§1.2.4#1Ἔσται δὲ καὶ ἐν τῷ ἐσχάτῳ σχήματι διὰ ψευδῶν ἀληθές, καὶ ἀμφοτέρων ψευδῶν οὐσῶν ὅλων καὶ ἐπί τι ἑκατέρας, καὶ τῆς μὲν ἑτέρας ἀληθοῦς ὅλης τῆς δʼ ἑτέρας ψευδοῦς, καὶ τῆς μὲν ἐπί τι ψευδοῦς τῆς δʼ ὅλης ἀληθοῦς, καὶ ἀνάπαλιν, καὶ ὁσαχῶς ἄλλως ἐγχωρεῖ μεταλαβεῖν τὰς προτάσεις.
And in the last figure also a true [conclusion] will be [drawn] through false [premises], both when both premises are wholly false, and when each is partially false, and when one is wholly true and the other wholly false, and when one is partially false and the other wholly true, and vice versa, and in whatever other ways it is possible to vary the premises.
οὐδὲν γὰρ κωλύει μήτε τὸ Α μήτε τὸ Β μηδενὶ τῷ Γ ὑπάρχειν, τὸ μέντοι Α τινὶ τῷ Β ὑπάρχειν, οἷον οὔτʼ ἄνθρωπος οὔτε πεζὸν οὐδενὶ ἀψύχῳ ἕπεται, ἄνθρωπος μέντοι τινὶ πεζῷ ὑπάρχει.
For nothing prevents both A and B from belonging to no C, while A belongs to some B, for instance, neither man nor pedestrian follows any inanimate thing, but man belongs to some pedestrian.
ἐὰν οὖν ληφθῇ τὸ Α καὶ τὸ Β παντὶ τῷ Γ ὑπάρχειν, αἱ μὲν προτάσεις ὅλαι ψευδεῖς, τὸ δὲ συμπέρασμα ἄληθές.
If, therefore, it should be assumed that A and B belong to every C, the premises indeed are wholly false, but the conclusion is true.
ὡσαύτως δὲ καὶ τῆς μὲν στερητικῆς τῆς δὲ καταφατικῆς οὔσης.
And similarly also when one is negative and the other affirmative.
ἐγχωρεῖ γὰρ τὸ μὲν Β μηδενὶ τῷ Γ ὑπάρχειν, τὸ δὲ παντί, καὶ τὸ Α τινὶ τῷ Β μὴ ὑπάρχειν, οἷον τὸ μέλαν οὐδενὶ κύκνῳ, ζῷον δὲ παντί, καὶ τὸ ζῷον οὐ παντὶ μέλανι.
For it is possible for B to belong to no C, and [A] to every [C], while A does not belong to some B, for instance, black belongs to no swan, but animal to every, and animal does not belong to every black thing.
ὥστ᾿ ἂν ληφθῇ τὸ μὲν Β παντὶ τῷ Γ, τὸ δὲ Α μηδενί, τὸ τινὶ τῷ Β οὐχ ὑπάρξει· καὶ τὸ μὲν συμπέρασμα ἀληθές, αἱ δὲ προτάσεις ψευδεῖς.
So that if it should be assumed that B belongs to every C, and A to no C, [A] will not belong to some B; and the conclusion indeed is true, but the premises are false.
καὶ εἰ ἐπί τι ἑκατέρα ψευδής, ἔσται τὸ συμπέρασμα ἀληθές.
And if each is partially false, the conclusion will be true.
οὐδὲν γὰρ κωλύει καὶ τὸ Α καὶ τὸ Β τινὶ τῷ Γ ὑπάρχειν, καὶ τὸ Α τινὶ τῷ. Β, οἷον τὸ λευκὸν καὶ τὸ καλὸν τινὶ ζῴῳ ὑπάρχει, καὶ τὸ λευκὸν τινὶ καλῷ.
For nothing prevents both A and B from belonging to some C, and A to some B, for instance, white and beautiful belong to some animal, and white to some beautiful.
ἐὰν οὖν τεθῇ τὸ Α καὶ τὸ Β παντὶ τῷ Γ ὑπάρχειν, αἱ μὲν προτάσεις ἐπί τι ψευδεῖς, τὸ δὲ συμπέρασμα ἀληθές.
If, therefore, it should be assumed that A and B belong to every C, the premises indeed are partially false, but the conclusion is true.
καὶ στερητικῆς δὲ τῆς Γ τιθεμένης ὁμοίως.
