SCIENTIFIC DISCOVERY AND LOGICAL PROOF

By F. C. S. Schiller

§ 1. Among the obstacles to scientific progress a high place must certainly be assigned to the analysis of scientific procedure which Logic has provided. This analysis has not only been inadequate in itself, but has set itself a mistaken aim. It has not tried to describe the methods by which the sciences have actually advanced, and to extract from their experience the logical rules which might be used to regulate scientific progress, but has treated scientific discoveries almost entirely as illustrations of a preconceived ideal of proof, and so has freely rearranged the actual procedure in accordance with its prejudices. For the order of discovery there has been substituted an order of ‘proof’, and this substitution has been justified by the assumption that if discovery had taken the ideally best course, it would have coincided with the process of proof. It followed, of course, that the same logic would do for both, and that this logic was already in existence.

The damage thus inflicted upon Science was twofold. Not only were the logicians given a plausible excuse for persisting in their profound misapprehension of scientific inquiry and rendered incapable of giving any help or guidance in the solution of actual problems, but, what was much worse, the scientists themselves were misled about the nature of their operations.

The precise value of the service which a correct logical analysis of its procedure might have rendered to Science is perhaps open to dispute, though it must surely be beneficial to operate consciously, and with a full understanding of their nature, the methods which have been hit upon empirically; but even if logicians have commonly been too unfamiliar with the details of scientific problems to offer much practical advice, it would be difficult to overrate the mischiefs which must have resulted from referring scientists to an incorrect analysis of their actual procedures. For the attempt to justify by such a false ideal what they had actually done was bound to divert their attention from the methods that were actually effective and fruitful to others which were impracticable and sterile, to waste energy upon false aims and impossible ideals, and so to hamper scientists fatally in the exercise of their scientific rights and powers.

Hence it is not too much to say that the more deference men of science have paid to Logic, the worse it has been for the scientific value of their reasoning, while the less they have troubled to know about the theory of Science, the better it has been for their practice.

Fortunately for the world, however, the great men of science have usually been kept in salutary ignorance of the logical tradition and left to their own devices, by the accident that the historical organization of academic studies nearly everywhere confined ‘logic’ to the literary curriculum. Nevertheless, the moral of this situation is not that it is right for science to neglect logic and for logic to despise science, but that science should appeal from logic as it is to logic as it ought to be, and should insist on being provided with a reformed logic. For surely if a scientific education is to be more than a narrow and technical specialty, and is to exert a ‘liberalizing’ and broadening effect on the mind, it ought to include a study of scientific method in its generality and a certain understanding of the intellectual instruments by which all others are operated and constructed.

The whole evidence for these contentions it will not, of course, be possible to marshal within the limits of this essay, but the systematic criticism to which the whole traditional logic has been subjected in my Formal Logic381 may perhaps absolve me from the duty of substantiating them exhaustively. It may suffice to indicate the extent of the scientific grievance against ‘logic’ by drawing up a list of problems in the logic of science which the traditional logic has misconceived, and then to select for fuller treatment a palmary example of the radical discrepancy between the two.

The traditional logic may be convicted of having gravely misrepresented, (1) the value of classification and the formation of classes, scientific processes of which the real logic was only revealed by the Darwinian theory, (2) the function of definition, (3) the importance of analogy, (4) of hypothesis and (5) of fictions, (6) the incomplete dependence of scientific results on the ‘principles’ by which they are (apparently) obtained, (7) the formation of scientific ‘law’ and its relation to its ‘cases’, (8) the nature of causal analysis. Other important features of scientific procedure cannot be said to have been recognized at all, e.g. (9) the problem of determining what is relevant to an inquiry and what practically must be, and safely may be, excluded, (10) the methods and justification of selection, (11) the essentially experimental nature of all thought and consequent inevitableness of risk, (12) the necessity of so conceiving ‘truth’ and ‘error’ that it is possible to discriminate between them, and (13) the need for an inquiry into meaning and into the conditions of its communication.

I

§ 2. The most instructive, however, of the discrepancies between ‘logic’ and scientific procedure will appear if we compare the logical notion of proof with the scientific process of discovery, and examine how far it can afford any means of regulating, stimulating, or even apprehending the latter. We shall find that the logical theory of ‘proof’ has no bearing on the scientific process of discovery, is not related to what the sciences call proof, and can only have a paralysing influence on any scientific activities which try to model themselves upon it. On the other hand, the study of the process of discovery will point to an important correction in the notion of logic.

§ 3. The scientific uselessness of the traditional logic should not, however, excite surprise. For what reason was there to expect that the theory of proof should turn out to be adequate, or even relevant, to scientific procedure? It had sprung from a totally different interest, proceeded on different assumptions, and aimed at different ends. It did not spring from interest in the exploration of nature, and did not aim at its prediction and control. Nor did it presuppose an incomplete system of knowledge which it was desired to extend and improve. It originated in a very special context, from the social need of regulating the practice of dialectical debate in the Greek schools, assemblies, and law-courts. It was necessary to draw up rules for determining which side had won, and which of the points that had been scored were good.

These were the aims Greek logic set itself, and successfully achieved. But the impress of this origin remains stamped all over it, and the accounts given of logical proof ever since have retained essential features of Greek dialectics.

Thus it was assumed that science could start from principles, as indisputable as are the current meanings of words in a dialectical debate, and the end of the whole theory of proof was always conceived as being to secure the conviction (ἔλεγχος) of one party to a dispute, who was to be definitely crushed by the triumphant cogency of a syllogistic demonstration, while the more real and fruitful analogy between scientific inquiry and debate, viz. that there is always another side, to which also it is well to listen, was unfortunately obscured by Aristotle’s discovery of the syllogistic form and its show of conclusiveness. But for the purpose of apprehending scientific procedure the syllogism is a snare: by putting scientific reasoning into syllogisms, the difference between the true and the false views is made to appear qualitative and absolute, instead of being a quantitative question of more or less of scientific value. Thus dogmatism is fostered at the expense of progressiveness, and the mistake is committed of approaching the discovery of truth in a party spirit. Hence its dialectical origin has become fons et origo malorum for logic.

§ 4. It is true that this mistake is very old, and has grown deeply into the fabric of logic. For Aristotle had no sooner worked out the classic formulation of the rules of dialectical proof than he proceeded to extend their scope by applying them to the theory of science, in the Posterior Analytics. His instinct in so doing was sound enough; for there is no better verification of a theory than its capacity to bear extension to analogous cases. And of course if this extension had been successful, it would have supported the belief that the theory of discovery could profitably be amalgamated with that of proof.

Unfortunately, however, the verification only seemed to be successful. Aristotle chose to exemplify his theory of scientific proof from the mathematical sciences. His choice was natural enough, because they were the only sciences which had reached any considerable development in his day, and they had, moreover, an apparent necessity and universality and a fascinating appearance of exactness. But he had unwittingly chosen the most difficult and deceptive exemplification of scientific procedure. Because the mathematical sciences were in a relatively advanced condition they seemed to lend themselves to his design. He could there find terms whose meaning, and principles whose truth, was no longer in dispute. They could in consequence be argued from with as much assurance as debaters could assume the recognized meanings of words. And the fact that results seemed to follow from mathematical definitions and premisses which were not merely verbal, shed a delusive glory on the forms of dialectical proof by which they had been reached. Hence it easily escaped notice that the logical superiority of mathematics was an achievement, not a datum. Just because the mathematical sciences were very ancient, their origins had been forgotten, and with them the tentative gropings which had first selected, and subsequently confirmed, their principles. They had become immediately certain and ‘self-​evident’, and no one was disposed to dispute them. On this psychological fact the whole theory of logical proof was erected.

Again, it was natural to suppose that the true nature of scientific knowing must be revealed in its most perfect specimens: no one stopped to reflect that even so the real difficulties of making a science are more keenly felt and more easily seen in the nascent stage than in one which has victoriously overcome them, and has rewritten its history in the assurance of its prosperous issue.

Lastly, the subtle ambiguity which pervades all mathematical reasoning, according as its terms are taken as pure or as applied, was overlooked entirely—with the disastrous result that the universality, certainty, and exactness pertaining (hypothetically) to the ideal creations of ‘pure’ mathematics were erroneously transferred to their ‘applied’ counterparts. To this day logicians are found to argue that real space is homogeneous because it is convenient in Euclidean geometry to abstract from the multitudinous deformations to which bodies moving through it are subjected, and to leave them to be treated by physics;382 nor are they aware of any lack of ‘exactness’ and discrimination when they identify the ideal triangle with the figures they draw on the blackboard.

