§1.1.5#1Ὅταν δὲ τὸ αὐτὸ τῷ μὲν παντὶ τῷ δὲ μηδενὶ ὑπάρχῃ, ἢ ἑκατέρῳ παντὶ ἢ μηδενί, τὸ μὲν σχῆμα τὸ τοιοῦτον καλῶ δεύτερον, μέσον δὲ ἐν αὐτῷ λέγω τὸ κατηγορούμενον ἀμφοῖν, ἄκρα δὲ καθʼ ὧν λέγεται τοῦτο, μεῖζον δὲ ἄκρον τὸ πρὸς τῷ μέσῳ κείμενον·
But whenever the same thing belongs to all of one and to none of the other, or to all of each or to none of each, I call such a figure the second; and in it I call the middle that which is predicated of both, and the extremes those of which this is said, and the major extreme that which lies near the middle, and the minor that which is further from the middle.
ἔλαττον δὲ τὸ πορρωτέρω τοῦ μέσου. τίθεται δὲ τὸ μέσον ἔξω μὲν τῶν ἄκρων, πρῶτον δὲ τῇ θέσει.
And the middle is placed outside the extremes, and first in position.
τέλειος μὲν οὖν οὐκ ἔσται συλλογισμὸς οὐδαμῶς ἐν τούτῳ τῷ σχήματι, δυνατὸς δʼ ἔσται καὶ καθόλου καὶ μὴ καθόλου τῶν ὅρων ὄντων.
A perfect syllogism, then, will not be possible in any way in this figure, but a syllogism will be possible both when the terms are universal and when they are not universal.
καθόλου μὲν οὖν ὄντων ἔσται συλλογισμὸς ὅταν τὸ μέσον τῷ μὲν παντὶ τῷ δὲ μηδενὶ ὑπάρχῃ, ἄν πρὸς ὁποτερῳοῦν ᾖ τὸ στερητικόν· ἄλλως δʼ οὐδαμῶς.
Now, when the terms are universal, there will be a syllogism whenever the middle belongs to all of one and to none of the other, whichever of the two the privative is related to, but in no other way.
κατηγορείσθω γὰρ τὸ Μ τοῦ μὲν Ν μηδενός, τοῦ δὲ Ξ παντός.
For let M be predicated of no N, but of every Ξ.
ἐπεὶ οὖν ἀντιστρέφει τὸ στερητικόν, οὐδενὶ τῷ Μ ὑπάρξει τὸ Ν· τὸ δέ γε Μ παντὶ τῷ Ξ ὑπέκειτο· ὥστε τὸ Ν οὐδενὶ τῷ Ξ· τοῦτο γὰρ δέδεικται πρότερον.
Since, then, the privative converts, N will belong to no M; but M was assumed to belong to every Ξ; so that N belongs to no Ξ; for this has been shown before.
πάλιν εἰ τὸ Μ τῷ μὲν Ν παντὶ τῷ δὲ Ξ μηδενί, οὐδὲ τὸ Ξ τῷ οὐδενὶ ὑπάρξει (εἰ γὰρ τὸ Μ οὐδενὶ τῷ Ξ, οὐδὲ τὸ Ξ οὐδενὶ τῷ Μ· τὸ δέ γε Μ παντὶ τῷ Ν ὑπῆρχεν· τὸ ἄρα Ξ οὐδενὶ τῷ Ν ὑπάρξει· γεγένηται γὰρ πάλιν τὸ πρῶτον σχῆμα)· ἐπεὶ δὲ ἀντιστρέφει τὸ στερητικόν, οὐδὲ τὸ Ν οὐδενὶ τῷ Ξ ὑπάρξει, ὥστʼ ἔσται ὁ αὐτὸς συλλογισμός.
Again, if M belongs to every N, but to no Ξ, neither will Ξ belong to any N (for if M belongs to no Ξ, neither will Ξ belong to any M; but M belonged to every N; therefore Ξ will belong to no N; for the first figure has come about again); and since the privative converts, neither will N belong to any Ξ, so that there will be the same syllogism.
ἔστι δὲ δεικνύναι ταῦτα καὶ εἰς τὸ ἀδύνατον ἄγοντας.
It is also possible to show these things by leading to the impossible.
ὅτι μὲν οὖν γίνεται συλλογισμὸς οὕτως ἐχόντων τῶν ὅρων, φανερόν, ἀλλʼ οὐ τέλειος· οὐ γὰρ μόνον ἐκ τῶν ἐξ ἀρχῆς ἀλλὰ καὶ ἐξ ἄλλων ἐπιτελεῖται τὸ ἀναγκαῖον.
That a syllogism, then, comes about when the terms are so related, is clear, but not a perfect one; for the necessity is completed not only from the things assumed at the beginning, but also from other things.
ἐὰν δὲ τὸ Μ παντὸς τοῦ Ν καὶ τοῦ Ξ κατηγορῆται, οὐκ ἕσται συλλογισμός.
But if M is predicated of every N and of every Ξ, there will not be a syllogism.
ὅροι τοῦ ὑπάρχειν οὐσία—ζῷον—ἄνθρωπος τοῦ μὴ ὑπάρχειν οὐσία—ζῷον—ἀριθμός· μέσον οὐσία.
