§1.2.3#1Ἐν δὲ τῷ μέσῳ σχήματι πάντως ἐγχωρεῖ διὰ ψευδῶν ἀληθὲς συλλογίσασθαι, καὶ ἁμφοτέρων τῶν προτάσεων ὅλων ψευδῶν λαμβανομένων καὶ ἐπί τι ἑκατέρας, καὶ τῆς μὲν ἀληθοῦς τῆς δὲ ψευδοῦς οὔσης ὁποτερασοῦν ψευδοῦς τιθεμένης, καὶ ἐν τοῖς καθόλου καὶ ἐπὶ τῶν ἐν μέρει συλλογισμῶν.
In the middle figure, it is thoroughly possible to syllogize what is true through false premises, both when both premises are assumed to be wholly false, and when each is false in part, and when one is true and the other false, whichever of them is placed as false, both in universal and in particular syllogisms.
εἰ γὰρ τὸ Α τῷ μὲν Β μηδενὶ ὑπάρχει τῷ δὲ Γ παντί, οἷον ζῷον λίθῳ μὲν οὐδενὶ ἵππῳ δὲ παντί, ἐὰν ἐναντίως τεθῶσιν αἰ προτάσεις καὶ ληφθῇ τὸ Α τῷ μὲν Β παντὶ τῷ δὲ Γ μηδενί, ἐκ ψευδῶν ὅλων τῶν προτάσεων ἀληθές ἔσται τὸ συμπέρασμα.
For if A belongs to no B and to every C (for instance, animal belongs to no stone and to every horse), if the premises should be assumed contrariwise and it should be assumed that A belongs to every B and to no C, the conclusion will be true from premises that are wholly false.
ὁμοίως δὲ καὶ εἰ τῷ μὲν Β παντὶ τῷ δὲ Γ μηδενὶ ὑπάρχει τὸ Α· ὁ γὰρ αὐτὸς ἔσται συλλογισμός.
And similarly also if A belongs to every B and to no C; for there will be the same syllogism.
Πάλιν εἰ ἡ μὲν ἑτέρα ὅλη ψευδὴς ἡ δʼ ἑτέρα ὅλη ἀληθής· οὐδὲν γὰρ κωλύει τὸ Α καὶ τῷ β καὶ τῷ Γ παντὶ ὑπάρχειν, τὸ μέντοι Β μηδενὶ τῷ Γ, οἷον τὸ γένος τοῖς μὴ ὑπʼ ἄλληλα εἴδεσιν.
Again, if one premise is wholly false and the other wholly true; for nothing prevents A from belonging to the whole of both B and C, while B belongs to no C, for instance, a genus in relation to species not subordinate to one another.
τὸ γὰρ ζῷον καὶ ἵππῳ παντὶ καὶ ἀνθρώπῳ, καὶ οὐδεὶς ἄνθρωπος ἵππος.
For animal belongs both to every horse and to every man, and no man is a horse.
ἐὰν οὖν ληφθῇ τῷ μὲν παντὶ τῷ δὲ μηδενὶ ὑπάρχειν, ἡ μὲν ὅλη ψευδὴς ἔσται ἡ δʼ ὅλη ἀληθής, καὶ τὸ συμπέρασμα ἀληθὲς πρὸς ὁποτερῳοῦν τεθέντος τοῦ στερητικοῦ.
If, therefore, it should be assumed to belong to the one wholly and to the other not at all, the one premise will be wholly false and the other wholly true, and the conclusion will be true, whichever of the premises the negative is placed with.
καὶ εἰ ἡ ἑτέρα ἐπί τι ψευδής, ἡ δʼ ἑτέρα ὅλη ἀληθής.
And if the one is false in part, and the other wholly true.
ἐγχωρεῖ γὰρ τὸ Α τῷ μὲν Β τινὶ ὑπάρχειν τῷ δὲ Γ παντί, τὸ μέντοι Β μηδενὶ τῷ Γ, οἷον ζῷον λευκῷ μὲν τινὶ κόρακι δὲ παντί, καὶ τὸ λευκὸν οὐδενὶ κόρακι.
For it is possible for A to belong to some B and to every C, while B belongs to no C, for instance, animal belongs to some white thing and to every raven, and white belongs to no raven.
ἐὰν οὖν ληφθῇ τὸ Α τῷ μὲν Β μηδενὶ τῷ δὲ Γ ὅλῳ ὑπάρχειν, ἡ μὲν Β πρότασις ἐπί τι ψευδής, ἡ δʼ Α Γ ὅλη ἀληθής, καὶ τὸ συμπέρασμα ἀληθές.
If, therefore, it should be assumed that A belongs to no B and to the whole of C, the premise B (AB) indeed will be false in part, and AC wholly true, and the conclusion true.
καὶ μετατιθεμένου δὲ τοῦ στερητικοῦ ὡσαύτως· διὰ γὰρ τῶν αὐτῶν ὅρων ἡ ἀπόδειξις.
And similarly also when the negative is transposed; for the proof is through the same terms.
καὶ εἰ ἡ καταφατικὴ πρότασις ἐπί τι ψευδής, ἡ δὲ στερητικὴ ὅλη ἀληθής.
And if the affirmative premise is false in part, and the negative wholly true.
οὐδὲν γὰρ κωλύει τὸ τῷ μὲν Β τινὶ ὑπάρχειν τῷ δὲ Γ ὅλῳ μὴ ὑπάρχειν, καὶ τὸ Β μηδενὶ τῷ Γ, οἷον τὸ ζῷον λευκῷ μὲν τινὶ πίττῃ δʼ οὐδεμιᾷ, καὶ τὸ λευκὸν οὐδεμιᾷ πίττῃ.
For nothing prevents [A] from belonging to some B and belonging to no C, and B from belonging to no C, for instance, animal belongs to some white thing and to no pitch, and white belongs to no pitch.
ὥστʼ ἐὰν ληφθῇ τὸ ὅλῳ τῷ Β ὑπάρχειν τῷ δὲ Γ μηδενί, ἡ μὲν Β ἐπί τι ψευδής, ἡ δʼ Γ ὅλη ἀληθής, καὶ τὸ συμπέρασμα ἀληθές.
So that if it should be assumed to belong to the whole of B and to no C, the premise B (AB) indeed will be false in part, and C (AC) wholly true, and the conclusion true.
καὶ εἰ ἀμφότεραι αἱ προτάσεις ἐπί τι ψευδεῖς, ἔσται τὸ συμπέρασμα ἀληθές.
And if both premises are false in part, the conclusion will be true.
ἐγχωρεῖ γὰρ τὸ Α καὶ τῷ β καὶ τῷ Γ τινὶ ὑπάρχειν, τὸ δὲ Β μηδενὶ τῷ Γ, οἷον ζῷον καὶ λευκῷ τινὶ καὶ μέλανί τινι, τὸ δὲ λευκὸν οὐδενὶ μέλανι.
For it is possible for A to belong both to some B and to some C, while B belongs to no C, for instance, animal belongs both to some white thing and to some black thing, but white belongs to no black thing.
ἐὰν οὖν ληφθῇ τὸ Α τῷ μὲν Β παντὶ τῷ δὲ Γ μηδενί, ἄμφω μὲν αἱ προτάσεις ἐπί τι ψευδεῖς, τὸ δὲ συμπέρασμα ἀληθές.
If, therefore, it should be assumed that A belongs to every B and to no C, both premises indeed will be false in part, but the conclusion true.
ὁμοίως δὲ καὶ μετατεθείσης τῆς στερητικῆς διὰ τῶν αὐτῶν ὅρων.
And similarly also when the negative is transposed, through the same terms.