So also Aristot. Analyt. Poster. I. iv. p. 73, b. 32: τὸ καθόλου δὲ ὑπάρχει τότε, ὅταν ἐπὶ τοῦ τυχόντος καὶ πρώτου δεικνύηται.
Aristotle passes next to Affirmatives, both Universal and Particular. First, if A can be predicated of all B, then B can be predicated of some A; for if B cannot be predicated of any A, then (by the rule for the Universal Negative) neither can A be predicated of any B. Again, if A can be predicated of some B, in this case also, and for the same reason, B can be predicated of some A.[18] Here the rule for the Universal Negative, supposed already established, is applied legitimately to prove the rules for Affirmatives. But in the first case, that of the Universal, it fails to prove some in the sense of not-all or some-at-most, which is required; whereas, the rules for both cases can be proved by Induction, like the formula about the Universal Negative. When we come to the Particular Negative, Aristotle lays down the position, that it does not admit of being necessarily converted in any way. He gives no proof of this, beyond one single exemplification: If some animal is not a man, you are not thereby warranted in asserting the converse, that some man is not an animal.[19] It is plain that such an exemplification is only an appeal to Induction: you produce one particular example, which is entering on the track of Induction; and one example alone is sufficient to establish the negative of an universal proposition.[20] The converse of a Particular Negative is not in all cases true, though it may be true in many cases.
[18] Aristot. Analyt. Prior. I. ii. p. 25, a. 17-22.
[19] Ibid. p. 25, a. 22-26.
[20] Though some may fancy that the rule for converting the Universal Negative is intuitively known, yet every one must see that the rule for converting the Universal Affirmative is not thus self-evident, or derived from natural intuition. In fact, I believe that every learner at first hears it with great surprise. Some are apt to fancy that the Universal Affirmative (like the Particular Affirmative) may be converted simply. Indeed this error is not unfrequently committed in actual reasoning; all the more easily, because there is a class of cases (with subject and predicate co-extensive) where the converse of the Universal Affirmative is really true. Also, in the case of the Particular Negative, there are many true propositions in which the simple converse is true. A novice might incautiously generalize upon those instances, and conclude that both were convertible simply. Nor could you convince him of his error except by producing examples in which, when a true proposition of this kind is converted simply, the resulting converse is notoriously false. The appeal to various separate cases is the only basis on which we can rest for testing the correctness or incorrectness of all these maxims proclaimed as universal.
From one proposition taken singly, no new proposition can be inferred; for purposes of inference, two propositions at least are required.[21] This brings us to the rules of the Syllogism, where two propositions as premisses conduct us to a third which necessarily follows from them; and we are introduced to the well-known three Figures with their various Modes.[22] To form a valid Syllogism, there must be three terms and no more; the two, which appear as Subject and Predicate of the conclusion, are called the minor term (or minor extreme) and the major term (or major extreme) respectively; while the third or middle term must appear in each of the premisses, but not in the conclusion. These terms are called extremes and middle, from the position which they occupy in every perfect Syllogism — that is in what Aristotle ranks as the First among the three figures. In his way of enunciating the Syllogism, this middle position formed a conspicuous feature; whereas the modern arrangement disguises it, though the denomination middle term is still retained. Aristotle usually employs letters of the alphabet, which he was the first to select as abbreviations for exposition;[23] and he has two ways (conforming to what he had said in the first chapter of the present treatise) of enunciating the modes of the First figure. In one way, he begins with the major extreme (Predicate of the conclusion): A may be predicated of all B, B may be predicated of all C; therefore, A may be predicated of all C (Universal Affirmative). Again, A cannot be predicated of any B, B can be predicated of all C; therefore, A cannot be predicated of any C (Universal Negative). In the other way, he begins with the minor term (Subject of the conclusion): C is in the whole B, B is in the whole A; therefore, C is in the whole A (Universal Affirmative). And, C is in the whole B, B is not in the whole A; therefore, C is not in the whole A (Universal Negative). We see thus that in Aristotle’s way of enunciating the First figure, the middle term is really placed between the two extremes,[24] though this is not so in the Second and Third figures. In the modern way of enunciating these figures, the middle term is never placed between the two extremes; yet the denomination middle still remains.
[21] Analyt. Prior. I. xv. p. 34, a. 17; xxiii. p. 40, b. 35; Analyt. Poster. I. iii. p. 73, a. 7.
[22] Aristot. Analyt. Prior. I. iv. p. 25, b. 26, seq.
[23] M. Barthélemy St. Hilaire (Logique d’Aristote, vol. ii. p. 7, n.), referring to the examples of Conversion in chap. ii., observes:— “Voici le prémier usage des lettres représentant des idées; c’est un procédé tout à fait algébrique, c’est à dire, de généralisation. Déjà , dans l’Herméneia, ch. 13, § 1 et suiv., Aristote a fait usage de tableaux pour représenter sa pensée relativement à la consécution des modales. Il parle encore spécialement de figures explicatives, liv. 2. des Derniers Analytiques, ch. 17, § 7. Vingt passages de l’Histoire des Animaux attestent qu’il joignait des dessins à ses observations et à ses théories zoologiques. Les illustrations pittoresques datent donc de fort loin. L’emploi symbolique des lettres a été appliqué aussi par Aristote à la Physique. Il l’avait emprunté, sans doute, aux procédés des mathématiciens.�
We may remark, however, that when Aristotle proceeds to specify those combinations of propositions which do not give a valid conclusion, he is not satisfied with giving letters of the alphabet; he superadds special illustrative examples (Analyt. Prior. I. v. p. 27, a. 7, 12, 34, 38).