Chapter 6: The Falsity of Its Deductivist Premiss

[ Acknowledgements | Introduction | Part One | Part Two | Part Three | Appendix ]


(i) The relation of deductivism to certain other theses

If Hume's argument for inductive scepticism is, as I have tried to show it is, valid but with false conclusions, it must have a false premiss. Deductivism (6) is one of the premisses, and in this chapter I try to show (section (iv)) that it is false. A necessary preliminary, however, both to the arguments against deductivism, and to the discussion in section (ii) of the currency of the thesis among philosophers, is a clear view of the logical relation of deductivism to other theses. And first, what is the relation between deductivism and inductive scepticism; between, that is, the statements of logical probability (6), `P(h, e.t) = P(h, t) if the argument from e to h is invalid', and (8), `P(h, e.t) = P(h, t) if the argument from e to h is inductive'?

It will be evident that both of these propositions could be true. It should also be equally evident that both could be false. But it is also possible for deductivism to be true and inductive scepticism false. For there is nothing in the content of (6) itself to tell us that the class of invalid arguments includes all arguments from observed to unobserved instances of empirical predicates. Inductive scepticism (8) follows from (6) conjoined with the thesis of inductive fallibilism (9), not from (6) alone. Finally, the falsity of deductivism is consistent with the truth of inductive scepticism. For there could be some invalid arguments from e to h such that P(h, e.t) != P(h, t), without there being, among these arguments, any inductive ones. The logical relation, consequently, between deductivism and inductive scepticism, is independence.

This result, even though half of it---the non-deducibility of (8) from (6)---was made clear in Chapter 3, is somewhat surprising. One expects the relation between deductivism and inductive scepticism to be closer, and more positive, than independence. For this mistaken expectation, the influence of the Carnapian usage of `inductive' may be partly responsible. For `inductive' in that usage is synonymous with `invalid', and then, of course, (6) and (8), far from being independent, are obviously equivalent.

There are, however, much deeper causes than this for the tendency to think that the two theses are more closely related than they are in reality. One is the fact that deductivism is widely read and correctly, even if indistinctly, believed to be a premiss in the only argument which is known for inductive scepticism (viz. Hume's). The second is the fact that the thesis which, when conjoined with (6), does allow the derivation of (8), viz. inductive fallibilism (9), is one which has become second nature to us. (That it has become so, must be ascribed to the influence of Hume. Cf. Chapters 7--8 below). A third fact which helps to obscure the independence of deductivism and inductive scepticism, is the following: that inductive inferences are the only ones of which philosophers nowadays are vividly aware, both that they are all invalid, and that the natural assessments of them are in many cases extremely favorable. This state of affairs must, again, be ascribed partly to the influence of Hume; but not wholly. It must partly be ascribed to the circumstances that very few philosophers or logicians have ever taken much interest in statistical inference, or in the kinds of inference which men constantly make in connection with games of chance. All of the inferences which I called `Bernoullian', for example, (Chapter 1, section (ix)), are invalid, although the natural assessment of many of them is extremely favorable; yet compared with the philosophical literature concerning inductive inference, the philosophical literature concerning Bernoullian inference is almost non-existent. And the effect of this circumstance is that the only class of inferences, which is immediately recognized by most philosophers as being doomed by deductivism to sceptical assessment, is that which concerns inductive inferences. This, I suggest, is the main reason why we are apt mistakenly to suppose that there is a close connection between deductivism and inductive scepticism.

In fact, however, they are logically independent. In particular, inductive scepticism is not `a special case of' deductivism: it is not part of the content of the deductivism thesis. This fact will be important in section (iv) below; as will the fact that the potential sceptical consequences of deductivism (once it is conjoined with appropriate judgements of invalidity), are by no means conjoined to induction.

I turn now to the relation between deductivism (6), and what I earlier called (Chapter 1 section (ii)) `the fundamental thesis of logical probability': the thesis, that is, that one argument may be more conclusive than another even though both are invalid.

This thesis and deductivism (6) cannot both be false. For suppose (6) false. Then there exists an argument from e to h such that P(h, e.t) < 1 yet P(h, e.t) != P(h, t). But if P(h, e.t) < 1 then (by the obvious principles of logical probability [1]), P(h, t) < 1. Whence two arguments exist, viz. from e to h and from t to h, both invalid but of unequal logical probability. Thus, if deductivism is false then the fundamental thesis of the theory of logical probability is true.

