§1.1.21Ἐὰν δὲ ἡ μὲν ὑπάρχειν ἡ δʼ ἐνδέχεσθαι σημαίνῃ τῶν προτάσεων, τὸ μὲν συμπέρασμα ἔσται ὅτι ἐνδέχεται καὶ οὐχ ὅτι ὑπάρχει, συλλογισμὸς δʼ ἔσται τὸν αὐτὸν τρόπον ἐχόντων τῶν ὅρων ὃν καὶ ἐν τοῖς πρότερον.
If one of the premises signifies belonging and the other being possible, the conclusion will be that it is possible and not that it belongs, and there will be a syllogism when the terms are related in the same way as in the previous cases.
ἔστωσαν γὰρ πρῶτον· κατηγορικοί, καὶ τὸ μὲν παντὶ τῷ Γ ὑπαρχέτω, τὸ δὲ Β παντὶ ἐνδεχέσθω ὑπάρχειν.
For let them first be affirmative, and let [A] belong to all Γ, and let B be possible to belong to all [Γ].
ἀντιστραφέντος οὖν τοῦ Β Γ τὸ πρῶτον ἔσται σχῆμα, καὶ τὸ συμπέρασμα ὅτι ἐνδέχεται τὸ τινὶ τῷ Β ὑπάρχειν· ὅτε γὰρ ἡ ἑτέρα τῶν προτάσεων ἐν τῷ πρώτῳ σχήματι σημαίνοι ἐνδέχεσθαι, καὶ τὸ συμπέρασμα ἦν ἐνδεχόμενον.
Then, when B-Γ is converted, the first figure will come about, and the conclusion will be that [A] is possible to belong to some B; for when one of the premises in the first figure signifies being possible, the conclusion also was possible.
ὁμοίως δὲ καὶ εἰ τὸ μὲν Γ ὑπάρχειν τὸ δὲ Α Γ ἐνδέχεσθαι, καὶ εἰ τὸ μὲν Α στερητικὸν τὸ δὲ Β Γ κατηγορικόν, ὑπάρχοι δʼ ὁποτερονοῦν, ἀμφοτέρως ἐνδεχόμενον ἔσται τὸ συμπέρασμα· γίνεται γὰρ πάλιν τὸ πρῶτον σχῆμα, δέδεικται δʼ ὅτι τῆς ἑτέρας προ· τάσεως ἐνδέχεσθαι σημαινούσης ἐν αὐτῷ καὶ τὸ συμπέρασμα ἔσται ἐνδεχόμενον.
And similarly also if [B] belongs to Γ and A-Γ is possible, and if [A-Γ] is privative and B-Γ is affirmative, whichever of the two is belonging, in both cases the conclusion will be possible; for again the first figure comes about, and it has been shown that when one of the premises in it signifies being possible, the conclusion also will be possible.
εἰ δὲ τὸ στερητικὸν τεθείη πρὸς τὸ ἔλαττον ἄκρον, ἢ καὶ ἄμφω ληφθείη στερητικά, διʼ αὐτῶν μὲν τῶν· κειμένων οὐκ ἔσται συλλογισμός, ἀντιστραφέντων δʼ ἔσται, καθάπερ ἐν τοῖς πρότερον.
But if the privative should be assumed in relation to the minor term, or if both should be assumed as privative, from the assumed premises themselves there will be no syllogism, but when they are converted there will be, just as in the previous cases.
Εἰ δʼ ἡ μὲν καθόλου τῶν προτάσεων ἡ δʼ ἐν μέρει, κατηγορικῶν μέν οὐσῶν ἀμφοτέρων, ἢ τῆς μὲν καθόλου στερητικῆς τῆς δʼ ἐν μέρει καταφατικῆς, ὁ αὐτὸς τρόπος ἔσται τῶν συλλογισμῶν· πάντες γὰρ περαίνονται διὰ τοῦ πρώτου σχήματος.
But if one of the premises is universal and the other particular, when both are affirmative, or when the universal is privative and the particular is affirmative, the method of the syllogisms will be the same; for they are all completed through the first figure.
ὥστε φανερὸν ὅτι τοῦ ἐνδέχεσθαι καὶ οὐ τοῦ ὑπάρχειν ἔσται ὁ συλλογισμός.
So it is clear that the syllogism will be of being possible and not of belonging.
εἰ δʼ ἡ μὲν καταφατικὴ καθόλου ἡ δὲ στερητικὴ ἐν μέρει, διὰ τοῦ ἀδυνάτου ἔσται ἡ ἀπόδειξις.
But if the affirmative is universal and the privative is particular, the proof will be through impossibility.
ὑπαρχέτω γὰρ τὸ μὲν Β παντὶ τῷ Γ, τὸ δὲ ἐνδεχέσθω τινὶ τῷ Γ μὴ ὑπάρχειν· ἀνάγκη δὴ τὸ Α ἐνδέχεσθαι τινὶ τῷ Β μὴ ὑπάρχειν.
For let B belong to all Γ, and let [A] be possible not to belong to some Γ; then it is necessary that A is possible not to belong to some B.
εἰ γὰρ παντὶ τῷ Β τὸ ὑπάρχει ἐξ ἀνάγκης, τὸ δὲ Β παντὶ τῷ Γ κεῖται ὑπάρχειν, τὸ Α παντὶ τῷ ἐξ ἀνάγκης ὑπάρξει· τοῦτο γὰρ δέδεικται πρότερον.
For if [A] belongs to all B of necessity, and B is assumed to belong to all Γ, A will belong to all [Γ] of necessity; for this has been shown before.
ἀλλʼ ὑπέκειτο τινὶ ἐνδέχεσθαι μὴ ὑπάρχειν.
But it was assumed that [A] is possible not to belong to some [Γ].
Ὅταν δʼ ἀδιόριστοι ἢ ἐν μέρει ληφθῶσιν ἀμφότεραι, οὐκ ἔσται συλλογισμός.
But when both are taken as indefinite or as particular, there will be no syllogism.
ἀπόδειξις δʼ ἡ αὐτὴ ἣ καὶ ἐν τοῖς πρότερον, καὶ διὰ τῶν αὐτῶν ὅρων.
The proof is the same as in the previous cases, and through the same terms.