§1.1.16#1Ὅταν δʼ ἡ μὲν ἐξ ἀνάγκης ὑπάρχειν ἡ δʼ ἐνδέχεσθαι σημαίνῃ τῶν προτάσεων, ὁ μὲν συλλογισμὸς ἔσται τὸν αὐτὸν τρόπον ἐχόντων τῶν ὅρων, καὶ τέλειος ὅταν πρὸς τῷ ἐλάττονι ἄκρῳ τεθῇ τὸ ἀναγκαῖον·
When one of the premises signifies belonging of necessity and the other being possible, a syllogism will come about when the terms are related in the same way, and it will be perfect when the necessary [premise] is placed in relation to the minor term.
τὸ δὲ συμπέρασμα κατηγορικῶν μὲν ὄντων τῶν ὅρων τοῦ ἐνδέχεσθαι καὶ οὐ τοῦ ὑπάρχειν ἔσται, καὶ καθόλου καὶ μὴ καθόλου τιθεμένων, ἐὰν δʼ ᾖ τὸ μὲν καταφατικὸν τὸ δὲ στερτητικόν, ὅταν μὲν ᾖ τὸ καταφατικὸν ἀναγκαῖον, τοῦ ἐνδέχεσθαι καὶ οὐ τοῦ μὴ ὑπάρχειν, ὅταν δὲ τὸ στερτητικόν, καὶ τοῦ ἐνδέχεσθαι μὴ ὑπάρχειν καὶ τοῦ μὴ ὑπάρχειν, καὶ καθόλου καὶ μὴ καθόλου τῶν ὅρων ὄντων·
And the conclusion, when the premises are affirmative, will be of being possible and not of belonging, whether they are assumed as universal or not universal. But if one is affirmative and the other privative, when the affirmative is necessary, [the conclusion] will be of being possible and not of not belonging; but when the privative is [necessary, the conclusion] will be both of being possible not to belong and of not belonging, whether the terms are universal or not universal.
τὸ δʼ ἐνδέχεσθαι ἐν τῷ συμπεράσματι τὸν αὐτὸν τρόπον ληπτέονὅνπερ καὶ ἐν τοῖς πρότερον.
And being possible in the conclusion must be assumed in the same way as in the previous cases.
τοῦ δʼ ἐξ ἀνάγκης μὴ ὑπάρχειν οὐκ ἔσται συλλογισμός· ἕτερον γὰρ τὸ μὴ ἐξ ἀνάγκης ὑπάρχειν καὶ τὸ ἐξ ἀνάγκης μὴ ὑπάρχειν.
But there will be no syllogism of not belonging of necessity; for to not belong of necessity is different from not belonging of necessity.
Ὅτι μὲν οὖν καταφατικῶν ὄντων τῶν ὅρων οὐ γίνεται τὸ συμπέρασμα ἀναγκαῖον, φανερόν.
It is clear, then, that when the premises are affirmative, the conclusion does not become necessary.
ὑπαρχέτω γὰρ τὸ μὲν Α παντὶ τῷ Β ἐξ ἀνάγκης, τὸ δὲ Β ἐνδεχέσθω παντὶ τῷ Γ. ἔσται δὴ συλλογισμὸς ἀτελὴς ὅτι ἐνδέχεται τὸ παντὶ τῷ Γ ὑπάρχειν.
For let A belong to all B of necessity, and let B be possible to belong to all Γ. There will indeed be an imperfect syllogism that it is possible to belong to all Γ.
ὅτι δʼ ἀτελής, ἐκ τῆς ἀποδείξεως δῆλον· τὸν αὐτὸν γὰρ τρόπον δειχθήσεται ὅνπερ κἀπὶ τῶν πρότερον.
And that it is imperfect is clear from the proof; for it will be demonstrated in the same way as in the previous cases.
πάλιν τὸ μὲν ἐνδεχέσθω παντὶ τῷ Β, τὸ δὲ Β παντὶ τῷ Γ ὑπαρχέτω ἐξ ἀνάγκης.
Again, let [A] be possible to belong to all B, and let B belong to all Γ of necessity.