The two theses are not, however, inconsistent. Their consistency would be proved if it were proved that two arguments, from e1 to h1 and from e2 to h2, can satisfy the following conditions: that P(h1, e1.t) < 1 and P(h2, e2.t) < 1; that P(h1, e1.t) = P(h1, t) and P(h2, e2.t) = P(h2, t); and that P(h1, e1.t) != P(h2, e2.t). For the first and third conditions would ensure the truth of the fundamental thesis of logical probability; while the first and second would ensure that deductivism has not been negated. Two arguments can satisfy these conditions. For let h1 be Fa.Ga, h2 be Fa; and let e1 and e2 both be Hb. Then all the three conditions are satisfied. Thus deductivism and the fundamental thesis of the theory of logical probability can both be true, although they cannot both be false. In short they are sub-contraries.

This result is, like the earlier one in this section, somewhat surprising. One would rather expect deductivism to be, if not the contradictory, at least a contrary to the fundamental thesis of the theory of logical probability. And, again as in the earlier case, there is a deep reason for this mistake, one which even goes far to excuse it. This is the fact that the deductivism which is a suppressed premiss of Hume's arguments admits of another interpretation than (6), and one according to which it is indeed inconsistent with the fundamental thesis of the theory of logical probability. For it would be very natural to interpret `All invalid arguments are unreasonable' as meaning that all invalid arguments have the same degree of conclusiveness. And in that case, of course, the logical relation in question is in fact contradiction. This interpretation is in fact not only natural but almost inevitable before the identification of Hume's argument is carried as far as it has been carried in Chapter 4 above [2]. Once it has been carried so far, however, the attribution to Hume of any version of deductivism stronger than (6) is not necessary to make his argument valid, and therefore would be unjustified.

In fact, however, the logical relation between the fundamental thesis of the theory of logical probability, and Hume's suppressed deductivist premiss, is sub-contrariety. This fact will be made use of in sections (ii) and (iv) below.


(ii) The currency of deductivism

The object of this section is to show that the currency of deductivism is deep, wide, and long.

It must be admitted at the outset, however, that the thesis (6) has no explicit currency whatever. No philosopher has actually asserted that, if the argument from e to h is invalid, P(h, e.t) = P(h, t). Even its less specific version, `All invalid arguments are unreasonable', has no currency on the surface of philosophy.

Still, we were able to show, in Chapters 2--4, that deductivism, both in the less specific version and in the form of (6), really was subscribed to by Hume, for one. And, especially now, when Hume's influence and reputation is so great, it ought to be possible for a philosopher to show, without saying, that his assessment of the conclusiveness of invalid arguments is no different from Hume's. There are, in fact, several ways in which he can do so.

One way in which a contemporary philosopher can show his deductivism is the way Hume showed his. That is, by proceeding as though, when he has proved an argument invalid, he has done all that is necessary to exclude any favorable assessment of its degree of conclusiveness. This is not, as described, an infallible indication of deductivism. Conjoined with other indications, however, or given sufficient explicitness (as there was in Hume), this indication can be decisive in individual cases.

Another way is, by a philosopher's subscribing to inductive scepticism. This indicates deductivism, not because of any logical connection between the two theses; for there is none, as we saw in the preceding section. What makes it such an indication, and even one which is in practice infallible to date, is the fact (also mentioned in the preceding section) that the only known argument for inductive scepticism is Hume's; an argument which does have deductivism as a premiss.

A third way in which a contemporary philosopher can show his commitment to deductivism is by rejecting the fundamental thesis of the theory of logical probability. That a philosopher does reject this thesis may, indeed, be not easy to establish with finality. But if he does, then (to avoid imputing inconsistency to him) we must impute deductivism to him. For these two theses, we have seen, cannot both be false.

Each of these indications of deductivism, I think it will be admitted, is given by some contemporary philosophers. A wholesale rejection of the theory of logical probability, for example, is certainly not an unknown attitude, or even an uncommon one, among the philosophical profession. Nor is inductive scepticism without its adherents, if what was said in Chapter 5 section (i) is true. And if this last indication is in practice infallible, as I have said, then deductivism has some currency, at any rate, among contemporary philosophers: viz. at least as much currency as inductive scepticism has. But since it has no surface currency, deductivism has some deep currency at present, as it had with Hume.

