§1.1.45#2Τῶν δʼ ἐν τῷ τελευταίῳ σχήματι συλλογισμῶν εἷς μόνος οὐκ ἀναλύεται εἰς τὸ πρῶτον, ὅταν μὴ καθόλου τεθῇ τὸ στερητικόν, οἱ δʼ ἄλλοι πάντες ἀναλύονται.
Of the syllogisms in the last figure, only one is not analyzed into the first, namely, when the negative is not posited as universal, but all the others are analyzed.
κατηγορείσθω γὰρ παντὸς τοῦ Γ ὁ Α καὶ τὸ Β· οὐκοῦν ἀντιστρέψει τὸ Γ πρὸς ἑκάτερον ἐπὶ μέρους·
For let A and B be predicated of all C; then C will convert with each of them particularly, and therefore will belong to some B.
ὑπάρχει ἄρα τινὶ τῷ Β. ὥστ᾿ ἔσται τὸ πρῶτον σχῆμα, εἰ τὸ μὲν Α παντὶ τῷ Γ, τὸ δὲ Γ τινὶ τῷ Β. καὶ εἰ τὸ μὲν παντὶ τῷ Γ, τὸ δὲ Β τινί, ὁ αὐτὸς λόγος· ἀντιστρέφει γὰρ πρὸς τὸ Β τὸ Γ. ἐὰν δὲ τὸ μὲν Β παντὶ τῷ Γ, τὸ δὲ Α τινὶ τῷ Γ, πρῶτος ὅρος. θετέος τὸ Β·
Thus there will be the first figure, if A belongs to all C, and C to some B. And if one belongs to all C, and the other to some C, it is the same argument; for C converts with B. But if B belongs to all C, and A to some C, we must posit B as the first term; for B belongs to all C, and C to some A, so that B belongs to some A.
τὸ γὰρ Β παντὶ τῷ Γ, τὸ δὲ Γ τινὶ τῷ Α, ὥστε τὸ Β τινὶ τῷ Α. ἐπεὶ δʼ ἀντιστρέφει τὸ ἐν μέρει, καὶ τὸ Α τινὶ τῷ ὑπάρξει.
And since the particular affirmative converts, A will also belong to some B.
καὶ εἰ στερητικὸς ὁ συλλογισμός, καθόλου τῶν ὅρων ὄντων, ὁμοίως ληπτέον. ὑπαρχέτω γὰρ τὸ Β παντὶ τῷ Γ, τὸ δὲ Α μηδενί·
And if the syllogism is negative, the terms being universal, it must be taken in the same way.
οὐκοῦν τινὶ τῷ Β ὑπάρξει τὸ Γ, τὸ δὲ Α οὐδενὶ τῷ Γ, ὥστʼ ἔσται μέσον τὸ Γ. ὁμοίως δὲ καὶ εἰ τὸ μὲν στερητικὸν καθόλου, τὸ δὲ κατηγορικὸν ἐν μέρει· τὸ μὲν γὰρ Α οὐδενὶ τῷ Γ, τὸ δὲ Γ τινὶ τῶν Β ὑπάρξει.
For let B belong to all C, and A to none; then C will belong to some B, and A to no C, so that C will be the middle. Similarly also if the negative is universal, and the affirmative particular; for A will belong to no C, and C will belong to some of the B's.
ἐὰν δʼ ἐν μέρει ληφθῇ τὸ στερητικόν, οὐκ ἔσται ἀνάλυσις, οἷον εἰ τὸ μὲν Β παντὶ τῷ Γ, τὸ δὲ τινὶ μὴ ὑπάρχει· ἀντιστραφέντος γὰρ τοῦ Β Γ ἀμφότεραι αἱ προτάσεις ἔσονται κατά μέρος.
But if the negative is taken as particular, there will be no analysis, for example, if B belongs to all C, and A does not belong to some; for when B and C are converted both premises will be particular.
