The exclusions through which, in the application of this method, the proof of the non-excluded alternative is effected are frequently justified by a form of proof usually known as Indirect Proof. The proposition 'S2 is P2' is assumed to be true, and it is then shown that on this assumption the proposition 'Sn is Pn' must also be accepted as true. Hence, if the proposition 'Sn is Pn' contradicts a proposition already proved to be true, if follows by the Modus Tollens that 'S2 is P2'cannot be true. Thus the main argument in Indirect Proof runs as follows : 2 If S2 is P2, then Sn is Pn. ... S2 is P2. The argument through which an indirect proof is effected is known as a reductio ad absurdum, the absurdity to which the argument reduces us being that of self-contradiction. The Reductio ad Absurdum has been utilized, under the name of Indirect Reduction or Reductio per impossibile, to test the correctness of inferences drawn in the 'imperfect' figures of the Syllogism (Figs. II., III., and IV.), on the assumption that inferences drawn in the first or 'perfect' figure, according to the Dictum de Omni et Nullo, could be accepted as correct. The process, however, is applicable to syllogistic reasoning only so far as this is treated as a truth-inference, so that the premisses and conclusion can legitimately be characterized as true or false. Indirect Reduction, then, may be defined as a proof, effected by means of a syllogism in the first figure, that the truth of certain conclusions drawn in the 'imperfect' figures follows with logical necessity upon the truth of their premisses, because, if those premisses are true, the contradictories of those conclusions are necessarily false. The traditional Logic, following Aristotle,* who in Direct Reduction made use of Conversion only and did not recognize Obversion, † singled out Baroco and Bocardo as the most suitable forms of Syllogism for illustrating this process of Indirect Reduction. We may therefore take one of these-Baroco-and use it for the illustration of the method. If it is suggested that the inference in Baroco is not correctso the method argues then in that case, though we assume the premisses to be true, the truth of the conclusion 'Some S's are-not P's' will not necessarily follow. That is, granted the truth of the premisses 'All P's are M's' and 'Some S's are-not M's,' still 'Some S's are-not P's' may be false. But since the denial of a proposition is logically equivalent to the affirmation of its contradictory, it follows that the proposition 'All S's are P's' may be true. Let us, for the sake of the argument, assume that it is true. Taking † Cf. above, pp. 189-192. * Vide An., Pr., A., c. 45, p. 516, 1, 2. this proposition, 'All S's are P's,' as the minor premiss of a syllogism, and combining it with our original major premiss, 'All P's are M's,' we can at once draw the conclusion 'All S's are M's' in the standard or 'perfect' figure. But this conclusion contradicts our original minor premiss 'Some S's are-not M's.' Now the premisses of the original syllogism are both, ex hypothesi, true. The statement 'All S's are M's' is therefore false. Hence, since the form of reasoning in Fig. I. is admittedly valid, one at least of the two premisses which necessitated this conclusion must be false; for, if both were true, then, by the Principle of Identity, the conclusion 'All S's are M's' would also be true. But of these two premisses, 'All P's are M's,' being our original major premiss, is, ex hypothesi, true. Hence the assumed statement 'All S's are P's' must be false, and its contradictory, 'Some S's are-not P's,' must be true. But this is the conclusion of our original syllogism in Baroco. Thus we see that in Baroco, granted the truth of the premisses, the truth of the conclusion necessarily follows. We have thus shown that Baroco is a valid form of Syllogism. Any other of the valid forms may be justified in a precisely similar way. The Method of Proof assumes a peculiar form whenever the premisses from which the demonstrandum is deducible are reversible or simply convertible. In this case these requisite premisses may be discovered by means of a process of regressive analysis based upon the assumption that the demonstrandum is true. We may show that, on this assumption, certain consequences necessarily follow which are already known to be true. We may then reason back, in Sorites form, from these known truths to the demonstrandum. Thus, let D be the demonstrandum. The regressive analysis will then take some such form as this : If D is true, C is true. Now, if the premisses are of such a character as permits us to argue from the truth of the consequent to that of the antecedent-i.e., if the premisses of the sorites are reversible we can present the proof as follows, in the form of a series of hypothetical syllogisms in the Modus Ponens : A is true. But if A is true, B is true. But if B is true, C is true. But if C is true, D is true. C. INDUCTION AND 'INDUCTIVE INFERENCE.' (1) The Meaning of Induction.* It is essential to note the wide sense in which we are proposing to use the term 'Induction.' It is not unusual to identify Induction with the first stage in the whole process of Scientific Explanation, the stage which starts with the observation of facts and terminates in the formulation of some hypothesis. Were this nomenclature adopted, Induction, Deduction, Verification would be the three successive stages in a complete scientific explanation. But if we do adopt this nomenclature, we must cease to talk of Inductive Logic, and must speak, instead, of the Logic of Scientific Explanation. For it is Scientific Explanation, qua completed process, which alone is governed by the fundamental principle of Fidelity to Relevant Fact. So far as the goal of a reasoning-process is the mere formulation of an undeveloped, unverified hypothesis, it is surely unreasonable to contend that it aims at not transgressing the evidence of fact. Does it not embody a tendency to go beyond the facts rather than not to go beyond them ? Assuming, then, that we accept 'Fidelity to Relevant Fact' as the Inductive Principle, the use of the word Induction in its narrower sense of a tentative passage from particulars to universals is, strictly speaking, illegitimate. We cannot be faithful to fact by reposing on untested generalizations from experience and dispensing with the test of Verification. In the wider and legitimate sense of the term, 'Induction' covers the whole process of Scientific Explanation, the formulation and verification of Hypothesis, a process which ends only when, through its various methods, it