But I believe that this greatly understates the position, and that deductivism is in fact a thesis to which not just some but most contemporary philosophers are committed. In saying this, I go not by the first indication, since that is not decisive; nor by the second, since although inductive scepticism is a decisive indicator, the inductive sceptics are certainly not in a majority; but by the third indication, i.e. rejection of the theory of logical probability.

A deep division exists among contemporary philosophers, between those who are cultivators of the theory of logical probability or (in the Carnapian sense) `inductive logic', and those who are not. Among the latter the very name `inductive logic' excites contempt or indignation, and the thing itself is regarded, I think it is no exaggeration to say, as an attempt to place a fig-leaf of respectability over the naked illogicality of mankind. The attempt is considered as in itself unworthy of a philosopher, and as one the futility of which has in any case long since been exposed [3]. As one of these philosophers vividly expressed it, inductive logic is simply `the hot air of certain Renaissance publicists'.

This division among philosophers clearly must be regarded as a division between those who accept, and those who reject, the fundamental thesis of the theory of logical probability. There can be no doubt, however, which of the two groups is the more numerous. But to reject the fundamental thesis of logical probability is to be committed to deductivism. Consequently most contemporary philosophers are committed to that thesis, and the currency of deductivism, though inexplicit, is very wide.

It is also long, since it extends at least from Hume's time to ours. For it will hardly be suggested that during that time deductivism has enjoyed only an intermittent currency among philosophers. In fact, before the emergence of the theory of logical probability in the present century, there is nothing [4] which one could point to as being even an implicit and indirect rejection of deductivism.

If most contemporary philosophers are deductivists, then since they are all or almost all inductive fallibilists (cf. Chapter 7 section (ii)), the majority ought also to be inductive sceptics. This of course is not what one finds in fact. But if what I have said or will say concerning the currency of the three theses is true, this fact can only be interpreted as showing that most philosophers have not yet brought their philosophy of induction into a consistent state.


(iii) On arguments for deductivism

Before advancing arguments against deductivism, we ought to consider what has been or could be said in its favor. First, then, what arguments have actually been advanced for deductivism?

None whatever, as far as I can discover. This is in fact an inevitable consequence of the kind of currency which deductivism has long enjoyed and still widely enjoys. Hume, we know, was so far from arguing for this thesis, that he never even stated it; and the currency of deductivism, we have seen, is no more explicit in contemporary philosophers who nevertheless give every indication of embracing it. Its invariable role is that of a suppressed premiss, and if there are any other beliefs on which it is in fact grounded, they are still further than it is from ever being explicitly stated.

Sometimes, of course, it is possible, and even easy, to say what the other beliefs are, which have disposed a philosopher to accept a certain thesis, even though the thesis has never been expressly defended by any arguments. But even this is not possible in the present case. At least, I am quite unable to suggest what if any premisses, still more deeply suppressed than deductivism, have disposed philosophers to accept that thesis.

There is only one possible exception, and this is an argument so absurd that it is only with diffidence that I will now suggest that Hume and others have been at all influenced by it. The argument in essence is this. `Deductivism is true, because it conflicts with the favorable assessments which men naturally make of many invalid inferences (e.g. inductive ones)'. Or still more bluntly: `Deductivism is true, because such inaccurate assessors of the conclusiveness of inferences as men are naturally think it false'.

The argument is not entirely without foundation, because of course many natural beliefs, whether about the conclusiveness of arguments, or about matters of fact, are false. But the argument still seems, and of course really is, absurd. It cannot follow from the fact that a certain statement of logical probability is the natural assessment of the class of inferences in question, that it is false. For it is the natural assessment of Barbara, or modus ponens, for example, that they are of the highest possible degree of conclusiveness; and of course they really are so. The above argument for deductivism is therefore at any rate not valid. And since deductivism itself makes an unfavorable assessment of the conclusiveness of every other argument than a valid one, further criticism of the argument is not necessary. It is worthwhile, however, to point out its invalidity in another way. This is, that the deductivist assessment of an invalid inference is by no means the only alternative, even if we suppose that the natural assessment of it is favorable and false. Thus, if the invalid argument from e to h is inductive, for example, and the natural assessment of it is P(h, e.t) > P(h, t), this assessment may be false without the deductivist assessment being true. For the `counter-inductive' assessment, P(h, e.t) < P(h, t), may be true if the natural assessment is false, and in that case the deductivist equality is false also.