Φανερὸν δὲ καὶ ὅτι πρὸς τὸ ἀναλύειν εἰς ἄλληλα τὰ σχήματα ἡ πρὸς τῷ ἐλάττονι ἄκρῳ πρότασις ἀντιστρεπτέα ἐν ἀμφοτέροις τοῖς σχήμασι· ταύτης γὰρ μετατιθεμένης ἡ μετάβασις ἐγίνετο.
It is also clear that for the purpose of analyzing the figures into one another, the premise at the minor extreme must be converted in both figures; for it was by this being transposed that the transition was made.
Τῶν δʼ ἐν τῷ μέσῳ σχήματι ἅτερος μὲν ἀναλύεται, ἅτερος δʼ οὐκ ἀναλύεται, εἰς τὸ τρίτον.
Of the syllogisms in the middle figure, one is analyzed into the third, and the other is not.
ὅταν μὲν γὰρ ᾖ τὸ καθόλου στερητικόν, ἀναλύεται.
For when the negative is universal, it is analyzed.
εἰ γὰρ τὸ Α μηδενὶ τῷ Β, τῷ δὲ Γ τινί, ἀμφότερα ὁμοίως ἀντιστρέφει πρὸς τὸ Α, ὥστε τὸ μὲν β οὐδενὶ τῷ Α, τὸ δὲ Γ τινί·
For if A belongs to no B, and to some C, both convert in the same way with A, so that B belongs to no A, and C to some A; therefore A is the middle.
μέσον ἄρα τὸ Α. ὅταν δὲ τὸ Α παντὶ τῷ Β, τῷ δὲ Γ τινὶ μὴ ὑπάρχῃ, οὐκ ἔσται ἀνάλυσις· οὐδετέρα γὰρ τῶν προτάσεων ἐκ τῆς ἀντιστροφῆς καθόλου.
But when A belongs to all B, and does not belong to some C, there will be no analysis; for neither of the premises becomes universal from the conversion.
Καὶ οἱ ἐκ τοῦ τρίτου δὲ σχήματος ἀναλυθήσονται εἰς τὸ μέσον, ὅταν ᾖ καθόλου τὸ στερητικόν, οἷον εἰ τὸ μηδενὶ τῷ Γ, τὸ δὲ Β τινὶ ἢ παντί.
And those from the third figure will also be analyzed into the middle when the negative is universal, for example, if A belongs to no C, and B to some or to all.
καὶ γὰρ τὸ Γ τῷ μὲν Α οὐδενί, τῷ δὲ Β τινὶ ὑπάρξει.
For C will belong to no A, and to some B.
ἐὰν δʼ ἐπὶ μέρους ᾖ τὸ στερητικόν, οὐκ ἀναλυθήσεται· οὐ γὰρ δέχεται ἀντιστροφὴν τὸ ἐν μέρει ἀποφατικόν.
But if the negative is particular, it will not be analyzed; for the particular negative does not admit of conversion.
Φανερὸν οὖν ὅτι οἱ αὐτοὶ συλλογισμοὶ οὐκ ἀναλύονται ἐν τούτοις τοῖς σχήμασιν οἵπερ οὐδʼ εἰς τὸ πρῶτον ἀνελύοντο, καὶ ὅτι εἰς τὸ πρῶτον· σχῆμα τῶν συλλογισμῶν ἀναγομένων οὗτοι μόνοι διὰ τοῦ ἀδυνάτου περαίνονται.
It is clear, therefore, that the very same syllogisms are not analyzed in these figures as were not analyzed into the first, and that when the syllogisms are reduced to the first figure, these alone are proved through the impossible.
Πῶς μὲν οὖν δεῖ τοὺς συλλογισμοὺς ἀνάγειν, καὶ ὅτι ἀναλύεται τὰ σχήματα εἰς ἄλληλα, φανερὸν ἐκ τῶν εἰρημένων.
How, then, we must reduce syllogisms, and that the figures are analyzed into one another, is clear from what has been said.