§1.2.10Ἐπὶ δὲ τοῦ τρίτου σχήματος ὅταν μὲν ἐναντίως ἀντιστρέφηται τὸ συμπέρασμα, οὐδετέρα τῶν προτάσεων ἀναιρεῖται κατʼ οὐδένα τῶν συλλογισμῶν ὅταν δʼ ἀντικειμένως, ἀμφότεραι καὶ ἐν ἅποσιν.
In the third figure, when the conclusion is converted contrarily, neither of the premises is refuted in any of the syllogisms; but when contradictorily, both are refuted in all of them.
δεδείχθω γὰρ τὸ Α τινὶ τῷ Β ὑπάρχον, μέσον δʼ εἰλήφθω τὸ Γ, ἔστωσαν δὲ καθόλου αἰ προτάσεις.
For let A be shown to belong to some B, and let C be taken as the middle, and let the premises be universal.
οὐκοῦν ἐὰν ληφθῇ τὸ Α τινὶ τῷ Β μὴ ὑπάρχειν, τὸ δὲ Β παντὶ τῷ Γ οὐ γίνεται συλλογισμὸς τοῦ Α καίτοῦ Γ. οὐδʼ εἰ τὸ Α τῷ μὲν Β τινὶ μὴ ὑπάρχει, τῷ δὲ Γ παντί, οὐκ ἔσται τοῦ Β καὶ τοῦ Γ συλλογισμός.
Therefore, if A is assumed not to belong to some B, and B to all C, there is no syllogism of A and C. Nor if A does not belong to some B, but belongs to all C, will there be a syllogism of B and C.
ὁμοίως δὲ δειχθήσεται καὶ εἰ μὴ καθόλου αἰ προτάσεις.
And in like manner will it be shown also if the premises are not universal.
ἢ γὰρ ἀμφοτέρας ἀνάγκη κατὰ μέρος εἶναι διὰ τῆς ἀντιστροφῆς, ἢ τὸ καθόλου πρὸς τῷ ἐλάττονι ἄκρῳ γίνεσθαι· οὕτω δʼ οὐκ ἦν συλλογισμὸς οὔτʼ ἐν τῷ πρώτῳ σχήματι οὔτʼ ἐν τῷ μέσῳ.
For it is necessary either that both should become particular through the conversion, or that the universal should be at the minor extreme; but in this way there was no syllogism either in the first figure or in the middle.
ἐὰν δʼ ἀντικειμένως ἀντιστρέφηται, αἰ προτάσεις ἀναιροῦνται ἀμφότεραι.
But if it is converted contradictorily, both premises are refuted.
εἰ γὰρ τὸ Α μηδενὶ τῷ Β, τὸ δὲ Β παντὶ τῷ Γ, τὸ Α οὐδενὶ τῷ πάλιν εἰ τὸ Α τῷ μὲν Β μηδενί, τῷ δὲ Γ παντί, τὸ Β οὐδενὶ τῷ Γ. καὶ εἰ ἡ ἑτέρα μὴ καθόλου, ὡσαύτως.
For if A belongs to no B, and B to all C, A belongs to no C. Again, if A belongs to no B, but to all C, B belongs to no C. And if the one premise is not universal, likewise.
εἰ γὰρ τὸ Α μηδενὶ τῷ Β, τὸ δὲ Β τινὶ τῷ Γ, τὸ Α τινὶ τῷ Γ οὐχ ὑπάρξει· εἰ δὲ τὸ Α τῷ μὲν Β μηδενί, τῷ δὲ Γ παντί, οδενὶ τῷ Γ τὸ Β.
Ὁμοίως δὲ καὶ εἰ στερητικὸς ὁ συλλογισμός.
For if A belongs to no B, and B to some C, A will not belong to some C; but if A belongs to no B, and to all C, B belongs to no C. And in like manner also if the syllogism is negative.
δεδείχθω γὰρ τὸ Α τινὶ τῷ Β μὴ ὑπάρχον, ἔστω δὲ κατηγορικὸν μὲν τὸ Β Γ, ἀποφατικὸν δὲ τὸ Α Γ· οὕτω γὰρ ἐγίνετο ὁ συλλογισμός.
For let A be shown not to belong to some B, and let BC be affirmative, and AC negative; for in this way the syllogism was produced.
ὅταν μὲν οὖν τὸ ἐναντίον ληφθῇ τῷ συμπεράσματι, οὐκ ἔσται συλλογισμός.
When, therefore, the contrary to the conclusion is assumed, there will be no syllogism.
εἰ γὰρ τὸ Α τινὶ τῷ Β, τὸ δὲ Β παντὶ τῷ Γ, οὐκ ἦν συλλογισμὸς τοῦ καὶ τοῦ Γ. οὐδʼ εἰ τὸ τινὶ τῷ Β, τῷ δὲ Γ μηδενί, οὐκ ἦν τοῦ Β καὶ τοῦ Γ συλλογισμός.