There are at least hints of the above argument in Hume: as when for example he writes that `When I give the preference to one set of arguments above another, I do nothing but decide from my feeling concerning the superiority of their influence' [5]. This at least suggests that, if I believe a certain comparative inequality, then what I believe is sufficiently discredited by the mere fact of my believing it. And contemporary philosophers must at least be suspected of having been influenced by essentially the same argument. For it is certainly often said to be `just a fact about us', that we make the favorable assessments that we do of some invalid inferences, (certain inductive ones, for example). Now, when this is said, the use of the word `just' in this context must mean that the inferences in question do not, in reality, have the degree of conclusiveness which men naturally take them to have. Yet does not this little word, with that momentous meaning, creep into philosophy sometimes, in contexts where all that has been previously established is that it is a fact about us, that we assess some invalid arguments favorably? I think that it does. If so, then the argument mentioned above has not been entirely without effect in disposing the minds of philosophers towards deductivism.

With this possible exception there appear to be no arguments whatever in favor of the deductivist thesis [6].

Even if actual arguments for deductivism are wanting, it will be of interest in itself, and of importance when we come to advance some arguments against deductivism, to consider now what arguments for deductivism there could be (in the future, say). Given the kind of proposition that deductivism is, and also the specific content which distinguishes it from other propositions of that kind, how is the range of possible arguments in favor of it restricted?

First, from the content of deductivism it will be evident that any argument for it must be a valid one. Second, there can be no arguments for deductivism from deductive logic. Even allowing, as one must, for some indeterminacy as to the boundaries of deductive logic, this remains true. For the arguments of deductivism must be valid ones, and if there were valid arguments for it from propositions belonging to deductive logic, then there would be some branch of deductive logic of which deductivism is simply a theorem. But of course there is no such branch of deductive logic.

This ought to be obvious, yet in fact the point needs emphasis. For it seems to be often supposed that deductivism somehow has the authority of deductive logic in its favor. It is very likely of course (as was said in note 6 above) that between deductive logic, and the currency of deductivism, there is a causal, historical, connection. But that there should be any (valid) arguments from the former to the latter, is precluded by the kind of proposition that deductivism (6) is. It is a certain statement of logical probability, and neither a judgement of validity nor a judgement of invalidity. Consequently, anyone who wishes to defend it, as much as one who wishes to attack it, must first of all leave the ground of deductive logic.

Third: there can be no arguments for deductivism from experience. For the only statements of logical probability (we saw in Chapter 1 section (vii)), of which the truth can be validly inferred from observational premisses, are judgements of invalidity. But deductivism is not a judgement of invalidity. It does not itself assert, concerning any given argument, that that argument falls within the class of arguments about which it generalizes.

What deductivism does assert, concerning every member of the class of invalid arguments, is a certain comparative equality. But we saw in Chapter 1 section (vii) that statements of logical probability of this kind cannot be proved true by valid inferences from any premisses other than ones which include other statements of logical probability. If such a statement is to be proved, therefore, it can only be from premisses one at least of which is of a kind that can be directly discovered to be true only intuitively. A fourth restriction, consequently, on arguments for deductivism, is this: that their premisses must contain at least one statement of a kind that can be directly discovered to be true only intuitively.

These restrictions are severe. They are not more so, however, than the restrictions which we will find applying to possible arguments against deductivism. Nor are they (as the arguments on the other side will show), so severe as to prevent there being still a great fund of arguments which could be urged in favor of deductivism.

It must none the less be admitted that the task of finding arguments for deductivism is a daunting one, because of the extremely great strength of that thesis. Not, of course, that it belongs to the strongest kind of statement of logical probability; deductivism (6) is not a numerical equality. But it is both immensely general in its subject and extremely specific in its predicate. The class of invalid inferences is immensely wide and heterogeneous; and to say, of every such argument from e to h, that its logical probability is exactly the same as the initial logical probability of h, is to say a very great deal. It is hard indeed to think of any propositions which, while being equal to the task of entailing this one, could possibly recommend themselves to philosophers.

It will hardly need to be added, finally, that arguments for deductivism are nevertheless needed. There are, of course, very many statements of logical probability of which the truth is known without inference, directly. But it will scarcely be seriously maintained that (6), `P(h, e.t) = P(h, t) if the argument from e to h is invalid', is one of those.


