[ Acknowledgements | Introduction | Part One | Part Two | Part Three | Appendix ]
The argument identified above was not expected by its author to have the effect of producing in his readers belief in its sceptical conclusion. The confidence which men (and some animals) repose in inductive inference, Hume thought, has its roots in a level of organic existence much too deep to be disturbed by philosophical arguments. Even more generally, Hume says, it is characteristic of `arguments [...] merely sceptical [...] that they admit of no answer and produce no conviction' [1]. Whether or not these generalizations are true, it seems to have been true, at least during the eighteenth and nineteenth centuries, that Hume's argument had no success in bringing about belief in inductive scepticism, or even in commanding attention to it.
The situation now, however, is very different. The vulgar, of course, still suppose that the inductive sciences are the offspring of observation and reasonable inference. But among the learned it is no longer the eccentricity of a solitary thinker, but rather a well-marked though minor tendency of thought, to suppose that on the contrary the sciences owe their existence to men's attachment to a kind of inference which sets logic or reason at defiance. This supposition is not made widely, deeply, or steadily, even among philosophers. But that it is often made, must be obvious to every reader of recent philosophy. This fact is just one expression of the enormous influence which, in the last fifty years, Hume's philosophy of science has acquired.
Of all contemporary philosophers of science, it is Karl Popper who gives his readers the general impression of coming closest to a full and clear acceptance of Hume's inductive scepticism. That this general impression is correct, is confirmed, in the light of Chapters 2--4 above, by an interesting passage in one of the new appendices added to the English translation of his classic book [2]. Here Popper discusses a certain statement of logical probability which is an obvious special case of what I have identified as being Hume's inductive scepticism (8): viz. `P(Fa, Fb.t) = P(Fa, t)' (in which F is an observational predicate). And this proposition Popper regards as being both true and the upshot of `Hume's criticism of induction' [3].
What would constitute an `answer to Hume' concerning induction? More specifically, what would constitute a proof of the falsity of his inductive scepticism? It is not possible to answer these questions before one has a clear and correct answer to the prior question, `What is Hume's inductive scepticism?'. My answer to this is at least clear: viz. (8) `P(h, e.t) = P(h, t) for all inductive arguments from e to h'. And if it is also correct, then an answer to the above question is not far to seek. A proof is at least a valid argument from true premisses. What is needed, then, is at least a valid argument from true premisses to some conclusion inconsistent with (8).
But, as we saw in Chapter 1 section (vii), the falsity of a statement of logical probability can be proved empirically---validly derived, that is, from observational premisses---only under very special conditions. One of these was, that the statement of logical probability in question be a judgement of validity. But inductive scepticism is not a judgement of validity. Consequently its falsity can be proved, if at all, only in the way in which the falsity of a false statement of logical probability can be proved. That is, by a valid derivation of a statement of logical probability inconsistent with it, from true premisses which must include other statements of logical probability.
Clearly, however, not every such derivation proves the falsity of a statement inconsistent with its conclusion. For that, the premisses must in addition be obviously true. Now, in considering possible proofs of the falsity of (8), this requirement presents no difficulty with respect to the premisses which are principles of logical probability. Any principle of logical probability is certainly sufficiently obvious to function as a premiss in a proof of falsity of a statement of logical probability. But what of those premisses which are themselves statements of logical probability? How obvious must the truth of one statement of logical probability be, before it can properly be used as a premiss in proving the falsity of other such statements?
I will not attempt a general answer to this question, but content myself with advancing a certain condition as being at any rate sufficient. A statement of logical probability will have been proved false, I suggest, if its falsity has been shown to follow from premisses which contain (apart from principles) only the weakest of all numerical inequalities, viz. judgements of invalidity, all of which concern arguments of extreme simplicity and obvious invalidity. More specifically still, a statement of logical probability will have been proved false, I suggest, if its falsity has been shown to follow from two judgements of regularity, P(h1, t) < 1 and P(h2, t) < 1, where h1 and h2 are contingent propositions free from quantifiers.
I believe that this will be accepted as a sufficient condition for premisses to be obvious enough to be used to prove the falsity of a statement of logical probability. It will be hard indeed, otherwise, to see how any false statement of logical probability could ever be proved false. Yet, as was observed in Chapter 1 section (vii), they certainly can be.
