[19] Ibid. II. xvi. p. 64, b. 30-35: καὶ γὰρ εἰ ὅλως μὴ συλλογίζεται, καὶ εἰ δι’ ἀγνωστοτέρων ἢ ὁμοίως ἀγνώστων, καὶ εἰ διὰ τῶν ὑστέρων τὸ πρότερον· ἡ γὰρ ἀπόδειξις ἐκ πιστοτέρων τε καὶ προτέρων ἐστιν.… τὰ μὲν δι’ αὑτῶν πέφυκε γνωρίζεσθαι, τὰ δὲ δι’ ἄλλων.
Distinct from all these three, however, Aristotle singles out and dwells upon another mode of error, which he calls Petitio Principii. Some truths, the principia, are by nature knowable through or in themselves, others are knowable only through other things. If you confound this distinction, and ask or assume something of the latter class as if it belonged to the former, you commit a Petitio Principii. You may commit it either by assuming at once that which ought to be demonstrated, or by assuming, as if it were a principium, something else among those matters which in natural propriety would be demonstrated by means of a principium. Thus, there is (let us suppose) a natural propriety that C shall be demonstrated through A; but you, overlooking this, demonstrate B through C, and A through B. By thus inverting the legitimate order, you do what is tantamount to demonstrating A through itself; for your demonstration will not hold unless you assume A at the beginning, in order to arrive at C. This is a mistake made not unfrequently, and especially by some who define parallel lines; for they give a definition which cannot be understood unless parallel lines be presupposed.[20]
[20] Analyt. Prior. II. xvi. p. 64, b. 33-p. 65, a. 9. Petere principium is, in the phrase of Aristotle, not τὴν ἀρχὴν αἰτεῖσθαι, but τὸ ἐν ἀρχῇ αἰτεῖσθαι or τὸ ἐξ ἀρχῆς αἰτεῖσθαι (xvi. p. 64, b. 28, 34).
When the problem is such, that it is uncertain whether A can be predicated either of C or of B, if you then assume that A is predicable of B, you may perhaps not commit Petitio Principii, but you certainly fail in demonstrating the problem; for no demonstration will hold where the premiss is equally uncertain with the conclusion. But if, besides, the case be such, that B is identical with C, that is, either co-extensive and reciprocally convertible with C, or related to C as genus or species, — in either of these cases you commit Petitio Principii by assuming that A may be predicated of B.[21] For seeing that B reciprocates with C, you might just as well demonstrate that A is predicable of B, because it is predicable of C; that is, you might demonstrate the major premiss by means of the minor and the conclusion, as well as you can demonstrate the conclusion by means of the major and the minor premiss. If you cannot so demonstrate the major premiss, this is not because the structure of the syllogism forbids it, but because the predicate of the major premiss is more extensive than the subject thereof. If it be co-extensive and convertible with the subject, we shall have a circular proof of three propositions in which each may be alternately premiss and conclusion. The like will be the case, if the Petitio Principii is in the minor premiss and not in the major. In the First syllogistic figure it may be in either of the premisses; in the Second figure it can only be in the minor premiss, and that only in one mode (Camestres) of the figure.[22] The essence of Petitio Principii consists in this, that you exhibit as true per se that which is not really true per se.[23] You may commit this fault either in Demonstration, when you assume for true what is not really true, or in Dialectic, when you assume as probable and conformable to authoritative opinion what is not really so.[24]
[21] Ibid. p. 65, a. 1-10.
[22] Ibid. p. 65, a. 10: εἰ οὖν τις, ἀδήλου ὄντος ὅτι τὸ Ἀ ὑπάρχει τῷ Γ, ὁμοίως δὲ καὶ ὅτι τῷ Β, αἰτοῖτο τῷ Β ὑπάρχειν τὸ Ἀ, οὕπω δῆλον εἰ τὸ ἐν ἀρχῇ αἰτεῖται, ἀλλ’ ὅτι οὐκ ἀποδείκνυσι, δῆλον· οὐ γὰρ ἀρχὴ ἀποδείξεως τὸ ὁμοίως ἄδηλον. εἰ μέντοι τὸ Β πρὸς τὸ Γ οὕτως ἔχει ὥστε ταὐτὸν εἶναι, ἢ δῆλον ὅτι ἀντιστρέφουσιν, ἢ ὑπάρχει θάτερον θατέρῳ, τὸ ἐν ἀρχῇ αἰτεῖται. καὶ γὰρ ἄν, ὅτι τῷ Β τὸ Ἀ ὑπάρχει, δι’ ἐκείνων δεικνύοι, εἰ ἀντιστρέφοι. νῦν δὲ τοῦτο κωλύει, ἀλλ’ οὐχ ὁ τρόπος. εἰ δὲ τοῦτο ποιοῖ, τὸ εἰρημένον ἂν ποιοῖ καὶ ἀντιστρέφοι ὡς διὰ τριῶν.
This chapter, in which Aristotle declares the nature of Petitio Principii, is obscure and difficult to follow. It has been explained at some length, first by Philoponus in the Scholia (p. 192, a. 35, b. 24), afterwards by Julius Pacius (p. 376, whose explanation is followed by M. B. St. Hilaire, p. 288), and by Waitz, (I. p. 514). But the translation and comment given by Mr. Poste appear to me the best: “Assuming the conclusion to be affirmative, let us examine a syllogism in Barbara:—
All B is A.
All C is B.
∴ All C is A.
And let us first suppose that the major premiss is a Petitio Principii; i.e. that the proposition All B is A is identical with the proposition All C is A. This can only be because the terms B and C are identical. Next, let us suppose that the minor premiss is a Petitio Principii: i.e. that the proposition All C is B is identical with the proposition All C is A. This can only be because B and A are identical. The identity of the terms is, their convertibility or their sequence (ὑπάρχει, ἕπεται). This however requires some limitation; for as the major is always predicated (ὑπάρχει, ἕπεται) of the middle, and the middle of the minor, if this were enough to constitute Petitio Principii, every syllogism with a problematical premiss would be a Petitio Principii.â€� (See the Appendix A, pp. 178-183, attached to Mr. Poste’s edition of Aristotle’s Sophistici Elenchi.)
Compare, about Petitio Principii, Aristot. Topic. VIII. xiii. p. 162, b. 34, in which passage Aristotle gives to the fallacy called Petitio Principii a still larger sweep than what he assigns to it in the Analytica Priora. Mr. Poste’s remark is perfectly just, that according to the above passage in the Analytica, every syllogism with a problematical (i.e. real as opposed to verbal) premiss would be a Petitio Principii; that is, all real deductive reasoning, in the syllogistic form, would be a Petitio Principii. To this we may add, that, from the passage above referred to in the Topica, all inductive reasoning also (reasoning from parts to whole) would involve Petitio Principii.