has made sure that its tentative explanation does not go beyond the evidence of the facts. (2) The So-called 'Inductive Inference.' No account of Induction, however introductory in character, can dispense with an allusion to the much-abused term 'Inductive Inference.' We hold, for our part, to the simple conviction that there is only one fundamental type of logical Inference-that, namely, which consists in rendering explicit what is implied in a system of given premisses, through the sole help of the principle of logical Validity. We consequently view the term 'Inductive Inference' as a misnomer. Inference may be Formal or Deductive; may be drawn, that is, from Formal premisses in the light of a strictly abstract validity-interest, or else from material grounds in the light of a genuine truth-interest. But in either case it is a strictly logical process governed exclusively by the Law of Identity and the Law of Non-Contradiction. Now, in inductive procedure, the only inferential stage of this kind is that of the deductive development and application of Hypothesis. Hence 'Inductive Inference' is either a misnomer for Deductive Inference, or it is the name for a type of thinking which is not exclusively governed by the Law of Logical Validity-that is, by the above-named Laws of Thought. * See also footnote, p. 316. It is in this latter sense, of a heuristic, tentative suggestion or supposition, that the term is customarily used. Thus, having observed that a large number of instances of a certain class have the mark x, we are said to infer (inductively) that all the instances of that class may be found to possess the mark x. So again, in the case of what is known as 'Analogical Inference,' there is a precisely similar use of the word 'Inference' in the sense of a tentative, though it may be a well-grounded, suggestion. We are accustomed to say that, since A resembles B in many important respects, we infer (by Analogy) that it will resemble it in some further respect also. The meaning we have given to the term 'Inference' prevents us, however, from making use of it to designate any tentative form of argument, whether enumerative, analogical, or of any other kind. We therefore, somewhat reluctantly, renounce the use of the familiar and time-honoured term 'Inductive Inference,' and with it the use of such cognate expressions as 'probable' and 'analogical' inference. The Theory of Induction, as we conceive it, is Induction without Inductive Inference. As a general substitute for 'inference' in this sense, we propose to use the term 'conjecture.' Thus, on the ground that certain S's are P's, we conjecture that all S's will be found to be P's; and, on the basis of the many important resemblances between the Earth and Mars, we conjecture that Mars will resemble the Earth in being inhabited also. In this way we hope to avoid confusion, though we cannot hope either to satisfy the ear or to uproot the inbred prejudice in favour of drawing 'inferences' from grounds which do not imply but only suggest them.* * It may be worth while, at this point, to draw attention to the ambiguity attaching to the present use of the more natural term 'Inference.' As currently used, it denotes now a Formal inference, now a deductive inference, now a deduction, now a complete induction, now some form of tentative guess-work culminating in a hypothesis. It also denotes now a process, now a product. Thus, the conclusion of a syllogism is frequently spoken of as an inference from the premisses, whilst the process of disimplicating the conclusion from the premisses is also referred to as an inference. Our own use of the term 'Inference is intended to refer exclusively to processes of strictly valid reasoning-that is, reasoning in which the conclusion follows from the premiss or premisses with logical necessity. CHAPTER XXXVIII. XI. (ii.) HYPOTHESIS. J. S. MILL defines a Hypothesis as follows: 'An hypothesis is any supposition which we make (either without actual evidence, or on evidence avowedly insufficient) in order to endeavour to deduce from it conclusions in accordance with facts which are known to be real.'* Not every supposition, therefore, can rank as a Hypothesis. A Hypothesis is a supposition made in view of a truth-interest. It is a supposition which (1) admits of being developed into its consequences, and (2) requires and admits of verification.↑ What Mill thus defines is, in fact, the legitimate Scientific Нуроthesis. To be legitimate a Hypothesis has one essential condition to satisfy: it must be verifiable. But to be verifiable it must be adequately developable. A legitimate hypothesis, again, is identical with a working hypothesis in the widest sense of that term. For a working hypothesis (e.g., Electricity behaves as though it were a fluid'; 'Vegetable mould is due to the action of earthworms') is a hypothesis that works-works, that is, by attempting to explain the facts. A successful working hypothesis is a hypothesis that works well. To work well, a hypothesis must be both resourceful and fruitful. To be resourceful, it must be rooted directly (or indirectly through the medium of a general working idea) in a reasoned system or science. To be fruitful, it must be capable of continually extending its sphere of verification, and of bringing more and more facts under scientific control. A working hypothesis is, as a rule, closely allied with what we may suitably call a working idea. This is the germinal conception out of which the true working hypothesis is shaped. The working hypothesis is developed out of the working idea not by being deduced from it with logical necessity (the 'idea' would in that case be only a more fundamental working hypothesis), but by processes of imaginative construction of a purely tentative kind. The working idea stands to the working hypothesis thus developed from it in a relation somewhat analogous to that in which the subject of discourse stands to the particular proposition through which it is at any moment being developed.§ It stands for the relatively * 'A System of Logic, Book III., ch. xiv., § 4. † The term 'Verification' is, in its current use, ambiguous. Ordinarily, as here. it means a 'test'-a test that may result in disproving the hypothesis. In its stricter use, 'Verification' is the process which pro tanto confirms the truth of a hypothesis. Only some hypotheses would, in this sense of the term, admit of being verified. Similar remarks apply to the use of the term 'verifiable.' + For a more radical criterion of the legitimacy of a hypothesis, see the chapter on the Inductive Postulate. § Cf. pp. 118, 119. |