Mr. Poste’s explanation of this difficult passage brings into view the original and valuable exposition made by Mr. John Stuart Mill of the Functions and Logical Value of the Syllogism. — System of Logic, Book II. ch. iii. sect 2:— �It must be granted, that in every syllogism, considered as an argument to prove the conclusion, there is a Petitio Principii,� &c.
Petitio Principii, if ranked among the Fallacies, can hardly be extended beyond the first of the five distinct varieties enumerated in the Topica, VIII. xiii.
[23] Analyt. Prior. II. xvi. p. 65, a. 23-27: τὸ γὰρ ἐξ ἀρχῆς τί δύναται, εἴρηται ἡμῖν, ὅτι τὸ δι’ αὑτοῦ δεικνύναι τὸ μὴ δι’ αὑτοῦ δῆλον. — τοῦτο δ’ ἔστι, τὸ μὴ δεικνύναι.
The meaning of some lines in this chapter (p. 65, a. 17-18) is to me very obscure, after all the explanations of commentators.
[24] Ibid. p. 65, a. 35; Topic. VIII. xiii. p. 162, b. 31.
We must be careful to note, that when Aristotle speaks of a principium as knowable in itself, or true in itself, he does not mean that it is innate, or that it starts up in the mind ready made without any gradual building up or preparation. What he means is, that it is not demonstrable deductively from anything else prior or more knowable by nature than itself. He declares (as we shall see) that principia are acquired, and mainly by Induction.
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]
[25] Analyt. Prior. II. xvii. p. 65, b. 4: ὅταν ἀναιρέθῃ τι δεικτικως διὰ τῶν Α, Β, Γ, &c.; xviii. 66, a. 17: ἢ γὰρ ἐκ τῶν δύο προτάσεων ἢ ἐκ πλειόνων πᾶς ἐστὶ συλλογισμός· εἰ μὲν οὖν ἐκ τῶν δύο, τούτων ἀνάγκη τὴν μὲν ἑτέραν ἢ καὶ ἀμφοτέρας εἶναι ψευδεῖς· &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.
[26] Ibid. II. xvii. p. 65, b. 38, b. 14, p. 66, a. 2, 7: τὸ μὴ παρὰ τοῦτο συμβαίνειν τὸ ψεῦδος — τοῦ μὴ παρὰ τὴν θέσιν εἶναι τὸ ψεῦδος — οὐ παρὰ τὴν θέσιν συμβαίνει τὸ ψεῦδος — οὐκ ἂν εἴη παρὰ τὴν θέσιν.
Instead of the preposition παρά, Aristotle on two occasions employs διά â€” οὕτω γὰρ ἔσται διὰ τὴν ὑπόθεσιν — p. 65, b. 33, p. 66, a. 3.