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Hume's inductive fallibilism asserts that all inductive inferences are invalid, and `incurably' so, in the sense that they remain invalid under any observational additional to their premisses; or abbreviated, (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'. His inductive scepticism asserts that all inductive arguments are unreasonable, or (8) `P(h, e.t) = P(h, t) if the argument from e to h is inductive'. Now, when we consider these propositions by themselves, it is obvious that neither the truth nor the falsity of inductive fallibilism requires either the truth or the falsity of inductive scepticism. The two propositions are logically independent.
This was not obvious to Hume, since (as was remarked in Chapter 3 section (iii)), he sometimes treats these two very different propositions as though they were equivalent. I do not think (despite Professor Mossner's admission mentioned above) that there is any need to question Hume's honesty on this account. A certain tendency to conflate inductive fallibilism with inductive scepticism was to be expected in view of one of those conflicts of purpose in Hume which Professor Passmore has so well portrayed [1]. When he undertook to examine `that evidence which assures us of any real existence and matter of fact' [2], Hume, it is clear, wanted to do two different things. On the one hand he wanted to stand forth as a champion of sober experimental science, against what he regarded as the parties of credulity and dogmatism. But, on the other hand, he wanted to search that topic to the bottom, as a philosopher does, applying to all inferences as he goes along the most rigorous standard of conclusiveness; and in this frame of mind a man is indifferent to the politics of the republic of letters, and in particular does not stop to consider whether his investigations will injure his own party therein. The first motive was bound to incline Hume towards inductive fallibilism; the second, towards inductive scepticism.
But more than that: consider how the two theses are logically related, when taken in conjunction with other propositions which Hume assumed. When inductive fallibilism is taken along with the deductivist thesis (6), it does entail inductive scepticism. And when inductive scepticism is taken along with the regularity thesis (4), it does entail the fallibility of inductive inference. But (4) and (6) are, as we have seen, propositions which Hume took so much for granted that, although he certainly assumed them in his argument for scepticism, he left them unstated. Nothing could be more natural, consequently, than that he should at least sometimes write as though scepticism and fallibilism about induction were the same thing.
What makes the difference between them all the more palpable to us, is that at the present time the currency of these two theses is so very different. Inductive scepticism, as I have said in Chapter 5 section (i), has acquired in the last fifty years a certain, very limited, currency; but the triumph of inductive fallibilism has been complete.
To say that inductive inferences are all incurably invalid is another way of saying that there is a permanent possibility of falsity in even the best-confirmed experimental generalizations and predictions. This is certainly a belief having extremely wide currency at the present time. Philosophers, I think, almost without exception accept it; but not only they. It has been absorbed into the common-sense philosophy of science which most educated men now share. It is taken for granted by many intelligent schoolboys. This conviction, moreover, is a living one: it is believed that there is a real possibility that any given scientific generalization or prediction might have to be given up in the light of subsequent experience. Finally, it is a steady conviction; not one which is, as it were, put on just for certain special intellectual occasions. The currency of inductive fallibilism, in other words, is in every respect the opposite of such currency as inductive scepticism can claim to have.
Matters did not always stand so, as I will soon have occasion to emphasize. But at present the incurable invalidity of inductive inference is so well recognized that there is a tendency to make invalidity part of what is meant by calling an inference `inductive'. In Carnap's usage of `inductive', to say that inductive inferences are all invalid, is trivial in precisely the same way as `Bachelors are all unmarried'. But (as was remarked in Chapter 1 section (ix)), that usage is still not the main one. In the mainstream of usage, as also here, it will be recalled, `inductive inference' has no evaluative meaning. (Here it is simply a translation of Hume's `probable arguments', which at the relevant places in Hume's text simply means, as we say, `arguments from observed to unobserved instances of empirical predicates'). On this understanding, the thesis of inductive fallibilism at any rate could be a non-trivial truth.
That it is in any case true is something which can be learnt from Hume's argument for inductive scepticism. For from Hume's premisses (e), (f), and the regularity premiss (4), it follows, as we have seen, that: all inductive inferences are invalid as they stand, and the addition to their premisses which is necessary in order to turn them into valid inferences is a proposition not deducible from necessary truths, and not deducible, either, from observational premisses without such an addition to them as would make the inference to that proposition circular. And this complex result reduces, we saw in Chapter 3 section (ii), to the statement that all inductive inferences are invalid and remain so under observational additions to their premisses.
But, I have argued briefly in Chapter 1 section (vi), the regularity premiss, that there can be no demonstrative arguments for a matter of fact, or (4) `P(h, t) < 1 for all contingent h', is true. Likewise, I have argued briefly in Chapter 2 section (iii), that Hume's premiss (e) is true: that all inductive inferences are invalid as they stand and can be turned into valid ones only by the addition of the Resemblance Thesis to their premisses. For the truth of (f), `The Resemblance Thesis is contingent', I now argue briefly as follows: from (e), in conjunction with an assumption which I have already employed in the `reduction' referred to above. This is, that a proposition which is needed to turn an invalid argument with a contingent conclusion into a valid one, is contingent.
