The ‘working’ of hypotheses is by no means simple and unambiguous. It admits of infinite gradations in amount and kind, and the ‘truth’ which is implicated in ‘working’ is nothing essentially but an index of its logical value, and may vary in quantity between values which cannot be psychologically discriminated from zero and from 100% or 1 (= ‘absolute’ certainty). It is crude, therefore, to confront a scientific hypothesis with the rigid alternative ‘either (absolutely) true, or (utterly) false’; its ‘truth’ really rests on its greater value, as compared with its competitors. Its value, then, is a question of more or less. The more extensively, conveniently, and economically a hypothesis works, the more value has it, i.e. the more likely is it to be called ‘true’, and to be supposed true absolutely: the more continuously and successfully the test of working has been applied to a doctrine, the greater the confidence and affection with which it is regarded, and the greater the presumption that it will continue to approve itself as true.
But, as we anticipated in [§ 24] (s.f.), it is vain to expect to establish any absolute truth by this method. It provides truth with ever-growing probability, but never with absolute certainty. For, however well a theory works, the thought that one may hereafter be found to work better can never logically be excluded. Even if every one alive were perfectly satisfied, and no one could imagine any improvement in an accepted truth—and these conditions are by no means often realized—such psychological considerations would not disprove the logical possibility that the best known was not the best absolutely, and logic would continue to distinguish between a truth that was absolute, and one liable to one billionth chance of error. The latter chance could be disregarded for all practical and scientific purposes, and would not have the slightest psychological effect on the confidence with which the truth was regarded; but logically it would still be there. Science, therefore, has to resign itself to the conclusion that its method cannot conceivably attain to absolute truth, and to make the best of it.
§ 27. Curiously enough this conclusion is fully confirmed by Formal Logic. It prides itself on pointing out that there is a formal fallacy involved in establishing truth by ‘working’. The essence of this method is to argue that if a theory is found to work (after the proper precautions have been taken), it is true. If e.g. the events anticipated by a theory occur, and nothing occurs that could not be anticipated, it grows more and more probable until it convinces every one. But ought it logically to have done this? The logician declares emphatically, it ought not. For the argument suffers from an incurable flaw, which has been recorded as a ‘fallacy’ for over 2,000 years. It is a flagrant ‘affirmation of the consequent’; symbolically, it argues that if A is, B is, but B is, ∴ A is. Now this is not ‘cogent’ or ‘valid’. That A is can be proved only from the premiss ‘only if A is, B is’, i.e. if A is the only theory which will account for the observed consequences. But this the fallacious method did not assert, and indeed could not assert. For that the best known is the best absolutely never can be proved (cf. [§ 26]); and even if they happened to be identical, and we had somehow stumbled upon an absolute truth, we should never know that this was so.
§ 28. To the logician this fact only seems to prove the superiority of his conception of ‘proof’. He infers, consistently enough, that no inductive reasoning from ‘facts’, no verification of hypotheses by events, can possibly amount to proof. What he seeks to impress upon his pupils is that verification is not proof and can never lead to it.
He considers himself entitled to look down upon science accordingly, its evidence, its methods, and its reasonings, and to contrast them with the absoluteness of his own ideal of demonstration. He upholds its validity in spite of all the failures of the sciences to realize it. As a rule he seems willing to grant that some mathematical proofs amount to logical demonstration;[398] but if pressed he would confess that scientific truth was only probable, whereas certain metaphysical truths, such as the law of contradiction, alone were absolutely certain.
The scientist, of course, is not in a position to deny that the nature of his truth is such as has been stated: but he should not attempt to do so. He should content himself with scientific truth, and contend that at its best it is good enough for any one. And he can carry the war into Africa by a vigorous counter-attack.
(1) He can deny—for the reasons stated in [§ 13]—that the logician’s formal ‘proof’ is as cogent and formally valid as the latter supposes, and show that after a conclusion has been ‘proved’ true, it has still to come true before it can be trusted to be ‘true’.
(2) He can point out that there is a serious lacuna in the logician’s plea for his notion of ‘proof’. The logician has assumed that the only alternative to his belief in absolutely certain premisses is complete scepticism, arguing that it must be possible to start from certainty, because otherwise no knowledge would be possible at all. He then urged ‘but there clearly is knowledge—the sciences attest it’, and consistently inferred that absolutely certain premisses must be obtainable. The more or less obvious failure of his attempts to explain their genesis by ‘self-evidence’, ‘intuition’, ‘necessities of thought’, &c. ([§ 15]), could not deter him from clinging to his belief, because the principles themselves seemed to him to be inevitable and to admit of no alternative.
In fact, however, there is a via media between scepticism and absolutism, and science safely pursues it, though logic has overlooked it. It is not necessary to start with absolutely certain premisses, because it is possible to adopt premisses hypothetically, to take them as true for the argument’s sake and for the purposes of the inquiry, to experiment with them, and to revise them in the light of the results of such experiments. Thus their value may be judged and established, after their adoption, by the experimental results, and they may come to depend logically upon these, and not upon the processes (analogies, suggestions, guesses, fancies, &c.) which led to their adoption. If they show themselves capable of advancing the science and solving its problems, confidence in their ‘truth’ increases progressively, and their initial assumption is justified. They cease to be ‘hypotheses’ and become ‘facts’, and even ‘principles’ beyond dispute. If they fail to ‘work’, they may be discarded in favour of others which are tried in their turn and similarly tested. Hence it is not true that what is uncertain to begin with must always remain so, nor is it hard to understand that hypothesis, willingness to believe, and belief may be the psychological forerunners of logical proof, which, nevertheless, rests not upon them, but upon the solid value of the results subsequently reached by their means. The certainty of scientific premisses then admits of indefinite growth, which at some point or other will overpower even the most obstinately sceptical temper. This point naturally lies at a greater distance from the starting-point for some minds than for others, but when it is reached, and when the last doubts and scruples have been overcome, the triumphant truth will feel absolutely certain, and to all intents and purposes will function as such. But the ‘practical certainty’ thus achieved will still be distinguishable in thought from the absolute certainty which logical theory mistakenly demanded. And logicians, from Plato downwards,[399] will be convicted of having failed to allow for the possibility that the certainty of premisses and principles may be a fruit of continuous experience and experiment, and to perceive that this is the method the sciences have actually employed. In short, necessary (needed) ‘truths’ need not be regarded as ‘a priori’, if it is seen how hypotheses are consolidated by experience.
(3) The scientist can deny that the ideal case, contemplated with so much satisfaction by the logician, can ever occur in actual knowing. He can point out that if the logical apparatus of demonstration is to work, it must be supplied with premisses that are absolutely true. But whence is the logician to obtain them? The ‘self-evident’ principles and ‘necessary’ axioms, for which so much has been claimed, have been shown ([§ 15]) to be highly disputable, and are themselves in need of support and verification. The truths which the sciences supply abundantly are all products of the method to which he takes exception. There are no scientific truths which have not to be, and have not been, verified, and if verification is logically vicious, and cannot amount to proof, they are not absolutely true. But if the premisses of a demonstration are not absolutely true, neither can its conclusion be. What then becomes first of the value, and ultimately of the ‘validity’, of an ideal of proof which can never be exemplified by actual reasoning, and serves only to condemn it?