferences, say, and still be a factual one. Suppose, for example, that by calling predictive-inductive inferences `unreasonable', Hume had meant this: `that every predictive-inductive inferrer will in fact almost always, in the long run, arrive at a false conclusion when his premisses are true'. This is a proposition which satisfies the four desiderata mentioned above for interpretations of (j); yet it is a factual proposition.
It is possible, however, to show with finality that (j) is not factual. But let us first show the same of the earlier conclusion (d). That conclusion, `All a priori inferences are unreasonable', was as we have seen inferred by Hume from the three following premisses. (a) `Whatever is intelligible is possible'; (b) `All a priori inferences are such that the supposition, that the premiss is true and the conclusion false, is intelligible'; and the unstated deductivist premiss, `All invalid arguments (i.e. arguments such that the truth of the premiss and falsity of the conclusion is possible) are unreasonable'. Now, the inference of Hume from these premisses to the conclusion (d) is a valid one. But not one of those three premisses is a proposition of a factual kind. Their conjunction, however (it will be assumed here, for the sake of Hume's argument), is consistent. Consequently the conclusion (d) which they entail cannot be a factual proposition either.
An exactly similar argument proves the non-factual character of (j). Hume's grounds for the conclusion, `All predictive-inductive inferences are unreasonable', consist, as we have seen, of two stated and two unstated premisses. The stated ones were: (e) `All inductive inferences are invalid, and are such that in order to turn them into valid inferences it is necessary to add the Resemblance Thesis to their premisses'; and (f) `The Resemblance Thesis is contingent'. The unstated premisses were, first: `All arguments from necessarily true premisses to contingent conclusions are invalid'; and second, the deductivist premiss, `All invalid arguments are unreasonable'. Now, moreover, from these four premisses, the conclusion (j) which Hume drew from them really does follow. (For the first three suffice for fallibilism, which with deductivism suffices for scepticism). But not one of those premisses is a proposition of a factual kind. Their conjunction, however (it will be assumed here, for the sake of Hume's argument), is consistent. Consequently the sceptical conclusion (j) `All predictive-inductive inferences are unreasonable', which these premisses entail, cannot be a factual proposition either.
This is the most important result to which one is led by my identification of Hume's argument for (predictive-)inductive scepticism. It is the key to my evaluation of that argument in Part Three below. No evaluation, moreover, of Hume's argument is likely to be correct if it does not set out from the fact, ascertainable from the texts themselves, that his sceptical conclusion can be validly inferred, as it happens historically to have been inferred by Hume, from premisses which are exclusively non-factual. From premisses, to be specific, which consist of: two judgements of invalidity (viz. (e) and the first suppressed premiss); one attribution (f) of modal status to a certain proposition; and the premiss, which no one could mistake for a factual one, of deductivism.
There is another ground on which any factual interpretation of the conclusion (j) can be excluded. Although, as I have remarked earlier, Hume never expressly discussed the universal-inductive inference, he is always considered to have done so in effect, and to have known that he had done so, in the course of his discussion of the predictive-inductive inference. For it is very obvious, and certainly was obvious to Hume, that his argument for a sceptical conclusion is entirely unaffected if we substitute `universal-' for `predictive-' in the conclusion (j) of stage 2; if, in other words, that stage of his argument is adapted so as to concern inductive inferences with conclusions such as `All flames are hot'. But now, as has also been remarked, (j) is itself a universal proposition. Hence if it were also factual, i.e. itself just such another proposition as `All flames are hot', then Hume's sceptical conclusion would say, concerning itself, that it is `not reason' which ever `engages' us to believe it. And that, certainly, is the precise opposite of what Hume thought! On this ground too, therefore, a factual interpretation of Hume's sceptical conclusion is excluded.
To determine positively what kind of proposition (j) is, let us again use stage 1 of Hume's argument, and its conclusion (d), as a stalking-horse. That (d), `All a priori inferences are unreasonable', is not a factual proposition, as we know. There is indeed, as was pointed out in the earlier section, a factual proposition about the a priori inference which Hume repeatedly affirms. This was, that knowledge of its premiss (`This is a flame') does not suffice to produce any belief in its conclusion (`This is hot'), in our minds as a matter of fact. And by contrast with this factual proposition, it is not hard to see what Hume was doing in stage 1 of his argument. He was considering the same inference, not now from the point of view of fact, but from the point of view of reason. He was asking not, how do we infer from `This is a flame', but how would we, `if reason determined us?' [4]. In other words, Hume was asking: what degree of belief, if any, in the conclusion of an a priori inference, would be produced by a knowledge of its premiss, in a completely rational inferrer?
Any answer to that question would be an assessment of the degree of conclusiveness of the a priori inference; that is, a certain statement of logical probability. The conclusion (d) is Hume's answer to it. Consequently (d) is a statement of logical probability.