The argument to be given in the next section, against Hume's inductive scepticism (8), is one which satisfies the above condition. As far as I know, this argument is new. But there is another argument, which I think is also a valid one from true premisses to a conclusion inconsistent with (8), which has been known, in a sense, for a very long time. That argument is Laplace's `inversion of Bernoulli's Theorem'. I say it has been known for a long time `in a sense', because although there is a vague tradition to the effect that Laplace's argument refutes inductive scepticism, the argument has never to my knowledge been presented in such a way as to make clear the nature and content of either its premisses or its conclusion. The main obstacle to such a presentation, beyond doubt, is the unparalleled amount of confusion which surrounds the nature and content of the `Theorems' of Bernoulli themselves. These propositions have been conceived in the most various ways possible: as mathematical truths; as empirical laws; as principles of logical probability; and again as statements of factual probability. (Textual support for each of the last three interpretations, for example, can be found in no less a place than Part V of Keynes's Treatise). If, though only if, the last-mentioned of these interpretations of Bernoulli's Theorems is correct [4], then the nature and validity of Laplace's famous `inversion' argument becomes clear. And if the `Theorem' in question is true, then its inversion is a valid argument from one true statement of logical probability to another which is inconsistent with inductive scepticism (8). Even on those two suppositions, however, Laplace's argument would not be a proof of the falsity of (8), at least under the condition advanced above as being sufficient for such a proof. For its premiss would be an extremely strong statement of logical probability, about a rather complex argument; unlike the premisses of the argument to be given.
(The propositions which are principles of logical probability are numbered (P1) etc.; those which are statements, (S1) etc.)
We assume the following principles.
(P1) The conjunction principle: P(q.r, p) = P(q, p) * P(r, p.q).
(P2) The negation principle: P(q, p) = 1 - P(~q, p).
(P3) The equivalence principle (that logically equivalent propositions can be substituted for one another salva probabilitate in statements and in principles of logical probability).
(P4) The lower-limit principle, P(q, p) >= 0.
Now, let the predicate F be observational. Then the argument from Fa to Fb.Fa is inductive. From (P1) it follows that:
(P5) P(Fa.(Fb.Fa), t) = P(Fa, t) * P(Fb.Fa, Fa.t).
Whence, with (P3), it follows that:
(P6) P(Fb.Fa, t) = P(Fa, t) * P(Fb.Fa, Fa.t).
From (P2) and (P3),
(P7) If P(Fb.Fa, t) = 0 then P(~Fb v ~Fa, t) = 1.
From (P2), (P4) and (P6),
(P8) If P(Fb.Fa, t) != 0, then (each term in (P6) is > 0 and <= 1, and) either P(Fa, t) = 1 or P(Fb.Fa, Fa.t) > P(Fb.Fa, t).
Consequently, from (P7) and (P8),
(P9) Either P(~Fb v ~Fa, t) = 1 or P(Fa, t) = 1 or P(Fb.Fa, Fa.t) > F(Fb.Fa, t).
But
(S1) P(~Fb v ~Fa, t) < 1
and
(S2) P(Fa, t) < 1.
Consequently
(S3) P(Fb.Fa, Fa.t) > P(Fb.Fa, t).
But, since Hume's inductive scepticism (8) entails that P(Fb.Fa, Fa.t) = P(Fb.Fa, t), (S3) is inconsistent with Hume's inductive scepticism.
Apart from the principles (P1)-(P4), the only premisses used in this argument are the statements of logical probability (S1) and (S2) [5]. Each of these is a weak statement, a numerical inequality, and at that a numerical inequality of the weakest kind: a judgement of invalidity. In fact, (S1) and (S2) are mere judgements of regularity, each of which says, concerning an argument from tautological premisses to a (non-quantified) contingent conclusion, just that its degree of conclusiveness is at any rate less than the highest possible degree. Consequently von Thun's argument is a proof of the falsity of Hume's inductive scepticism.
