[37] Analyt. Prior. I. viii. p. 29, a. 32; xiii. p. 32, a. 20-36: τὸ γὰρ ἀναγκαῖον ὁμωνύμως ἐνδέχεσθαι λέγομεν. In xiv. p. 33, b. 22, he excludes this equivocal meaning of τὸ ἐνδεχόμενον — δεῖ δὲ τὸ ἐνδέχεσθα λαμβάνειν μὴ ἐν τοῖς ἀναγκαίοις, ἀλλὰ κατὰ τὸν εἰρημένον διορισμόν. See xiii. p. 32, a. 33, where τὸ ἐνδέχεσθαι ὑπάρχειν is asserted to be equivalent to or convertible with τὸ ἐνδέχεσθαι μὴ ὑπάρχειν; and xix. p. 38, a. 35: τὸ ἐξ ἀνάγκης οὐκ ἦν ἐνδεχόμενον. Theophrastus and Eudemus differed from Aristotle about his theory of the Modals in several points (Scholia ad Analyt. Priora, pp. 161, b. 30; 162, b. 23; 166, a. 12, b. 15, Brand.). Respecting the want of clearness in Aristotle about τὸ ἐνδεχόμενον, see Waitz’s note ad p. 32, b. 16. Moreover, he sometimes uses ὑπάρχον in the widest sense, including ἐνδεχόμενον and ἀναγκαῖον, xxiii. p. 40, b. 24.
[38] Analyt. Prior. I. xv. p. 34, b. 7.
Having finished with the Modals, Aristotle proceeds to lay it down, that all demonstration must fall under one or other of the three figures just described; and therefore that all may be reduced ultimately to the two first modes of the First figure. You cannot proceed a step with two terms only and one proposition only. You must have two propositions including three terms; the middle term occupying the place assigned to it in one or other of the three figures.[39] This is obviously true when you demonstrate by direct or ostensive syllogism; and it is no less true when you proceed by Reductio ad Impossibile. This last is one mode of syllogizing from an hypothesis or assumption:[40] your conclusion being disputed, you prove it indirectly, by assuming its contradictory to be true, and constructing a new syllogism by means of that contradictory together with a second premiss admitted to be true; the conclusion of this new syllogism being a proposition obviously false or known beforehand to be false. Your demonstration must be conducted by a regular syllogism, as it is when you proceed directly and ostensively. The difference is, that the conclusion which you obtain is not that which you wish ultimately to arrive at, but something notoriously false. But as this false conclusion arises from your assumption or hypothesis that the contradictory of the conclusion originally disputed was true, you have indirectly made out your case that this contradictory must have been false, and therefore that the conclusion originally disputed was true. All this, however, has been demonstration by regular syllogism, but starting from an hypothesis assumed and admitted as one of the premisses.[41]
[39] Ibid. xxiii. p. 40, b. 20, p. 41, a. 4-20.
[40] Ibid. p. 40, b. 25: ἔτι ἢ δεικτικῶς ἢ ἐξ ὑποθέσεως· τοῦ δ’ ἐξ ὑποθέσεως μέρος τὸ διὰ τοῦ ἀδυνάτου.
[41] Ibid. p. 41, b. 23: πάντες γὰρ οἱ διὰ τοῦ ἀδυνάτου περαίνοντες τὸ μὲν ψεῦδος συλλογίζονται, τὸ δ’ ἐξ ἀρχῆς ἐξ ὑποθέσεως δεικνύουσιν, ὅταν ἀδύνατόν τι συμβαίνῃ τῆς ἀντιφάσεως τεθείσης.
It deserves to be remarked that Aristotle uses the phrase συλλογισμὸς ἐξ ὑποθέσεως, not συλλογισμὸς ὑποθετικός. This bears upon the question as to his views upon what subsequently received the title of hypothetical syllogisms; a subject to which I shall advert in a future [note].
Aristotle here again enforces what he had before urged — that in every valid syllogism, one premiss at least must be affirmative, and one premiss at least must be universal. If the conclusion be universal, both premisses must be so likewise; if it be particular, one of the premisses may not be universal. But without one universal premiss at least, there can be no syllogistic proof. If you have a thesis to support, you cannot assume (or ask to be conceded to you) that very thesis, without committing petitio principii, (i.e. quæsiti or probandi); you must assume (or ask to have conceded to you) some universal proposition containing it and more besides; under which universal you may bring the subject of your thesis as a minor, and thus the premisses necessary for supporting it will be completed. Aristotle illustrates this by giving a demonstration that the angles at the base of an isosceles triangle are equal; justifying every step in the reasoning by an appeal to some universal proposition.[42]
[42] Analyt. Prior. I. xxiv. p. 41, b. 6-31. The demonstration given (b. 13-22) is different from that which we read in Euclid, and is not easy to follow. It is more clearly explained by Waitz (p. 434) than either by Julius Pacius or by M. Barth. St. Hilaire (p. 108).
Again, every demonstration is effected by two propositions (an even number) and by three terms (an odd number); though the same proposition may perhaps be demonstrable by more than one pair of premisses, or through more than one middle term;[43] that is, by two or more distinct syllogisms. If there be more than three terms and two propositions, either the syllogism will no longer be one but several; or there must be particulars introduced for the purpose of obtaining an universal by induction; or something will be included, superfluous and not essential to the demonstration, perhaps for the purpose of concealing from the respondent the real inference meant.[44] In the case (afterwards called Sorites) where the ultimate conclusion is obtained through several mean terms in continuous series, the number of terms will always exceed by one the number of propositions; but the numbers may be odd or even, according to circumstances. As terms are added, the total of intermediate conclusions, if drawn out in form, will come to be far greater than that of the terms or propositions, multiplying as it will do in an increasing ratio to them.[45]