In regard to the case here put forward as illustration, Aristotle has an observation which shows his anxiety to maintain the characteristic principles of the Syllogism; one of which principles he had declared to be--That nothing less than three terms and two propositions, could warrant the inferential step from premisses to conclusion. In the present case he assumed, If A exists, then B must exist; giving only one premiss as ground for the inference. This (he adds) does not contravene what has been laid down before; for A in the case before us represents two propositions conceived in conjunction.[6] Here he has given the type of hypothetical reasoning; not recognizing it as a variety _per se_, nor following it out into its different forms (as his successors did after him), but resolving it into the categorical syllogism.[7] He however conveys very clearly the cardinal principle of all hypothetical inference--That if the antecedent be true, the consequent must be true also, but not _vice versâ_; if the consequent be false, the antecedent must be false also, but not _vice versâ_. [Footnote 6: Analyt. Prior. II. ii. p. 53, b. 16-25. [Greek: to\ ou)=n A ô(/sper e(\n kei=tai, du/o prota/seis sullêphthei=sai.]] [Footnote 7: Aristotle, it should be remarked, uses the word [Greek: katêgoriko/s], not in the sense which it subsequently acquired, as the antithesis of [Greek: u(pothetiko/s] in application to the proposition and syllogism, but in the sense of affirmative as opposed to [Greek: sterêtiko/s].] 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. [Footnote 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] [Footnote 9: Ibid. II. v.-viii. p. 57, b. 18-p. 59, a. 35.]