And similarly also when the negative [premise] is assumed.
οὐδὲν γὰρ κωλύει τὸ μὲν τινὶ τῷ Γ μὴ ὑπάρχειν, τὸ δὲ Β τινὶ ὑπάρχειν, καὶ τὸ Α τῷ Β μὴ παντὶ ὑπάρχειν, οἷον τὸ λευκὸν τινὶ ζῴῳ οὐχ ὑπάρχει, τὸ δὲ καλὸν τινὶ ὑπάρχει, καὶ τὸ λευκὸν οὐ παντὶ καλῷ.
For nothing prevents one from not belonging to some C, while B belongs to some, and A from not belonging to every B, for instance, white does not belong to some animal, but beautiful belongs to some, and white does not belong to every beautiful.
ὥστʼ ἂν ληφθῇ τὸ μὲν Α μηδενὶ τῷ Γ, τὸ δὲ Β παντί, ἀμφότεραι μὲν αἱ προτάσεις ἐπί τι ψευδεῖς, τὸ δὲ συμπέρασμα ἀληθές.
So that if it should be assumed that A belongs to no C, and B to every C, both premises indeed are partially false, but the conclusion is true.
Ὡσαύτως δὲ καὶ τῆς μὲν ὅλης ψευδοῦς τῆς δʼ ὅλης ἀληθοῦς λαμβανομένης.
And similarly also when one is wholly false and the other wholly true.
ἐγχωρεῖ γὰρ καὶ τὸ καὶ τὸ Β παντὶ τῷ Γ ἕπεσθαι, τὸ μέντοι Α τινὶ τῷ Β μὴ ὑπάρχειν, οἷον ζῷον καὶ λευκὸν παντὶ κύκνῳ ἕπεται, τὸ μέντοι ζῷον οὐ παντὶ ὑπάρχει λευκῷ.
For it is possible for both [A] and B to follow every C, while A does not belong to some B, for instance, animal and white follow every swan, but animal does not belong to every white thing.
τεθέντων οὖν ὅρων τοιούτων, ἐὰν ληφθῇ τὸ μὲν Β ὅλῳ τῷ Γ ὑπάρχειν, τὸ δὲ Α ὅλῳ μὴ ὑπάρχειν. ἡ μὲν Β Γ ὅλη ἔσται ἀληθής, ἡ δὲ Α Γ ὅλη ψευδής, καὶ τὸ συμπέρασμα ἀληθές.
If, therefore, such terms being assumed, it should be assumed that B belongs to the whole of C, and A to none of C, the premise BC indeed will be wholly true, and AC wholly false, and the conclusion true.
ὁμοίως δὲ καὶ εἰ τὸ μὲν Β Γ ψεῦδος, τὸ δὲ Α Γ ἀληθές· οἱ γὰρ αὐτοὶ ὅροι πρὸς τὴν ἀπόδειξιν.
And similarly also if BC is false, and AC true; for the same terms [serve] for the proof.
ἀλλὰ καὶ εἰ ἀμφότεραι λαμβάνοιντο καταφατικαί.
Moreover, also if both should be assumed as affirmative.
οὐδὲν γὰρ κωλύει τὸ μὲν Β παντὶ τῷ Γ ἕπεσθαι, τὸ δὲ ὅλῳ μὴ ὑπάρχειν, καὶ τὸ τινὶ τῷ Β ὑπάρχειν, οἷον κύκνῳ παντὶ ζῷον, μέλαν δʼ οὐδενὶ κύκνῳ, καὶ τὸ μέλαν ὑπάρχει τινὶ ζῴῳ.
For nothing prevents B from following every C, and [A] from belonging to none of [C], while [A] belongs to some B, for instance, animal follows every swan, but black follows no swan, and black belongs to some animal.
ὥστʼ ἂνληφθῇ τὸ Α καὶ τὸ Β παντὶ τῷ Γ ὑπάρχειν, ἡ μὲν Β Γ ὅλη ἀληθής, ἡ δὲ Γ ὅλη ψευδής, καὶ τὸ συμπέρασμα ἀληθές.
So that if it should be assumed that A and B belong to every C, the premise BC indeed is wholly true, and the premise [A]C is wholly false, and the conclusion is true.
ὁμοίως δὲ καὶ τῆς Γ ληφθείσης ἀληθοῦς· διὰ γὰρ τῶν αὐτῶν ὅρων ἡ ἀπόδειξις.
And similarly also when the premise [A]C is assumed as true; for the proof is through the same terms.