§2.1.17Ἐν δὲ τοῖς μὴ ἀτόμως ὑπάρχουσιν, ὅταν μὲν διὰ τοῦ οἰκείου μέσου γίνηται τοῦ ψεύδους ὁ συλλογισμός, οὐχ οἷόν τε ἀμφοτέρας ψευδεῖς εἶναι τὰς προτάσεις, ἀλλὰ μόνον τὴν πρὸς τῷ μείζονι ἄκρῳ. (λέγω δʼ οἰκεῖον μέσον διʼ οὗ γίνεται τῆς ἀντιφάσεως ὁ συλλογισμός. ) ὑπαρχέτω γὰρ τὸ τῷ Β διὰ μέσου τοῦ Γ ἐπεὶ οὖν ἀνάγκη τὴν Γ Β καταφατικὴν λαμβάνεσθαι συλλογισμοῦ γινομένου, δῆλον ὅτι ἀεὶ αὕτη ἔσται ἀληθής·
But in those which do not belong without middle, when the syllogism of the falsehood is made through the appropriate middle, it is not possible for both premises to be false, but only the one at the major term. (I call "appropriate middle" that through which the syllogism of the contradiction is made.) For let A belong to B through the middle C; since, then, it is necessary for the C-B premise to be taken as affirmative if there is to be a syllogism, it is clear that this will always be true; for it is not transposed.
οὐ γὰρ ἀντιστρέφεται. ἡ δὲ Γ ψευδής· ταύτης γὰρ ἀντιστρεφομένης ἐναντίος γίνεται ὁ συλλογισμός.
But the C-A premise is false; for when this is transposed the contrary syllogism is produced.
ὁμοίως δὲ καὶ εἰ ἐξ ἄλλης συστοιχίας ληφθείη τὸ μέσον, οἷον τὸ Δ εἰ καὶ ἐν τῷ ὅλῳ ἐστι καὶ κατὰ τοῦ Β κατηγορεῖται παντός· ἀνάγκη γὰρ τὴν μὲν Δ Β πρότασιν μένειν, τὴν δʼ ἑτέραν ἀντιστρέφεσθαι, ὥσθ᾿ ἡ μὲν ἀεὶ ἀληθής, ἡ δʼ ἀεὶ ψευδής.
And similarly if the middle should be taken from another coordinate series, as D, if indeed it is both in the whole and is predicated of all B; for it is necessary for the D-B premise to remain, and for the other to be transposed, so that one is always true, and the other always false.
καὶ σχεδὸν ἥ γε τοιαύτη ἀπάτη ἡ αὐτή ἐστι τῇ διὰ τοῦ οἰκείου μέσου.
And nearly such deception is the same as that through the appropriate middle.
ἐὰν δὲ μὴ διὰ τοῦ οἰκείου μέσου γίνηται ὁ συλλογισμός, ὅταν μὲν ὑπὸ τὸ Α ᾖ τὸ μέσον, τῷ δὲ Β μηδενὶ ὑπάρχῃ, ἀνάγκη ψευδεῖς εἶναι ἀμφοτέρας.
But if the syllogism is not made through the appropriate middle, when the middle is under A, but belongs to no B, it is necessary for both premises to be false.
ληπτέαι γὰρ ἐναντίως ἢ ὡς ἔχουσιν αἱ προτάσεις, εἰ μέλλει συλλογισμὸς ἔσεσθαι· οὕτω δὲ λαμβανομένων ἀμφότεραι γίνονται ψευδεῖς.
For they must be taken in the contrary way to how they are, if indeed there is to be a syllogism; and when they are taken in this way, both become false.
οἷον εἰ τὸ μὲν ὅλῳ τῷ Δ ὑπάρχει, τὸ δὲ Δ μηδενὶ τῶν Β· ἀντιστραφέντων γὰρ τούτων συλλογισμός τʼ ἔσται καὶ αἱ προτάσεις ἀμφότεραι ψευδεῖς.
As when A belongs to the whole of D, but D belongs to none of the B; for when these are transposed there will be a syllogism and both premises will be false.
ὅταν δὲ μὴ ᾖ ὑπὸ τὸ τὸ μέσον, οἷον τὸ Δ, ἡ μὲν Α Δ ἀληθὴς ἔσται, ἡ δὲ Δ Β ψευδής.
But when the middle, as D, is not under A, the A-D premise will be true, and the D-B false.
ἡ μὲν γὰρ Α Δ ἀληθής, ὅτι οὐκ ἦν ἐν τῷ Α τὸ Δ, ἡ δὲ Δ ψευδής, ὅτι εἰ ἦν ἀληθής, κἂν τὸ συμπέρασμα ἦν ἀληθές· ἀλλʼ ἦν ψεῦδος.
For the A-D is true because D was not in A, but the D-B is false because if it were true, the conclusion would also be true; but it was false.
Διὰ δὲ τοῦ μέσου σχήματος γινομένης τῆς ἀπάτης, ἀμφοτέρας μὲν οὐκ ἐνδέχεται ψευδεῖς εἶναι τὰς προτάσεις ὅλας (ὅταν γὰρ τὸ Β ὑπὸ τὸ Α, οὐδὲν ἐνδέχεται τῷ μὲν παντὶ τῷ δὲ μηδενὶ ὑπάρχειν, καθάπερ ἐλέχθη καὶ πρότερον), τὴν ἑτέραν δʼ ἐγχωρεῖ, καὶ ὁποτέραν ἔτυχεν.