This argument is not only sufficient but may actually be more than is necessary to refute Hume's inductive scepticism: for it should be noticed that it would still refute Hume's inductive scepticism even if that thesis were not (8), but any one of a certain number of propositions considerably weaker than (8). Suppose that my interpretation of Hume's argument in Chapter 4 above was mistaken, and that Hume was not in fact discussing the logical probability of inductive arguments. Suppose that the property of inductive arguments which Hume was assessing, was not their logical probability, but their `rationality'; that is, some non-empirical magnitude, belonging to arguments, but distinct from degree of conclusiveness. (That there are such magnitudes, is certain: for example, what Keynes called the `weight' of arguments) [6]. On this supposition, Hume's sceptical conclusion about all arguments from e to h which are inductive would be, not the statement of logical probability (8), but the `statement of rationality': R(h, e.t) = R(h, t)'. Now, of course, von Thun's argument does not refute this `proposition', because so far almost nothing has been said to make the concept of rationality determinate; we do not yet know what properties we are to credit this magnitude with. What von Thun's argument shows is that if rationality has all the properties of---that is, is identical with---logical probability, then Hume's inductive scepticism is false. What may not be obvious is that his argument would suffice to refute Hume's inductive scepticism, as long as `rationality' were like logical probability in the following two respects: that it is a strictly monotonic function of logical probability (i.e. that P(q, p) > P(s, r) if and only if R(q, p) > R(s, r)); and that the rationality of an argument does not attain its upper limit if the argument has tautological premisses and a non-quantified contingent conclusion. For under these conditions our present premisses (P1)-(P4) will entail a disjunctive analogue of (P9) for rationality; and its first two disjuncts will still be able to be negated by true judgements of `non-maximal rationality'. Then it will follow, corresponding to the present (S3), that R(Fb.Fa, Fa.t) > R(Fb.Fa, t); and thus even the more cautious or indeterminate version of Hume's inductive scepticism which was suggested above will have been refuted. This consideration is important, because it shows that von Thun's argument still finds its mark even when considerable allowance has been made for error on my part in deciding what Hume's inductive scepticism is.
Another feature of von Thun's argument which is worthy of notice is this: that by it we arrive at a very interesting result from principles of logical probability alone, before any statements of logical probability have been asserted [7]. This is the principle (P9). For what (P9) shows is that Hume's inductive scepticism requires a violation of regularity. In other words, (to use the terminology of Chapter 1 section (viii) above), (P9) shows that if one departs from the natural assessment of the conclusiveness of inductive inferences in the direction of scepticism, one must make another, unanticipated, departure from natural assessments, only this time in the direction of credulity, concerning certain other (non-inductive) inferences. And not just an assessment of those inferences which is slightly more favorable than most men would make; for the cost of Hume's inductive scepticism, (P9) shows, is that one must ascribe, either to an argument from a tautology to `This is a flame', say, or to an argument from a tautology to `This or that is not a flame', the highest possible degree of conclusiveness!
More generally, (P9) is interesting for the light it throws on a suggestion which is sometimes made, though on what grounds is not usually clear, that `inductive logic' in the Carnapian sense requires the admission of `the synthetic a priori': that is, requires the thesis that at least some factual propositions can be known a priori. Now, to assert that propositions such as `This is a flame' or `This or that is not a flame' can be inferred from a tautology with the highest possible degree of conclusiveness, would certainly be to say that they can be known a priori. Consequently (P9) has the merit of showing that it is Hume's inductive scepticism, not its denial, which requires the admission of the synthetic a priori.
Von Thun's argument is a proof that inductive scepticism (8) is false. That it is such a proof is entirely independent, of course, of the question who, if any one, has held (8), or on what grounds. But (8) was in fact held by Hume, and it is especially interesting, in the light of von Thun's argument, to recall the grounds on which Hume held it.
For on the one hand, the only premisses of von Thun's argument (apart from some principles of logical probability and some extremely obvious judgements of validity) were the two judgements of regularity, (S1) and (S2). And on the other hand, it will be recalled, one of the (suppressed) premisses of Hume's argument for inductive scepticism was the regularity premiss (4) `P(h, t) < 1 for all contingent h'. But this means that von Thun's argument against Hume's inductive scepticism (8) has for its main premiss two propositions which are deducible from a proposition which Hume not only accepted, but actually employed as a premiss in his argument for (8). Both Hume's argument for (8), and von Thun's argument against it, are valid. Consequently von Thun's argument, as well as showing the falsity of Hume's conclusion, also shows that Hume's premisses, conjoined with some of the principles of logical probability, are inconsistent.
Apart from the deductivist premiss (6), it will be recalled, Hume's only premisses were (e), (f) and the regularity premiss (4), which together entail, and entail no more than, inductive fallibilism (9). Thus the strongest conjunction of statements of logical probability to which Hume's premisses commit him is the conjunction of (4), (6) and (9). ((4) P(h, t) < 1 for all contingent h; (6) P(h, e.t) = P(h, t) if the argument from e to h is invalid; (9) P(h, e1.t) < 1 and P(h, e1.e2.t) < 1 if the argument from e1 to h is inductive and e2 is observational). Thus, this conjunction of statements of logical probability is shown by the von Thun argument to be inconsistent with the principles of logical probability.
