§1.1.10Ἐπὶ δὲ τοῦ δευτέρου σχήματος, εἰ μὲν ἡ στερητικὴ πρότασίς ἐστιν ἀναγκαία, καὶ τὸ συμπέρασμα ἔσται ἀναγκαῖον, εἰ δʼ ἡ κατηγορική, οὐκ ἀναγκαῖον.
In the second figure, if the privative premise is necessary, the conclusion will also be necessary; but if the affirmative is necessary, the conclusion will not be necessary.
ἔστω γὰρ πρῶτον ἡ στερητικὴ ἀναγκαία, καὶ τὸ Α τῷ μὲν Β μηδενὶ ἐνδεχέσθω, τῷ δὲ Γ ὑπαρχέτω μόνον.
For let the privative premise first be necessary, and let A be possible to belong to no B, but to Γ let it belong only.
ἐπεὶ οὖν ἀντιστρέφει τὸ στερητικόν, οὐδὲ τὸ Β τῷ οὐδενὶ ἐνδέχεται· τὸ δὲ παντὶ τῷ Γ ὑπάρχει, ὥστʼ οὐδενὶ τῷ Γ τὸ Β ἐνδέχεται· τὸ γὰρ Γ ὑπὸ τὸ Α ἐστίν.
Since, then, the privative premise converts, B also is not possible to belong to any A; but A belongs to every Γ, so that B is not possible to belong to any Γ; for Γ is under A.
ὡσαύτως δὲ καὶ εἰ πρὸς τῷ Γ τεθείη τὸ στερητικόν· εἰ γὰρ τὸ Α μηδενὶ τῷ Γ ἐνδέχεται, οὐδὲ τὸ Γ οὐδενὶ τῷ Α ἐγχωρεῖ· τὸ δὲ παντὶ τῷ Β ὑπάρχει, ὥστʼ οὐδενὶ τῷ Β τὸ Γ ἐνδέχεται· γίνεται γὰρ τὸ πρῶτον σχῆμα πάλιν.
Similarly also if the privative premise is placed in relation to Γ; for if A is not possible to belong to any Γ, Γ also is not possible to belong to any A; but A belongs to every B, so that Γ is not possible to belong to any B; for it becomes the first figure again.
οὐκ ἄρα οὐδὲ τὸ Β τῷ Γ· ἀντιστρέφει γὰρ ὁμοίως.
Therefore B does not belong to Γ either; for it converts in the same way.
Εἰ δὲ ἡ κατηγορικὴ πρότασίςἐστιν ἀναγκαία, οὐκ ἔσται τὸ συμπέρασμα ἀναγκαῖον.
But if the affirmative premise is necessary, the conclusion will not be necessary.
ὑπαρχέτω γὰρ τὸ παντὶ τῷ Β ἐξ ἀνάγκης, τῷ δὲ Γ μηδενὶ ὑπαρχέτωμόνον.
For let A belong to every B of necessity, and to Γ let it belong to none only.
ἀντιστραφέντος οὖν τοῦ στερητικοῦ τὸ πρῶτον γίνεται σχῆμα· δέδεικται δʼ ἐν τῷ πρώτῳ ὅτι μὴ ἀναγκαίας οὔσης τῆς πρὸς τὸ μεῖον στερητικῆς οὐδὲ τὸ συμπέρασμα ἔσται ἀναγκαῖον, ὥστʼ οὐδʼ ἐπὶ τούτων ἔσται ἐξ ἀνάγκης.
When the privative premise is converted, then, the first figure comes about; but it has been shown in the first figure that when the privative premise in relation to the minor term is not necessary, the conclusion will not be necessary either, so that in these cases too it will not be of necessity.
ἔτι δʼ εἰ τὸ συμπέρασμά ἐστιν ἀναγκαῖον, συμβαίνει τὸ Γ τινὶ τῷ Α μὴ ὑπάρχειν ἐξ ἀνάγκης.
Further, if the conclusion is necessary, it results that Γ does not belong to some A of necessity.
εἰ γὰρ τὸ Β τῷ Γ μηδενὶ ὑπάρχει ἐξ ἀνάγκης, οὐδὲ τὸ Γ τῷ Β οὐδενὶ ὑπάρξει ἐξ ἀνάγκης.
For if B belongs to no Γ of necessity, Γ also will belong to no B of necessity.
τὸ δέ γε Β τινὶ τῷ Α ἀνάγκη ὑπάρχειν, εἴπερ καὶ τὸ Α παντὶ τῷ Β ἐξ ἀνάγκης ὑπῆρχεν.
But B must belong to some A of necessity, if A belonged to every B of necessity.
ὥστε τὸ Γ ἀνάγκη τινὶ τῷ Α μὴ ὑπάρχειν.
Therefore Γ must not belong to some A of necessity.
ἀλλʼ οὐδὲν κωλύει τὸ Α τοιοῦτον ληφθῆναι ᾧ παντὶ τὸ Γ ἐνδέχεται ὑπάρχειν.