§ 5. After its apparent success in analysing mathematical procedure there was no more disputing the supremacy of the theory of ‘proof’. The facts that its field of application was soon found to be much narrower than that of science, and that it failed egregiously to apply to the procedures of the (openly) empirical sciences, and a fortiori could not justify them, if they were noticed at all, were held merely to show that these sciences stood on a low level of thought, which from the loftier standpoint of logic could be contemplated only with contempt; if they required help and got none, so much the worse for them. Accordingly the whole theory of science was so interpreted, and the whole of logic was so constructed, as to lead up to the ideal of demonstrative science, which in its turn rested on a false analogy which assimilated it to the dialectics of ‘proof’. Does not this mistake go far to account for the neglect of experience and the unprogressiveness of science for nearly 2,000 years after Aristotle?

§ 6. Yet the deplorable consequences of this error should not render us unjust. The influence of Aristotelian logic on the theory of science was natural, and in a sense deserved. For Aristotelian logic is perhaps the mightiest discovery any man has achieved single-​handed. Its might is sufficiently attested by the length of its reign. Euclidean geometry alone is comparable with it, and Euclid owed far more to his predecessors than Aristotle. Moreover, the Aristotelian logic may be said to have achieved its purpose. It was able to regulate dialectical discussion. The syllogism did determine whether a disputant had proved his case, and for any one who had accepted its assumptions its decision was final, while even its severest critics had to admit that it was an indisputable fact, the interpretation of which was a real problem.

Unfortunately, there is not yet any agreement among logicians about the solution of this problem. Aristotle’s own analysis did not go back far enough: he stopped short at the Dictum de Omni and the reduction of syllogisms in the second and third figures to the first. He did not penetrate to the ultimate assumptions which were implied in the dialectical purpose and social function of the syllogism. But the truth is that syllogistic reasoning presupposes quite a number of conventions which Aristotle did not state, and which can hardly be said to have been adequately recognized since.

§ 7. (1) The first of these may be called the Fixity of Terms. Syllogistic reasoning manifestly depends on the assumption that the terms occurring in it have meanings sufficiently stable to stand transplantation from one context to another; for only so can they establish connexions between one context and another. Thus a syllogism in Barbara argues that because all M is P and all S is M, all S must be P. But it can do this ‘validly’ only if M, its middle term, remains immutably itself, and is the same in both premisses. Doubt, dispute, or confute this assumption, and the cogency of the syllogism as a form of ‘proof’ is overthrown at once. If the sense in which M is P is not the same as that in which S is M, the syllogism breaks in two, and its conclusion becomes precarious. Raise the question of how far reality conforms to this assumption, and you get at once a subtle problem of the applicability of the syllogistic form to the case in hand, which is precisely analogous to the question whether a theorem of pure mathematics is applicable to the behaviour of a real thing. In either case the cogency of the ‘proof’ which establishes the conclusion is impaired and ceases to be unconditional. The conclusion of a ‘valid’ syllogism will only follow if the middle term can be known to be unambiguous, and if the objects designated by the terms do not change rapidly enough to defeat the inference. And that this is the case can usually be ascertained only by actual experience. The conclusion, therefore, cannot be simply deduced; it has actually to come true, before we can be sure that the reasoning was sound. Absolutely a priori proof thus becomes impossible, if the assumption of the fixity of terms is contested: all proof becomes, in a sense, empirical.

Nevertheless, experience shows that the fixity of terms, though not a ‘fact’, is a valid ‘fiction’: in ordinary discussion the terms may usually be taken as fixed enough to render valid syllogisms common. An ordinary debate proceeds upon the assumption that the meaning of the terms involved is fixed, and cannot be varied arbitrarily. To science, however, this assumption does not apply without restriction. In a progressive science the meaning of terms often develops so rapidly that such verbal reasoning does not suffice. Hence the mere occurrence of verbal contradictions in a scientific reasoning is no proof that the argument is unsound. It may show merely that its terms are growing.

It should be observed further that this same assumption is implied in the fundamental ‘laws of thought’ on which the traditional logic rests. Indeed, the notorious ‘Law of Identity’ seems to be merely another statement of it. It is usually formulated as ‘A is A’, but in its actual logical use it is really the assumption that ‘everything is what it is called’. It is, of course, anything but self-​evident that ‘A’ is A, but unless the S, M, and P of the syllogism are rightly so called, the syllogism will not hold. Similarly, the Law of Contradiction collapses at once if the terms to which it is applied are allowed to change. The inability of ‘A’ both to be B and not to be B vanishes if ‘A’ is not fixed and may change its habits. And of course the real things known to science all change, and are fixed only by a fiction. Hence every application of the logical convention to real things may be challenged: it involves a fiction and takes a risk, and both of these may be bad. But the traditional logic ignores both the risk and the fiction and the lack of cogency in its attitude.

§ 8. (2) It is a further presupposition of the syllogism that the meaning of its terms is known. When a discussion is begun the parties to it are supposed to understand each other, and not to have first to find out and form the meaning of the terms they use. This assumption also is roughly true in ordinary debate, and its convenience is manifest. If things are rightly named, and if this feat has been accomplished once for all—presumably by Adam and Eve before they were turned out of Paradise for trying to know too much—we shall escape many of the most trying difficulties of scientific inquiry. We need no longer trouble whether the best names have been given, and whether a name good for one purpose is equally good for another, nor need we inquire whether our names may not unite what is alien on account of a superficial likeness, or separate what is akin on account of a superficial difference.

In science, on the other hand, the assumption that we know what meanings our terms can convey is not made as a matter of course. We may begin with roughly labelling objects of interest, and then inquiries may be conducted into, e.g., ‘electricity’, ‘elements’, ‘life’, ‘species’, &c., in the hope of settling what these terms shall mean, and of finding out more about their meaning, and without making the assumption that whatever new facts are discovered about them must conform to our preconceptions and confirm our nomenclature. Thus to a man of science it will not be cogent to argue that because an ‘element’ is (by definition) an ultimate form of matter which cannot be broken up, and ‘radium’ breaks up, ‘radium’ is not an ‘element’, or that because ‘species’ are eternal forms, and the Darwinian theory claims that they are not immutable, it can be dismissed as involving the ‘contradiction’ that a ‘species’ is not a species. Thus the best syllogisms lose their cogency so soon as a question is raised whether the verbal identity of their terms is an adequate guarantee of the real identity of the things they are applied to.

§ 9. (3) It is a further presupposition of the logician’s conception of ‘proof’ that absolute truths exist, and that in the ideal demonstration they form the premisses from which the conclusion follows. This presupposition is not stated, and is not implied in the form of the syllogism. For a syllogism is no less ‘valid’ if its premisses are true only hypothetically, and not absolutely. Indeed, it is not thought to impair the ‘validity’ of a syllogism that its premisses should be utterly false. At any rate we can reason quite as well with hypotheses and probabilities as with absolute truths, and this is in fact what we usually do, whether or not we are aware that our premisses are conditional and hypothetical. This ordinary practice, however, is resented by the traditional logic. For if our premisses are only hypothetically true, how can they lead to conclusions which can be declared absolutely true? And if our conclusions are not absolutely true, how can they be certain? Are they not bound to remain infected with the doubts which beset their premisses?383 As we value the certainty of our conclusions, therefore, absolutely true and certain premisses must be procured. If they cannot be procured, even the best formal proofs will remain hypothetical, and all truth will become dependent on experience. For if nothing is true absolutely, and every truth has originated humbly in a guess that has grown into a successful hypothesis, it can always be suggested that after all it may benefit by a little more verification. It may be true enough psychologically and for practical purposes, but it does not realize the ideal of ‘logical certainty’.