If these arguments are correct, then (e), (f), and (4) are all true. Then their consequence, inductive fallibilism, must also be true.
Supposing inductive fallibilism true, the best way to show that it is a non-trivial truth would be to show that there is something in men which strongly disposes them to deny it, or at least to disregard it. For we would then have shown that inductive fallibilism needs to be emphasized, as a corrective to a deep-seated tendency to error. We would a fortiriori have shown that a proof of inductive fallibilism, such as Hume provided, and as distinct from mere assertion of it, is a valuable contribution to the philosophy of science.
Is there, then, anything in us which puts us in need of a reminder of inductive fallibilism? Yes: at least if Hume is right, there are in fact two distinct sources of a powerful tendency in men to disregard the truth of inductive fallibilism.
First, pride---`the pride of mere human reasoners', as Hume says [3]. Or more accurately, that complex of mental traits, of which pride is merely a prominent usual component, which sometimes causes men of science to treat a certain universal generalization, say, which is confessedly grounded in the end only on observational evidence, as though no subsequent empirical evidence could possibly refute it. Presumably, other common components of that state of mind are `the search for certainty', and ordinary intellectual unimaginativeness.
Valid inferences all have, and they alone have, the characteristic that their logical probability is unaffected under all possible additions to their premisses. That is, both in the Keynesian and in the ordinary technical sense of `irrelevant', e2 is irrelevant to h relative to e1, for all e2, if and only if the argument from e1 to h is valid. And it must be admitted that for a man to behave as though all observational e2 were necessarily irrelevant to a favored theory h---i.e. as though the argument from e1 to h were valid---when in fact that argument is inductive, is no unheard-of occurrence in the history of science.
The mental traits which can cause such an overestimation, by a scientist, of the conclusiveness of certain inductive inferences, sometimes, moreover, triumph over all opposing tendencies among the wider body of educated men. Then an entire period of inductive over-confidence sets in. An example of such a period, and probably both the most extreme and the most important, is that roughly from 1700 to 1900: the period of Newtonian supremacy. In order to see whether inductive fallibilism is a corrective that is ever needed, it will be instructive to reflect briefly on two striking features of the intellectual climate of that period.
On the one hand, it is a disturbing feature of the period that observation and experiment are said, repeatedly and emphatically, to be the only kind of evidence on the basis of which an empirical generalization or prediction can acquire a claim to our belief. Everything else---revelation, sacred tradition, Aristotle, universal assent, the fitness of things, inner light, pure reason---is denied all authority in empirical science, by almost every characteristic writer of the period. We, of course, have great difficulty in entering into imaginative sympathy with a state of mind in which these self-denying ordinances would require a painful effort. Yet they plainly did require that, and it can only be a deficiency in our historical imagination which inclines us to think otherwise. This fact on its own should prepare us for the possibility that we might be guilty of a similar deficiency in historical imagination, if we think that inductive fallibilism can never be a needed corrective.
On the other hand, however, a second feature of this period is that, during it, the successes of Newtonian mechanics and gravitation theory were such as to overwhelm, in almost every informed mind, any lingering doubt as to their precise and universal truth. Even very early in the eighteenth century, confidence that space, time, and matter had universally the properties which Newton attributed to them was so wide and deep as to consign the doubts even of a Leibniz and a Berkeley to a long oblivion. And by the end of that century, Kant could erect a whole philosophy just in order to explain how empirical knowledge of such universality and finality was possible. (It is part of Kant's lasting merit that he perceived that this is a problem; but then he had been awakened by the great inductive fallibilist). Or consider the writer whose influence predominated in the philosophy of science in Britain in the second half of the nineteenth century. J.S.Mill insists that ultimately the evidence for scientific generalizations can only be observational. Yet he plainly also believes that observational evidence can be such, and actually has been such, as to place at least some universal generalizations, viz. at least those of Newtonian mechanics and gravitation, beyond the possibility of falsity. Mill does, indeed, write for example that `we must hold even our strongest convictions with an opening left in our minds for the reception of facts which contradict them [...]' [4]. But even this is with him a maxim of the ethics of intellectual life in general, rather than specifically a concession to inductive fallibilism. It is evident to any reader of Mill that he has no living conviction of the possibility that subsequent experience might require changes in the Newtonian framework.
It should go without saying that these are brush-strokes of the broadest kind, and are subject to countless qualifications and corrections; even geographical ones. For example, for reasons not entirely intellectual, confidence in the finality of the Newtonian generalizations was more pronounced in Britain, on the whole, than elsewhere. But taken in the large, the philosophy of science prevailing during this period does have these two characteristics: the Baconian emphasis on the sole authority for empirical science of observational evidence, and the conviction that the Newtonian generalizations had been placed, by the evidence in their favor, beyond the reach of subsequent correction. If so, then there are periods, since the pax Newtoniana was one, when a reminder is needed of the truth of inductive fallibilism. It is, in fact, only one lifetime ago that `the pride of mere human reasoners' last needed such a reminder.
