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] [Footnote 39: Ibid. xxiii. p. 40, b. 20, p. 41, a. 4-20.] [Footnote 40: Ibid. p. 40, b. 25: [Greek: e)/ti ê)\ deiktikô=s ê)\ e)x u(pothe/seôs; tou= d' _e)x u(pothe/seôs_ me/ros to\ dia\ tou= a)duna/tou.]] [Footnote 41: Ibid. p. 41, b. 23: [Greek: pa/ntes ga\r oi( dia\ tou= a)duna/tou perai/nontes to\ me\n pseu=dos sullogi/zontai, to\ d' e)x a)rchê=s _e)x u(pothe/seôs_ deiknu/ousin, o(/tan a)du/nato/n ti sumbai/nê| tê=s a)ntipha/seôs tethei/sês.] It deserves to be remarked that Aristotle uses the phrase [Greek: sullogismo\s _e)x u(pothe/seôs_], not [Greek: sullogismo\s u(pothetiko/s]. 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.]