§ 10. This ideal Logic has formulated from the first. Aristotle already was not content with merely analysing the form of reasoning; he aspired to formulate the norm of scientific demonstration. The ‘demonstrative syllogism’, which he held to be the form of truly scientific reasoning, differs from the formal syllogism in two essential respects. Its premisses are absolutely true, and its middle term states the real ‘cause’, which connects its terms and is not merely a ratio cognoscendi. The reasoning proceeds, therefore, from premisses which are unambiguous, true, and certain, i.e. necessarily true and absolutely certain. Nor does the conclusion lose any of this excellence. Logic puts on a fine air of modesty, and merely claims that the syllogistic form is a guarantee that no truth can be lost on the way from the premisses to the conclusion in a ‘valid’ argument. If, therefore, our thought is properly arranged, our conclusion will be as true and certain as were its premisses, and no man will be able to gainsay it. It is the great beauty and merit of the syllogistic form that it is an arrangement which gives us this guarantee.

It was natural, therefore, that throughout the history of logic enormous importance should be attached to the acquisition of unquestionable starting-​points. For the possession of ‘valid forms’ was not enough. It only insured against loss of truth, it did not provide for its acquisition. It seemed, however, to imply that truth could only be generated out of truth, and handed down from the premisses to the conclusion. Hence the insistent demand for assured starting-​points, self-​evident ‘principles’, which the infallible method of syllogistic deduction might conduct to equally certain conclusions.

In reality, however, this demand for certainty was extra-​logical: it is not required for the purpose of analysing reasoning. For it is just as easy to reason from doubtful and probable premisses as from certainties, nor need the doubt in the reasoner’s mind affect the form of the reasoning. If, however, there is an imperative desire for certainty, it must be somehow gratified by logic. And there seemed to be no way of doing so except by ascribing absolute truth and certainty to the initial principles of science.

Of course it was covertly assumed that certainty could only be reached by starting from certainty, and that no possibility of a growth of assurance in the progress of the reasoning could be entertained. In a sense this assumption was correct (cf. §§ 27, 28), because it is true that the gradual verification of scientific truths does not render them absolute; but it led to neglect of all methods which appeared to start with premisses initially doubtful and hardening into certainties by gradual confirmation. No doubt it was not strictly impossible to reason from premisses not known to be true, but such reasoning was despised as ‘dialectical’, and no inquiry was made into the frequency of its occurrence in actual science. Why, then, waste time upon so unworthy a procedure, instead of fixing one’s whole attention upon the truly logical ideal, the absolute proof of absolute truth? Let us maintain, rather, the old Aristotelian384 conviction that the truly scientific syllogism proceeds from premisses that are true and underivative (because ‘self-​evident’) and inerrant, and demonstrates its conclusion with ineluctable necessity! Thus the attainment of absolute truth was unobtrusively smuggled in as the aim of reasoning, and became an integral feature of the ideal of ‘demonstration’.

§ 11. From the standpoint of the scientific inquirer, however, this whole theory of proof is open to the gravest objections. He finds first that it is impracticable, being composed throughout of counsels of perfection with which he cannot comply, and then that, even if he could, they would be perfectly useless, and destructive of his aims.

(1) It strikes him at once that the Fixity of Terms is an obvious fiction. He will of course be aware, from his scientific experience, that fictions have their uses and are often indispensable; but he will know also that not all fictions are useful, and that the adoption of a fiction has in each case to be justified by its usefulness. Moreover, it is not so much its immediate and prospective use which justifies it, though this yields the usual motive for its adoption, as the ulterior uses ascertained ex post facto by experience.

He will ask, therefore, for evidence that an absolute fixity of terms is the vital necessity for logic it is declared to be. He will admit, of course, the familiar arguments for a certain stability of meanings which have come down from the days of Plato, but he will suggest that a relative fixity of terms is quite sufficient to content them. He will point out that in a progressive science any absolute fixity in its terms is precluded by the very progress of the science. For the terms in use must somehow manage to convey the growing knowledge they are employed to ‘fix’. The term ‘gas’, for example, must not be tied down to the meaning Van Helmont desired to convey when he invented it; it must incorporate all that physics has discovered about ‘gases’ ever since. Similarly, when Darwinism transforms the notion of ‘species’, and the discovery of radio-​activity that of ‘atom’, these developments of meaning must be recognized as perfectly proper. To object to these conceptions as modern science uses them, on the ground that, because to Plato and Aristotle species were eternal and immutable, a ‘species’ that changes cannot be truly a species, or that because an ‘atom’ is etymologically ‘indivisible’, it becomes an impossible self-​contradiction when it is made up out of ‘electrons’, will seem to him to reveal only the fatuous pedantry of an utterly unscientific mind.

§ 12. (2) If he is acquainted with psychology, he will perceive also that the fiction of the fixity of terms is subject to a further restriction. It is not only in science as such—for all sciences must be conceived as progressive—that the fixity of terms cannot be made absolute: a real fixity is strictly inconceivable for and in every human mind. For every term that is actually used to convey a meaning must be held to form part of a new truth,385 i.e. of a truth that was not previously in being. It is not a question of principle whether the truth is supposed to be new only to the person to whom it is addressed, or claims to be new to all, i.e. to science. For no judgement would be made unless it had something new to say.386 Hence every real judgement, as opposed to the verbal formulas which are called judgements in the logic-books, more or less modifies the meaning of its terms. If it succeeds in being a real judgement and a new truth, it establishes a new and previously unknown relation between its subject and its predicate. ‘S’ is henceforth an S-which-can-have-P-predicated-of-it, and ‘P’ a P-which-can-be-predicated-of-S. Thus both the psychological associations and the logical associates of S and P are changed. That logicians should not have noticed so obvious a fact can be attributed only to their inveterate habit of not using in their illustrations real judgements intended to cope with actual problems, but operating with their verbal skeletons, which are not being used by any one to convey his meaning, and so do not have any actual meaning.

Clearly, then, no science can interpret the fixity of terms quite literally. Or rather, it can only interpret it literally—as a matter of the literal integrity of the words that may convey a meaning. But in a scientific inquiry the convention of formal logic must be reversed; the fixity of terms must be understood not to be absolute, but to be merely ad hoc and sufficient to convey a definite meaning, which it is desired to develop. Accordingly it must always be assumed that the results of an inquiry are to modify its terms, and that it is permissible, and indeed inevitable, to develop their meaning, so long as they remain capable of expressing and conveying the new truth. We must come to every inquiry with a willingness to learn and to expand our terms. The Fixity of Terms, as it is tacitly presupposed in the traditional logic, is a scientific blunder of the gravest kind.

§ 13. (3) To renounce it, however, entails further consequences. It appears to undermine the whole notion of formal validity. For if we admit in principle that the meaning of terms depends vitally on that of the judgement in which they occur, how can we continue to rely absolutely on the mere verbal identity of its terms to hold together a syllogism? In any syllogism the middle term, M, may have one shade of meaning in relation to P, another in relation to S. It may be quite right to call M P in one connexion, and to call S M in another; and yet, when the two assertions are put together, they may lead to a conclusion which is an error or an absurdity. The man who (in his laboratory) would rightly declare that ‘all salt is soluble in water’ and (at his dinner table) as properly hold that ‘all Cerebos is salt’, could not combine these assertions to draw the conclusion that ‘all Cerebos is soluble in water’, without finding that the facts confuted his anticipation.

No doubt, when this had happened, he might explain it, ex post facto (if he knew logic), by alleging a hidden ‘ambiguity of the middle term’. We need not here discuss whether it is fair to treat as an inherent ambiguity what is really a juxtaposition of shades of meaning which were relative to different purposes and right in their original contexts, thus manufacturing a fallacy by selecting the premisses: the important thing is that the logician should be driven to admit that any middle term may become ambiguous in this way when a syllogism is constructed, and that this completely stultifies his assumption that the verbal identity of the middle guarantees the real identity of the objects to which it refers.387 If we call two things, which are and must be different if they are to be two, both ‘M’, we necessarily take the risk that the differences are irrelevant for the purpose of our argument. We may legitimately assume this, but if we do, our hypothesis has to be confirmed in fact; it is naïve to think that the verbal identity of the terms is quite enough. If, then, actual identity cannot be absolutely guaranteed, if there is always a possibility that the same term when put into a syllogism and used in reasoning may develop an ambiguity and become effectively two, it is evident that no amount of formal validity will safeguard the truth of a conclusion, even when the premisses are in themselves severally true. The syllogistic form is convicted of losing truth which it started from, and this is the very thing it boasted it could never do. Moreover, its coercive ‘cogency’ is exploded: whoever wishes to deny a ‘valid’ conclusion after admitting its premisses, has merely to suggest that by putting the premisses together a fatal ambiguity has been generated in the middle term.

