Next to _Petitio Principii_, Aristotle indicates another fallacious or erroneous procedure in dialectic debate; misconception or misstatement of the real grounds on which a conclusion rests--_Non per Hoc_. You may impugn the thesis (set up by the respondent) directly, by proving syllogistically its contrary or contradictory; or you may also impugn it indirectly by _Reductio ad Absurdum_; _i.e._ you prove by syllogism some absurd conclusion, which you contend to be necessarily true, if the thesis is admitted. Suppose you impugn it in the first method, or directly, by a syllogism containing only two premisses and a conclusion: _Non per Hoc_ is inapplicable here, for if either premiss is disallowed, the conclusion is unproved; the respondent cannot meet you except by questioning one or both of the premisses of your impugning syllogism.[25] But if you proceed by the second method or indirectly, _Non per Hoc_ may become applicable; for there may then be more than two premisses, and he may, while granting that the absurd conclusion is correctly made out, contend that the truth or falsehood of his thesis is noway implicated in it. He declares (in Aristotle's phrase) that the absurdity or falsehood just made out does not follow as a consequence from his thesis, but from other premisses independent thereof; that it would stand equally proved, even though his thesis were withdrawn.[26] In establishing the falsehood or absurdity you must take care that it shall be one implicated with or dependent upon his thesis. It is this last condition that he (the respondent) affirms to be wanting.[27] [Footnote 25: Analyt. Prior. II. xvii. p. 65, b. 4: [Greek: o(/tan a)naire/thê| ti deiktikôs dia\ tô=n A, B, G], &c.; xviii. 66, a. 17: [Greek: ê)\ ga\r e)k tô=n du/o prota/seôn ê)\ e)k pleio/nôn pa=s e)sti\ sullogismo/s; ei) me\n ou)=n e)k tô=n du/o, tou/tôn a)na/gkê tê\n me\n e(te/ran ê)\ kai\ a)mphote/ras ei)=nai pseudei=s;] &c. Whoever would understand this difficult chapter xvii., will do well to study it with the notes of Julius Pacius (p. 360), and also the valuable exposition of Mr. Poste, who has extracted and illustrated it in Appendix B. (p. 190) of the notes to his edition of the Sophistici Elenchi. The six illustrative diagrams given by Julius Pacius afford great help, though the two first of them appear to me incorrectly printed, as to the brackets connecting the different propositions.] [Footnote 26: Ibid. II. xvii. p. 65, b. 38, b. 14, p. 66, a. 2, 7: [Greek: to\ mê\ _para\ tou=to_ sumbai/nein to\ pseu=dos--tou= mê\ _para\ tê\n the/sin_ ei)=nai to\ pseu=dos--ou) _para\ tê\n the/sin_ sumbai/nei to\ pseu=dos--ou)k a)\n ei)/ê _para\ tê\n the/sin_.] Instead of the preposition [Greek: para/], Aristotle on two occasions employs [Greek: dia/--ou(/tô ga\r e)/stai _dia\ tê\n u(po/thesin_]--p. 65, b. 33, p. 66, a. 3.