§1.1.16#2ἐὰν δὲ πρὸς τῷ ἐλάττονι ἄκρῳ τεθῇ τὸ στερητικόν, ὅταν μὲν ἐνδέχεσθαι σημαίνῃ, συλλογισμὸς ἔσται διὰ τῆς ἀντιστροφῆς, καθάπερ ἐν τοῖς πρότερον, ὅταν δὲ μὴ ἐνδέχεσθαι, οὐκ ἔσται.
But if the privative premise is placed in relation to the minor term, when it signifies being possible, a syllogism will come about through conversion, just as in the previous cases; but when it does not [signify being possible], there will not be.
οὐδʼ ὅταν ἄμφω μὲν τεθῇ στερητικά, μὴ ᾖ δʼ ἐνδεχόμενον τὸ πρὸς τὸ ἔλαττον.
Nor will there be when both are assumed as privative, and the premise in relation to the minor term is not possible.
ὅροι δʼ οἱ αὐτοί, τοῦ μὲν ὑπάρχειν λευκόν–ζῷον–χιών, τοῦ δὲ μὴ ὑπάρχειν λευκόνζῷον–πίττα.
The terms are the same: for belonging, "white—animal—snow"; for not belonging, "white—animal—pitch".
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 στερητικὸν ἀναγκαῖον, καὶ τὸ συμπέρασμα ἔσται τοῦ μὴ ὑπάρχειν.
For when the privative premise is necessary, the conclusion will also be of not belonging.
οἷον εἰ τὸ μὲν Α μηδενὶ τῷ Β ἐνδέχεται ὑπάρχειν, τὸ δὲ Β τινὶ τῷ Γ ἐνδέχεται ὑπάρχειν, ἀνάγκη τὸ Α τινὶ τῷ Γ μὴ ὑπάρχειν.
For example, if A is possible to belong to no B, and B is possible to belong to some Γ, it is necessary for A to not belong to some Γ.
εἰ γὰρ παντὶ ὑπάρχει, τῷ δὲ Β μηδενὶ ἐνδέχεται, οὐδὲ τὸ Β οὐδενὶ τῷ ἐνδέχεται ὑπάρχειν.
For if it belongs to all, and is possible to belong to no B, B is also possible to belong to no A.
ὥστʼ εἰ τὸ Α παντὶ τῷ Γ ὑπάρχει, οὐδενὶ τῷ Γ τὸ Β ἐνδέχεται.
So if A belongs to all Γ, B is possible to [belong] to no Γ.
ἀλλʼ ὑπέκειτό τινι ἐνδέχεσθαι.
But it was assumed to be possible [to belong] to some.
ὅταν δὲ τὸ ἐν μέρει καταφατικὸν ἀναγκαῖον ᾖ, τὸ ἐν τῷ στερητικῷ συλλογισμῷ, οἷον τὸ Β Γ, ἢ τὸ καθόλου τὸ ἐν τῷ κατηγορικῷ, οἷον τὸ Β, οὐκ ἔσται τοῦ ὑπάρχειν συλλογισμός.
But when the particular affirmative premise is necessary (the one in the privative syllogism, such as B [belonging to some] Γ), or the universal in the affirmative syllogism (such as [A belonging to all] B), there will not be a syllogism of belonging.
ἀπόδειξις δʼ ἡ αὐτὴ ἣ καὶ ἐπὶ τῶν πρότερον.
And the proof is the same as in the previous cases.
ἐὰν δὲ τὸ μὲν καθόλου τεθῇ πρὸς τὸ ἔλαττον ἄκρον, ἢ καταφατικὸν ἢ στερητικόν, ἐνδεχόμενον, τὸ δʼ ἐν μέρει ἀναγκαῖον [πρὸς τῷ μείονι ἄκρῳ], οὐκ ἔσται συλλογισμός (ὅροι δὲ τοῦ μὲν ὑπάρχειν ἐξ ἀνάγκης ζῷον–λευκόν–ἄνθρωπος, τοῦ δὲ μὴ ἐνδέχεσθαι ζῷον–λευκόν–ἱμάτιον)·
But if the universal is placed in relation to the minor term, whether affirmative or privative, as possible, and the particular is necessary [in relation to the minor term], there will not be a syllogism (the terms for belonging of necessity are "animal—white—human", and for not being possible, "animal—white—cloak").
ὅταν δʼ ἀναγκαῖον ᾖ τὸ καθόλου, τὸ δʼ ἐν μέρει ἐνδεχόμενον, στερητικοῦ μὲν ὄντος τοῦ καθόλου τοῦ μὲν ὑπάρχειν ὅροι ζῷον–λευκόν–κόραξ, τοῦ δὲ μὴ ὑπάρχειν ζῷον–λευκόν–πίττα, καταφατικοῦ δὲ τοῦ μὲν ὑπάρχειν ζῷον–λευκόν–κύκνος, τοῦ δὲ μὴ ἐνδέχεσθαι ζῷον–λευκόν–χιών.
But when the universal is necessary and the particular is possible, if the universal is privative, the terms for belonging are "animal—white—crow", and for not belonging, "animal—white—pitch"; but if it is affirmative, for belonging, "animal—white—swan", and for not being possible, "animal—white—snow".
οὐδʼ ὅταν ἀδιόριστοι ληφθῶσιν αἱ προτάσεις ἢ ἀμφότεραι κατὰ μέρος, οὐδʼ οὕτως ἔσται συλλογισμός.
Nor when the premises are assumed as indefinite or both as particular, not even in this way will there be a syllogism.
ὅροι δὲ κοινοὶ τοῦ μὲν ὑπάρχειν ζῷον–λευκόν–ἄνθρωπος, τοῦ δὲ μὴ ὑπάρχειν ζῷον–λευκόν–ἄψυχον.
The common terms are: for belonging, "animal—white—human"; for not belonging, "animal—white—soulless".
καὶ γὰρ τὸ ζῷον τινὶ λευκῷ καὶ τὸ λευκὸν ἀψύχῳ τινὶ καὶ ἀναγκαῖον ὑπάρχειν καὶ οὐκ ἐνδέχεται ὑπάρχειν.
For indeed, for animal to belong to some white, and for white to belong to some soulless, is both necessary to belong and not possible to belong.
κἀπὶ τοῦ ἐνδέχεσθαι ὁμοίως, ὥστε πρὸς ἅπαντα χρήσιμοι οἱ ὅροι.
And similarly in the case of being possible, so that the terms are useful for all cases.
Φανερὸν οὖν ἐκ τῶν εἰρημένων ὅτι ὁμοίως ἐχόντων τῶν ὅρων ἔν τε τῷ ὑπάρχειν καὶ ἐν τοῖς ἀναγκαίοις γίνεταί τε καὶ οὐ γίνεται συλλογισμός, πλὴν κατὰ μὲν τὸ ὑπάρχειν τιθεμένης τῆς στερητικῆς προτάσεως τοῦ ἐνδέχεσθαι ἦν ὁ συλλογισμός, κατὰ δὲ τὸ ἀναγκαῖον τῆς στερητικῆς καὶ τοῦ ἐνδέχεσθαι καὶ τοῦ μὴ ὑπάρχειν.
It is clear, then, from what has been said, that when the terms are related in the same way, a syllogism comes about and does not come about similarly both in the case of belonging and in the case of necessary [premises]—except that, in the case of belonging, when the privative premise was assumed, the syllogism was of being possible, whereas in the case of the necessary [premise], when the privative was assumed, [the syllogism is] both of being possible and of not belonging.