§1.1.19Ἐὰν δʼ ἡ μὲν ἐξ ἀνάγκης ἡ δʼ ἐνδέχεσθαι σημαίνῃ τῶν προτάσεων, τῆς μὲν στερητικῆς ἀναγκαίας οὔσης ἔσται συλλογισμός, οὐ μόνον ὅτι ἐνδέχεται μὴ ὑπάρχει, ἀλλὰ καὶ ὅτι οὐχ ὑπάρχει, τῆς δὲ καταφατικῆς οὐκ ἔσται.
If one of the premises signifies necessity and the other being possible, when the privative premise is necessary there will be a syllogism, not only that it is possible not to belong, but also that it does not belong, but when the affirmative is necessary there will not be.
κείσθω γὰρ τὸ τῷ μὲν Β ἐξ ἀνάγκης μηδενὶ ὑπάρχειν, τῷ δὲ πσαντὶ ἐνδέχεσθαι.
For let A be assumed to belong to no B of necessity, and to be possible to belong to all Γ.
ἀντιστραφείσης οὖν τῆς στερητικῆς οὐδὲ τὸ Β τῷ Α οὐδενὶ ὑπάρξει· τὸ δὲ παντὶ τῷ Γ ἐνεδέχετο· γίψεται δὴ πάλιν διὰ τοῦ· πρώτου σχήματος ὁ συλλογισμός ὅτι τὸ Β τῷ Γ ἐνδέχεται μηδενὶ ὑπάρχειν.
Through the conversion of the privative premise, then, B will belong to no A of necessity; and A was possible to belong to all Γ; thus a syllogism comes about again through the first figure that B is possible to belong to no Γ.
ἅμα δὲ δῆλον ὅτι οὐδʼ ὑπάρξει τὸ Β οὐδενὶ τῷ Γ. κείσθω γὰρ ὑπάρχειν·
And at the same time it is clear that B will not belong to any Γ. For let us assume that it does belong.
οὐκοῦν εἰ τὸ τῷ Β μηδενὶ ἐνδέχεται, τὸ δὲ Β ὑπάρχει τινὶ τῷ Γ, τὸ Α τῷ Γ τινὶ οὐκ ἐνδέχεται· ἀλλὰ παντὶ ὑπέκειτο ἐνδέχεσθαι.
Then, if A is possible to belong to no B, and B belongs to some Γ, A is not possible to belong to some Γ; but it was assumed to be possible to belong to all Γ.
τὸν αὐτὸν δὲ τρόπον δειχθήσεται καὶ εἰ πρὸς τῷ Γ τεθείη τὸ στερητικόν.
And in the same way it will be shown also if the privative is assumed in relation to Γ.
Πάλιν ἔστω τὸ κατηγορικὸν ἀναγκαῖον, θάτερον δʼ ἐνδεχόμενον, καὶ τὸ Α τῷ μὲν Β ἐνδεχέσθω μηδενί, τῷ δὲ Γ παντὶ ὑπαρχέτω ἐξ ἀνάγκης.
Again, let the affirmative be necessary, and the other possible, and let A be possible to belong to no B, and belong to all Γ of necessity.
οὕτως οὖν ἐχόντων τῶν ὅρων οὐδεὶς ἔσται συλλογισμός.
When the terms are thus related, then, there will be no syllogism.
συμβαίνει γὰρ τὸ Β τῷ Γ ἐξ ἀνάγκης μὴ ὑπάρχειν.
For it results that B does not belong to Γ of necessity.
ἔστω γὰρ τὸ μὲν Α λευκόν, ἐφʼ ᾧ δὲ τὸ Β ἄνθρωπος, ἐφʼ ὧ δὲ τὸ Γ κύκνος.
For let A be white, B human, and Γ swan.
τὸ δὴ λευκὸν κύκνῳ μὲν ἐξ ἀνάγκης ὑπάρχεῖ, ἀνθρώπῳ δʼ ἐνδέχεται μηδενί καὶ ἄνθρωπος οὐδενὶκύκνῳ ἐξ ἀνάγκης.
White belongs to swan of necessity, and is possible to belong to no human; and human does not belong to any swan of necessity.
ὅτι μὲν οὖν τοῦ ἐνδέχεσθαι οὐκ ἔστι συλλογισμός, φανερόν· τὸ γὰρ ἐξ ἀνάγκης οὐκ ἦν ἐνδεχόμενον.
That there is no syllogism of the possible, then, is clear; for the necessary was not possible.