It seems at present as though inductive fallibilism has been absorbed into the thought of educated men for good. If this really is so, then there is indeed one sense in which inductive fallibilism has become, or is becoming, trivial: the sense in which any very general, simple, logico-philosophical truth, once perceived as true by all educated men, is trivial. But that is a sense in which it is perfectly possible for a proposition to be trivial and still be one of the great intellectual conquests of the human race. Some other judgements of invalidity are in that sense, and in that sense alone, trivial yet also truths of great importance. For example, `P(x is F, x is G and all F are G) < 1', the invalidity of undistributed middle for logically independent predicates.
Inductive fallibilism itself, however, forbids us to repose an entire confidence in the permanence of men's belief in inductive fallibilism. It is possible, in every sense of `possible', that the future history of science may contain further period of inductive over-confidence. But even if it should turn out that the last fallibilist reminder ever needed was that which was administered to Newtonian over-confidence at the beginning of the century, still there is something else in us which stands in need of the fallibilist reminder, and which is quite distinct from intellectual pride, and far less amenable to correction.
This is what, to distinguish it from the `scientific' inductive over-confidence just discussed, may be called `organic' inductive over-confidence. The former is occasional, and affects only the educated; the latter is (at least comparatively) permanent, deep-seated, and common to the learned and the vulgar alike. It is displayed, in particular, in the over-assessment which men, as well as some of the lower animals, make of the conclusiveness of the predictive-inductive inference. That is (to return to our stock example), of the inference from `This is a flame, and all the many flames observed in the past have been hot' to `This is hot'.
Let us compare the degree of conclusiveness which it is natural to ascribe to this inference, with that which it is natural to ascribe to some other: for example, to the non-inductive inference from `This die is about to be thrown, and is a fair die marked in the usual way', to `This throw will not result in a "four"'. Now men naturally ascribe a high degree of conclusiveness to this Bernoullian inference, as they do to the predictive-inductive one. But there is no natural tendency whatever to ascribe to it the highest possible degree of conclusiveness; i.e. to mistake it for a valid inference. On the contrary, everyone recognizes that, in relation to its premisses, its conclusion is a proper subject of betting. Everyone can easily, by introspection, discriminate between the degree of conclusiveness which he naturally ascribes to this inference, and that which he naturally ascribes to modus ponens or to Barbara, for example.
The same is true, of course, of many inductive inferences, but, if Hume is right, not of all; and in particular, not of the predictive-inductive inference. For with us, Hume says, that kind of inference is `entirely free from doubt and uncertainty' [5]; it leaves `no room for doubt or opposition' [6]. A man who knows the premiss of a predictive-inductive inference believes the conclusion `with the last degree of assurance, and regards his past experience as a full proof' [7] of the hotness of the next flame. And surely Hume is right? In organisms such as we are, the characteristic effect of a long uniform experience of flames as being hot, and the observation of a new flame, is to produce a mental state in which we cannot at all recognize `This is hot' as a proper subject of betting. We have no living conviction of the possibility of the falsity of that conclusion. (This is a factual claim which any of us can test---I admit only rather roughly and not quite directly---by determining how willing we are to put a hand deliberately into the next flame we see). The degree of conclusiveness which men naturally ascribe to the predictive-inductive inference is not introspectively discriminable from that which they naturally ascribe to Barbara or modus ponens.
I agree with Hume, then, on the factual question as to what degree of conclusiveness men ascribe to the predictive-inductive inference. But Hume is right too on the non-factual question, as to what degree of conclusiveness that kind of inference really has; at least, qua inductive fallibilist, as distinct from inductive sceptic, he is right. For inferences of that kind are what inductive fallibilism says they are, and what Barbara and modus ponens are not: invalid, and consequently not of the highest possible degree of conclusiveness.
Men, therefore, are inveterate over-estimators of the conclusiveness of the predictive-inductive inference. However closely we may approach the ideal of the completely rational inferrer in connection with some inferences, including the Bernoullian inference about the die and even some inductive ones, we regularly fall short of that ideal in connection with some other inductive inferences. Inductive fallibilism, consequently, is needed as a standing reminder that even if predictive-inductive inferences are more conclusive than Hume's inductive scepticism says they are, still they are less conclusive than all of us every day take them to be.
If there were no inductive over-confidence, either scientific or organic, then indeed inductive fallibilism, though true, would not be a truth we needed to be reminded of, or even taught at all; there would be nothing in us for inductive fallibilism to `bite on', as it were. But there is inductive over-confidence, both scientific and organic, and, in consequence, inductive fallibilism is important, as well as true; and a proof of it, such as Hume's, is therefore a proof of an important truth.
[1] In his Hume's Intentions (Cambridge, 1952).
[2] Enquiry, p.26.
[3] In the passage quoted in Chapter 3 from the Letter from a Gentleman. But of course the theme is a recurrent one with Hume. Cf. e.g. Enquiry, pp.161--2.
[4] A System of Logic (8th edn., London 1843), p.376.
[5] Treatise, p.124. My italics.
[6] Enquiry, p.56 n. My italics.
[7] Ibid., p.110. Hume's italics.
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