Similarly (j), we know, is not a factual proposition. There is indeed, as we have seen, a factual proposition about the predictive-inductive inference which Hume repeatedly affirms. This was, that knowledge of its premisses (unlike knowledge of the premisses of a priori inferences) is sufficient for a certain degree of belief in, and even for `the last degree of assurance' about, the conclusion `This is hot'. Now by contrast with this factual proposition, it is clear that in stage 2 of his argument, Hume is considering the predictive-inductive inference, not from the point of view of fact, but from the point of view of reason. He is asking here, not how men do infer from `This is a flame and all of the many flames observed in the past have been hot', but how would they, if reason were `the guide of life' [5]. In other words, Hume was asking: what degree of belief, if any, in the conclusion of a predictive-inductive inference, would accompany knowledge of its premisses in a completely rational inferrer?
But any answer to that question would be an assessment of the degree of conclusiveness of the predictive-inductive inference; that is, a certain statement of logical probability. The conclusion (j) is Hume's answer to it. Consequently (j), his predictive-inductive scepticism, is a statement of logical probability.
The next question, given that this is the kind of proposition that (j) is, concerns the specific content of (j). What assessment of the conclusiveness of predictive-inductive inferences does Hume's sceptical conclusion make? More particularly, is it possible to capture the specifically sceptical content of (j) in one of the more definite of the various types of statements of logical probability which we distinguished in Chapter 1 above?
The more definite types of statements of logical probability, it will be recalled, are those that I called respectively numerical and comparative equalities and inequalities. It would strain credibility if I were to try to show that Hume's conclusion (j) is really a certain numerical equality. But it can indeed be shown, what is perhaps scarcely less surprising, that there is a certain comparative equality which embodies a large part at least of the sceptical content of Hume's conclusion.
It is to be observed, first, that although Hume's two conclusions (d) and (j), about the a priori and the predictive-inductive inference respectively, are both adverse, only the second is sceptical: that is, shockingly contrary to common belief, and unfavorable to men's pretensions to knowledge. Men in general, as we have seen that Hume stresses, tacitly make not only a favorable `classificatory' assessment of the conclusiveness of the predictive-inductive inference, but they in effect ascribe to it the highest possible degree of conclusiveness. And Hume's conclusion, whatever precise content we are to attach to the epithet `unreasonable', is certainly some assessment of the predictive-inductive inference which is sharply contrary to that common one. But his assessment (d) of the conclusiveness of a priori inference, viz. that it is unreasonable, is, on the other hand, no different from that which other men make---as Hume himself stresses, and as is in any case obvious.
Now, what is that assessment? What degree of conclusiveness do men in general agree in ascribing to the a priori inference? To this, Hume's answer is quite clear, at least in comparative terms. Men, he says, are no more inclined, by the first flame they observe, towards belief in `This is hot', than they are by the total absence of experience of any kind [6]; though of course not less either. It is only experience of the constant conjunction of heat with flame that, on the observation of a further flame, produces belief in `This is hot'. Men in general, then, according to Hume, regard the a priori inference as nothing more or less conclusive as an inference to the same conclusion from premisses which are entirely devoid of (what experience alone can supply) factual content. In other words, abbreviating in the usual way, Hume ascribed to men in general the following statement of logical probability:
(1) P(This is hot, This is a flame.t) = P(This is hot, t)
And about the a priori inference, as we have seen, Hume's conclusion (d) agrees with what other men believe; he was not intending to ascribe more conclusiveness (or, of course, less) to the a priori inference than others do! We must, therefore, ascribe to Hume, as being part at least of what he means by calling the a priori inference `unreasonable', the statement of logical probability (1) [7].
Next, it is certain, quite independently of any interpretation or even translation of Hume's argument, that what he concluded in (j) about the predictive-inductive inference was at any rate the same as he had concluded earlier in (d) about the a priori inference. Once (d) and (j) are identified as being statements of logical probability, therefore, there is at least one further comparative equality which we must ascribe to Hume. That is, that the degree of conclusiveness of the predictive-inductive inference is the same as that of the a priori inference. Or (abbreviating the primary propositions in an obvious way, and for the sake of brevity abridging the `many' observed flames to two), we must ascribe to Hume the statement of logical probability:
(2) P(Hot a, Flame a.Hot b.Flame b.Hot c.Flame c.t) = P(Hot a, Flame a.t)
This proposition would perhaps serve as expressing the content of (j). But a still better suggestion emerges if we take (2) along with the earlier statement of logical probability which Hume endorsed, and which when similarly abbreviated reads:
(1) P(Hot a, Flame a.t) = P(Hot a, t)
For (1) and (2) now entail:
(2) P(Hot a, Flame a.Hot b.Flame b.Hot c.Flame c.t) = P(Hot a, t)
And here indeed is a statement of logical probability which clearly expresses the sceptical content of Hume's conclusion (j). Or rather, since (3) may not exhaust the meaning of saying that the predictive-inductive inference is `unreasonable', we must say that (3) is at least part of the content of Hume's sceptical conclusion, and a part of it big enough to partake of the sceptical character of the whole. Hume's predictive-inductive scepticism, then, is a judgement of `irrelevance' in Keynes's sense: in particular, it asserts the irrelevance of experience, as we may say, to the initial probability of predictive conclusions.
