§1.1.6#1Ἐὰν δὲ τῷ αὐτῷ τὸ μὲν παντὶ τὸ δὲ μηδενὶ ὑπάρχῃ, ἢ ἄμφω παντὶ ἢ μηδενί, τό μὲν σχῆμα τὸ τοιοῦτον καλῶ τρίτον, μέσον δʼ ἐν αὐτῷ λέγω καθʼ οὗ ἄμφω τὰ κατηγορούμενα, ἄκρα δὲ τὰ κατηγορούμενα, μεῖζον δʼ ἄκρον τὸ πορρώτερον τοῦ μέσου, ἔλαττον δὲ τὸ ἐγγύτερον.
But if to the same thing one belongs to all and the other to none, or both to all or to none, I call such a figure the third; and by the middle in it I mean that of which both the predicates are predicated, and by extremes the predicates, and by the major extreme that which is further from the middle, and by the minor that which is nearer.
τίθεται δὲ τὸ μέσον ἔξω μὲν τῶν ἄκρων, ἔσχατον δὲ τῇ θέσει.
And the middle is placed outside the extremes, and last by position.
τέλειος μὲν οὖν οὐ γίνεται συλλογισμὸς οὐδʼ ἐν τούτῳ τῷ σχήματι, δυνατὸς δʼ ἔσται καὶ καθόλου καὶ μὴ καθόλου τῶν ὅρων ὄντων πρὸς τὸ μέσον.
Now, a perfect syllogism does not come about in this figure either, but it will be possible whether the terms are universal or not universal in relation to the middle.
Καθόλου μὲν οὖν ὄντων, ὅταν καὶ τὸ καὶ τὸ Ρ παντὶ τῷ Σ ὑπάρχῃ, ὅτι τινὶ τῷ Ρ τὸ ὑπάρξει ἐξ ἀνάγκης· ἐπεὶ γὰρ ἀντιστρέφει τὸ κατηγορικόν, ὑπάρξει τὸ Σ τινὶ τῷ Ρ, ὥστʼ ἐπεὶ τῷ μὲν Σ παντὶ τὸ Π, τῷ δὲ Ρ τινὶ τὸ Σ, ἀνάγκη τὸ Π τινὶ τῶ Ρ ὑπάρχειν·
Now, when they are universal, and both Π and Ρ belong to every Σ, Π will of necessity belong to some Ρ; for since the affirmative premise converts, Σ will belong to some Ρ, so that since Π belongs to every Σ, and Σ to some Ρ, Π must of necessity belong to some Ρ; for a syllogism comes about through the first figure.
γίνεται γὰρ συλλογισμὸς διὰ τοῦ πρώτου σχήματος. ἔστι δὲ καὶ διὰ τοῦ ἀδυνάτου καὶ τῷ ἐκθέσθαι ποιεῖν τὴν ἀπόδειξιν· εἰ γὰρ ἄμφω παντὶ τῷ Σ ὑπάρχει, ἂν ληφθῇ τι τῶν Σ οἷον τὸ Ν, τούτῳ καὶ τὸ Π καὶ τὸ Ρ ὑπάρξει, ὥστε τινὶ τῷ Ρ τὸ Π ὑπάρξει.
And it is also possible to make the demonstration both through the impossible and by exposition; for if both belong to every Σ, if some Σ is taken, e.g. Ν, both Π and Ρ will belong to this, so that Π will belong to some Ρ.
καὶ ἂν τὸ μὲν Ρ παντὶ τῷ Σ, τὸ δὲ Π μηδενὶ ὑπάρχῃ, ἔσται συλλογισμὸς ὅτι τὸ τινὶ τῷ Ρ οὐχ ὑπάρξει ἐξ ἀνάγκης· ὁ γὰρ αὐτὸς τρόπος τῆς ἀποδείξεως ἀντιστραφείσης τῆς Ρ Σ προτάσεως.
And if Ρ belongs to every Σ, but Π to no Σ, there will be a syllogism that Π will of necessity not belong to some Ρ; for there is the same method of demonstration when the Ρ-Σ premise is converted.
δειχθείη δʼ ἂν καὶ διὰ τοῦ ἀδυνάτου, καθάπερ ἐπὶ τῶν πρότερον.
And it would also be shown through the impossible, as in the previous cases.
ἐὰν δὲ τὸ μέν Ρ μηδενὶ τὸ δὲ παντὶ ὑπάρχῃ τῷ Σ, οὐκ ἔσται συλλογισμός.
