§1.1.15#1Ἐὰν δʼ ἡ μὲν ὑπάρχειν ἡ δʼ ἐνδέχεσθαι λαμβάνηται τῶν προτάσεων, ὅταν μὲν ἡ πρὸς τὸ μεῖζον ἄκρον ἐνδέχεσθαι σημαίνῃ, τέλειοί τʼ ἔσονται πάντες οἱ συλλογισμοὶ καὶ τοῦ ἐνδέχεσθαι κατὰ τὸν εἰρημένον διορισμόν, ὅταν δʼ ἡ πρὸς τὸ ἔλαττον, ἀτελεῖς τε πάντες, καὶ οἱ στερητικοὶ τῶν συλλογισμῶν οὐ τοῦ κατὰ τὸν διορισμὸν ἐνδεχομένου, ἀλλὰ τοῦ μηδενὶ ἢ μὴ παντὶ ἐξ ἀνάγκης ὑπάρχειν·
one of the premises is assumed to belong and the other to be possible, when the premise with the major extreme signifies being possible, all the syllogisms will be perfect and of possibility according to the said definition; but when the premise with the minor extreme [signifies being possible], all of them will be imperfect, and the privative of the syllogisms will be not of the possible according to the definition, but of belonging of necessity to none or not to all.
εἰ γὰρ μηδενὶ ἢ μὴ παντὶ ἐξ ἀνάγκης, ἐνδέχεσθαί φαμεν καὶ μηδενὶ καὶ μὴ παντὶ ὑπάρχειν.
For if it is necessary for it to belong to none or not to all, we say that it is possible for it to belong both to none and not to all.
ἐνδεχέσθω γὰρ τὸ παντὶ τῷ Β, τὸ δὲ Β παντὶ τῷ Γ κείσθω ὑπάρχειν.
For let A be possible to belong to all B, and let B be assumed to belong to all Γ.
ἐπεὶ οὖν ὑπὸ τὸ Β ἐστὶ τὸ Γ, τῷ δὲ Β παντὶ ἐνδέχεται τὸ Α, φανερὸν ὅτι καὶ τῷ Γ παντὶ ἐνδέχεται.
Since, then, Γ is under B, and A is possible to all B, it is clear that it is also possible to all Γ.
γίνεται δὴ τέλειος συλλογισμός·
Thus a perfect syllogism comes about.
ὁμοίως δὲ καὶ στερητικῆς οὔσης τῆς Β προτάσεως, τῆς δὲ Β Γ καταφατικῆς, καὶ τῆς μὲν ἐνδέχεσθαι τῆς δʼ ὑπάρχειν λαμβανομένης, τέλειος ἔσται συλλογισμὸς ὅτι τὸ ἐνδέχεται μηδενὶ τῷ Γ ὑπάρχειν.
Similarly also when the premise AB is privative, and BΓ is affirmative, and one is assumed as possible and the other as belonging, there will be a perfect syllogism that A is possible to belong to no Γ.
Ὅτι μὲν οὖν τοῦ ὑπάρχειν τιθεμένου πρὸς τὸ ἔλαττον ἄκρον τέλειοι γίγνονται συλλογισμοί, φανερόν·
That, then, when the belonging is placed with the minor extreme, perfect syllogisms come about, is clear.
ὅτι δʼ ἐναντίως ἔχοντος ἔσονται συλλογισμοί, διὰ τοῦ ἀδυνάτου δεικτέον.
But that there will be syllogisms when they are in the contrary relation must be shown through the impossible.
ἅμα δʼ ἔσται δῆλον καὶ ὅτι ἀτελεῖς· ἡ γὰρ δεῖξις οὐκ ἐκ τῶν εἰλημμένων προτάσεων.
And at the same time it will be clear that they are imperfect; for the proof is not from the assumed premises.
πρῶτον δὲ λεκτέον ὅτι εἰ τοῦ Α ὄντος ἀνάγκη τὸ Β εἶναι, καὶ δυνατοῦ ὄντος τοῦ δυνατὸν ἔσται καὶ τὸ Β ἐξ ἀνάγκης.
But first it must be said that if, when A is, it is necessary for B to be, also when A is possible, B will also be possible of necessity.
ἔστω γὰρ οὕτως ἐχόντων τὸ μὲν ἐφʼ ᾧ τὸ Α δυνατόν, τὸ δʼ ἐφʼ ᾧ τὸ Β ἀδύνατον.
For let, when they are in this way, A be possible, and B be impossible.
