§1.2.11#2Πάλιν ὑποκείσθω τὸ τινὶ τῷ Β ὑπάρχειν, εἰλήφθω δὲ τὸ Γ παντὶ τῷ Α. ἀνάγκη οὖν τὸ Γ τινὶ τῷ Β ὑπάρχειν.
Again, let it be assumed that [A] belongs to some B, and let Γ be taken to belong to all A. It is necessary, therefore, that Γ belongs to some B.
τοῦτο δʼ ἔστω ἀδύνατον, ὥστε ψεῦδος τὸ ὑποτεθέν.
And let this be impossible, so that the assumption is false.
εἰ δʼ οὕτως, ἀληθὲς τὸ μηδενὶ ὑπάρχειν.
If so, it is true that [A] belongs to no [B].
ὁμοίως δὲ καὶ εἰ στερητικὸν ἐλήφθη τὸ Γ Α. εἰ δʼ ἡ πρὸς τῷ Β εἴληπται πρότασις, οὐκ ἔσται συλλογισμός.
And similarly if Γ-A was taken as negative. But if the premise in relation to B has been taken, there will be no syllogism.
ἐὰν δὲ τὸ ἐναντίον ὑποτεθῇ, συλλογισμὸς μὲν ἔσται καὶ τὸ ἀδύνατον, οὐ δείκνυται δὲ τὸ προτεθέν.
But if the contrary is assumed, there will be a syllogism and the impossibility, but what was proposed is not demonstrated.
ὑποκείσθω γὰρ παντὶ τῷ Β τὸ ὑπάρχειν, καὶ τὸ Γ τῷ Α εἰλήφθω παντί.
For let it be assumed that [A] belongs to all B, and let Γ be taken to belong to all A.
οὐκοῦν ἀνάγκη τὸ Γ παντὶ τῷ Β ὑπάρχειν.
It is necessary, then, that Γ belongs to all B.
τοῦτο δʼ ἀδύνατον, ὥστε ψεῦδος τὸ παντὶ τῷ Β τὸ ὑπάρχειν.
But this is impossible, so that it is false that [A] belongs to all B.
ἀλλʼ οὔπω γε ἀναγκαῖον, εἰ μὴ παντί, μηδενὶ ὑπάρχειν.
But it is not yet necessary, if it does not belong to all, that it belongs to none.
ὁμοίως δὲ καὶ εἰ πρὸς τῷ Β ληφθείη ἡ ἑτέρα πρότασις· συλλογισμὸς μὲν γὰρ ἔσται καὶ τὸ ἀδύνατον, οὐκ ἀναιρεῖται δʼ ἡ ὑπόθεσις· ὥστε τὸ ἀντικείμενον ὑποθετέον.
And similarly also if the other premise is taken in relation to B; for there will be a syllogism and the impossibility, but the assumption is not destroyed; so that the contradictory must be assumed.
Πρὸς δὲ τὸ μὴ παντὶ δείξαι ὑπάρχον τῷ Β τὸ Α, ὑποθετέον παντὶ ὑπάρχειν·
In order to show that A does not belong to all B, we must assume that it belongs to all.
εἰ γὰρ τὸ Α παντὶ τῷ Β καὶ τὸ παντὶ τῷ Α, τὸ Γ παντὶ τῷ Β, ὥστ’ εἰ τοῦτο ἀδύνατον, ψεῦδος τὸ ὑποτεθέν.
For if A belongs to all B, and [Γ] to all A, Γ belongs to all B, so that if this is impossible, the assumption is false.
ὁμοίως δὲ καὶ εἰ πρὸς τῷ Β ἐλήφθη ἡ ἑτέρα πρότασις.
And similarly if the other premise was taken in relation to B.
καὶ εἰ στερητικὸν ἦν τὸ Γ Α, ὡσαύτως· καὶ γὰρ οὕτω γίνεται συλλογισμός.
And if Γ-A was negative, likewise; for in this way too a syllogism is produced.
ἐὰν δὲ πρὸς τῷ Β ᾖ τὸ στερητικόν, οὐδὲν δείκνυται.
But if the negative is in relation to B, nothing is demonstrated.
