§1.1.15#3Ἐὰν δὲ τὸ στερητικὸν τεθῇ πρὸς τὸ ἔλαττον ἄκρον ἐνδέχεσθαι σημαῖνον, ἐξ αὐτῶν μὲν τῶν εἰλημμένων προτάσεων οὐδεὶς ἔσται συλλογισμός, ἀντιστραφείσης δὲ τῆς κατὰ τὸ ἐνδέχεσθαι προτάσεως ἔσται, καθάπερ ἐν τοῖς πρότερον.
But if the privative [premise] is placed in relation to the minor term, signifying being possible, from the assumed premises themselves there will be no syllogism, but if the premise concerning possibility is converted, there will be, just as in the previous cases.
ὑπαρχέτω γὰρ τὸ παντὶ τῷ Β, τὸ δὲ Β ἐνδεχέσθω μηδενὶ τῷ Γ. οὕτω μὲν οὖν ἐχόντων τῶν ὅρων οὐδὲν ἔσται ἀναγκαῖον·
For let A belong to all B, and let B be possible to belong to no Γ.
ἐὰν δʼ ἀντιστραφῇ τὸ Β Γ καὶ ληφθῇ τὸ Β παντὶ τῷ Γ ἐνδέχεσθαι, γίνεται συλλογισμὸς ὥσπερ πρότερον· ὁμοίως γὰρ ἔχουσιν οἱ ὅροι τῇ θέσει.
When the terms are related in this way, nothing necessary will result; but if BΓ is converted and B is assumed to be possible to belong to all Γ, a syllogism comes about just as before; for the terms are in the same relation by their position.
τὸν αὐτὸν δὲ τρόπον καὶ στερητικῶν ὄντων ἀμφοτέρων τῶν διαστημάτων, ἐὰν τὸ μὲν Β μὴ ὑπάρχειν, τὸ δὲ Β Γ μηδενὶ ἐνδέχεσθαι σημαίνῃ· διʼ αὐτῶν μὲν γὰρ τῶν εἰλημμένων οὐδαμῶς γίνεται τὸ ἀναγκαῖον, ἀντιστραφείσης δὲ τῆς κατὰ τὸ ἐνδέχεσθαι προτάσεως ἔσται συλλογισμός.
In the same manner also when both intervals are privative, if the B [premise] signifies not belonging, and the BΓ signifies being possible to belong to no [Γ]; for from the assumed premises themselves the necessary in no way comes about, but if the premise concerning possibility is converted, there will be a syllogism.
εἰλήφθω γὰρ τὸ μὲν μηδενὶ τῷ Β ὑπάρχειν, τὸ δὲ Β ἐνδέχεσθαι μηδενὶ τῷ Γ. διὰ μὲν οὖν τούτων οὐδὲν ἀναγκαῖον· ἐὰν δὲ ληφθῇ τὸ Β παντὶ τῷ Γ ἐνδέχεσθαι, ὅπερ ἐστὶν ἀληθές, ἡ δὲ Β πρότασις ὁμοίως ἔχῃ, πάλιν ὁ αὐτὸς ἔσται συλλογισμός.
For let A be assumed to belong to no B, and B to be possible to belong to no Γ. Therefore through these nothing necessary results; but if B is assumed to be possible to belong to all Γ (which is true), and the premise AB is in the same relation, again there will be the same syllogism.
ἐὰν δὲ μὴ ὑπάρχειν τεθῇ τὸ Β παντὶ τῷ Γ καὶ μὴ ἐνδέχεσθαι μὴ ὑπάρχειν, οὐκ ἔσται συλλογισμὸς οὐδαμῶς, οὔτε στερητικῆς οὔσης οὔτε καταφατικῆς τῆς Α Β προτάσεως.
But if B is assumed not to belong to all Γ, and not to be possible not to belong, there will in no way be a syllogism, whether the premise AB is privative or affirmative.
ὅροι δὲ κοινοὶ τοῦ μὲν ἐξ ἀνάγκης ὑπάρχειν λευκόν–ζῷον–χιών, τοῦ δὲ μὴ ἐνδέχεσθαι λευκόν–ζῷον–πίττα. Φανερὸν οὖν ὅτι καθόλου τῶν ὅρων ὄντων, καὶ τῆς μὲν ὑπάρχειν τῆς δʼ ἐνδέχεσθαι λαμβανομέντης τῶν προτάσεων, ὅταν ἡ πρός τὸ ἔλαττον ἄκρον ἐνδέχεσθαι λαμβάνηται πρότασις, ἀεὶ γίνεται συλλογισμός, πλὴν ὁτὲ μὲν ἐξ αὐτῶν ὁτὲ δʼ ἀντιστραφείσης τῆς προτάσεως.
Common terms for belonging of necessity are "white-animal-snow," and for not being possible "white-animal-pitch." It is clear, then, that when the terms are universal, and one of the premises is assumed to belong and the other to be possible, whenever the premise in relation to the minor term is assumed to be possible, a syllogism always comes about, except that sometimes it is from the premises themselves and sometimes from the premise being converted.
πότε δὲ τούτων ἑκάτερος καὶ διὰ τίνʼ αἰτίαν, εἰρήκαμεν.
And when each of these comes about, and for what reason, we have said.
Ἐὰν δὲ τὸ μὲν καθόλου τὸ δʼ ἐν μέρει ληφθῇ τῶν διαστημάτων, ὅταν μὲν τὸ πρὸς τὸ μεῖον ἄκρον καθόλου τεθῇ καὶ ἐνδεχόμενον, εἴτʼ ἀποφατικὸν εἴτε καταφατικόν, τὸ δʼ ἐν μέρει καταφατικόν καὶ ὑπάρχον, ἔσται συλλογισμὸς τέλειος, καθάπερ καὶ καθόλου τῶν ὅρων ὄντων.