But when the deception is made through the middle figure, it is not possible for both premises to be false as wholes (for when B is under A, it is not possible for anything to belong to all of one but to none of the other, as was also said before), but it is possible for one of them to be, and whichever it may happen to be.
εἰ γὰρ τὸ Γ καὶ τῷ Α καὶ τῷ Β ὑπάρχει, ἐὰν ληφθῇ τῷ μὲν Α ὑπάρχειν τῷ δὲ Β μὴ ὑπάρχειν, ἡ μὲν Γ Α ἀληθὴς ἔσται, ἡ δʼ ἑτέρα ψευδής.
For if C belongs both to A and to B, if it is assumed to belong to A but not to B, the C-A premise will be true, and the other false.
πάλιν δʼ εἰ τῷ μὲν Β ληφθείη τὸ ὑπάρχον, τῷ δὲ Α μηδενί, ἡ μὲν Γ Β ἀληθὴς ἔσται, ἡ δʼ ἑτέρα ψευδής.
And again, if what belongs is assumed of B, but to no A, the C-B premise will be true, and the other false.
Ἐὰν μὲν οὖν στερητικὸς ᾖ τῆς ἀπάτης ὁ συλλογισμός, εἴρηται πότε καὶ διὰ τίνων ἔσται ἡ ἀπάτη· ἐὰν δὲ καταφατικός, ὅταν μὲν διὰ τοῦ οἰκείου μέσου, ἀδύνατον ἀμφοτέρας εἶναι ψευδεῖς· ἀνάγκη γὰρ τὴν Γ Β μένειν, εἴπερ ἔσται συλλογισμός, καθάπερ ἐλέχθη καὶ πρότερον.
If, then, the syllogism of the deception is privative, it has been said when and through what things the deception will be; but if it is affirmative, when it is through the appropriate middle, it is impossible for both premises to be false; for it is necessary for C-B to remain, if indeed there is to be a syllogism, as was also said before.
ὥστε ἡ Α Γ ἀεὶ ἔσται ψευδής· αὕτη γάρ ἐστιν ἡ ἀντιστρεφομέντη.
So that A-C will always be false; for this is the one that is transposed.
ὁμοίως δὲ καὶ εἰ ἐξ ἄλλης συστοιχίας λαμβάνοιτο τὸ μέσον, ὥσπερ ἐλέχθη καὶ ἐπὶ τῆς στερητικῆς ἀπάτης· ἀνάγκη γὰρ τὴν μὲν Δ Β μένειν, τὴν δʼ Δ ἀντιστρέφεσθαι, καὶ ἡ ἀπάτη ἡ αὐτὴ τῇ πρότερον.
And similarly if the middle should be taken from another coordinate series, just as was said also in the case of the privative deception; for it is necessary for the D-B to remain, and for the A-D to be transposed, and the deception is the same as the former.
ὅταν δὲ μὴ διὰ τοῦ οἰκείου, ἐὰν μὲν ᾖ τὸ Δ ὑπὸ τὸ Α, αὕτη μὲν ἔσται ἀληθής, ἡ ἑτέρα δὲ ψευδής· ἐγχωρεῖ γὰρ τὸ Α πλείοσιν ὑπάρχειν ἃ οὐκ ἔστιν ὑπʼ ἄλληλα.
But when it is not through the appropriate middle, if D is under A, this indeed will be true, and the other false; for it is possible for A to belong to several things which are not under one another.
ἐὰν δὲ μὴ ᾖ τὸ Δ ὑπὸ τὸ Α, αὕτη μὲν ἀεὶ δῆλον ὅτι ἔσται ψευδής (καταφατικὴ γὰρ λαμβάνεται), τὴν δὲ Δ Β ἐνδέχεται καὶ ἀληθῆ εἶναι καὶ ψευδῆ· οὐδὲν γὰρ κωλύει τὸ μὲν Α τῷ Δ μηδενὶ ὑπάρχειν, τὸ δὲ Δ τῷ Β παντί, οἷον ζῷον ἐπιστήμῃ, ἐπιστήμη δὲ μουσικῇ.
But if D is not under A, this indeed will always clearly be false (for it is taken as affirmative), but the D-B can be both true and false; for nothing prevents A from belonging to no D, but D to all B, as animal to science, and science to music.
οὐδʼ αὖ μήτε τὸ Α μηδενὶ τῶν Δ μήτε τὸ Δ μηδενὶ τῶν Β.
Ποσαχῶς μὲν οὖν καὶ διὰ τίνων ἐγχωρεῖ γίνεσθαι τὰς κατὰ συλλογισμόν ἀπάτας ἔν τε τοῖς ἀμέσοις καὶ ἐν τοῖς διʼ ἀποδείξεως, φανερόν.
Nor again does anything prevent A from belonging to no D, nor D to no B. How many ways, then, and through what things deception according to syllogism can occur both in immediate things and in those through demonstration, is clear.