Aristotle reminds us once more of what he had before said, that in the Second and Third figures, not all varieties of conclusion are possible, but only some varieties; accordingly, when we are reducing a piece of reasoning to the syllogistic form, the nature of the conclusion will inform us which of the three figures we must look for. In the case where the question debated relates to a definition, and the reasoning which we are trying to reduce turns upon one part only of that definition, we must take care to look for our three terms only in regard to that particular part, and not in regard to the whole definition.[84] All the modes of the Second and Third figures can be reduced to the First, by conversion of one or other of the premisses; except the fourth mode (Baroco) of the Second, and the fifth mode (Bocardo) of the Third, which can be proved only by Reductio ad Absurdum.[85]
[84] Analyt. Prior. I. xlii., xliii. p. 50, a. 5-15. I follow here the explanation given by Philoponus and Julius Pacius, which M. Barthélemy St. Hilaire adopts. But the illustrative example given by Aristotle himself (the definition of water) does not convey much instruction.
[85] Ibid. xlv. p. 50, b. 5-p. 51, b. 2.
No syllogisms from an Hypothesis, however, are reducible to any of the three figures; for they are not proved by syllogism alone: they require besides an extra-syllogistic assumption granted or understood between speaker and hearer. Suppose an hypothetical proposition given, with antecedent and consequent: you may perhaps prove or refute by syllogism either the antecedent separately, or the consequent separately, or both of them separately; but you cannot directly either prove or refute by syllogism the conjunction of the two asserted in the hypothetical. The speaker must ascertain beforehand that this will be granted to him; otherwise he cannot proceed.[86] The same is true about the procedure by Reductio ad Absurdum, which involves an hypothesis over and above the syllogism. In employing such Reductio ad Absurdum, you prove syllogistically a certain conclusion from certain premisses; but the conclusion is manifestly false; therefore, one at least of the premisses from which it follows must be false also. But if this reasoning is to have force, the hearer must know aliunde that the conclusion is false; your syllogism has not shown it to be false, but has shown it to be hypothetically true; and unless the hearer is prepared to grant the conclusion to be false, your purpose is not attained. Sometimes he will grant it without being expressly asked, when the falsity is glaring: e.g. you prove that the diagonal of a square is incommensurable with the side, because if it were taken as commensurable, an odd number might be shown to be equal to an even number. Few disputants will hesitate to grant that this conclusion is false, and therefore that its contradictory is true; yet this last (viz. that the contradictory is true) has not been proved syllogistically; you must assume it by hypothesis, or depend upon the hearer to grant it.[87]
[86] Ibid. xliv. p. 50, a. 16-28.
[87] Analyt. Prior. I. xliv. p. 50, a. 29-38. See above, xxiii. p. 40, a. 25.
M. Barthélemy St. Hilaire remarks in the note to his translation of the Analytica Priora (p. 178): “Ce chapitre suffit à prouver qu’Aristote a distingué très-nettement les syllogismes par l’absurde, des syllogismes hypothétiques. Cette dernière dénomination est tout à fait pour lui ce qu’elle est pour nous.� Of these two statements, I think the latter is more than we can venture to affirm, considering that the general survey of hypothetical syllogisms, which Aristotle intended to draw up, either never was really completed, or at least has perished: the former appears to me incorrect. Aristotle decidedly reckons the Reductio ad Impossibile among hypothetical proofs. But he understands by Reductio ad Impossibile something rather wider than what the moderns understand by it. It now means only, that you take the contradictory of the conclusion together with one of the premisses, and by means of these two demonstrate a conclusion contradictory or contrary to the other premiss. But Aristotle understood by it this, and something more besides, namely, whenever, by taking the contradictory of the conclusion, together with some other incontestable premiss, you demonstrate, by means of the two, some new conclusion notoriously false. What I here say, is illustrated by the very example which he gives in this chapter. The incommensurability of the diagonal (with the side of the square) is demonstrated by Reductio ad Impossibile; because if it be supposed commensurable, you may demonstrate that an odd number is equal to an even number; a conclusion which every one will declare to be inadmissible, but which is not the contradictory of either of the premisses whereby the true proposition was demonstrated.
Here Aristotle expressly reserves for separate treatment the general subject of Syllogisms from Hypothesis.[88]
[88] The expressions of Aristotle here are remarkable, Analyt. Prior. I. xliv. p. 50, a. 39-b. 3: πολλοὶ δὲ καὶ ἕτεροι περαίνονται ἐξ ὑποθέσεως, οὓς ἐπισκέψασθαι δεῖ καὶ διασημῆναι καθαρῶς. τίνες μὲν οὖν αἱ διαφοραὶ τούτων, καὶ ποσαχῶς γίνεται τὸ ἐξ ὑποθέσεως, ὕστερον ἐροῦμεν· νῦν δὲ τοσοῦντον ἡμῖν ἔστω φανερόν, ὅτι οὐκ ἔστιν ἀναλύειν εἰς τὰ σχήματα τοὺς τοιούτους συλλογισμούς. καὶ δι’ ἣν αἰτίαν, εἰρήκαμεν.
Syllogisms from Hypothesis were many and various, and Aristotle intended to treat them in a future treatise; but all that concerns the present treatise, in his opinion, is, to show that none of them can be reduced to the three Figures. Among the Syllogisms from Hypothesis, two varieties recognized by Aristotle (besides οἰ διὰ τοῦ ἀδυνάτου) were οἱ κατὰ μετάληψιν and οἱ κατὰ ποιότητα. The same proposition which Aristotle entitles κατὰ μετάληψιν, was afterwards designated by the Stoics κατὰ πρόσληψιν (Alexander ap. Schol. p. 178, b. 6-24).