For if A belongs to some B, and B to all C, there was no syllogism of A and C. Nor if A belongs to some B, and to no C, was there a syllogism of B and C.
ὥστε οὐκ ἀναιροῦνται αἱ προτάσεις.
So that the premises are not refuted.
ὅταν δὲ τὸ ἀντικείμενον, ἀναιροῦνται.
But when the contradictory is assumed, they are refuted.
εἰ γὰρ τὸ Α παντὶ τῷ Β καὶ τὸ Β τῷ Γ, τὸ Α παντὶ τῷ Γ· ἀλλʼ οὐδενὶ ὑπῆρχεν.
For if A belongs to all B, and B to C, A belongs to all C; but it belonged to none.
πάλιν εἰ τὸ παντὶ τῷ Β, τῷ δὲ Γ μηδενί, τὸ Β οὐδενὶ τῷ Γ· ἀλλὰ παντὶ ὑπῆρχεν.
Again, if A belongs to all B, and to no C, B belongs to no C; but it belonged to all.
ὁμοίως δὲ δείκνυται καὶ εἰ μὴ καθόλου εἰσὶν αἰ προτάσεις.
And in like manner is it shown also if the premises are not universal.
γίνεται γὰρ τὸ Γ καθόλου τε καὶ στερητικόν, θάτερον δʼ ἐπὶ μέρους καὶ κατηγορικόν.
For C becomes universal and negative, and the other particular and affirmative.
εἰ μὲν οὖν τὸ Α παντὶ τῷ Β, τὸ δὲ Β τινὶ τῷ Γ, τὸ Α τινὶ τῷ Γ συμβαίνει· ἀλλʼ οὐδενὶ ὑπῆρχεν.
If, then, A belongs to all B, and B to some C, it results that A belongs to some C; but it belonged to none.
πάλιν εἰ τὸ παντὶ τῷ Β, τῷ δὲ Γ μηδενί, τὸ Β οὐδενὶ τῷ Γ· ἔκειτο δὲ τινί.
Again, if A belongs to all B, and to no C, B belongs to no C; but it was assumed to belong to some.
εἰ δὲ τὸ τινὶ τῷ Β καὶ τὸ Β τινὶ τῷ Γ, οὐ γίνεται συλλογισμός· οὐδʼ εἰ τὸ τινὶ τῷ Β, τῷ δὲ Γ μηδενί, οὐδʼ οὕτως.
But if A belongs to some B, and B to some C, there is no syllogism; nor if A belongs to some B, and to no C, not even thus.
ὥστʼ ἐκείνως μὲν ἀναιροῦνται, οὕτω δʼ οὐκ ἀναιροῦνται αἱ προτάσεις.
So that in the former way the premises are refuted, but in the latter way they are not.
Φανερὸν οὖν διὰ τῶν εἰρημένων πῶς ἀντιστρεφομένου τοῦ συμπεράσματος ἐν ἑκάστῳ σχήματι γίνεται συλλογισμός, καὶ πότʼ ἐναντίος τῇ προτάσει καὶ πότʼ ἀντικείμενος, καὶ ὅτι ἐν μὲν τῷ πρώτῳ σχήματι διὰ τοῦ μέσου καὶ τοῦ ἐσχάτου γίνονται οἱ συλλογισμοί, καὶ ἡ μὲν πρὸς τῷ ἐλάττονι ἄκρῳ ἀεὶ διὰ τοῦ μέσου ἀναιρεῖται, ἡ δὲ πρὸς τῷ μείζονι διὰ τοῦ ἐσχάτου· ἐν δὲ τῷ δευτέρῳ διὰ τοῦ πρώτου καὶ τοῦ ἐσχάτου, ἡ μὲν πρὸς τῷ ἐλάττονι ἄκρῳ ἀεὶ διὰ τοῦ πρώτου σχήματος, ἡ δὲ πρὸς τῷ μείζονι διὰ τοῦ ἐσχάτου· ἐν δὲ τῷ τρίτῳ διὰ τοῦ πρώτου καὶ διὰ τοῦ μέσου, καὶ ἡ μὲν πρὸς τῷ μείζονι διὰ τοῦ πρώτου ἀεί, ἡ δὲ πρὸς τῷ ἐλάττονι διὰ τοῦ μέσου.
It is clear, then, from what has been said, how a syllogism is produced in each figure when the conclusion is converted, and when contrarily to the premise and when contradictorily, and that in the first figure the syllogisms are produced through the middle and the last figure, and the premise at the minor extreme is always refuted through the middle, and that at the major through the last; but in the second figure through the first and the last, the premise at the minor extreme always through the first figure, and that at the major through the last; and in the third figure through the first and through the middle, and the premise at the major always through the first, and that at the minor through the middle.