(iv) Some arguments against deductivism

The nature of the deductivist thesis, and its specific content, we have seen, impose restrictions on the range of possible arguments for it. They similarly impose restrictions on the range of possible arguments against it.

First, arguments against deductivism will have to be valid ones. This restriction flows from the specific content of deductivism, as did the same restriction on arguments for it, though not in the same way. Clearly, arguments against deductivism will need to be of a high degree of conclusiveness; but to claim this for some invalid arguments against deductivism would evidently be to beg the question against the deductivist.

Second, there can be no arguments against deductivism `from the theory of logical probability', if that phrase is used, as it sometimes is, to refer just to the principles of logical probability. For deductivism (6) is a certain statement of logical probability, and therefore nothing inconsistent with it can be validly inferred from the principles, unaided by statements, of logical probability.

Third, there can be no arguments against deductivism from experience. For (as we saw in Chapter 1 section (vii)) the only statements of logical probability, of which the falsity can be validly inferred from observational premisses, are certain judgements of validity. But deductivism is not a judgement of validity.

Any argument against deductivism will evidently be an argument for a certain comparative inequality. But we saw in Chapter 1 section (vii) that statements of logical probability of this kind cannot be proved true by valid inference from any premisses other than ones which include other statements of logical probability. If such a statement is to be proved, therefore, it can only be from premisses at least one of which is of a kind that can be directly discovered to be true only intuitively. A fourth restriction, consequently, on arguments against deductivism is this: that their premisses must contain at least one statement of a kind that can be directly discovered to be true only intuitively.

From the preceding section it will be recalled that this restriction is one which applies also to possible arguments for deductivism (6). This symmetry is inevitable, since arguments against (6) will conclude with a comparative inequality, arguments for it with a comparative equality, and these two kinds of statements of logical probability are alike in that knowledge of them rests ultimately on intuition. But the symmetry also has a very important consequence. For it means that the issue between deductivism and its denial must be decided by arguments, the premisses of which cannot be discovered to be true without reliance sooner or later on intuitive assessments of the conclusiveness of inferences.

It is no objection, therefore, to the arguments about to be given against deductivism, that they each contain at least one premiss the truth of which could not be discovered without a mediate or immediate reliance on intuitive assessment of logical probability. No other arguments, in fact, are possible in the case, and arguments in favor of deductivism, if any could be found, would be ones which rested on no different foundation.

The following are some arguments against deductivism.

Deductivism asserts that if the argument from e to h is invalid, P(h, e.t) = P(h, t). But this is false.

For (a) let h be `Socrates is mortal'; let e be `Socrates is a man and all of the many men observed in the past have been mortal'.

Then

(S1) P(h, e.t) < 1,

but

(S2) P(h, e.t) != P(h, t).

Again (b) let h be `Socrates is mortal'; let e be `Socrates is a man born in Greece in the fifth century B.C. and 95 per cent of all men born in Greece in that century are mortal'. Then

(S1) P(h, e.t) < 1,

but

(S2) P(h, e.t) != P(h, t).

Again (c) let h be `Socrates is not mortal'; let e be `Socrates is a man born in Greece in the fifth century B.C. and all men born in Greece in that century are mortal'. Then

(S1) P(h, e.t) < 1,

but

(S2) P(h, e.t) != P(h, t).

Suppose that deductivism (6) is true. Then let there be two arguments, both invalid, from the same premiss e, to two different conclusions h1 and h2. Then, by the hypothesis, P(h1, e.t) = P(h1, t) and P(h2, e.t) = P(h, t), and by the symmetry of irrelevance (in Keynes's sense) [7], P(e, h1.t) = P(e, t) = P(e, h2.t). Thus deductivism entails that if both of two arguments from the same premiss to different conclusions are invalid, the corresponding `inverse' arguments have the same logical probability. But this is false.

For (d) let e be `5 per cent of the present Australian bird population are white'; let h1 be `5 per cent of the many observed birds Australian birds are white'; and let h2 be `95 per cent of the many observed Australian birds are white'. Then

(S1) P(h1, e.t) < 1,

and

(S2) P(h2, e.t) < 1,

but

(S3) P(e, h1) != P(e, h2).