Hume's sceptical conclusion (3) is a comparative equality. No comparative equality, we saw in Chapter 1 section (vii), can be proved empirically. If a comparative equality is validly inferred, therefore, it can only be from premisses some at least of which are themselves statements of logical probability. Hume's conclusion was validly inferred from his premisses. Hence some of them too, as well as his conclusion, must be statements of logical probability.
One such is the first suppressed premiss of the argument: `No contingent proposition can be validly inferred from necessarily true premisses'. Clearly, this is a judgement of invalidity. It asserts the invalidity of every argument from necessarily true premisses to contingent conclusions. Consequently this premiss has, as at least part of its content, the statement of logical probability:
(4) If h is contingent, P(h, t) < 1
This is a very general statement of initial logical probability, and an especially interesting one. For it will be recognized (from Chapter 1 section (vi)), as being Carnap's requirement, for adequate measures of logical probability, that they be `regular'. I will accordingly refer to (4), which will be very important later in the evaluation of Hume's argument, as `the Regularity premiss' of that argument.
From the Regularity premiss and Hume's two stated premisses, the most that follows, we have seen, is that all predictive-inductive inferences are invalid, and that the inferences which result when their premisses are supplemented by further observational ones are likewise invalid. This, the fallibilist consequence, is clearly, as has been remarked earlier, a general judgement of invalidity; or rather, it is two such judgements. It too, therefore, is a statement of logical probability, viz.:
(5) If h and e1 are such that the inference from e1 to h is predictive-inductive, P(h, e1.t) < 1; and if e2 is observational, P(h, e1.e2.t) < 1
In order to make Hume's argument from the fallibilist consequence to his sceptical conclusion a valid one, we found it necessary to ascribe to him a further unstated premiss, that of deductivism: `All invalid arguments are unreasonable'. But if Hume's conception of an unreasonable inference is correctly specified, at least in part, as it has been specified in (1) and (3) above, then part at least of the content of his deductivist premiss is contained in the following statement of logical probability:
(6) If e and h are such that e does not entail h, P(h, e.t) = P(h, t)
It is obvious, as was remarked earlier, that stage 2 of Hume's argument is altogether unaffected if we substitute in its conclusion (j) `universal-inductive' for `predictive-inductive' inference; that is, if we make that stage of the argument concern, instead of predictive-inductive inferences, the class of inferences of which `All of the many observed flames have been hot, so, all flames are hot' is a paradigm. It is equally obvious that Hume knew this, and that he intended the sceptical conclusion which he drew about predictive-inductive inferences to be drawn also about universal-inductive ones. If, then, the nature and at least part of the content of his predictive-inductive scepticism are correctly represented by (3) above, then we must also ascribe to Hume a `universal-inductive scepticism', i.e.:
(7) If e and h are such that the inference from e to h is universal-inductive, P(h, e.t) = P(h, t)
But of course even (3) and (7) together do not exhaust the range of the scepticism about inductive inference which Hume intended his argument to lead to. It is not as though he thought that there are other inductive inferences which are more conclusive than either the predictive- or the universal-inductive inference. His scepticism extended, and of course has always been understood to extend, to `all reasoning from experience' [8]. Hume's inductive scepticism, therefore, is general inductive scepticism; i.e. the judgement of irrelevance, much more general than (3) or (7):
(8) If e and h are such that the inference from e to h is inductive, P(h, e.t) = P(h, t)
For the same reason, Hume's inductive fallibilism is not confined, any more than his inductive scepticism is, to the special case of the predictive-inductive inference. We ought therefore to ascribe to him the judgement of invalidity which generalizes (5) in just the same way as (8) generalizes (3) and (7); that is, general inductive fallibilism:
(9) If h and e1 are such that the inference from e1 to h is inductive, P(h, e1.t) < 1; and if e2 is observational, P(h, e1.e2.t) < 1
When, therefore, Hume's argument for inductive scepticism is generalized in a way in which he certainly intended it to be generalized, it can be summed up as follows. Hume drew the sceptical conclusion:
(8) For all inductive arguments from e to h, P(h, e.t) = P(h, t)
from premisses which entail at most, about inductive arguments, the fallibilist consequence:
(9) P(h, e.t) < 1
taken together with the deductivist assumption:
(6) For all invalid arguments from e to h, P(h, e.t) = P(h, t)
[1] See, for example, Abstract, p.293; Enquiry, p.42; Treatise, p.87.
[2] Enquiry, p.110.
[3] Treatise, p.97 n.
[4] Treatise, p.89.
[5] Abstract, p.294.
[6] All the versions of stage 1 of Hume's argument will bear this statement out. The clearest expression by Hume, however, of his ascription to men in general of the proposition (1) about to be mentioned, will be found on pp.29--30 of the Enquiry.
[7] It was, therefore, no accident of terminology, but on the contrary an expression of something Hume believed (viz. (1)), that he should have called the a priori inference an inference `without', or `before we have had' experience, notwithstanding that its premiss is an observation-statement.
[8] Abstract, p.294.
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