§ 14. (4) The assumption that everything has been named rightly, and is what it is called, will scarcely commend itself to the scientific researcher. He will know from much painful experience that language only embodies the knowledge which has been acquired up to date, and too often is only a compendium of popular errors. Hence in any research which really breaks new ground the existing terminology will always prove inadequate, and new technical terms have usually to be devised in order to embody the new knowledge. The reason is obvious. Ex hypothesi we are inquiring farther into the subject, because our knowledge is felt to be insufficient. Accordingly the probable defects of the terminology we are initially forced to use must be borne in mind: we may expect it to omit what is unknown, to misdescribe and to classify wrongly what is partially known, putting together what does not belong together and separating what does, emphasizing the unimportant and slurring over the important, and generally failing to provide the mind with words that give it a real apprehension of the objects under inquiry. Hence the tacit assumption of Aristotelian logic that the terms reasoned with are fully known, that adequate notions are already extant, that truth has merely to be disentangled by a verbal criticism of existing opinions, and has not to be discovered outright, is false; nor can any argument from a verbal identity be taken as final.

§ 15. (5) But of all the assumptions lurking in the theory of proof, the belief that reasoning can and should start from certainty will seem the falsest and most pernicious to the man of science. For it means that we are committed to a search for absolutely certain premisses as a preliminary to every inquiry, and proscribes consciously hypothetical, i.e. truly experimental, reasoning altogether, or at least condemns it as incapable of leading to certainty. This search, however, will either be perfunctory and uncritical, if it accepts false claims to certainty; or else vain, if it is conscientious. For every attempt to prove a conclusion absolutely demands two absolutely true premisses; hence the more we try to prove, the more we have to prove, and our search grows the more endless and futile, the longer it is continued. An immutable basis of absolutely certain truths, therefore, for reasoning to start from, is nowhere to be found. In no science is it possible to start with truths that are absolutely certain. In every science the initial ‘facts’ are doubtful; they are alleged, but not yet approved. They embody only unsystematic observation and prescientific experience of the subject, and so are probably the products of inaccurate observation, bad interpretation, false preconceptions, and popular superstitions. To acquire any considerable scientific value, such material has to be thoroughly revised and refined.

The validity of methods and the certainty of ‘principles’ are no more assured than the ‘facts’, initially. Every science has to work out its own appropriate methods experimentally; even if it borrows methods from another, it has to find out how and how far they apply to a new subject. Neither does a science acquire its principles by divine revelation; even if they fell from heaven ready-​made, it would insist on testing the authenticity of the revelation. But philosophers have been extremely reluctant to admit that the certainty of principles is a gradual growth: for over 2,000 years they have been endeavouring to discover some way of securing an infallibility to principles which would render them independent of the working of the sciences which use them. But if their labours have proved anything, it is that no such way can be found.

(a) They have recognized many principles as ‘self-evident’, and equipped the mind with a variety of ‘faculties’, expressly invented to enable it to apprehend the ‘self-​evident’ inerrantly. But they have not been able to agree upon a list of self-​evident principles,388 nor even to find any truth whose claims to self-​evidence have not been denied by competent critics. Nor have they been able to define their notion of ‘self-​evidence’ itself; they cannot discriminate between the sound ‘logical’ self-​evidence, which they conceived to guarantee truth, and its merely ‘psychological’ ‘mimic’, which is certainly much commoner, and becomes more intense and extensive the more unsound is the mind that ‘apprehends’ it.389 Hence an unprejudiced observer has no reason to put the ‘intuitions’ of philosophers and the ‘faculties’ which apprehend them on a higher cognitive level than those of women or even lunatics. They all impose themselves psychologically; but this proves nothing as to their logical value, and science has to test them just the same.

(b) The principles which are said to be necessary or logical ‘presuppositions’ all turn out to be hypothetical when they are examined. They are needed, no doubt, to solve the problem in hand, if the particular way it is formulated is taken for granted. But if either the order or the formulation of problems is altered, they cease to be either ‘necessary’ or ‘presuppositions’. For example, the ‘axiom of parallels’, alias ‘Euclid’s postulate’, is a necessary presupposition of geometry, if the existence of parallels is assumed. But if we prefer it, we can just as well (with Aristotle) make it our axiomatic ‘presupposition’ that the interior angles of a triangle are equal to two right angles, and can then deduce the existence of parallels. I.e. Euclid might have deduced what he assumed, and assumed what he deduced. If, moreover, we do not desire to construct a Euclidean geometry at all, we can deny both presuppositions, and proceed from alternative postulates, which lead to the various metageometries. The only things, in short, which all scientific principles presuppose are the desire to construct a science, and the desire to construct it in a particular way, which is simplest, or easiest, or most systematic, or most in accordance with the reigning prejudices. But these desires are the very things which the logician’s account of principles always omits to mention.

Again, the whole of Kant’s scheme of a priori presuppositions in the theory of knowledge rests upon an arbitrary assumption, viz. that mental data are to be conceived as originally discrete and are therefore in need of ‘synthesis’. But it is just as possible to conceive an analysis of knowledge which starts from the ‘presupposition’ of a continuum or flux, and proceeds to trace out the principles by means of which this continuum is broken up into a world of apparently distinct things and processes. Nor is it possible to say in advance of experience which of such ‘presuppositions’ is going to be more convenient and more conducive to scientific progress.

(c) It demands a high and rare degree of philosophic insight to perceive that very many principles are neither certain, necessary, nor probable, but simply methodological. Whether we think them true or not, we adopt them because of their eminent convenience. If they turn out to be false, candour compels us to call them methodological fictions; but they continue in use. Our belief in the trustworthiness of memory is a good example. For though we often find that our memory has played us tricks, we continue to accept as true what we ‘distinctly remember’. If no limitations to the truth-​claim of such assumptions are discovered, enthusiasts will probably insist on promoting them to the rank of indisputable ‘axioms’, and hail them as absolute truths. But their scientific value is not thereby enhanced, and the cautious will eschew such exaggerations. For there is no real reason why the scientific rank of principles should not rest openly and entirely on their actual services, and why a ‘methodological assumption’ should not rank higher than a ‘self-​evident truth’. For the latter is at most a fact of our mental organization which nothing has so far turned up in nature to set at naught, and as such a fact it is itself a thing to marvel at rather than an explanation of other things. The scientific spirit will always hesitate to acquiesce in the limits which are set to inquiry by sheer brute facts, and if the absolute truth of certain principles were merely an ultimate fact which could neither be impugned nor explained, this would go far to make these principles appear unintelligible and would be a constant challenge to dispense with them, or somehow to evade them. A principle, then, should always be prepared to state the reasons a science had for adopting it: only the reasons will appear from the actual working of the science. They will involve a reference forward to the facts it copes with, not back to higher principles or to any claim that proves itself by its self-assertion.

(d) Indisputable principles, then, are not consonant with the spirit of inquiry: it will gladly let them go, if it can attain truth and advance knowledge in other ways. It will not shrink even from repudiating the ideal of absolutely true and demonstrated truth, if it can be realized only by sacrificing the progressiveness of science; nor will it be dismayed to find that this ideal is unrealizable. For when the inquirer reflects upon his own procedure, he finds that it points to a radically different ideal, and that the existence of absolute truths would only be a hindrance and a restriction upon his endeavour (cf. § 28 (4)).

II

§ 16. Before, however, we attempt to delineate the logical ideal of the discoverer, it will be necessary to encounter a serious objection which protests on principle against such an undertaking, and urges that discovery by its very nature must elude logical treatment. It is contended, in the supposed interests of logic, that discovery is a process so inherently and incurably psychological that no logical account can ever be given of it. Discoveries are windfalls, and come as ‘happy thoughts’ to the gifted geniuses that make them, in a manner neither they nor any one else can account for or describe: they are therefore logically fortuitous, and to set forth the ideal of proof by which the truth of discoveries is tested is all that need, or can, be the concern of logic.