But if Ρ belongs to no Σ, and Π to every Σ, there will not be a syllogism.
ὅροι τοῦ ὑπάρχειν ζῷον–ἵππος–ἄνθρωπος, τοῦ μὴ ὑπάρχειν ζῷονἄψυχον–ἄνθρωπος.
Terms for belonging: animal—horse—man; for not belonging: animal—inanimate—man.
οὐδʼ ὅταν ἄμφω κατὰ μηδενὸς τοῦ Σ λέγηται, οὐκ ἔσται συλλογισμός.
Nor when both are said of no Σ, will there be a syllogism.
ὅροι τοῦ ὑπάρχειν ζῷον–ἵπποςἄ—ψυχον, τοῦ μὴ ὑπάρχειν ἄνθρωπος–ἵππος–ἄψυχον· μέσον ἄψυχον.
Terms for belonging: animal—horse—inanimate; for not belonging: man—horse—inanimate; middle: inanimate.
φανερὸν οὖν καὶ ἐν τούτῳ τῷ σχήματι πότʼ ἔσται καὶ πότʼ οὐκ ἔσται συλλογισμός καθόλου τῶν ὅρων ὄντων.
It is clear, then, in this figure too, when there will and when there will not be a syllogism when the terms are universal.
ὅταν μὲν γὰρ ἀμφότεροι οἱ ὅροι ὦσι κατηγορικοί, ἔσται συλλογισμός ὅτι τινὶ ὑπάρχει τὸ ἄκρον τῷ ἄκρῳ, ὅταν δὲ στερητικοί, οὐκ ἔσται.
For when both terms are affirmative, there will be a syllogism that the extreme belongs to some of the other extreme; but when they are privative, there will not be.
ὅταν δʼ ὁ μὲν ᾖ στερητικὸς ὁ δὲ καταφατικός, ἐὰν μὲν ὁ μείζων γένηται στερητικὸς ἅτερος δὲ καταφατικός, ἔσται συλλογισμός ὅτι τινὶ οὐχ ὑπάρχει τὸ ἄκρον τῷ ἄκρῳ, ἐὰν δʼ ἀνάπαλιν, οὐκ ἔσται.
But when one is privative and the other affirmative, if the major becomes privative and the other affirmative, there will be a syllogism that the extreme does not belong to some of the other extreme; but if vice versa, there will not be.
Ἐὰν δʼ ὁ μὲν ᾖ καθόλου πρὸς τὸ μέσον ὁ δʼ ἐν μέρει, κατηγορικῶν μὲν ὄντων ἀμφοῖν ἀνάγκη γίνεσθαι συλλογισμόν, ἂν ὁποτεροσοῦν ᾖ καθόλου τῶν ὅρων.
And if one is universal in relation to the middle and the other particular, both being affirmative, a syllogism must come about, whichever of the terms is universal.
εἰ γὰρ τὸ μὲν Ρ παντὶ τῷ Σ τὸ δὲ Π τινί, ἀνάγκη τὸ Π τινὶ τῷ Ρ ὑπάρχειν.
For if Ρ belongs to every Σ and Π to some Σ, Π must of necessity belong to some Ρ.
ἐπεὶ γὰρ ἀντιστρέφει τὸ καταφατικόν, ὑπάρξει τὸ Σ τινὶ τῷ Π, ὥστʼ ἐπεὶ τὸ μὲν Ρ παντὶ τῷ Σ, τὸ δὲ Σ τινὶ τῷ Π, καὶ τὸ Ρ τινὶ τῷ Π ὑπάρξει· ὥστε τὸ τινὶ τῷ Ρ. πάλιν εἰ τὸ μὲν Ρ τινὶ τῷ Σ τὸ δὲ παντὶ ὑπάρχει, ἀνάγκη τὸ τινὶ τῷ Ρ ὑπάρχειν·
For since the affirmative converts, Σ will belong to some Π, so that since Ρ belongs to every Σ, and Σ to some Π, Ρ will also belong to some Π; so that Π will belong to some Ρ.
ὁ γὰρ αὐτός τρόπος τῆς ἀποδείξεως.
Again, if Ρ belongs to some Σ and Π to every Σ, Π must of necessity belong to some Ρ; for there is the same method of demonstration.
ἔστι δʼ ἀποδεῖξαι καὶ διὰ τοῦ ἀδυνάτου καὶ τῇ ἐκθέσει, καθάπερ ἐπὶ τῶν πρότερον.
And it is also possible to demonstrate both through the impossible and by exposition, as in the previous cases.