Again (e) let e be `5 per cent of the many observed Australian birds are white'; let h1 be `5 per cent of the present Australian bird population are white'; and let h2 be `95 per cent of the present Australian bird population are white'. Then

(S1) P(h1, e.t) < 1,

and

(S2) P(h2, e.t) < 1,

but

(S3) P(e, h1) != P(e, h2).

Again (f) let e be `All Australian native swans alive at present are white'; let h1 be `Some Australian native birds alive at present are white and not swans'; and let h2 be `Some Australian native swans alive at present are not white'. Then

(S1) P(h1, e.t) < 1,

and

(S2) P(h2, e.t) < 1,

but

(S3) P(e, h1) != P(e, h2).

Objection is very likely to be made to the arguments (a) and (d) along the following lines: that these arguments have among their premisses some of the natural favorable assessments of certain inductive inferences, and that consequently they beg the question against deductivism.

But as we saw in section (i) above that the sceptical assessment of induction (8) is not part of the content of deductivism (6). Consequently, to assert such contraries of inductive scepticism (8) as the proposition (S2) in argument (a), or the proposition (S3) in (d), is not to deny any part of what deductivism (6) affirms. These propositions would indeed beg the question, if they were advanced as premisses of an argument against inductive scepticism. But in the present chapter the thesis which is in question is not inductive scepticism but the logically independent one of deductivism. So the inference of my imaginary objector is invalid.

His premiss is also false. It is not true that (S2) in argument (a), or (S3) in (d), asserts the natural favorable assessment of a certain inductive inference. For neither of these two propositions says which of the two arguments mentioned in it has the greater logical probability. Each merely denies, what deductivism conjoined with the preceding judgements of invalidity entails, that their logical probabilities are equal. In other words, the premisses of the two arguments (a) and (d) are entirely consistent with the unnatural (counter-inductive) assessment, P(h, e.t) < P(h, t) in connection with (a), or P(e, h1) < P(e, h2) in connection with (d).

To our imaginary objector we may reply, therefore, that arguments (a) and (d) would not beg the question against deductivism even if their premisses did include some of the natural favorable assessments of certain inductive inferences; but that in fact they do not.

What is of course true, and all the truth that lies behind the objection just answered, is that arguments (a) and (d) ought not to be regarded as the reduction of deductivism to absurdity by a deductivist who is also an inductive sceptic. Against such a deductivist (Hume, for example), one would have to rely on arguments other than (a) and (d): for example, on the arguments (b), (c), (e), and (f) above. But (a) and (d) are not on this account any worse as arguments against deductivism: which is what they are here offered as being. And it would be idle, of course, to try to construct any argument which would be at once a reductio ad absurdum of a thesis T, and recognizable as such by every adherent of T regardless of what other thesis T' he might happen also to maintain.

It will be evident that the arguments given above against deductivism could easily be added to. For the arguments (a)-(f) all concern one or other of just three kinds of invalid inference: inductive inference, Bernoullian inference, and inferences which are not only invalid but have a conclusion actually inconsistent with their premiss. Even within this limited range, all the arguments given are from singular statements of logical probability only, and therefore only a small fragment of the arguments against deductivism which naturally arise from a consideration of even the above three kinds of invalid inference. There would be little point, however, in adding other arguments to this kind, since their premisses would be likely to meet with acceptance where and only where the premisses of the arguments already given do so.

There is, however, one other argument against deductivism which it will be worthwhile to add, because its premisses are sufficiently different from those of any of the arguments (a)-(f). This is an adaptation of the argument of von Thun against inductive scepticism.

We affirm two judgements of regularity, P(Fa,t) < 1 and P(~Fb v ~Fa, t) < 1, as well as certain obvious judgements of validity mentioned in the preceding chapter, note 5. These of course are simply the premisses of the von Thun argument, and from them follows the falsity of inductive scepticism (8). But inductive scepticism is a consequence, as Hume argued, of deductivism (6), conjoined with the fallibilist premiss that for all inductive arguments from e to h, P(h,e.t) < 1. Hence we need only further affirm this inductive-fallibilist thesis, and the falsity of Hume's other premiss, deductivism, follows.

The arguments given above, despite their being only a small selection from those available, suffice to do what any number of arguments against deductivism would in the end have to be content with doing. For deductivism (6) is so comprehensive and indiscriminating an unfavorable assessment of invalid inferences, that all that arguments against it can do is this: to `assemble reminders', (in Wittgenstein's phrase), of the number and variety of statements of logical probability, of which the truth is intuitively obvious, which would all have to be denied if deductivism were admitted. The arguments given disclose only a corner of this field; but they will serve to suggest how immense a field it is.

