§1.1.46#2Καὶ ἐπὶ πολλῶν δέ, ὧν τοῖς μὲν ὑπάρχει τοῖς δʼ οὐχ ὑπάρχει ταὐτόν, ἡ μὲν ἀπόφασις ὁμοίως ἀληθεύοιτʼ ἄν, ὅτι· οὐκ ἔστι λευκὰ πάντα ἢ ὅτι οὐκ ἔστι λευκὸν ἕκαστον·
And in many cases where the same thing belongs to some but not to others, the negation would be equally true, namely, that not all things are white, or that not each thing is white; but that each thing is not-white, or all things are not-white, is false.
ὅτι δ᾿ ἐστὶν οὐ λευκὸν ἕκαστον ἢ πάντα ἐστὶν οὐ λευκά, ψεῦδος ὁμοίως δὲ καὶ τοῦ ἔστι πᾶν ζῷον λευκόν οὐ τὸ ἔστιν οὐ λευκὸν ἅπαν ζῷον ἀπόφασις (ἄμφω γὰρ ψευδεῖς), ἀλλὰ τὸ οὐκ ἔστι πᾶν ζῶον λευκόν. Ἐπεὶ δὲ δῆλον ὅτι ἕτερον σημαίνει τὸ ἔστιν οὐ λευκόν καὶ οὐκ ἔστι λευκόν, καὶ τὸ μὲν κατάφασις τὸ δʼ ἀπόφασις, φανερὸν ὡς οὐχ ὁ αὐτὸς τρόπος τοῦ δεικνύναι ἑκάτερον, οἷον ὅτι ὃ ἂν ᾖ ζῷον οὐκ ἔστι λευκὸν ἢ ἐνδέχεται μὴ εἶναι λευκόν, καὶ ὅτι ἀληθὲς εἰπεῖν μὴ λευκόν· τοῦτο γάρ ἐστιν εἶναι μὴ λευκόν. ἀλλὰ τὸ μὲν ἀληθὲς εἰπεῖν ἔστι λευκόν εἴτε μὴ λευκόν ὁ αὐτὸς κατασκευαστικῶς γὰρ ἄμφω διὰ τοῦ πρώτου δείκνυται σχήματος·
Similarly, too, the negation of "every animal is white" is not "every animal is not-white" (for both are false), but "not every animal is white." Since it is clear that "it is not-white" and "it is not white" signify different things, and the former is an affirmation while the latter is a negation, it is evident that the method of proving each is not the same—for instance, that whatever is an animal is not white (or can not-be white), and that it is true to say "not-white", for this is "to be not-white." But the method for "it is true to say it is white" and "it is true to say it is not-white" is the same; for both are proved constructively through the first figure.
τὸ γὰρ ἀληθὲς τῷ ἔστιν ὁμοίως τάττεται· τοῦ γὰρ ἀληθὲς εἰπεῖν λευκὸν οὐ τὸ ἀληθὲς εἰπεῖν μὴ λευκὸν ἀπόφασις, ἀλλὰ τὸ μὴ ἀληθὲς εἰπεῖν λευκόν. εἰ δὴ ἔσται ἀληθὲς εἰπεῖν ὃ ἂν ᾖ ἄνθρωπος μουσικὸν εἶναι ἢ μὴ μουσιὸν εἶναι, ὃ ἂν ᾖ ζῷον ληπτέον ἢ εἶναι μουσικὸν ἢ εἶναι μὴ μουσικόν, καὶ δέδεικται.
For "it is true" is arranged similarly to "is"; for the negation of "it is true to say white" is not "it is true to say not-white", but "it is not true to say white." If, then, it is to be true to say of whatever is a man that he is musical or is not-musical, one must assume that whatever is an animal is either musical or is not-musical, and it has been proved.
τὸ δὲ μὴ εἶναι μουσικὸν ὃ ἂν ᾖ ἄνθρωπος, ἀνασκευαστικῶς δείκνυται κατὰ τοὺς εἰρτημένους τρόπους τρεῖς.
But that whatever is a man is not musical is proved destructively according to the three methods already mentioned.
Ἁπλῶς δʼ ὅταν οὕτως ἔχῃ τὸ καὶ τὸ Β ὥσθ᾿ ἅμα μὲν τῷ αὐτῷ μὴ ἐνδέχεσθαι, παντὶ δὲ ἐξ ἀνάγκης θάτερον, καὶ πάλιν τὸ Γ καὶ τὸ Δ ὡσαύτως, ἕπηται δὲ τῷ Γ τὸ Α καὶ μὴ ἀντιστρέφῃ, καὶ τῷ Β τὸ Δ ἀκολουθήσει καὶ οὐκ ἀντιστρέφει· καὶ τὸ μὲν καὶ Δ ἐνδέχεται τῷ αὐτῷ, τὸ δὲ Β καὶ οὐκ ἐνδέχεται.
Generally speaking, whenever [A] and B are so related that they cannot belong to the same thing at the same time, but one of them must belong to everything of necessity, and again C and D are related in the same way, and A follows C but not conversely, then D will follow B, and not conversely; and [A] and D can belong to the same thing, but B and [C] cannot.
πρῶτον μὲν οὖν ὅτι τῷ Β τὸ Δ ἕπεται, ἐνθένδε φανερόν.
First, then, that D follows B is clear from the following.
ἐπεὶ γὰρ παντὶ τῶν Γ Δ θάτερον ἐξ ἀνάγκης, ᾧ δὲ τὸ Β, οὐκ ἐνδέχεται τὸ Γ διὰ τὸ συνεπιφέρειν τὸ Α, τὸ δὲ καὶ Β μὴ ἐνδέχεσθαι τῷ αὐτῷ, φανερὸν ὅτι τὸ Δ ἀκολουθήσει.