But nothing prevents A from being taken as such that Γ is possible to belong to all of it.
ἔτι κἂν ὅρους ἐκθέμενον εἴη δεῖξαι ὅτι τὸ συμπέρασμα οὐκ ἔστιν ἀναγκαῖον ἁπλῶς, ἀλλὰ τούτων ὄντων ἀναγκαῖον.
Further, it would also be possible to show by setting out terms that the conclusion is not necessary absolutely, but necessary when these premises are.
οἶον ἔστω τὸ Α ζῷον, τὸ δὲ Β ἄνθρωπος, τὸ δὲ Γ λευκόν, καὶ αἱ προτάσεις ὁμοίως εἰλήφθωσαν· ἐνδέχεται γὰρ τὸ ζῷον μηδενὶ λευκῷ ὑπάρχειν.
For example, let A be animal, B man, and Γ white, and let the premises be taken in the same way; for it is possible for animal to belong to no white thing.
οὐχ ὑπάρξει δὴ οὐδʼ ὁ ἄνθρωπος οὐδενὶ λευκῷ, ἀλλʼ οὐκ ἐξ ἀνάγκης· ἐνδέχεται γὰρ ἄνθρωπον γενέσθαι λευκόν, οὐ μέντοι ἕως ἂν ζῷον μηδενὶ λευκῷ ὑπάρχῃ.
Man, then, will not belong to any white thing either, but not of necessity; for it is possible for a man to become white, though not as long as animal belongs to no white thing.
ὥστε τούτων μὲν ὄντων ἀναγκαῖον ἔσται τὸ συμπέρασμα, ἀπλῶς δʼ οὐκ ἀναγκαῖον.
So when these premises are, the conclusion will be necessary, but not necessary absolutely.
Ὁμοίως δʼ ἕξει καὶ ἐπὶ τῶν ἐν μέρει συλλογισμῶν.
It will be the same also in the case of particular syllogisms.
ὅταν μὲν γὰρ ἡ στερητικὴ πρότασις καθόλου τʼ ᾖ· καὶ ἀναγκαία, καὶ τὸ συμπέρασμα ἔσται ἀναγκαῖον· ὅταν δὲ ἡ κατηγορικὴ καθόλου, ἡ δὲ στερητικὴ κατὰ μέρος, οὐκ ἔσται τὸ συμπέρασμα ἀναγκαῖον.
For when the privative premise is universal and necessary, the conclusion will also be necessary; but when the affirmative is universal and the privative is particular, the conclusion will not be necessary.
ἔστω δὴ πρῶτον ἡ στερητικὴ καθόλου τε καὶ ἀναγκαία, καὶ τὸ Α τῷ μὲν Β μηδενὶ ἐνδεχέσθω ὑπάρχειν, τῷ δὲ Γ τινὶ ὑπαρχέτω.
Let the privative premise first be universal and necessary, and let A be possible to belong to no B, and let it belong to some Γ.
ἐπεὶ οὖν ἀντιστρέφει τὸ στερητικόν, οὐδὲ τὸ Β τῷ Α οὐδενὶ ἐνδέχοιτʼ ἂν ὑπάρχειν· τὸ δέ γε Α τινὶ τῷ Γ ὑπάρχει, ὥστʼ ἐξ ἀνάγκης τινὶ τῷ Γ οὐχ ὑπάρξει τὸ Β. πάλιν ἔστω ἡ κατηγορικὴ καθόλου τε καὶ ἀναγκαία, καὶ κείσθω πρὸς τῷ Β τὸ κατηγορικόν.
Since, then, the privative premise converts, B also would not be possible to belong to any A; but A belongs to some Γ, so that of necessity B will not belong to some Γ.
εἰ δὴ τὸ παντὶ τῷ Β ἐξ ἀνάγκης ὑπάρχει, τῷ δὲ Γ τινὶ μὴ ὑπάρχει, ὅτι μὲν οὐχ ὑπάρξει τὸ Β τινὶ τῷ Γ, φανερόν, ἀλλʼ οὐκ ἐξ ἀνάγκης·
Again, let the affirmative premise be universal and necessary, and let the affirmative be placed in relation to B.
οἱ γὰρ αὐτοὶ ὅροι ἔσονται πρός τὴν ἀπόδειξιν: οἵπερ ἐπὶ τῶν καθόλου συλλογισμῶν.
If, then, A belongs to every B of necessity, and does not belong to some Γ, that B will not belong to some Γ is clear, but not of necessity; for the same terms will be for the proof as in the case of universal syllogisms.
ἀλλʼ οὐδʼ εἰ τὸ στερητικὸν ἀναγκαῖόν ἐστιν ἐν μέρει ληφθέν, οὐκ ἔσται τὸ συμπέρασμα ἀναγκαῖον· διὰ γὰρ τῶν αὐτῶν ὅρων ἡ ἀπόδειξις.
Nor if the privative is assumed to be necessary as particular will the conclusion be necessary; for the proof is through the same terms.