Men, Horses, Mules, &c., are long-lived (major).
Men, Horses, Mules, &c., are bile-less (minor).
And, inasmuch as the subject of the minor proposition is co-extensive with the predicate (which, if quantified according to Hamilton’s phraseology, would be, All bile-less animals), so that the proposition admits of being converted simply, — the middle term will become the subject of the conclusion, All bileless animals are long-lived.
[54] Analyt. Prior. II. xxiii. p. 68, b. 27: δεῖ δὲ νοεῖν τὸ Γ τὸ ἐξ ἁπάντων τῶν καθ’ ἕκαστον συγκείμενον· ἡ γὰρ ἐπαγωγὴ διὰ πάντων.
[55] Analyt. Prior. II. xxiii. p. 68, p. 23: εἰ οὖν ἀντιστρέφει τὸ Γ τῷ Β, καὶ μὴ ὑπερτείνει τὸ μέσον, ἀνάγκη τὸ Α τῷ Β ὑπάρχειν.
Julius Pacius translates this: “Si igitur convertatur τὸ Γ cum B, nec medium excedat, necesse est τὸ Α τῷ Β inesse.â€� These Latin words include the same grammatical ambiguity as is found in the Greek original: medium, like τὸ μέσον, may be either an accusative case governed by excedat, or a nominative case preceding excedat. The same may be said of the other Latin translations, from Boethius downwards.
But M. Barthélemy St. Hilaire in his French translation, and Sir W. Hamilton in his English translation (Lectures on Logic, Vol. II. iv. p. 358, Appendix), steer clear of this ambiguity. The former says: “Si donc C est réciproque à B, et qu’il ne dépasse pas le moyen, il est nécessaire alors que A soit à B:â€� to the same purpose, Hamilton, l. c. These words are quite plain and unequivocal. Yet I do not think that they convey the meaning of Aristotle. In my judgment, Aristotle meant to say: “If then C reciprocates with B, and if the middle term (B) does not stretch beyond (the minor C), it is necessary that A should be predicable of B.â€� To show that this must be the meaning, we have only to reflect on what C and B respectively designate. It is assumed that C designates the sum of individual bile-less animals; and that B designates the class or class-term bile-less, that is, the totality thereof. Now the sum of individuals included in the minor (C) cannot upon any supposition overpass the totality: but it may very possibly fall short of totality; or (to state the same thing in other words) the totality may possibly surpass the sum of individuals under survey, but it cannot possibly fall short thereof. B is here the limit, and may possibly stretch beyond C; but cannot stretch beyond B. Hence I contend that the translations, both by M. Barthélemy St. Hilaire and Sir W. Hamilton, take the wrong side in the grammatical alternative admissible under the words καὶ μὴ ὑπερτείνει τὸ μέσον. The only doubt that could possibly arise in the case was, whether the aggregate of individuals designated by the minor did, or did not, reach up to the totality designated by the middle term; or (changing the phrase) whether the totality designated by the middle term did, or did not, stretch beyond the aggregate of individuals designated by the minor. Aristotle terminates this doubt by the words: “And if the middle term does not stretch beyond (the minor).â€� Of course the middle term does not stretch beyond, when the terms reciprocate; but when they do not reciprocate, the middle term must be the more extensive of the two; it can never be the less extensive of the two, since the aggregate of individuals cannot possibly exceed totality, though it may fall short thereof.
I have given in the text what I think the true meaning of Aristotle, departing from the translations of M. Barthélemy St. Hilaire and Sir W. Hamilton.
[56] Analyt. Prior. II. xxiii. p. 68, b. 30-38: ἔστι δ’ ὁ τοιοῦτος συλλογισμὸς τῆς πρώτης καὶ ἀμέσου προτάσεως· ὧν μὲν γάρ ἐστι μέσον, διὰ τοῦ μέσου ὁ συλλογισμός, ὧν δὲ μή ἐστι, δι’ ἐπαγωγῆς. — φύσει μὲν οὖν πρότερος καὶ γνωριμώτερος ὁ διὰ τοῦ μέσου συλλογισμός, ἡμῖν δ’ ἐναργέστερος ὁ διὰ τῆς ἐπαγωγῆς.
From Induction he proceeds to Example. You here take in (besides the three terms, major, middle, and minor, of the Syllogism) a fourth term; that is, a new particular case analogous to the minor. Your purpose here is to show — not, as in the ordinary Syllogism, that the major term is predicable of the minor, but, as in the Inductive Syllogism — that the major term is predicable of the middle term; and you prove this conclusion, not (as in the Inductive Syllogism) through the minor term, but through the new case or fourth term analogous to the minor.[57] Let A represent evil or mischievous; B, war against neighbours, generally; C, war of Athens against Thebes, an event to come and under deliberation; D, war of Thebes against Phokis, a past event of which the issue is known to have been signally mischievous. You assume as known, first, that A is predicable of D, i.e. that the war of Thebes against Phokis has been disastrous; next, that B is predicable both of C and of D, i.e. that each of the two wars, of Athens against Thebes, and of Thebes against Phokis, is a war of neighbours against neighbours, or a conterminous war. Now from the premiss that A is predicable of D, along with the premiss that B is predicable of D, you infer that A is predicable of the class B, or of conterminous wars generally; and hence you draw the farther inference, that A is also predicable of C, another particular case under the same class B. The inference here is, in the first instance, from part to whole; and finally, through that whole, from the one part to another part of the same whole. Induction includes in its major premiss all the particulars, declaring all of them to be severally subjects of the major as predicate; hence it infers as conclusion, that the major is also predicable of the middle or class-term comprising all these particulars, but comprising no others. Example includes not all, but only one or a few particulars; inferring from it or them, first, to the entire class, next, to some new analogous particular belonging to the class.[58]
[57] Ibid. II. xxiv. p. 68, b. 38: παραδεῖγμα δ’ ἐστὶν ὅταν τῷ μέσῳ τὸ ἄκρον ὑπάρχον δειχθῇ διὰ τοῦ ὁμοίου τῷ τρίτῳ.