Certainly the great majority of deductive logicians have taken up some such attitude towards the process of discovery. Aristotle contents himself with a bare mention of ‘sagacity’ (ἀγχίνοια), which is defined as the instantaneous apprehension of the suitable middle term for constructing a demonstrative syllogism.390 When one recollects the weary centuries of painful effort and continual failure which elapsed while the élite of the human race were seeking for clues to, e.g., the mysteries of disease and of physical happenings, before they hit upon the notions of microbes and the mechanical theory, this naïve underestimate of the most difficult and essential of scientific procedures sounds like a mockery. Yet the whole Aristotelian school pass over the problem as lightly. They all seem to believe that, while it is merely low cunning to make a discovery, it is a real proof of mental capacity to arrange it ‘in logical order’ after it has been made, and to show how far short it falls of the logical ideal. Even the inductive logicians may be said to have participated in this attitude. For they were not more anxious to propound methods of discovery than to contend that their conclusions were just as rigidly proved and just as formally valid as those of syllogisms. They did not see that they were thereby accepting the demonstrative ideal of proof and giving away their own; what they should have shown was that this ideal was utterly nugatory, and that their own methods could never conduct to ‘proof’, but only to something vastly superior.

§ 17. In spite, however, of this wonderful consensus of logicians the above argument depends essentially on a confusion. It has confused two things which are perfectly distinct, the actual procedure of the individual discoverer, and the generalized description of the attitude of mind and procedures of discoverers, as they appear to subsequent logical reflection. Both present problems to the logician, but the problems are not the same. To anticipate the process of actual discovery may well be left to the prophets; it will transcend the powers of logic and indeed of any science, unless it be individual psychology, if it exists, or history, if it be a science.391 It may readily be admitted that anecdotes about the bath which fomented in the mind of Archimedes the idea of specific gravity, and the streets of Syracuse through which he ran and cried ‘Heureka!’, or about the apple-tree which shed its fruit upon Newton’s receptive head, and stimulated his brain to frame the law of universal gravitation, are beneath the dignity of science. Their narration belongs to history, which can go as deeply into their details as the scale of the history and the purpose of the historian demand; but the particular circumstances of a particular discovery may well be treated as ‘accidental’, and be smoothed out of the scientific record. But why does it follow that no common features can be traced in these histories of discovery, and that there cannot be compiled out of a sufficient number of them a generalized account of what appears to be the ‘essential’, i.e. really relevant, procedure of discoverers, which may serve as a guide and model to subsequent discoverers? Why should this be more difficult than to describe the method of lion-​hunting from the records of lion hunts, or the treatment of a disease from the history of a number of cases? Indeed, it would seem that the thing has been done. Any discoverer may reflect upon his own discoveries, and, like Poincaré,392 formulate the method he has found successful. And if discoverers are not all perfectly unique in their methods, important uniformities will probably be found by comparing the methods of a number of discoverers.

Why again should it be assumed that the general account thus extracted from a retrospective study of discoveries must at once coincide with the logical ‘ideal of proof’? Why should it even point to this, or be related to it otherwise than by contrast? Surely the possibility should be discussed that there are two procedures for logic to consider, of which the one describes how human knowers, starting from what they believe themselves to know, set about it to fortify and extend their knowledge, while the other moves on a superhuman plane and describes, with Platonic fervour, how ideal demonstration, descending from absolutely certain principles, moulds into a closed and inexpugnable system all the truths which are deducible from these and alone intelligible. The two accounts must be distinct, for they have different starting-​points and work upon different material. Nor need they ever have any point of contact. For it may well be that human knowing never attains to an absolute certainty and a completed system, while deductive proof never condescends to notice mundane fact.

This was certainly so in the first rapturous vision of a priori ‘proof’ which solaced Plato amid the elusiveness and opacity of the flow of happenings. The deduction of the intelligible order of the ideal ‘Forms’ from their supreme ground and (sole!) premiss in the ‘Idea of the Good’ stopped short of facts and events at the laws of minimum generality,393 and recognized in all the happenings of the sensible world an ineradicable taint of ‘not-being’ which rendered their stability impossible and their prediction vain. Aristotle similarly distinguished between the procedure which started from the notiora nobis, the apparent facts of perception, and that which began with the notiora naturae, the self-​evident principles which could form the ultimate premisses of demonstrations. But that these two methods must somehow coincide was assumed rather than proved, in a way that should have discredited the doctrine. For Aristotle also was not able to explain how ‘science’, being of ‘universals’, could apply to particulars, which nevertheless he would not with Plato stigmatize as ‘unreal’, while the ascent from the sensible fact to the ‘universal’, which was called the ‘induction’ of the ‘principle’, is hardly validated by the naïve allegation of a mental faculty of ‘intuitive reason’ (νου̑ς) endowed with the special function of apprehending principles in their particular exemplifications. It is high time, therefore, that this whole assumption that a necessary congruity exists between the logic of discovery and of proof should be subjected to a thorough examination.

III

§ 18. Such an examination will speedily establish that the mental attitude of the discoverer is, and must be, quite different from that of the prover.

In the first place, the discoverer is not in possession of the knowledge he covets. It is for him a desire, an aspiration, an aim to be attained. Proof, on the other hand, presupposes knowledge. Not only must the demonstrator know the assured truths he uses as premisses, not only must he have a supply of absolutely certain truths if his proof is not to remain hypothetical (§ 9), but he must already know the conclusion he exhibits. He cannot be ignorant, like the discoverer, of the result he is to arrive at. He is not engaged in discovering new truth, he is only showing how it follows from old truths. His retrospective contemplation has merely to retrace the history of its attainment, or rather to rearrange it in the more pleasing order which he calls ‘logical’. This order is not that in which it was discovered, nor even that in which it could be discovered. For there are such things as necessary errors, indispensable artifices, and indefensible fictions, and the way to a truth often lies through them. Thus from time immemorial mathematicians have represented the continuous by the discrete, quantities by numbers, knowing full well what fictions their practice involved. Again, mathematical calculation of shapes, areas, and motions necessarily presupposes the fictions that bodies have the ideal and regular forms to which they ‘approximate’, and that their ‘mass’ is concentrated at their (ideal) ‘centre of gravity’. It is more than doubtful whether the notion of an ‘evolution’ of species could ever have been reached, except by starting from the false notion of the fixity of species, or whether the true nature of the mobility and development of meanings could have been understood except by correcting the Platonic theory of immutable and eternal ‘universals’. To ‘proof’ all these incidents and accidents of the history of discovery are irrelevant; all that has to be done is to show that the new truth can be deduced from the old, and that a ‘logical connexion’ exists between them.

§ 19. Not only is this much easier to do than to make the discovery, but it is very much easier to follow. Any one can see the connexion once the data have been arranged in logical order. Hence the assumption that this order somehow represents the actual process in a perfected form is natural enough. But it leads to contempt for the procedure of discovery. The discovery is made to look so easy that it becomes impossible to appreciate its difficulty and its merit, and it seems astonishing that no one made it long before. For did not the ‘facts’ all but force it upon the dullest mind? Who could have failed to see that fossils must be (at least) as old as the rocks in which they are embedded, that obviously worked flints, similarly, attest the antiquity of man, that northern Europe is scratched all over with the marks of a gigantic glaciation? It is forgotten that these ‘facts’ were not there until there came a mind prepared to notice them. Hence none of these discoveries were in fact easy to make, and they were preceded by a long struggle of the human mind with false preconceptions and the illusory ‘facts’ which they had engendered.

Nor are discoveries easy to get recognized when they have been made. The persecutions to which discoverers of new truth are subjected always and everywhere (more or less) form as discreditable a chapter of human history as the persecution of moral reformers. Those may count themselves fortunate who are simply ignored. Hence everything has to be ‘discovered’ over and over again. Nothing new ever enters the world, just as nothing old ever passes away, without infinite pains and after a protracted struggle. One curious result of this inertia which deserves to rank among the great fundamental ‘laws’ of nature, is that when a discovery has finally won tardy recognition, it is usually found to have been anticipated, often with cogent reasons and in great detail. Darwinism, e.g., may be traced back through the ages to Heraclitus and Anaximander. Thus it is true that there is ‘nothing new under the sun’; but only because when a new truth first appears it does not prevail: when after a hundred repetitions it is at length recognized, it is no longer strictly new. Accordingly, the ‘discovery’ of a truth is only the beginning of its career, the first step by which it makes its way in the world, and still very distant from the crowning ‘proof’ with which logic complacently adorns it ex post facto, when it has ‘arrived’. The slowness and difficulty, then, with which the human race makes discoveries, and its blindness to the most obvious facts, if it happens to be unprepared or unwilling to see them, should suffice to show that there is something gravely wrong about the logician’s account of discovery.

