[6] Analyt. Prior. II. ii. p. 53, b. 16-25. τὸ οὖν Ἀ ὥσπερ ἓν κεῖται, δύο προτάσεις συλληφθεῖσαι.
[7] Aristotle, it should be remarked, uses the word κατηγορικός, not in the sense which it subsequently acquired, as the antithesis of ὑποθετικός in application to the proposition and syllogism, but in the sense of affirmative as opposed to στερητικός.
Having laid down the principle, that the conclusion may be true, though one or both the premisses are false, Aristotle proceeds, at great length, to illustrate it in its application to each of the three syllogistic figures.[8] No portion of the Analytica is traced out more perspicuously than the exposition of this most important logical doctrine.
[8] Analyt. Prior. II. ii.-iv. p. 53, b. 26-p. 57, b. 17. At the close (p. 57, a. 36-b. 17), the general doctrine is summed up.
It is possible (he then continues, again at considerable length) to invert the syllogism and to demonstrate in a circle. That is, you may take the conclusion as premiss for a new syllogism, together with one of the old premisses, transposing its terms; and thus you may demonstrate the other premiss. You may do this successively, first with the major, to demonstrate the minor; next, with the minor, to demonstrate the major. Each of the premisses will thus in turn be made a demonstrated conclusion; and the circle will be complete. But this can be done perfectly only in Barbara, and when, besides, all the three terms of the syllogism reciprocate with each other, or are co-extensive in import; so that each of the two premisses admits of being simply converted. In all other cases, the process of circular demonstration, where possible at all, is more or less imperfect.[9]
[9] Ibid. II. v.-viii. p. 57, b. 18-p. 59, a. 35.
Having thus shown under what conditions the conclusion can be employed for the demonstration of the premisses, Aristotle proceeds to state by what transformation it can be employed for the refutation of them. This he calls converting the syllogism; a most inconvenient use of the term convert (ἀντιστρέφειν), since he had already assigned to that same term more than one other meaning, distinct and different, in logical procedure.[10] What it here means is reversing the conclusion, so as to exchange it either for its contrary, or for its contradictory; then employing this reversed proposition as a new premiss, along with one of the previous premisses, so as to disprove the other of the previous premisses — i.e. to prove its contrary or contradictory. The result will here be different, according to the manner in which the conclusion is reversed; according as you exchange it for its contrary or its contradictory. Suppose that the syllogism demonstrated is: A belongs to all B, B belongs to all C; Ergo, A belongs to all C (Barbara). Now, if we reverse this conclusion by taking its contrary, A belongs to no C, and if we combine this as a new premiss with the major of the former syllogism, A belongs to all B, we shall obtain as a conclusion B belongs to no C; which is the contrary of the minor, in the form Camestres. If, on the other hand, we reverse the conclusion by taking its contradictory, A does not belong to all C, and combine this with the same major, we shall have as conclusion, B does not belong to all C; which is the contradictory of the minor, and in the form Baroco: though in the one case as in the other the minor is disproved. The major is contradictorily disproved, whether it be the contrary or the contradictory of the conclusion that is taken along with the minor to form the new syllogism; but still the form varies from Felapton to Bocardo. Aristotle shows farther how the same process applies to the other modes of the First, and to the modes of the Second and Third figures.[11] The new syllogism, obtained by this process of reversal, is always in a different figure from the syllogism reversed. Thus syllogisms in the First figure are reversed by the Second and Third; those in the second, by the First and Third; those in the Third, by the First and Second.[12]
[10] Schol. (ad Analyt. Prior. p. 59, b. 1), p. 190, b. 20, Brandis. Compare the notes of M. Barthélemy St. Hilaire, pp. 55, 242.
[11] Analyt. Prior. II. viii.-x. p. 59, b. 1-p. 61, a. 4.
[12] Ibid. x. p. 61, a. 7-15.