29 a τῶν προτάσεων ἀνάγκη τὸ Γ τινὶ τῷ Α μὴ ὑπάρχειν. ὁμοίως δὲ κἀπὶ τῶν ἑτέρων σχημάτων· ἀεὶ γὰρ γίγνεται διὰ τῆς ἀντιστροφῆς συλλογισμός. δῆλον δὲ καὶ ὅτι τὸ ἀδιόριστον ἀντὶ τοῦ κατηγορικοῦ τοῦ ἐν μέρει τιθέμενον τὸν αὐτὸν ποιήσει συλλογισμὸν ἐν ἅπασι τοῖς σχήμασιν.30
Φανερὸν δε καὶ ὅτι πάντες οἱ ἀτελεῖς συλλογισμοὶ τελειοῦνται διὰ τοῦ πρώτου σχήματος. ἢ γὰρ δεικτικῶς ἢ διὰ τοῦ ἀδυνάτου περαίνονται πάντες· ἀμφοτέρως δὲ γίγνεται τὸ πρῶτον σχῆμα, δεικτικῶς μὲν τελειουμένων, ὅτι διὰ τῆς ἀντιστροφῆς ἐπεραίνοντο πάντες, ἡ δ᾿ ἀντιστροφὴ τὸ πρῶτον 35ἐποίει σχῆμα, διὰ δὲ τοῦ ἀδυνάτου δεικνυμένων, ὅτι τεθέντος τοῦ ψευδοῦς ὁ συλλογισμὸς γίγνεται διὰ τοῦ πρώτου σχήματος· οἷον ἐν τῷ τελευταίῳ σχήματι, εἰ τὸ Α καὶ τὸ Β παντὶ τῷ Γ ὑπάρχει, ὅτι τὸ Α τινὶ τῷ Β ὑπάρχει· εἰ γὰρ μηδενί, τὸ δὲ Β παντὶ τῷ Γ, οὐδενὶ τῷ Γ τὸ Α· ἀλλ᾿ ἦν παντί. ὁμοίως δὲ καὶ ἐπὶ τῶν ἄλλων.29 b
Ἔστι δὲ καὶ ἀναγαγεῖν πάντας τοὺς συλλογισμοὺς εἰς τοὺς ἐν τῷ πρώτῳ σχήματι καθόλου συλλογισμούς. οἱ μὲν γὰρ ἐν τῷ δευτέρῳ φανερὸν ὅτι δι᾿ ἐκείνων τελειοῦνται, πλὴν οὐχ ὁμοίως πάντες, ἀλλ᾿ 5οἱ μὲν καθόλου τοῦ στερητικοῦ ἀντιστραφέντος, τῶν δ᾿ ἐν μέρει ἑκάτερος διὰ τῆς εἰς τὸ ἀδύνατον ἀπαγωγῆς· οἱ δ᾿ ἐν τῷ πρώτῳ οἱ κατὰ μέρος ἐπιτελοῦνται μὲν καὶ δι᾿ αὑτῶν, ἔστι δὲ καὶ διὰ
the premisses are converted it necessarily follows that C does not apply to some A.a Similarly too in the other figures, for we always get a syllogism by the process of conversion.b It is obvious also that in all the figures if the particular affirmative is replaced by the indefinite the result will be the same syllogism.
It is evident also that all imperfect syllogisms are All imperfect syllogisms are validated in the first figure. completed by means of the first figure. Tor all the conclusions are reached either by demonstration or by reduction ad impossibile, and in both cases we get the first figure: in the case of those which are completed by demonstration because, as we have seen, all the conclusions are reached by means of conversion, and the conversion produces the first figure; and in the case of those which are demonstrated by reduction ad impossibile because if a false premiss is assumed we get the syllogism by means of the first figure. E.g., in the last figure, if A and B apply to all C, we get a syllogismc to the effect that A applies to some B; for if it applies to no B, and B applies to all C, A applies to no C. But ex hypothesi it applies to all C. Similarly too in the other cases.
It is possible also to reduce all syllogisms to the All syllogisms reducible to the universal syllogisms of the first figure. universal syllogisms in the first figure. Those in the second figure are obviously completed by their help, but not all in a similar manner: the universal syllogisms are completed by the conversion of the negative statement, and each of the particular ones by a reduction ad impossibile. The particular syllogisms in the first figure are indeed completed by means of themselves, but it is possible also to prove them by means