εἰ οὖν τὸ μὲν δυνατόν, ὅτε δυνατὸν εἶναι, γένοιτʼ ἄν, τὸ δʼ ἀδύνατον, ὅτʼ ἀδύνατον, οὐκ ἂν γένοιτο, ἅμα δʼ εἴη τὸ δυνατὸν καὶ τὸ Β ἀδύνατον, ἐνδέχοιτʼ ἂν τὸ Α γενέσθαι ἄνευ τοῦ Β, εἰ δὲ γενέσθαι, καὶ εἶναι· τὸ γὰρ γεγονός, ὅτε γέγονεν, ἔστιν.
If, then, the possible, when it is possible to be, could come to be, and the impossible, when it is impossible, could not come to be, and at the same time A were possible and B impossible, it would be possible for A to come to be without B; and if to come to be, also to be; for that which has come to be, when it has come to be, is.
δεῖ δὲ λαμβάνειν μὴ μόνον ἐν τῇ γενέσει τὸ ἀδύνατον καὶ δυνατόν, ἀλλὰ καὶ ἐν τῷ ἀληθεύεσθαι καὶ ἐν τῷ ὑπάρχειν, καὶ ὁσαχῶς ἄλλως λέγεται τὸ δυνατόν· ἐν ἅπασι γὰρ ὁμοίως ἕξει.
But we must assume the impossible and possible not only in coming to be, but also in being true and in belonging, and in however many other ways the possible is spoken of; for in all of them it will be in the same way.
ἔτι τὸ ὄντος τοῦ Α τὸ Β εἶναι, οὐχ ὡς ἑνός τινος ὄντος τοῦ Α τὸ Β ἔσται δεῖ ὑπολαβεῖν· οὐ γὰρ ἔστιν οὐδὲν ἐξ ἀνάγκης ἑνός τινος ὄντος, ἀλλὰ δυοῖν ἐλαχίστοιν, οἷον ὅταν αἱ προτάσεις οὕτως ἔχωσιν ὡς ἐλέχθτη κατὰ τὸν συλλογισμόν.
Further, "when A is, B is" must not be understood as though when A is some single thing, B will be; for nothing is of necessity when some single thing is, but when there are at least two, as when the premises are in the way that was said concerning the syllogism.
εἰ γὰρ τὸ Γ κατὰ τοῦ Δ, τὸ δὲ Δ κατὰ τοῦ Ζ, καὶ τὸ Γ κατὰ τοῦ Ζ ἐξ ἀνάγκης·
For if Γ is predicated of Δ, and Δ of Z, Γ also is of necessity predicated of Z.
καὶ εἰ δυνατὸν ἑκάτερον, καὶ τὸ συμπέρασμα δυνατόν.
And if each of them is possible, the conclusion is also possible.
ὥσπερ οὖν εἴ τις θείη τὸ μὲν Α τὰς προτάσεις, τὸ δὲ Β τὸ συμπέρασμα, συμβαίνοι ἂν οὐ μόνον ἀναγκαίου τοῦ ὄντος ἅμα καὶ τὸ Β εἶναι ἀναγκαῖον, ἀλλὰ καὶ δυνατοῦ δυνατόν.
Just as, then, if one were to place A as the premises and B as the conclusion, it would result not only that when A is necessary, B is also necessary at the same time, but also that when A is possible, B is possible.
Τούτου δὲ δειχθέντος, φανερὸν ὅτι ψεύδους ὑποτεθέντος καὶ μὴ ἀδυνάτου καὶ τὸ συμβαῖνον διὰ τὴν ὑπόθεσιν; ψεῦδος ἔσται καὶ οὐκ ἀδύνατον.
And this being shown, it is clear that if a falsehood is assumed, but not an impossible one, the result through the hypothesis will also be false and not impossible.
οἷον εἰ τὸ ψεῦδος μέν ἐστι μὴ μέντοι ἀδύνατον, ὄντος δὲ τοῦ τὸ Β ἔστι, καὶ τὸ Β ἔσται ψεῦδος μὲν οὐ μέντοι ἀδύνατον.
For example, if A is false indeed but not impossible, and when A is, B is, B also will be false indeed but not impossible.
ἐπεὶ γὰρ δέδεικται ὅτι εἰ τοῦ Α ὄντος τὸ Β ἔστι, καὶ δυνατοῦ ὄντος τοῦ Α ἔσται τὸ Β δυνατόν, ὑπόκειται δὲ τὸ Α δυνατὸν εἶναι, καὶ τὸ ἔσται δυνατόν· εἰ γὰρ ἀδύνατον, ἅμα δυνατὸν ἔσται τὸ αὐτὸ καὶ ἀδύνατον.
For since it has been shown that if, when A is, B is, also when A is possible, B will be possible, and A is assumed to be possible, B also will be possible; for if it is impossible, the same thing will be possible and impossible at the same time.