This criticism of Hume's argument for inductive scepticism (8) is, of course, no more than an ad hominem one; i.e., an argument for (8), which was different from Hume's argument, would not necessarily be exposed to it. Just how serious a criticism it nevertheless is, may not be immediately obvious. For it might be thought, in particular, that it would be open to Hume to avoid inconsistency, while still maintaining (4), (6), and (9), simply by rejecting one or more of the principles of logical probability (P1)-(P4), which were used as premisses in von Thun's argument. But that is not so.
If there is someone who makes no statements of logical probability whatever, it may be possible for him to reject some or all of the principles of logical probability. It is no part of the object of this book, as was said in the beginning, to try to establish any of them; but in any case we are not here dealing with such a person. We are dealing with Hume, and he, as has been shown, is committed to a number of statements of logical probability. And no one who is thus committed is at liberty to reject the principles of logical probability.
The reason is, that without the principles of logical probability (as was said in Chapter 1 section (i)), statements of logical probability are `blind'. That is, it is the principles, and they alone, which inform us what the consequences are of any given statement of logical probability. Consequently any one who denied the principles, while making a statement of logical probability, would be committing himself, despite the appearances, to nothing determinate whatever. In such circumstances, in other words, the possibility would vanish of any criticism whatever of any given statements of logical probability.
In particular, without the principles, one could never be in a position to say that a given statement of logical probability had, as one of its consequences, the falsity of another; i.e., that two statements of logical probability are inconsistent. For there is, strictly, no inconsistency between, for example, P(A,B) = 1/2 and P(~A,B) = 1. There is only the inconsistency of their conjunction with the negation principle. There is, strictly, no inconsistency between P(A,B) = 1/2 and P(A,B) < 1/2; or again between (S3) P(Fb.Fa, Fa.t) > P(Fb.Fa, t), and the Humean P(Fb.Fa, Fa.t) = P(Fb, Fa.t). There is only the inconsistency of their conjunction with a certain principle of logical probability (viz. the uniqueness principle, that no argument has more than one degree of logical probability). In fact, inconsistency with one or more of the principles of logical probability is all that ever is or can be meant by speaking of two statements of logical probability as being inconsistent with one another.
Hume, then, having asserted some statements of logical probability, is not at liberty to reject the principles of logical probability [8]. Consequently, the full ad hominem effect of the von Thun argument is this: that, in the only sense in which statements of logical probability can be inconsistent, it shows that Hume's deductivism (6), regularity (4), and inductive fallibilism (9) (taken with the extremely obvious judgements of validity mentioned in note 5 above) are inconsistent statements of logical probability. For they entail both Hume's inductive scepticism (8), and a statement (S3) inconsistent with (8). Contrary, then, to the assumption which in Chapter 4 section (i) was tentatively made in Hume's favor, the premisses of his argument for inductive scepticism are actually inconsistent.
[1] Enquiry, p.155. Hume's italics.
[2] The Logic of Scientific Discovery (London, 1959), starred appendix vii, pp.367--70.
[3] Ibid. p.369. My italics. (Popper here writes statements of initial logical probability `unconditionally', and also uses unanalyzed, or at most subscripted, abbreviations for the primary propositions. Thus he writes the above judgement of irrelevance, for example (p.368), `P(a) = P(a, b)').
[4] As Carnap (Foundations, pp.498 ff.) simply assumes; although really a great deal of argument is needed in order to exclude other interpretations, and especially the view of Bernoulli's Theorem as principles of logical probability. The beginning of such argument is given in my `Misconditionalization', The Australasian Journal of Philosophy, vol.50, no.2, 1972.
[5] This is not strictly true. For in the step from (P5) to (P6) I have assumed the logical equivalence of Fa.(Fb.Fa) and Fb.Fa; and similarly in (P7) I have assumed the logical equivalence of ~(Fb.Fa) and ~Fb v ~Fa. Now, judgements of logical equivalences are conjunctions of judgements of validity, and are therefore themselves (conjunctions of) statements of logical probability. All of the four judgements of validity involved here, however, are so extremely obviously true, that to have made them explicit would have been to encumber the presentation of the argument to no purpose.
[6] Cf. Treatise, Ch.VI.
[7] Apart, that is, from the extremely obvious judgements of validity referred to in note 5 above.
[8] The last two paragraphs apply equally, of course, to factual probability. One who appears to assert some statements, while denying the principles, of factual probability, actually evacuates the content of the former.
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