§ 20. Quite apart from the difficulties which the psychological constitution and social organization of man put in the way of innovators, the making of a new truth which formulates a new ‘fact’ is also intrinsically anxious work. It is not merely that its maker can have no assurance that his enterprise will succeed, that he cannot start with a feeling of certainty from established truths, and be wafted by an irresistible wave of logical necessity to the safe haven of a predestined conclusion. He must start with a consciousness of ignorance and an all-​pervading feeling of doubt about every step of his inquiry. This doubt he should not, moreover, endeavour to disregard or to suppress; for it is the best guarantee that no way to the truth will be passed by in his explorations. Doubt, therefore, should be recognized on principle, and equipped with a technique of testing and experimentation: the inquirer should be proud that he has to feel his way in fear and trembling to the very end.

Yet his condition will not contravene Aristotle’s dictum that all inquiry and research proceed from knowledge previously acquired.394 In a sense he will still start from what he knows, or thinks he knows. For it is psychologically impossible to do anything else. The knowledge he believes himself to have cannot but affect all his ideas, and he cannot get away from it. His boldest speculations, his most hazardous hypotheses, will have some relation, however subtle and recondite, to the knowledge at his disposal. It will influence all his thoughts and guide his guesses. As he cannot divest himself of his knowledge and the ideas it has rendered familiar to him, he has to accept its limitations. His only problem is to use it as effectively as possible.

But it is clear that he cannot regard his knowledge with the same sort and amount of confidence as the believer in demonstrative proof. He must conceive himself as an explorer, and his attitude must be tentative throughout. Knowing that his premisses are questionable and only doubtfully true, he will recognize that his inferences are only probable, and stand in need of confirmation. As a rule he can, no doubt, find accepted truths to argue from; but these being relative to the existing state of knowledge are known to be subject to correction. Even where he has started with premisses of the most superior kind, which are generally deemed absolutely self-​evident and certain in themselves, he will still be conscious of a doubt whether they will prove to be the right premisses for his purpose. If they are not, their truth is irrelevant and will lead him astray. In no case, therefore, can he escape the responsibility of choosing the right ones from his limited stock of known truths and familiar ideas, as he contemplates the infinite expanse of possible discovery. In whatever direction he moves, the unknown lies before him; he may come upon surprises or be stopped by unsuspected obstacles. In short, there is nothing of the irresistible about his progress; it has not the faintest resemblance to the majestic march from inevitable premisses to a predestined conclusion which so fascinates us in the theory of proof.

§ 21. But, it may be said, all this is not enough. The differences in the attitudes assumed by the reasoner in discovery and in proof may be only psychological. They do not prove any real logical difference between them; the logician’s account may still be what the discoverer would acknowledge to have been his best course, if he could have seen it. It has, therefore, to be shown that the differences in question arise out of, and develop into, differences which are indisputably logical.

Thus, the ignorance which the inquirer feels is doubtless a psychological fact, but the lack of knowledge which engenders it is surely a logical fact of some importance. In general, the feelings of doubt, expectancy, and perplexity which beset the mind of the inquirer, and contrast so distinctly with the feelings of confidence, knowledge, certainty, and necessity which accompany a ‘proof’, originate in a logical fact. Every inquiry starts from a problem, of which the solution is not yet known. An inquiry is, as the name implies, a question, put, not to nature at large and at random, but to some part of it, which is taken to be relevant and to contain a possible answer to the inquirer’s question. Now this dependence of inquiry upon problems springs no doubt from the psychological fact that until there is something put before it the mind cannot get to work upon it; but it is surely a fact of the utmost logical significance, and it is astounding that the logical tradition should have slurred it over so completely.

Especially as in the very beginnings of logic some of the Greeks distinctly caught a glimpse of it. For, having started their reflection upon reasoning from a desire to regulate debate and to argue a case at law, they naturally noticed that there are two sides (at least) to every question. Accordingly, Protagoras appears to have taught systematically that there were always two reasonings (λόγοι) to be considered,395 Socrates, treated scientific inquiry as an extension of the art of cross-​examination, and Plato conceived the search for ideal truth as a ‘dialectical’ process, as a sort of dialogue of the soul with itself. Now this whole doctrine is equally good as logic and as psychology. It is profoundly true of the inquirer’s mind; he must be keenly alive, not only to the evidence for, but also to that against his working theory. But it is also true of the logical nature of inquiry that it is a process of determining which of the alleged ‘facts’ and of the theories to interpret them are real and true. Inquiry logically ‘presupposes’ a conflict between the data, and a dispute about them.

Unfortunately, however, the conception of scientific research as an inquiry lapses from the logical consciousness in consequence of Aristotle’s work. His discovery of the forms and formulas of demonstration overshadowed it, and restored the reign of dogma which is so congenial to the authorities everywhere.396 The true conception of inquiry does not revive again until our days, when Mr. Alfred Sidgwick and Professor John Dewey have endeavoured, not with the success they deserved, to reopen the eyes of logicians to the facts of the scientific situation.

§ 22. To conceive an inquiry as a question then is, we see, implicitly to conceive it as having a plurality of answers, all of which have to be examined. All these answers are initially hypotheses, and a choice has to be made between them. This renders the recognition of alternatives a paramount necessity for a logic of discovery, which can no longer dismiss them with a jejune chapter on ‘disjunctive propositions’. Their existence is no longer to be treated as an annoying complication which delays the progress of science, but must be taken to inhere in the logical nature of problems, and to be essential to their proper elucidation.

Logic, therefore, should regard it as its duty to inquire (1) how the inquirer is furnished with an adequate supply of theories for analysing and testing the apparent facts of his subject, (2) what methods are used to sift hypotheses and to select the more valuable, and (3) if it can, to add some hints as to how theories and methods ought to be handled.

(1) To the first question there is no exhaustive answer. No logic can guarantee that all the possible theories which concern the facts under inquiry will be available. They may not yet have occurred to any human mind, and may never do so. This alone ought to be considered a fatal objection to all methods which presuppose exhaustiveness, and are pressed by the logician upon the man of science. It ought to dispose of methods which demand that all the facts should be assembled before theorizing is begun, or that all the alternatives should be stated and the true one extracted by the successive elimination of the false ones, or that define a ‘cause’ as reciprocating with its ‘effect’, and assume that the true cause has been discovered when no other has been thought of, or that if a theory works we may take it that it alone will do so and is (absolutely) true. All these notions demand an impossible exhaustion of the alternatives, and try to convert a (psychological) failure to think of any more into a logical proof that there are no more. And they all regard the plurality of alternatives as a hindrance to be got rid of, and not as a safeguard and a help to proper inquiry.

Hence the real difficulty was not perceived, viz. that there is no formal guarantee that the supply of hypotheses for use upon the facts in any inquiry will be adequate. It may well be that for lack of a good working theory to go upon, all the theorizing on a subject proves vain and sterile. In the beginnings of all the sciences this sort of condition always exists and often lasts for centuries, and it is a main reason why some sciences make little progress even now.

Nevertheless, the difficulty is not in practice as fatal as it looks on paper. It is probable that the inquirer will in fact usually have a supply of alternatives to start from. For (a) he will naturally select a subject in which there are disputed points. And (b), what is even more important, human minds are naturally various: they put, therefore, different interpretations on the same facts and value them differently. Some are attracted by novelty, others by orthodoxy; some incline to one type of theory and method of inquiry, others to another. Hence in any inquiry upon which a number of minds are actively engaged, there will always be differences of opinion, and these will be most marked in the rapidly growing regions of every progressive science, which, like the growing cells in the trunk of a tree, are always on the outskirts. There will always be a conservative and a liberal party, even in science, and the clash between their views will always provide alternative solutions of problems, the comparative merits of which the inquirer can examine. But the sciences owe their progress largely to the man who raises new questions, and should provide for him in their organization.