But if one of the intervals is assumed as universal and the other as particular, whenever the one in relation to the major term is placed as universal and possible, whether privative or affirmative, and the other as particular, affirmative, and belonging, there will be a perfect syllogism, just as when the terms are universal.
ἀπόδειξις δʼ ἡ αὐτὴ ἣ καὶ πρότερον.
And the proof is the same as before.
ὅταν δὲ καθόλου μὲν ᾖ τὸ πρὸς τὸ μεῖζον ἄκρον, ὑπάρχον δέ καὶ μὴ ἐνδεχόμενον, θάτερον δʼ ἐν μέρει καὶ ἐνδεχόμενον, ἐάν τʼ ἀποφατικαὶ ἐάν τε καταφατικαὶ τεθῶσιν ἀμφότεραι, ἐάν τε ἡ μὲν ἀποφατικὴ ἡ δὲ καταφατική, πάντως ἔσται συλλογισμὸς ἀτελής.
But when the one in relation to the major term is universal, belonging and not possible, and the other is particular and possible, whether both are assumed as privative or both as affirmative, or one is privative and the other affirmative, in all cases there will be an imperfect syllogism.
πλὴν οἱ μὲν διὰ τοῦ ἀδυνάτου δειχθήσονται, οἱ δὲ καὶ διὰ τῆς ἀντιστροφῆς τῆς τοῦ ἐνδέχεσθαι, καθάπερ ἐν τοῖς πρότερον.
Except that some will be proved through the impossible, and others also through the conversion of the possible, just as in the previous cases.
ἔσται δὲ συλλογισμὸς διὰ τῆς ἀντιστροφῆς ὅταν ἡ μὲν καθόλου πρὸς τὸ μεῖζον ἄκρον τεθεῖσα σημαίνῃ τὸ ὑπάρχειν, ἡ δʼ ἐν μέρει στερητικὴ οὖσα τὸ ἐνδέχεσθαι λαμβάνῃ, οἷον εἰ τὸ μὲν παντὶ τῷ Β ὑπάρχει ἢ μὴ ὑπάρχει, τὸ δὲ Β τινὶ τῷ Γ ἐνδέχεται μὴ ὑπάρχειν· ἀντιστραφέντος γὰρ τοῦ Β Γ κατὰ τὸ ἐνδέχεσθαι γίνεται συλλογισμός.
And there will be a syllogism through conversion when the universal premise placed in relation to the major term signifies belonging, and the particular premise, being privative, assumes being possible, as for example if A belongs to all B or does not belong, and B is possible not to belong to some Γ; for when BΓ is converted according to possibility, a syllogism comes about.
ὅταν δὲ τὸ μὴ ὑπάρχειν λαμβάνῃ ἡ κατὰ μέρος τεθεῖσα, οὐκ ἔσται συλλογισμός.
But when the premise placed as particular assumes not belonging, there will be no syllogism.
ὅροι τοῦ μὲν ὑπάρχειν λευκόν–ζῷον–χιών, τοῦ δὲ μὴ ὑπάρχειν λευκόν–ζῷον–πίττα· διὰ γὰρ τοῦ ἀδιορίστου ληπτέον τὴν ἀπόδειξιν.
Terms for belonging are "white-animal-snow," and for not belonging "white-animal-pitch." For the proof must be assumed through the indefinite.
ἐὰν δὲ τὸ καθόλου τεθῇ πρὸς τὸ ἔλαττον ἄκρον, τὸ δʼ ἐν μέρει πρὸς τὸ μεῖζον, ἐάν τε στερητικὸν ἐάν τε καταφατικόν, ἐάν τʼ ἐνδεχόμενον ἐάν θʼ ὑπάρχον ὁποτερονοῦν, οὐδαμῶς ἔσται συλλογισμός.
And if the universal is placed in relation to the minor term, and the particular in relation to the major, whether privative or affirmative, and whether possible or belonging, whichever of the two, there will in no way be a syllogism.
Οὐδ᾿ ὅταν ἐν μέρει ἢ ἀδιόριστοι τεθῶσιν αἱ προτάσεις, εἴτʼ ἐνδέχεσθαι λαμβάνουσαι εἴθ᾿ ὑπάρχειν εἴτʼ ἐναλλάξ, οὐδʼ οὕτως ἔσται συλλογισμός.
Nor when the premises are assumed as particular or indefinite, whether they assume being possible or belonging or alternating, not even in this way will there be a syllogism.
ἀπόδειξις δʼ ἡ αὐτὴ ἥπερ κἀπὶ τῶν πρότερον.
And the proof is the same as in the previous cases.
ὅροι δὲ κοινοὶ τοῦ μὲν ὑπάρχειν ἐξ ἀνάγκης ζῷον–λευκόν–ἄνθραωπος, τοῦ δὲ μὴ ἐνδέχεσθαι ζῷον–λευκόν–ἱμάτιον. φανερὸν οὖν ὅτι τοῦ μὲν πρὸς τὸ μεῖζον ἄκρον καθόλου τεθέντος ἀεὶ γίνεται συλλογισμός, τοῦ δὲ πρὸς τὸ ἔλαττον οὐδέποτʼ οὐδενός.
Common terms for belonging of necessity are "animal-white-man," and for not being possible "animal-white-cloak." It is clear, then, that when the premise in relation to the major term is universal a syllogism always comes about, but when the one in relation to the minor term [is universal], never any.