ἀλλὰ μὴν οὐδὲ τοῦ ἀναγκαίου· τὸ γὰρ ἀναγκαῖον ἢ ἐξ ἀμφοτέρων ἀναγκαίων ἢ ἐκ τῆς στερητικῆς συνέβαινεν.
But indeed, neither is there of the necessary; for the necessary resulted either from both being necessary or from the privative.
ἔτι δὲ καὶ ἐγχωρεῖ τούτων κειμένων τὸ Β τῷ Γ ὑπάρχειν· οὐδὲν γὰρ κωλύει τὸ μὲν Γ ὑπὸ τὸ Β εἶναι, τὸ δὲ Α τῷ μὲν Β παντὶ ἐνδέχεσθαι, τῷ δὲ Γ ἐξ ἀνάγκης ὑπάρχειν, οἷον εἰ τὸ μὲν Γ εἴη ἐγρηγορός, τὸ δὲ Β ζῶον, τὸ δʼ ἐφʼ ᾧ τὸ Α κίνησις.
Furthermore, it is also admissible, with these assumed, that B belongs to Γ; for nothing prevents Γ from being under B, and A being possible to belong to all B, and belonging to Γ of necessity, as for instance if Γ were waking, B animal, and A motion.
τῷ μὲν γὰρ ἐγρηγορότι ἐξ ἀνάγκης κίνησις, ζῴῳ δὲ παντὶ ἐνδέχεται· καὶ πᾶν τὸ ἐγρηγορὸς ζῷον.
For to the waking motion belongs of necessity, and to all animal it is possible; and every waking thing is an animal.
φανερὸν οὖν ὅτι οὐδὲ τοῦ μὴ ὑπάρχειν, εἴπερ οὕτως ἐχόντων ἀνάγκη ὑπάρχειν.
It is clear, then, that there is no syllogism of not belonging either, since when they are thus related it is necessary to belong.
οὐδὲ δὴ τῶν ἀντικειμένων καταφάσεων, ὥστʼ οὐδεὶς ἔσται συλλογισμός.
Nor indeed of the opposite affirmations, so that there will be no syllogism.
ὁμοίως δὲ δειχθήσεται καὶ ἀνάπαλιν τεθείσης τῆς καταφατικῆς.
And similarly it will be shown also if the affirmative is assumed the other way around.
Ἐὰν δʼ ὁμοιοσχήμονες ὦσιν αἱ προτάσεις, στερητικῶν μὲν οὐσῶν ἀεὶ γίνεται συλλογισμὸς ἀντιστραφείσης τῆς κατὰ τὸ ἐνδέχεσθαι προτάσεως καθάπερ ἐν τοῖς πρότερον.
But if the premises are of the same form, when they are privative a syllogism always comes about through the conversion of the premise in relation to the possible, just as in the previous cases.
εἰλήφθω γὰρ τὸ Α τῷ μὲν Β ἐξ ἀνάγκης μὴ ὑπάρχειν, τῷ δὲ Γ ἐνδέχεσθαι μὴ ὑπάρχειν·
For let A be assumed to belong to no B of necessity, and to be possible not to belong to Γ.
ἀντιστραφεισῶν οὖν τῶν προτάσεων τὸ μὲν Β τῷ οὐδενὶ ὑπάρχει, τὸ δὲ Α παντὶτῷ Γ ἐνδέχεται· γίνεται δὴ τὸ πρῶτον σχῆμα.
When the premises are converted, then, B belongs to no A, and A is possible to belong to all Γ; thus the first figure comes about.
κἂν εἰ πρὸς τῷ Γ τεθείη τὸ στερητικόν, ὡσαύτως.
And if the privative is assumed in relation to Γ, likewise.
ἐὰν δὲ κατηγορικαὶ τεθῶσιν, οὐκ ἔσται συλλογισμός.
But if they are assumed as affirmative, there will be no syllogism.
τοῦ μὲν γὰρ μὴ ὑπάρχειν ἢ τοῦ ἐξ ἀνάγκης μὴ ὑπάρχειν φανερὸν ὅτι οὐκ ἔσται διὰ τὸ μὴ εἰλῆφθαι στερητικὴν πρότασιν μήτʼ ἐν τῷ ὑπάρχειν μήτʼ ἐν τῷ ἐξ ἀνάγκης ὑπάρχειν.
For that there will not be of not belonging or of not belonging of necessity is clear because no privative premise was assumed, either in belonging or in belonging of necessity.