§ 23. It should be noted further that if this feature in discovery were properly recognized and emphasized, it would have important educational and ethical effects. At present the study of logic can hardly be said to liberalize and broaden the mind or to improve the temper. So long as its chief interest is in a theory of absolute proof and complete certainty, it will tend to breed pedants and bigots. The effect would be very different if an adequate logic of discovery had imbued the mind with an ever-​present thought that every subject may and must be considered from several points of view, and that an inquirer should beware of letting his predilections and preconceptions blind him to possible alternatives. The logical attitude of inquiry, when fully understood, demands a tolerant and open mind, and excludes the narrow-​mindedness and dogmatism which the theory of proof has fostered by its pretence of showing that there was but one truth and one inevitable way of reaching it. Moreover, the necessity of continually choosing between a number of alternatives should cultivate a judicial temper, conducing to fair-​mindedness and consideration towards the views of others. For a mind which is in the habit of choosing between alternatives must be impressed by the facts that there is something to be said for the views it does not accept, that the view accepted is often not so very much superior to those rejected, and that new facts and new knowledge may always revive views which were supposed to be defunct.

Of course our natural dogmatism will take alarm at the flabby toleration of ideas which this attitude seems to imply. It will be objected that no one who can see the good and truth in beliefs he does not accept, can really be strenuous in upholding those he does. The full answer to this bigots’ argument can only be appreciated when the attitude of progressive science is fully understood (cf. § 33), but in general it may be pointed out that a power of first weighing alternatives, choosing the best and acting upon it strenuously, is precisely what life demands of us at every step. It should not, therefore, be impossible to compass it in science.

§ 24. (2) To the second question of § 22, viz. what are the methods used by the inquirer in sifting the alternative hypotheses in the field, and picking out the most valuable, the answer is comparatively easy. It is substantially the answer given by the pragmatist analysis of knowledge. That theory is preferred, and tends to be accepted as true, which for the time being works best. The formula looks simple, but needs more thinking out than its critics usually bestow upon it.

(a) It implies, of course, that all the alternatives (before the mind) ‘work’ more or less. They must be (or appear) scientifically plausible, and proffer a more or less satisfactory explanation of some or all of the admitted ‘facts’. This is why agencies like the Devil, who could once be extensively alleged to explain anything unusual, have dropped out of the purview of science.

(b) ‘Working’ must be conceived somewhat widely. Its primary appeal is to the accepted principles and recognized interests of the science; practically to ‘work’ means to conduce to the development of the science on the recognized lines, and the proper judges of what ‘working’ counts are the experts who cultivate each science.

(c) But there will often be complications due to certain disputable workings, of which the relevance is not yet established, and about these there will legitimately be differences of opinion. These should not be suppressed, but candidly argued out.

(d) Moreover, every new departure will be pro tanto disputable, because it will conflict more or less with the vested interests of the established doctrines. One great factor in the ‘working’ of a new truth is the extent to which it upsets, or is thought to upset, the old, and demands a reconstruction of beliefs, a correction of authorities, a revision of text-books, a renewal of plant, &c. Hence what works best in the abstract may not do so under the actual conditions. It may ‘pay’ a professor better to be ‘orthodox’ than to be an innovator, and he is usually quite alive to this, though it does not render him a good investment scientifically for the institution that appoints him. If then we looked at this side of the matter alone, the verdict would always go against the novelty. For very few new truths are fortunate enough to find the field free and unoccupied. Usually they have to spring up in a soil densely overgrown with a rank growth of prejudices, dogmas, and superstitions, to which the world is accustomed and even devoted. So they have to fight for an opening in which they can take root and grow up.

(e) The ‘working’, however, need not amount to a claim to represent ‘the’ truth. A discoverer may know that by reason of his deliberate use of fictions, his results have forfeited their claims to be strictly true; yet they may ‘work’ better than anything else in sight. The typical example here is, of course, mathematics. When physical objects are treated mathematically, they are identified by a fiction with the objects of pure mathematics, and it is only on this assumption that their behaviour can be calculated. They are, of course, vastly more than mathematical objects, but their surplus meaning becomes irrelevant wherever objects admit of mathematical treatment. And apart from the restriction of the claim to truth necessitated by the use of fictions, it should, of course, be recognized also that there are sound logical reasons for denying that truths which rest on their ‘workings’ can ever be ‘absolute’ (§ 26 s.f.). Their truth is pragmatic, and is optimi iuris only if pragmatism establishes that no other and no better truth exists.

(f) More specifically a very important form of working is the prediction of events. Knowledge of the future is an almost universal object of human desire, which men have sought to compass by fair means and foul, and the calculation of the future is the avowed aim of many scientific inquiries. Hence there is nothing more potent to dispose the mind to accept a theory than the success of the predictions it has led to. Yet here again this form of ‘working’ differs generically from ‘proof’. It is clear that prediction is not strictly proof. For predictions may be made with considerable accuracy by the aid of hypotheses which turn out to be false or impossible. Thus eclipses and other celestial events were predicted for centuries by means of the Ptolemaic astronomy, and they cannot be predicted even now with absolute accuracy. Indeed, physically speaking, absolute accuracy is unthinkable. No instrument and no organ of observation can be conceived to measure to more than a finite degree of accuracy, and the best value for any physical ‘fact’ will always be the mean of a number of good observations after all the accessible sources of error have been allowed for.

At no point, then, does the test of ‘working’ conduct to the notion that absolute truth is discoverable. But the right inference may be, not that the test is worthless, but that absolute truth is a chimera.

§ 25. (3) It cannot then be seriously disputed either that alternative hypotheses are always (more or less consciously) present to the mind of the inquirer, or that the working of a theory is in fact used, in all the sciences, to test its claim to be true. But does it follow that logic should bow to scientific fact and recognize these practices? Should it set itself to devise a technique for regulating the formation of hypotheses and the establishment of their truth by their working? It is here that the traditional logic demurs, and disputes begin. Nevertheless, strong reasons may be advanced for answering both questions in the affirmative.

(a) An abundance of hypotheses is a guarantee of great logical value that all the important facts will be properly observed. For it is evident that every theory will produce a certain bias in the observer. It will direct his attention upon those facts and those features which are relevant to his theory, and, more particularly, which support it. This is usually an advantage, because it helps him to select what is relevant to his inquiry from the chaos of events; but it will pari passu blind him to whatever does not seem to be related to, and to fit into, his theory. He will, therefore, fail to observe and to appreciate what will seem to him to have little or no scientific interest. And in so thinking he may be quite wrong.

The old theory of ‘induction,’ thought to get over this difficulty by saying, ‘Well, of course, all the facts must be observed’. It did not observe the fact that in practice this is impossible, and is never done. Nothing is observed but what the knowledge and preconceptions of the time make visible to the scientific eye. Of what is visible at any time only a small part seems worthy of the scientific microscope. Complete observation, therefore, of literally all the facts is scientifically impracticable.

As a logical ideal also this notion of all-​inclusiveness is absurd. If no inquiry could ever begin until all the facts had been assembled, how could anything be discovered until omniscience had been achieved, i.e. when there was nothing left to discover? For how are we to know that our assembly of ‘facts’ really is complete? And if literally all the facts have to be used as data in any inquiry, shall we not speedily find that every fact ramifies into infinity, and drags in the totality of reality, and a knowledge of all things present, past, and future? This ‘logical ideal’, therefore, renders inquiry impossible.

In point of fact the data of any inquiry are always a selection. They are such of the recognized facts as are thought to be relevant, i.e. to be truly ‘facts’ for the purpose in hand. But being a selection they involve us in the risk that we may have selected wrongly, and omitted what is important while admitting what is not. From this risk there is no escape. For we cannot effect a compromise by including merely so much of the facts as we can lay hold of. Not only does this yield no guarantee that everything that is needed has been included, but it may be a positive hindrance to try to include too much. For if our data grow into an unwieldy mass, they will not seem susceptible of any order or principle, and even the most penetrating inquirer will lose his way.

It is better, therefore, to give up altogether the idea of securing formal validity by postulating an all-​inclusive exhaustiveness. The obvious alternative is to operate simultaneously with a plurality of theories, each of which means a certain ordering of the ‘facts’ relatively to what seems a relevant and promising point of view. Each will involve a selection and induce a bias; but with any luck they will neutralize each other’s bias, and so will increase the probability that no really relevant fact has escaped notice. This will not satisfy the logical ‘ideal’, but in practice it means a good deal, and is enough for scientific progress. Of course it must be understood that the hypotheses employed are in a general way relevant to the problems and the condition of the sciences, and not random guesses. This proviso will cut down their exuberance even more than the limitations of the human imagination, which seems to be psychologically incapable of really departing very far from the suggestions of experience.