Even so, the group of arguments (a)-(f) is diversified enough to show that the sceptical consequences of accepting deductivism are not at all confined to inductive inferences. They show that, on the contrary, deductivism, conjoined with appropriate and true judgements of invalidity, entails sceptical equalities concerning even the most conclusive of statistical inferences, and even concerning inferences some of which have, while others have not, conclusions inconsistent with their premisses. And while the arguments (a)-(f) point out that deductivism generates not only inductive but statistical and other scepticisms, the argument adapted from von Thun points out that deductivism at the same time generates the extreme credulity of denying either inductive fallibilism (9) or else one of the judgements of regularity (S1) or (S2) of Chapter 5 section (iii).

The balance of arguments concerning deductivism, therefore, is this. We have indefinitely many, and various, reasons, each of them in itself sufficient, for thinking that it is false. And to be set against these, the reasons we have for thinking it true are---non-existent.

If deductivism is false, we saw in section (i) above, then the fundamental thesis of logical probability is true. Any argument against deductivism is at the same time, therefore, an argument for that thesis. And in fact the great fund of natural comparative inequalities, which deductivism (as arguments (a)-(f) show) would obliterate, constitutes most of the raw materials which the theory of logical probability exists to systematize. It is this mass of natural comparative assessments of the conclusiveness of inferences which enables the `inductive logician' in the Carnapian sense to regard with equanimity even so basic a scepticism as was expressed by Ramsey about the foundations of the theory of logical probability.

Ramsey wrote that

there really do not seem to be any such things as the probability relations [Keynes] describes. [Keynes] supposes that, at any rate in certain cases, they can be perceived; but speaking for myself I feel confident that this is not true. I do not perceive them [...]; moreover I shrewdly suspect that others do not perceive them either, because they are able to come to so very little agreement as to which of them relates any two given propositions. All we appear to know about them are certain general propositions, the laws of addition and multiplication; it is as if everyone knew the laws of geometry but no one could tell us whether any given object were round or square; and I find it hard to imagine how so large a body of general knowledge can be combined with so slender a stock of particular facts [8].

This criticism of Keynes depends for what plausibility it has on one's attention to being confined to numerical equalities. But once we extend our attention to the weaker kinds of statements of logical probability, and especially to comparative equalities and inequalities, then the number and variety of the assessments of logical probability which men are able to come to agreement about are simply overwhelming.


Footnotes

[1] e.g. by theorem 9, Keynes's Treatise, p.140.

[2] For this reason I did in fact mistakenly attribute the stronger version of deductivism to Hume in my article, `Deductivism', Australasian Journal of Philosophy, vol.48, no.1, 1970.

[3] It is even widely believed, appropriately enough, that it was Hume who exposed for all time the futility of this attempt. Cf. Appendix section (iii).

[4] Nothing, that is, within the arena of philosophy proper. But the classical theory of probability, or at least the school of Laplace, was very consciously such a rejection. Cf. Chapter 8 section (i) below.

[5] Treatise, p.103.

[6] If this is so, it poses an interesting historical question, how to account for the wide and long currency of deductivism among philosophers. Part at least of the answer to this question must presumably lie in the great age and prestige of deductive logic. This can scarcely be the whole answer, however, if only because Hume, and indeed the whole tradition of British empiricism, from Bacon to Mill, is openly contemptuous of deductive logic. Perhaps the key to this historical question is to be found in the influence of Euclid, rather than in that of Aristotle. The idea that the degree of conclusiveness of the arguments in Euclid's geometry is the standard at which an argument ought to aim in every branch of knowledge, including empirical science, has certainly exercised an immense influence. Against it, the thesis of inductive fallibilism (9), for example, which now seems so excessively obvious, has had in fact to wage a long struggle to acquire currency. (Cf. Chapters 7--8 below). Perhaps the currency of deductivism among philosophers is one facet of the influence of that idea.

[7] i.e. by the principle of logical probability that if P(q, r.p) = P(q, r) then P(p, r.q) = P(p, r) (provided q and r are consistent).

[8] F.P.Ramsey, Foundations of Mathematics (London, 1931), pp.161--2. My italics.


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