For since one of C and D belongs of necessity to everything, and C cannot belong to that to which B belongs (because C entails A, and [A] and B cannot belong to the same thing), it is clear that D will follow.
πάλιν ἐπεὶ τῷ Α τὸ Γ οὐκ ἀντιστρέφει, παντὶ δὲ τὸ Γ ἢ τὸ Δ, ἐνδέχεται τὸ καὶ τὸ Δ τῷ αὐτῷ ὑπάρχειν.
Again, since C does not convert with A, but either C or D belongs to everything, it is possible for [A] and D to belong to the same thing.
τὸ δέ γε Β καὶ τὸ Γ οὐκ ἐνδέχεται διὰ τὸ συνακολουθεῖν τῷ Γ τὸ Α· συμβαίνει γάρ τι ἀδύνατον.
But B and C cannot belong to the same thing because A accompanies C (for an impossibility would result).
φανερὸν οὖν ὅτι οὐδὲ τῷ Δ τὸ Β ἀντιστρέφει, ἐπείπερ ἐγχωρεῖ ἅμα τὸ Δ καὶ τὸ Α ὑπάρχειν.
It is clear, therefore, that B does not convert with D either, since it is possible for D and A to belong to the same thing at the same time.
Συμβαίνει δʼ ἐνίοτε καὶ ἐν τῇ τοιαύτῃ τάξει τῶν ὅρων ἀπατᾶσθαι διὰ τὸ μὴ τὰ ἀντικείμενα λαμβάνειν ὀρθῶς ὧνἀνάγκη παντὶ θάτερον ὑπάρχειν·
But it sometimes happens that in such an arrangement of terms one is deceived because of not taking correctly the opposites of which one or the other must belong to everything.
οἷον εἰ τὸ Α καὶ τὸ Β μὴ ἐνδέχεται ἅμα τῷ αὐτῷ, ἀνάγκη δʼ ὑπάρχειν, ᾧ μὴ θάτερον, θάτερον, καὶ πάλιν τὸ Γ καὶ τὸ Δ ὡσαύτως, ᾧ δὲ τὸ Γ, παντὶ ἕπεται τὸ Α. συμβήσεται γὰρ ᾧ τὸ Δ, τὸ Β. ὑπάρχειν ἐξ ἀνάγκης, ὅπερ ἐστὶ ψεῦδος.
For example, if A and B cannot belong to the same thing at the same time, but it is necessary that whichever one does not belong, the other does, and again C and D in the same way, and A follows everything to which C belongs. For it will result that B must belong to whatever D belongs, which is false.
εἰλήφθω γὰρ ἀπόφασις τῶν Α Β ἡ ἐφʼ ᾧ Ζ, καὶ πάλιν τῶν Γ Δ ἡ ἐφʼ ὧ Θ. ἀνάγκη δὴ παντὶ ἢ τὸ Α ἢ τὸ Ζ. ἢ γὰρ τὴν φάσιν ἢ τὴν ἀπόφασιν.
For let Z be taken as the negation of A from the pair A and B, and again H as the negation of C from the pair C and D. It is necessary then that either A or Z belongs to everything; for it must be either the affirmation or the negation.
καὶ πάλιν ἢ τὸ Γ ἢ τὸ Θ· φάσις γὰρ καὶ ἀπόφασις.
And again either C or H; for they are affirmation and negation.
καὶ ᾧ τὸ Γ, παντὶ τὸ Α ὑπόκειται.
And to whatever C belongs, A is assumed to belong to all of it.
ὥστε ᾧ τὸ Ζ, παντὶ τὸ Θ. πάλιν ἐπεὶ τῶν Ζ Β παντὶ θάτερον καὶ τῶν Θ Δ ὡσαύτως, ἀκολουθεῖ δὲ τῷ Ζ τὸ Θ, καὶ τῷ Δ ἀκολουθήσει τὸ Β· τοῦτο γὰρ ἴσμεν.
So that to whatever Z belongs, H belongs to all of it. Again, since one of Z and B belongs to everything, and similarly one of H and D, and H follows Z, B will also follow D; for we know this.
εἰ ἄρα τῷ Γ τὸ Α, καὶ τῷ Δ τὸ Β. τοῦτο δὲ ψεῦδος· ἀνάπαλιν γὰρ ἦν ἐν τοῖς οὕτως ἔχουσιν ἡ ἀκολούθησις.
Therefore, if A follows C, B will follow D. But this is false; for in things so related, the consequence was the other way around.
οὐ γὰρ ἴσως ἀνάγκη παντὶ τὸ Α ἢ τὸ Ζ, οὐδὲ τὸ Ζ ἢ τὸ Β. οὐ γάρ·
For it is not equally necessary that either A or Z belongs to everything, or that either Z or B does.
ἐστιν ἀπόφασις τοῦ Α τὸ Ζ. τοῦ γὰρ ἀγαθοῦ τὸ οὐκ ἀγαθὸν ἀπόφασις· οὐ ταὐτὸ δʼ ἐστὶ τὸ οὐκ ἀγαθὸν τῷ οὔτʼ ἀγαθὸν οὔτʼ οὐκ ἀγαθόν.
For Z is not the negation of A. For the negation of 'good' is 'not-good'; and 'not-good' is not the same as 'that which is neither good nor not-good'.
ὁμοίως δὲ καὶ ἐπὶ τῶν Γ Δ· αἱ γὰρ ἀποφάσεις αἰ εἰλημμέναι δύο εἰσίν.
Similarly also in the case of C and D; for the negations taken are two.