Aristotle · Prior Analytics §1.2.7

Conditions and Limits of Circular Proof in the Third Figure

Passage 58 of 123 · Greek

Summary

Discusses the possibility of circular proof in the third figure. If both premises are universal, circular proof is impossible because the conclusion is always particular. However, if one premise is universal and the other particular, the conditions under which circular proof becomes possible are demonstrated, depending on the arrangement of premises or the introduction of specific additional assumptions.

§1.2.7Ἐπὶ δὲ τοῦ τρίτου σχήματος ὅταν μὲν ἀμφότεραι αἰ προτάσεις καθόλου ληφθῶσιν, οὐκ ἐνδέχεται δεῖξαι διʼ ἀλλήλων· τὸ μὲν γὰρ καθόλου δείκνυται διὰ τῶν καθόλου, τὸ a δʼ ἐν τούτῳ συμπέρασμα ἀεὶ κατὰ μέρος, ὥστε φανερόν ὅτι ὅλως οὐκ ἐνδέχεται δεῖξαι διὰ τούτου τοῦ σχήματος τὴν καθόλου πρότασιν.

In the third figure, when both premises are assumed to be universal, it is not possible to prove them through each other; for the universal is proved through universals, but the a conclusion in this [figure] is always particular, so that it is clear that it is not possible at all to prove the universal premise through this figure.

Ἐὰν δʼ ἡ μὲν ἦ καθόλου ἡ δʼ ἐν μέρει. ποτὲ μὲν ἔσται ποτὲ δʼ οὐκ ἔσται.

But if one is universal and the other particular, sometimes it will be possible and sometimes it will not be.

ὅταν μὲν οὖν ἀμφότεραι κατηγορικαὶ ληφθῶσι καὶ τὸ καθόλου γένηται πρὸς τῷ ἐλάττονι ἄκρῳ, ἔσται, ὅταν δὲ πρὸς θατέρῳ, οὐκ ἔσται.

Therefore, when both are assumed to be affirmative and the universal is placed with the minor term, it will be possible, but when with the other, it will not be.

ὑπαρχέτω γὰρ τὸ Α παντὶ τῷ Γ, τὸ δὲ Β τινί· συμπέρασμα τὸ Α Β. ἐὰν οὖν ληφθῇ τὸ Γ παντὶ τῷ ὑπάρχειν, τὸ μὲν Γ δέδεικται τινὶ τῷ Β ὑπάρχον, τὸ δὲ Β τινὶ τῷ Γ οὐ δέδεικται.

For let A belong to every C, and B to some C; the conclusion is that A belongs to some B. If, then, C is assumed to belong to every B, C is proved to belong to some B, but B is not proved to belong to some C.

καίτοι ἀνάγκη, εἰ τὸ Γ τινὶ τῷ Β, καὶ τὸ Β τινὶ τῷ Γ ὑπάρχειν.

And yet it is necessary, if C belongs to some B, that B also belongs to some C.

ἀλλʼ οὐ ταὐτόν ἐστι τόδε τῶδε καὶ τόδε τῷδε ὑπάρχειν· ἀλλὰ προσληπτέον, εἰ τόδε τινὶ τῷδε, καὶ θάτερον τινὶ τῷδε.

But it is not the same for this to belong to that and that to this; rather, we must additionally assume that if this belongs to some of that, the other also belongs to some of this.

τούτου δὲ ληφθέντος οὐκέτι γίνεται ἐκ τοῦ συμπεράσματος καὶ τῆς ἑτέρας προτάσεως ὁ συλλογισμός.

But if this is assumed, the syllogism no longer is produced from the conclusion and the other premise.

εἰ δὲ τὸ Β παντὶ τῷ Γ, τὸ δὲ Α τινὶ τῷ Γ, ἔσται δεῖξαι τὸ Α Γ, ὅταν ληφθῇ τὸ μὲν Γ παντὶ τῷ Β ὑπάρχειν τὸ δὲ Α τινί.

But if B belongs to every C, and A to some C, it will be possible to prove the AC [premise] (i.e., that A belongs to some C), when C is assumed to belong to every B, and A to some [B].

