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.