§2.1.16#1Ἄγνοια δʼ ἡ μὴ κατʼ ἀπόφασιν ἀλλὰ κατὰ διάθεσιν λεγομένη ἔστι μὲν ἡ διὰ συλλογισμοῦ γινομέντη ἀπάτη, αὕτη δʼ ἐν μὲν τοῖς πρώτως ὑπάρχουσιν ἢ μὴ ὑπάρχουσι συμβαίνει διχῶς· ἢ γὰρ ὅταν ἁπλῶς ὑπολάβῃ ὑπάρχειν ἢ μὴ ὑπάρχειν, ἢ ὅταν διὰ συλλογισμοῦ λάβῃ τὴν ὑπόληψιν.
Ignorance—which is called not according to negation but according to disposition—is the error coming about through a syllogism, and this, in the case of things that belong or do not belong primarily, happens in two ways: either when one simply supposes belonging or not belonging, or when one gets the supposition through a syllogism.
τῆς μὲν οὖν ἁπλῆς ὑπολήψεως ἁπλῆ ἡ ἀπάτη, τῆς δὲ διὰ συλλογισμοῦ πλείους.
Of simple supposition, then, the error is simple, but of that through a syllogism, there are more.
μὴ ὑπαρχέτω γὰρ τὸ Α μηδενὶ τῷ Β ἀτόμως· οὐκοῦν ἐὰν συλλογίζηται ὑπάρχειν τὸ Α τῷ Β, μέσον λαβὼν τὸ Γ, ἠπατημένος ἔσται διὰ συλἀογισμοῦ.
For let A not belong to any B atomically; if, therefore, one should syllogize that A belongs to B, taking C as the middle term, one will have been deceived through a syllogism.
ἐνδέχεται μὲν οὖν ἀμφοτέρας τὰς προτάσεις εἶναι ψευδεῖς, ἐνδέχεται δὲ τὴν ἑτέραν μόνον.
Now, it is possible for both premises to be false, and it is possible for only one of them to be so.
εἰ γὰρ μήτε τὸ μηδενὶ τῶν Γ ὑπάρχει μήτε τὸ Γ μηδενὶ τῶν Β, εἴληπται δʼ ἐκατέρα ἀνάπαλιν, ἄμφω ψευδεῖς ἔσονται.
For if A belongs to none of the C's, and C to none of the B's, and each has been taken the other way around, both will be false.
ἐγχωρεῖ δʼ οὕτως ἔχειν τὸ Γ πρὸς τὸ Α καὶ Β ὥστε μήτε ὑπὸ τὸ εἶναι μήτε καθόλου τῷ Β. τὸ μὲν γὰρ Β ἀδύνατον εἶναι ἐν ὅλῳ τινί (πρώτως γὰρ ἐλέγετο αὐτῷ τὸ μὴ ὑπάρχειν), τὸ δὲ οὐκ ἀνάγκη πᾶσι τοῖς οὖσιν εἶναι καθόλου, ὥστʼ ἀμφότεραι ψευδεῖς.
And it is possible for C to be so related to A and B that it is neither under A nor universal to B. For it is impossible for B to be in any whole (since its not belonging was said to be primary), and it is not necessary for A to be universal to all existing things, so that both premises are false.
ἀλλὰ καὶ τὴν ἑτέραν ἐνδέχεται ἀληθῆ λαμβάνειν, οὐ μέντοι ὁποτέραν ἔτυχεν, ἀλλὰ τὴν Α Γ· ἡ γὰρ Γ Β πρότασις ἀεὶ ψευδὴς ἔσται διὰ τὸ ἐν μηδενὶ εἶναι τὸ Β, τὴν δὲ Α Γ ἐγχωρεῖ, οἷον εἰ τὸ Α καὶ τῷ Γ καὶ τῷ Β ὑπάρχει ἀτόμως (ὅταν γὰρ πρώτως κατηγορῆται ταὐτὸ πλειόνων, οὐδέτερον ἐν οὐδετέρῳ ἔσται).
But it is also possible to take one of them as true—not, however, whichever it happens to be, but the A-C premise; for the C-B premise will always be false because B is in nothing, but the A-C premise is possible, as when A belongs both to C and to B atomically (for when the same thing is predicated primarily of more than one thing, neither will be in the other).
διαφέρει δʼ οὐδέν, οὐδʼ εἰ μὴ ἀτόμως ὑπάρχει.
And it makes no difference even if it does not belong atomically.
Ἡ μὲν οὖν τοῦ ὑπάρχειν ἀπάτη διὰ τούτων τε καὶ οὕτω γίνεται μόνως (οὐ γὰρ ἦν ἐν ἄλλῳ σχήματι τοῦ ὑπάρχειν συλλογισμός), ἡ δὲ τοῦ μὴ ὑπάρχειν ἔν τε τῷ πρώτῳ καὶ ἐν τῷ μέσῳ σχήματι.
The error of belonging, then, comes about through these things and in this way alone (for there was no syllogism of belonging in another figure), but that of not belonging comes about both in the first and in the middle figure.
πρῶτον οὖν εἴπωμεν ποσαχῶς ἐν τῷ πρώτῳ γίνεται, καὶ πῶς ἐχουσῶν τῶν προτάσεων.
First, then, let us say in how many ways it comes about in the first, and how the premises are related.
ἐνδέχεται μὲν οὖν ἀμφοτέρων ψευδῶν οὐσῶν, οἷον εἰ τὸ καὶ τῷ Γ καὶ τῷ Β ὑπάρχει ἀτόμως· ἐὰν γὰρ ληφθῇ τὸ μὲν τῷ Γ μηδενί, τὸ δὲ Γ παντὶ τῷ Β, ψευδεῖς αἱ προτάσεις.
It is possible, then, when both are false, as when A belongs both to C and to B atomically; for if it is assumed that A belongs to no C, and C to all B, the premises are false.
ἐνδέχεται δὲ καὶ τῆς ἑτέρας ψευδοῦς οὔσης, καὶ ταύτης ὁποτέρας ἔτυχεν.
And it is possible also when one of them is false, and whichever of them it happens to be.
ἐγχωρεῖ γὰρ τὴν μὲν Γ ἀληθῆ εἶναι, τὴν δὲ Γ Β ψευδῆ, τὴν μὲν Α Γ ἀληθῆ ὅτι οὐ πᾶσι τοῖς οὖσιν ὑπάρχει τὸ Α, τὴν δὲ Γ Β ψευδῆ ὅτι ἀδύνατον ὑπάρχειν τῷ Β τὸ Γ, ᾧ μηδενὶ ὑπάρχει τὸ Α· οὐ γὰρ ἔτι ἀληθὴς ἔσται ἡ Γ πρότασις· ἅμα δέ, εἰ καὶ εἰσὶν ἀμφότεραι ἀληθεῖς, καὶ τὸ συμπέρασμα ἔσται ἀληθές.
For it is possible for the A-C premise to be true, and the C-B premise false—the A-C true because A does not belong to all existing things, and the C-B false because C cannot belong to B, to which A belongs to none; for then the A-C premise would no longer be true (and at the same time, if both were true, the conclusion also would be true).