εἰ γὰρ τὸ Γ παντὶ τῷ Β, τὸ δὲ Α τινὶ τῷ Β, ἀνάγκη τὸ Α τινὶ τῷ Γ ὑπάρχειν· μέσον τὸ Β. καὶ ὅταν ἢ ἡ μὲν κατηγορικὴ ἡ δὲ στερητική, καθόλου δʼ ἡ κατηγορική, δειχθήσεται ἡ ἑτέρα.

For if C belongs to every B, and A to some B, it is necessary for A to belong to some C; the middle term is B. And when one is affirmative and the other negative, and the affirmative is universal, the other premise will be proved.

ὑπαρχέτω γὰρ τὸ Β παντὶ τῷ Γ, τὸ δὲ τινὶ μὴ ὑπαρχέτω· συμπέρασμα ὅτι τὸ τινὶ τῷ Β οὐχ ὑπάρχει.

For let B belong to every C, and A not belong to some C; the conclusion is that A does not belong to some B.

ἐὰν ὖν προσληφθῇ τὸ Γ παντὶ τῷ Β ὑπάρχειν, ἀνάγκη τὸ Α τινὶ τῷ Γ μὴ ὑπάρχειν· μέσον τὸ Β. ὅταν δʼ ἡ στερητικὴ καθόλου γένηται, οὐ δείκνυται ἡ ἑτέρα, εἰ μὴ ὥσπερ ἐπὶ τῶν πρότερον, ἐὰν ληφθῇ, ᾧ τοῦτο τινὶ μὴ ὑπάρχει, θάτερον τινὶ ὑπάρχειν, οἷον εἰ τὸ μὲν Α μηδενὶ τῷ Γ, τὸ δὲ Β τινί· συμπέρασμα ὅτι τὸ τινὶ τῷ Β οὐχ ὑπάρχει.

If, then, C is additionally assumed to belong to every B, it is necessary for A not to belong to some C; the middle term is B. But when the negative premise is universal, the other premise is not proved, unless, as in the previous cases, it is assumed that to whatever this does not belong to some, the other belongs to some; for example, if A belongs to no C, and B to some C; the conclusion is that A does not belong to some B.

ἐὰν οὖν ληφθῇ, ᾧ τὸ Α τινὶ μὴ ὑπάρχει, τὸ Γ τινὶ ὑπάρχειν, ἀνάγκη τὸ Γ τινὶ τῷ Β ὑπάρχειν.

If, then, it is assumed that to whatever A does not belong to some, C belongs to some, it is necessary for C to belong to some B.

ἄλλως δʼ οὐκ ἔστιν ἀντιστρέφοντα τὴν καθόλου πρότασιν δεῖξαι τὴν ἑτέραν· οὐδαμῶς γὰρ ἔσται συλλογισμός.

But otherwise it is not possible to prove the other premise by converting the universal premise; for in no way will there be a syllogism.

Notes

59aτὸ δʼ ἐν τούτῳ συμπέρασμα ἀεὶ κατὰ μέρος — This refers to the logical property of the third figure (τὸ τρίτον σχῆμα), in which a universal conclusion cannot be derived from the premises; the conclusion is always particular (κατὰ μέρος). Consequently, circular proof (δεῖξις δι' ἀλλήλων) to prove a universal premise from a particular conclusion and another premise is inherently impossible in this figure.
10ἀλλʼ οὐ ταὐτόν ἐστι τόδε τῶδε καὶ τόδε τῷδε ὑπάρχειν — It states that the conversion (ἀντιστροφή) between "this (B) belonging to that (Γ)" and "that (Γ) belonging to this (B)" is not logically identical. While the particular affirmative "B belongs to some Γ" can be derived from the universal affirmative premise "Γ belongs to every B," the reverse (from particular to universal) does not generally hold, making an additional assumption necessary for circular proof.
25ἐὰν ληφθῇ, ᾧ τοῦτο τινὶ μὴ ὑπάρχει, θάτερον τινὶ ὑπάρχειν — The dative relative pronoun `ᾧ` includes its antecedent, meaning "to whatever." Here, it functions within the conditional `ἐάν` clause. Under the relationship between `τοῦτο` (this, i.e., A) and `θάτερον` (the other, i.e., Γ), it assumes that "to whatever A does not belong to some, Γ belongs to some." Only by introducing this complex assumption can an indirect syllogism be established.

Cite this passage

Aristotle, Prior Analytics §1.2.7. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg0086.tlg001.humanitext-grc2:1.2.7

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