Terms of belonging: substance—animal—man; of not belonging: substance—animal—number; middle: substance.
οὐδʼ ὅταν μήτε τοῦ μήτε τοῦ Ξ μηδενὸς κατηγορῆται τὸ Μ. ὅροι τοῦ ὑπάρχειν γραμμή—ζῷον—ἄνθρωπος, τοῦ μὴ ὑπάρχειν γραμμή— ζῷον—λίθος.
Nor when M is predicated of no N and of no Ξ. Terms of belonging: line—animal—man; of not belonging: line—animal—stone.
φανερὸν οὖν ὅτι ἂν ᾖ συλλογισμὸς καθόλου τῶν ὅρων ὄντων, ἀνάγκη τοὺς ὅρους ἔχειν ὡς ἐν ἀρχῇ εἴπομεν· ἄλλως γὰρ ἐχόντων οὐ γίνεται τὸ ἀναγκαῖον.
It is clear, then, that if there is a universal syllogism when the terms are universal, the terms must be related as we said at the beginning; for if they are otherwise related, the necessity does not come about.
Ἐὰν δὲ πρὸς τὸν ἕτερον ᾖ καθόλου τὸ μέσον, ὅταν μὲν πρὸς τὸν μείζω γένηται καθόλου ἢ κατηγορικῶς ἢ στερητικῶς, (πρὸς δὲ τὸν ἐλάττω κατὰ μέρος καὶ ἀντικειμένως τῷ καθόλου (λέγω δὲ τὸ ἀντικειμένως, εἰ μὲν τὸ καθόλου στερητικόν, τὸ ἐν. μέρει καταφατικόν· εἰ δὲ κατηγορικὸν τὸ καθόλου, τὸ ἐν μέρει στερτητικόν), ἀνάγκτη γίνεσθαι συλλογισμὸν στερητικὸν κατὰ μέρος.
But if the middle is universal in relation to one of the extremes, whenever it becomes universal in relation to the major, whether affirmatively or privatively, (and particular in relation to the minor and opposite to the universal (and by "opposite" I mean: if the universal is privative, the particular is affirmative; and if the universal is affirmative, the particular is privative)), there must be a particular privative syllogism.
εἰ γὰρ τὸ Μ τῷ μὲν μηδενὶ τῷ δὲ Ξ τινὶ ὑπάρχει, ἀνάγκη τὸ Ν τινὶ τῷ Ξ μὴ ὑπάρχειν.
For if M belongs to no N, but to some Ξ, N must not belong to some Ξ.
ἐπεὶ γὰρ ἀντιστρέφει τὸ στερητικόν, οὐδενὶ τῷ ὑπάρξει τὸ Ν· τὸ δέ γε Μ ὑπέκειτο τινὶ τῷ Ξ ὑπάρχειν· ὥστε τὸ Ν τινὶ τῷ Ξ οὐχ ὑπάρξει· γίνεται γὰρ συλλογισμὸς διὰ τοῦ πρώτου σχήματος.
For since the privative converts, N will belong to no M; but M was assumed to belong to some Ξ; so that N will not belong to some Ξ; for a syllogism comes about through the first figure.
πάλιν εἰ τῷ μὲν Ν παντὶ τὸ Μ, τῷ δὲ Ξ τινὶ μὴ ὑπάρχει, ἀνάγκη τὸ Ν τινὶ τῷ Ξ μὴ ὑπάρχειν· εἰ γὰρ παντὶ ὑπάρχει, κατηγορεῖται δὲ καὶ τὸ Μ παντὸς τοῦ Ν, ἀνάγκη τὸ παντὶ τῷ Ξ ὑπάρχειν· ὑπέκειτο δὲ τινὶ μὴ ὑπάρχειν.
Again, if M belongs to every N, but does not belong to some Ξ, N must not belong to some Ξ; for if it belongs to all, and M is also predicated of every N, M must belong to every Ξ; but it was assumed not to belong to some.
καὶ εἰ τὸ Μ τῷ μὲν Ν παντὶ ὑπάρχει τῷ δὲ Ξ μὴ παντί, ἔσται συλλογισμὸς ὅτι οὐ παντὶ τῷ Ξ τὸ Ν· ἀπόδειξις δʼ ἡ αὐτή.
And if M belongs to every N, but not to every Ξ, there will be a syllogism that N does not belong to every Ξ; and the proof is the same.
ἐὰν δὲ τοῦ μὲν Ξ παντὸς τοῦ δὲ Ν μὴ παντὸς κατηγορῆται, οὐκ ἔσται συλλογισμός.
But if M is predicated of every Ξ, but not of every N, there will not be a syllogism.
ὅροι ζῷνο–οὐσία–κόραξ, ζῷον–λευκόνκόραξ.
Terms: animal—substance—crow; animal—white—crow.
οὐδʼ ὅταν τοῦ μὲν Ξ μηδενός, τοῦ δὲ Ν τινός.
Nor when M is predicated of no Ξ, but of some N.
ὅροι τοῦ ὑπάρχειν ζῷον–οὐσία–μονάς, τοῦ μὴ ὑπάρχειν ζῷον–οὐσίαἐπιστήμη.
Terms of belonging: animal—substance—unit; of not belonging: animal—substance—science.