§1.1.18Εἰ δʼ ἡ μὲν ὑπάρχειν ἡ δʼ ἐνδέχεσθαι σημαίνει, τῆς μὲν κατηγορικῆς ὑπάρχειν τεθείσης τῆς δὲ στερητικῆς ἐνδέχεσθαι οὐδέποτʼ ἔσται συλλογισμός, οὔτε καθόλου τῶν ὅρων οὔτʼ ἐν μέρει λαμβανομένων (ἀπόδειξις δʼ ἡ αὐτὴ καὶ διὰ τῶν αὐτῶν ὅρων)·
But if one of the premises signifies belonging and the other being possible, when the affirmative is assumed as belonging and the privative as possible, there will never be a syllogism, whether the terms are assumed as universal or as particular (and the proof is the same and through the same terms).
ὅταν δʼ ἡ μὲν καταφατικὴ ἐνδέχεσθαι ἡ δὲ στερητικὴ ὑπάρχειν, ἔσται συλλογισμός.
But when the affirmative signifies being possible and the privative belonging, there will be a syllogism.
εἰλήφθω γὰρ τὸ Α τῷ μὲν Β μηδενὶ ὑπάρχειν, τῷ δὲ Γ παντὶ ἐνδέχεσθαι.
For let A be assumed to belong to no B, and to be possible to belong to all Γ.
ἀντιστραφέντος οὖν τοῦ στερητικοῦ τὸ Β τῷ Α οὐδενὶ ὑπάρξει· τὸ δὲ παντὶ τῷ Γ ἐνεδέχετο· γίνεται δὴ συλλογισμός ὅτι ἐνδέχεται τὸ Β μηδενὶ τῷ Γ διὰ τοῦ πρώτου σχήματος.
Through the conversion of the privative premise, then, B will belong to no A; and A was possible to belong to all Γ; thus a syllogism comes about through the first figure that B is possible to belong to no Γ.
ὁμοίως δὲ καὶ εἰ πρὸς τῷ Γ τεθείη τὸ στερητικόν.
And similarly also if the privative is assumed in relation to Γ.
ἐὰν δʼ ἀμ μὲν ὦσι στερητικαί, σημαίνῃ δʼ ἡ μὲν μὴ ὑπάρχειν ἡ δʼ ἐνδέχεσθαι, διʼ αὐτῶν μὲν τῶν εἰλημμένων οὐδὲν συμβαίνει ἀναγκαῖον, ἀντιστραφείσης δὲ τῆς κατὰ τὸ ἐνδέχεσθαι προτάσεως γίγνεται συλλογισμος ὅτι τὸ Β τῷ Γ ἐνδέχεται μηδενὶ ὑπάρχειν, καθάπερ ἐν τοῖς πρότερον· ἔσται γὰρ πάλιν τὸ πρῶτον σχῆμα.
But if both premises are privative, and one signifies not belonging and the other being possible, from the assumed premises themselves no necessary consequence occurs, but when the premise in relation to the possible is converted, a syllogism comes about that B is possible to belong to no Γ, just as in the previous cases; for there will be the first figure again.
ἐὰν δʼ ἀμφότεραι τεθῶσι κατηγορικαί, οὐκ ἔσται συλλογισμός.
But if both are assumed as affirmative, there will be no syllogism.
ὅροι τοῦ μὲν ὑπάρχειν ὑγίεια—ζῷον—ἄνθρωπος, τοῦ δὲ μὴ ὑπάρχειν ὑγίεια—ἵππος—ἄνθρωτπος.
The terms for belonging are health—animal—human, and for not belonging, health—horse—human.
Τὸν αὐτὸν δὲ τρόπον ἕξει κἀπὶ τῶν ἐν μέρει συλλογισμῶν.
And it will be the same way also in the case of particular syllogisms.
ὅταν μὲν γὰρ ᾖ τὸ καταφατικὸν ὑπάρχον, εἴτε κα· θόλου εἴτʼ ἐν μέρει ληφθέν, οὐδεὶς ἔσται συλλογισμός (τοῦτο δʼ ὁμοίως καὶ διὰ τῶν αὐτῶν ὅρων δείκνυται τοῖς πρότερον), ὅταν δὲ τὸ στερητικόν, ἔσται διὰ τῆς ἀντιστροφῆς, καθάπερ ἐν τοῖς πρότερον.
For when the affirmative is belonging, whether assumed as universal or as particular, there will be no syllogism (and this is shown in the same way and through the same terms as in the previous cases); but when the privative is, there will be a syllogism through conversion, just as in the previous cases.
πάλιν ἐὰν ἄμφω μὲν τὰ διαστήματα στερητικὰ ληφθῇ, καθόλου δὲ τὸ μὴ ὑπάρχειν, ἐξ αὐτῶν μὲν τῶν προτάσεων οὐκ ἔσται τὸ ἀναγκαῖον, ἀντιστραφέντος δὲ τοῦ ἐνδέχεσθαι καθάπερ ἐν τοῖς πρότερον ἔσται συλλογισμός.
Again, if both intervals are assumed as privative, and the not belonging is universal, from the premises themselves there will be no necessary consequence, but when the possible is converted, just as in the previous cases, there will be a syllogism.
ἐὲν δὲ ὑπάρχον· μὲν ᾖ τὸ στερητικόν, ἐν μέρει δὲ ληφθῇ, οὐκ ἔσται συλλογισμός, οὔτε καταφατικῆς οὔτε στερητικῆς οὔσης τῆς ἑτέρας προτάσεως.
But if the privative is belonging and is assumed as particular, there will be no syllogism, whether the other premise is affirmative or privative.
οὐδʼ ὅταν ἀμφότεραι ληφθῶσιν ἀδιόριστοι—ἢ καταφατικαὶ ἢ ἀποφατικαί—ἢ κατὰ μέρος.
Nor when both are assumed as indefinite—either affirmative or negative—or as particular.
ἀπόδειξις δʼ ἡ αὐτὴ καὶ διὰ τῶν αὐτῶν ὅρων.
And the proof is the same and through the same terms.