ἀλλὰ μὴν οὐδὲ τοῦ ἐνδέχεσθαι μὴ ὑπάρχειν· ἐξ ἀνάγκης γὰρ οὕτως ἐχόντων τὸ Β τῷ Γ οὐχ ὑπάρξει, οἷον εἰ τὸ μὲν Α τεθείη λευκόν, ἐφʼ δὲ τὸ Β κύκνος, τὸ δὲ Γ ἄνθρωπος.
But indeed, neither of being possible not to belong; for when they are thus related B will not belong to Γ of necessity, as for instance if A is assumed white, B swan, and Γ human.
οὐδέ γε τῶν ἀντικειμένων καταφάσεων, ἐπεὶ δέδεικται τὸ Β τῷ Γ ἐξ ἀνάγκης οὐχ ὑπάρχον.
Nor indeed of the opposite affirmations, since it has been shown that B does not belong to Γ of necessity.
οὐκ ἄρα γίνεται συλλογισμὸς ὅλως.
There is therefore no syllogism at all.
Ὁμοίως δʼ 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 καθόλου τε καὶ ἀναγκαῖον, ἀεὶ συλλογισμὸς ἔσται καὶ τοῦ ἐνδέχεσθαι καὶ τοῦ μὴ ὑπάρχειν (ἀπόδειξις δὲ διὰ τῆς ἀντιστροφῆς), ὅταν δὲ τὸ καταφατικόν, οὐδέποτε·
For when the privative is universal and necessary, there will always be a syllogism both of being possible and of not belonging (and the proof is through conversion); but when the affirmative is, there never will be.
τὸν αὐτὸν γὰρ τρόπον δειχθήσεται ὃν καὶ ἐν τοῖς καθόλου, καὶ διὰ τῶν αὐτῶν ὅρων.
For it will be shown in the same way as in the case of universal premises, and through the same terms.
οὐδʼ ὅταν ἀμφότεραι ληφθῶσι καταφατικαί· καὶ γὰρ τούτου ἡ αὐτὴ ἀπόδειξις ἣ καὶ πρότερον.
Nor when both are assumed as affirmative; for of this the proof is the same as before.
ὅταν δὲ ἀμφότεραι μὲν στερητικαί, καθόλου δὲ καὶ ἀναγκαία ἡ τὸ μὴ ὑπάρχειν σημαίνουσα, διʼ αὐτῶν μὲν τῶν εἰλημμένων οὐκ ἔσται τὸ ἀναγκαῖον, ἀντιστραφείσης δὲ τῆς κατὰ τὸ ἐνδέχεσθαι προτάσεως ἔσται συλλογισμός, καθάπερ ἐν τοῖς πρότερον.
But when both are privative, and the one signifying not belonging is universal and necessary, from the assumed premises themselves there will be no necessary consequence, but when the premise in relation to the possible is converted, there will be a syllogism, just as in the previous cases.
ἐὰν δʼ ἀμφότεραι ἀδιόριστοι ἢ ἐν μέρει τεθῶσιν, οὐκ ἔσται συλλογισμός.
But if both are assumed as indefinite or as particular, there will be no syllogism.
ἀπόδειξις δʼ ἡ αὐτὴ καὶ διὰ τῶν αὐτῶν ὅρων.
And the proof is the same and through the same terms.
Φανερὸν οὖν ἐκ τῶν εἰρημένων ὅτι τῆς μὲν στερητικῆς τῆς καθόλου τιθεμένης ἀναγκαίας ἀεὶ γίνεται συλλογισμὸς οὐ μόνον τοῦ ἐνδέχεσθαι μὴ ὑπάρχειν, ἀλλὰ καὶ τοῦ μὴ ὑπάρχειν, τῆς δὲ καταφατικῆς οὐδέποτε.
It is clear, then, from what has been said, that when the universal privative premise is assumed as necessary, there always comes about a syllogism not only of being possible not to belong, but also of not belonging, but when the affirmative is, never.
καὶ ὅτι τὸν αὐτὸν τρόπον ἐχόντων ἔν τε τοῖς ἀναγκαίοις καὶ ἐν τοῖς ὑπάρχουσι γίνεταί τε καὶ οὐ γίνεται συλλογισμός.
And that a syllogism comes about and does not come about in the same way as when they are in necessary and in belonging premises.
δῆλον δὲ καὶ ὅτι πάντες ἀτελεῖς οἱ συλλογισμοί, καὶ ὅτι τελειοῦνται διὰ τῶν προειρημένων σχημάτων.
And it is clear also that all these syllogisms are incomplete, and that they are completed through the aforementioned figures.