§ 26. When logic has recognized the use and value of ‘working’ as the test of truth, it must, however, make it clear to itself and to others both what precisely this test is, and what it can, and cannot, accomplish.

In the first place, it must be made clear that it is not a logical implication of the test that ‘whatever works is true’, and the reasons for disputing this dictum must be set forth. The fact is that we all have a strong psychological tendency to believe in the truth of what is found to work, without much criticism of the sort and extent of the ‘working’. But the logician should carefully investigate the various sorts of working that occur, and take special note of those which either do not themselves lay claim to full truth, or do not (ordinarily) have their claim conceded.

For example, ‘fictions’ are not supposed to be strictly true; but they may ‘work’ and be ‘as good as true’, or ‘pragmatically true’, or ‘sufficiently true for the purpose in hand’. They work, in fact, within limits; but these limits are known, and so they are not confused with full-​fledged truths, to the applicability of which there are no known limits.

The case of ‘methodological assumptions’ is more difficult and instructive, and is usually misconceived. In their case the existence of limits to their ‘working’ is either not known or not relevant, because they owe their adoption to their use and convenience in analysing and organizing a subject of inquiry. Thus the principle of Causation, the assumption that every event has a cause which determines it fully, is properly to be regarded as methodological. It declares merely that if we desire to calculate the course of events, it is scientifically convenient to treat events as if they had ‘causes’, from which their occurrence could be predicted, whether or not they have them in fact. This assumption may be purely methodological; it need not, and should not, be turned into a dogmatic, metaphysical denial that there may be indeterminate happenings. There may even be good reasons to suspect their occurrence, and indeterminism may be ultimately true, and yet scientific method may rightly ignore this possibility, because it would render the calculation of events impossible.397 Even an indeterminist then is fully entitled to reason as if events were determined, and to search for ‘causes’, for the purely methodological reason that this enables him to calculate events, and that after all they may be calculable. So long as they work for scientific purposes it is not, in the case of methodological principles, necessary to raise the question of their metaphysical truth.

The ‘lie’ again is a curious case of ‘working’. A lie, works, as a rule, only so long as it passes for truth, and is believed to have the meaning and value its author claims for it; when it is ‘found out’, it ceases to work. Hence it can both work and fail to work at the same time, according as it is, or is not, known to be a ‘lie’. Clearly nothing can be made of the lie logically, until this double aspect inherent in its nature is recognized; if the logician refuses to distinguish between the persons concerned in its making, acceptance, and rejection, it remains (like ‘error’ to Plato) an insoluble ‘contradiction’. It is, however, a mere prejudice to refuse to make these distinctions.

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?

(4) The ideal of absolute certainty may be repudiated altogether, even as an ideal, for sound scientific reasons. It may be shown that if it were possible it would be scientifically undesirable. For it would mean the creation of absolute bars to scientific progress. If truths existed which were absolutely certain, this would mean that nothing more could be learnt about them, and nothing could be done to strengthen their position. No experience, no inquiry, no experiment, could any longer affect them, and add to or detract from their value. They could not, therefore, form avenues to further knowledge. They would simply be stops which would arrest scientific inquiry. But how could such things form an ideal of scientific knowledge? How could it be in the spirit, and to the interest, of science to recognize them? They would merely be for science brute facts which it was forbidden to investigate. And must not science on principle hold out for the right to inquire into everything, to test every belief, however true it may seem? How, then, can it be the ideal of science to adopt an ideal which would stop inquiry?

Nor will it suffice in reply to point to the fact that the sciences continually assume the truth of the premisses they argue from. For though this is often a convenient assumption for the purpose in hand, it is one thing to assume the truth of premisses for the purposes of an inquiry, and quite another to assume it absolutely. For in the former case our assumption may be, and should be, accompanied by a consciousness that upon another and fitting occasion the premisses now assumed to be true may themselves be inquired into: to regard them, therefore, as absolute is to misinterpret their logical condition.

There are no good reasons, then, why the sciences should surrender to the arbitrary demands of the traditional logic, and sacrifice their practices which have been sanctified by the successes of 2,000 years to theories which sprang from a misunderstanding of scientific procedure, and have since lost all contact with it. The original mistake was pardonable, but it ought not to be regarded as an insult to logic to require it to understand the procedure by which the sciences actually progress.

§ 29. The scientist then should not be terrified by the charge that his ‘truths’ are ‘only probable’. For it is better to be satisfied with probabilities than to demand impossibilities and starve. Moreover, a high degree of probability means ‘practical certainty’, i.e. confidence enough to move to action. Such certainty so convinces and satisfies the mind that it cannot feel more certain about anything; the logical gap between it and absolute certainty is psychologically negligible. We are sacrificing, therefore, nothing but a superstition, nothing that has any value for us, by renouncing the demand for absolute truth and demonstrative ‘proof’, and we gain in return a charter of liberty. For to admit the essential progressiveness of scientific truth and its indefinite capacity for improvement means unlimited freedom to research into truths which are infinitely perfectible, because they are never ‘absolute’. The ideal of the infinite perfectibility of truth, and the infinite progressiveness of science, is more than an adequate substitute for the ‘logical ideal’ which is abandoned. For not only is it an ideal which works, but it really embodies a nobler aspiration than that which represented science as ‘resting’ in absolute perfection on fixed ‘foundations’ of ‘eternal’ truth. The sentiment which inspires this group of metaphors is given away by the word ‘rest’. A science that desires to rest is one that is unwilling to move and unable to advance. Fixed ‘foundations’ are needed only for standing firm and standing still, and it turns out that what is strictly meant by ‘eternal’ is not that truths last for ever, but that they are not related to ‘time’ at all, and so have really no application to ‘events’.400

On the other hand, a science which sincerely desires to progress needs fixed foundations as little as fixed ideas, and firm ground as little as assurances to ‘rest’ on. It needs only a starting-​point, or jumping-off place, whence it can plunge into the unharvested seas of the unknown. Now the essence of a starting-​point is to be a place you want to get away from, and its excellence lies in being such as to prompt you to leave it as easily and eagerly as possible. If, therefore, scientific ‘principles’ (ἀρχαί) are really to be starting-​points, they need not, and must not, be so comfortable and so deceptively similar to ‘absolute’ truths as to tempt the scientific spirit to repose. They should be tentative assumptions which are gladly abandoned in the hope of reaching something better, stepping-​stones to farther and higher things, which are valued for their consequences, and logically dependent on the conclusions to which they formed the premisses. The logic of science, therefore, has no reason to postulate stability or solidity for its initial principles: the most indispensable of them are only principles of method, and even of the tried and tested principles it arrives at the ‘validity’ (= strength) demanded is merely that they should be able to float the accumulated wealth of knowledge down the stream of time.

IV

§ 30. It is clear, then, that the time has come when Science should break decisively with the logical tradition, and proclaim a logic of its own which has always been implicit in its procedure. It must definitely declare that what it needs is not a logic which describes only the static relations of an unchanging system of knowledge, but one which is open to perceive motion, and willing to appreciate the dynamic process of a knowledge that never ceases to grow, and is never really stereotyped into a system. To show that such a logic is not inconceivable will be the endeavour of the concluding sections of this essay.

We have already had occasion to note many of the most important features of this logic. We have seen that logical, i.e. critical, reflection upon discovery must start from, and be guided by, the conception of a scientific problem with which the process of knowing experiments (§ 21). This problem has, of course, to be attacked with the existing resources of a science, i.e. with the knowledge it possesses up to date. These resources form the scientific capital which is necessarily risked in research if it is to yield interest. It comprises (a) approved principles, (b) known facts, and (c) established meanings of words. About each of them a little more may advantageously be said.

(a) We have seen (§ 15) that the principles of any science could not rightly be conceived as inscrutable, ultimate, absolute certainties of divine descent, and acknowledging no human ancestry. We saw that they could be understood only as hypotheses which reflection upon a problem had somehow suggested to an ingenious mind, which had been provisionally adopted in order to explore and organize a subject of inquiry, and had finally been verified and confirmed by their success (§ 15 (c), § 24).

The principles thus accepted by a science are often regarded as descriptive of fact when they are merely methodological and convenient,401 but this is a point of secondary importance. And even the most amply verified principles never quite lose their hypothetical character. So long as they are used, their meaning, scope, and truth are not absolutely fixed. They can be extended, restricted, and modified by the working of the principles.