[57] Analyt. Prior. I. xxviii. p. 44, b. 25-37.
In the survey prescribed, nothing is gained by looking out for predicates (of A and E) which are different or opposite: we must collect such as are identical, since our purpose is to obtain from them a suitable middle term, which must be the same in both premisses. It is true that if the list B (containing the predicates universally belonging to A) and the list F (containing the predicates universally belonging to E) are incompatible or contrary to each other, you will arrive at a syllogism proving that no A can belong to E. But this syllogism will proceed, not so much from the fact that B and F are incompatible, as from the other fact, distinct though correlative, that B will to a certain extent coincide with H (the list of predicates which cannot belong to E). The middle term and the syllogism constituted thereby, is derived from the coincidence between B and H, not from the opposition between B and F. Those who derive it from the latter, overlook or disregard the real source, and adopt a point of view merely incidental and irrelevant.[58]
[58] Ibid. p. 44, b. 38-p. 45, a. 22. συμβαίνει δὴ τοῖς οὕτως ἐπισκοποῦσι προσεπιβλέπειν ἄλλην ὁδὸν τῆς ἀναγκαίας, διὰ τὸ λανθάνειν τὴν ταὐτότητα τῶν Β καὶ τῶν Θ.
The precept here delivered — That in order to obtain middle terms and good syllogisms, you must study and collect both the predicates and the subjects of the two terms of your thesis — Aristotle declares to be equally applicable to all demonstration, whether direct or by way of Reductio ad Impossibile. In both the process of demonstration is the same — involving two premisses, three terms, and one of the three a suitable middle term. The only difference is, that in the direct demonstration, both premisses are propounded as true, while in the Reductio ad Impossibile, one of the premisses is assumed as true though known to be false, and the conclusion also.[59] In the other cases of hypothetical syllogism your attention must be directed, not to the original quæsitum, but to the condition annexed thereto; yet the search for predicates, subjects, and a middle term, must be conducted in the same manner.[60] Sometimes, by the help of a condition extraneous to the premisses, you may demonstrate an universal from a particular: e.g., Suppose C (the list of subjects to which A belongs as predicate) and G (the list of subjects to which E belongs as predicate) to be identical; and suppose farther that the subjects in G are the only ones to which E belongs as predicate (this seems to be the extraneous or extra-syllogistic condition assumed, on which Aristotle’s argument turns); then, A will be applicable to all E. Or if D (the list of predicates which cannot belong to A) and G (the list of subjects to which E belongs as predicate) are identical; then, assuming the like extraneous condition, A will not be applicable to any E.[61] In both these cases, the conclusion is more universal than the premisses; but it is because we take in an hypothetical assumption, in addition to the premisses.
[59] Ibid. I. xxix. p. 45, a. 25-b. 15.
[60] Ibid. I. xxix. p. 45, b. 15-20. This paragraph is very obscure. Neither Alexander, nor Waitz, nor St. Hilaire clears it up completely. See Schol. pp. 178, b., 179, a. Brandis.
Aristotle concludes by saying that syllogisms from an hypothesis ought to be reviewed and classified into varieties — ἐπισκέψασθαι δὲ δεῖ καὶ διελεῖν ποσαχῶς οἱ ἐξ ὑποθέσεως (b. 20). But it is doubtful whether he himself ever executed this classification. It was done in the Analytica of his successor Theophrastus (Schol. p. 179, a. 6, 24). Compare the note of M. Barthélemy St. Hilaire, p. 140.
[61] Analyt. Prior. I. xxix. p. 45, b. 21-30.
Aristotle has now shown a method of procedure common to all investigations and proper for the solution of all problems, wherever soluble. He has shown, first, all the conditions and varieties of probative Syllogism, two premisses and three terms, with the place required for the middle term in each of the three figures; next, the quarter in which we are to look for all the materials necessary or suitable for constructing valid syllogisms. Having the two terms of the thesis given, we must study the predicates and subjects belonging to both, and must provide a large list of them; out of which list we must make selection according to the purpose of the moment. Our selection will be different, according as we wish to prove or to refute, and according as the conclusion that we wish to prove is an universal or a particular. The lesson here given will be most useful in teaching the reasoner to confine his attention to the sort of materials really promising, so that he may avoid wasting his time upon such as are irrelevant.[62]
[62] Ibid. b. 36-xxx. p. 46, a. 10.