ἐὰν δὲ μὴ παντὶ ἀλλὰ τινὶ ὑπάρχειν ὑποτεθῇ, οὐ δείκνυται ὅτι οὐ παντὶ ἀλλʼ ὅτι οὐδενί.
But if it is assumed not to belong to all, but to belong to some, it is not demonstrated that it does not belong to all, but that it belongs to none.
εἰ γὰρ τὸ Α τινὶ τῷ Β, τὸ δὲ Γ παντὶ τῷ Α, τινὶ τῷ Β τὸ Γ ὑπάρξει.
For if A belongs to some B, and Γ to all A, Γ will belong to some B.
εἰ οὖν τοῦτʼ ἀδύνατον, ψεῦδος τὸ τινὶ ὑπάρχειν τῷ Β τὸ Α, ὥστ’ ἀληθὲς τὸ μηδενί.
If, then, this is impossible, it is false that A belongs to some B, so that it is true that it belongs to none.
τούτου δὲ δειχθέντος προσαναιρεῖται τὸ ἀληθές· τὸ γὰρ Α τῷ Β τινὶ μὲν ὑπῆρχε, τινὶ δʼ οὐχ ὑπῆρχεν.
But when this is demonstrated, the truth is additionally destroyed; for A did belong to some B, and did not belong to some.
ἔτι οὐδὲν παρὰ τὴν ὑπόθεσιν συμβαίνει ἀδύνατον· ψεῦδος γὰρ ἂν εἴη, εἴπερ ἐξ ἀληθῶν μὴ ἔστι ψεῦδος συλλογίσασθαι· νῦν δʼ ἐστὶν ἀληθές·
Moreover, nothing impossible happens because of the assumption; for [the conclusion] would be false, if indeed it is impossible to syllogize what is false from what is true; but now it is true; for belonging to some B is true.
ὑπάρχει γὰρ τὸ τινὶ τῷ Β. ὥστʼ οὐχ ὑποθετέον τινὶ ὑπάρχειν, ἀλλὰ παντί.
So we must not assume that it belongs to some, but to all.
ὁμοίως δὲ καὶ εἰ τινὶ μὴ ὑπάρχον· τῷ Β τὸ Α δεικνύοιμεν· εἰ γὰρ ταὐτὸ τὸ τινὶ μὴ ὑπάρχειν καὶ μὴ παντὶ ὑπάρχειν, ἡ αὐτὴ ἀμφοῖν ἀπόδειξις.
And similarly also if we were to demonstrate that A does not belong to some B; for if "not belonging to some" and "not belonging to all" are the same, the demonstration for both is the same.
Φανερὸν οὖν ὅτι οὐ τὸ ἐναντίον ἀλλὰ τὸ ἀντικείμενον ὑποθετέον ἐν ἅπασι τοῖς συλλογισμοῖς.
It is clear, therefore, that in all syllogisms not the contrary but the contradictory must be assumed.
οὕτω γὰρ τό τε ἀναγκαῖον ἔσται καὶ τὸ ἀξίωμα ἔνδοξον.
For in this way there will be necessity, and the axiom will be acceptable.
εἰ γὰρ κατὰ παντὸς ἡ' φάσις ἢ ἡ ἀπόφασις, δειχθέντος ὅτι οὐχ ἡ ἀπόφασις, ἀνάγκη τὴν κατάφασιν ἀληθεύεσθαι.
For if of everything either the assertion or the negation is true, when it is shown that the negation is not true, the assertion must be true.
πάλιν εἰ μὴ τίθησιν ἀληθεύεσθαι τὴν κατάφασιν, ἔνδοξον τὸ ἀξιῶσαι τὴν ἀπόφασιν.
Again, if one does not admit the assertion to be true, it is acceptable to require the negation.
τὸ δʼ ἐναντίον οὐδετέρως ἁρμόττει ἀξιοῦν· οὔτε γὰρ ἀναγκαῖον, εἰ τὸ μηδενὶ ψεῦδος, τὸ παντὶ ἀληθές, οὔτʼ ἔνδοξον ὡς εἰ θάτερον ψεῦδος, ὅτι θάτερον ἀληθές.
But to require the contrary is appropriate in neither way; for neither is it necessary, if "to none" is false, that "to all" is true, nor is it acceptable that if the one is false, the other is true.