ἔσται δὴ συλλογισμός ὅτι τὸ παντὶ τῷ Γ ἐνδέχεται ὑπάρχειν, ἀλλʼ οὐχ ὅτι ὑπάρχει, καὶ τέλειος, ἀλλʼ οὐκ ἀτελής· εὐθὺς γὰρ ἐπιτελεῖται διὰ τῶν ἐξ. ἀρχῆς προτάσεων.
There will indeed be a syllogism that [A] is possible to belong to all Γ, but not that it belongs, and it is perfect, not imperfect; for it is completed immediately through the premises from the beginning.
Εἰ δὲ μὴ ὁμοιοσχήμονες αἱ προτάσεις, ἔστω πρῶτον ἡ στερητικὴ ἀναγκαία, καὶ τὸ μὲν Α μηδενὶ ἐνδεχέσθω τῷ Β, τὸ δὲ Β παντὶ τῷ Γ ἐνδεχέσθω.
But if the premises are not of the same form, let first the privative be necessary, and let A be possible to belong to no B, and let B be possible to belong to all Γ.
ἀνάγκη δὴ τὸ Α μηδενὶ τῷ Γ ὑπάρχειν.
It is necessary indeed for A to belong to no Γ.
κείσθω γὰρ ὑπάρχειν ἢ παντὶ ἢ τινί· τῷ δὲ Β ὑπέκειτο μηδενὶ ἐνδέχεσθαι.
For let it be assumed to belong either to all or to some; but it was assumed to be possible to belong to no B.
ἐπεὶ οὖν ἀντιστρέφει τὸ στερητικόν, οὐδὲ τὸ Β τῷ Α οὐδενὶ ἐνδέχεται· τὸ δὲ τῶ Γ ἢ παντὶ ἢ τινὶ κεῖται ὑπάρχειν· ὥστ᾿ οὐδενὶ ἢ οὐ παντὶ τῷ Γ τὸ Β ἐνδέχοιτʼ ἂν ὑπάρχειν· ὑπέκειτο δὲ παντὶ ἐξ ἀρχῆς.
Since, then, the privative premise converts, B is also possible to belong to no A; but [A] is assumed to belong either to all or to some of Γ; so B would be possible to belong to no Γ or not to all Γ; but it was assumed from the beginning to [be possible to belong] to all.
φανερὸν δʼ ὅτι καὶ τοῦ ἐνδέχεσθαι μὴ ὑπάρχειν γίγνεται συλλογισμός, εἴπερ καὶ τοῦ μὴ ὑπάρχειν.
And it is clear that a syllogism of being possible not to belong also comes about, if indeed also one of not belonging.
πάλιν ἔστω ἡ καταφατικὴ πρότασις ἀναγκαία, καὶ τὸ μὲν Α ἐνδεχέσθω μηδενὶ τῷ Β ὑπάρχειν, τὸ δὲ Β παντὶ τῷ Γ ὑπαρχέτω ἐξ ἀνάγκης.
Again, let the affirmative premise be necessary, and let A be possible to belong to no B, and let B belong to all Γ of necessity.
ὁ μὲν οὖν συλλογισμὸς ἔσται τέλειος, ἀλλʼ οὐ τοῦ μὴ ὑπάρχειν ἀλλὰ τοῦ ἐνδέχεσθαι μὴ ὑπάρχειν· ἥ τε γὰρ πρότασις οὕτως ἐλήφθη ἡ ἀπὸ τοῦ μείζονος ἄκρου, καὶ εἰς τὸ ἀδύνατον οὐκ ἔστιν ἀγαγεῖν· εἰ γὰρ ὑποτεθείη τὸ Α τῷ Γ τινὶ ὑπάρχειν, κεῖται δὲ καὶ τῷ Β ἐνδέχεσθαι μηδενὶ ὑπάρχειν, οὐδὲν συμβαίνει διὰ τούτων ἀδύνατον.
The syllogism indeed will be perfect, but not of not belonging, but of being possible not to belong; for the premise from the major term was assumed in this way, and it is not possible to lead to the impossible; for if A were assumed to belong to some Γ, and it is also assumed to be possible to belong to no B, nothing impossible results through these.