§1.1.17Ἐν δὲ τῷ δευτέρῳ σχήματι ὅταν μὲν ἐνδέχεσθαι λαμβάνωσιν ἀμφότεραι αἱ προτάσεις, οὐδεὶς ἔσται συλλογισμός, οὔτε κατηγορικῶν οὔτε στερητικῶν τιθεμένων, οὔτε καθόλου οὔτε κατὰ μέρος·
In the second figure, when both premises are assumed as possible, there will be no syllogism, whether they are set as affirmative or privative, or as universal or particular.
ὅταν δὲ ἡ μὲν ὑπάρχειν ἡ δʼ ἐνδέχεσθαι σημαίνῃ, τῆς μὲν καταφατικῆς ὑπάρχειν σημαινούσης οὐδέποτʼ ἔσται, τῆς δὲ στερητικῆς τῆς καθόλου ἀεί.
But when one signifies belonging and the other being possible, if the affirmative signifies belonging, there will never be a syllogism, but if the universal privative does, there always will be.
τὸν αὐτὸν δὲ τρόπον καὶ. ὅταν ἡ μὲν ἐξ ἀνάγκης ἡ δʼ ἐνδέχεσθαι λαμβάνηται τῶν προτάσεων.
In the same way also when one of the premises is assumed as necessary and the other as possible.
δεῖ δὲ καὶ ἐν τούτοις λαμβάνειν τὸ ἐν τοῖς συμπεράσμασιν ἐνδεχόμενον ὥσπερ ἐν τοῖς πρότερον.
And in these cases too, one must assume the possible in the conclusions just as in the previous cases.
Πρῶτον οὖν δεικτέον ὅτι οὐκ ἀντιστρέφει τὸ ἐν τῷ ἐνδέχεσθαι στερητικόν, οἷον εἰ τὸ ἐνδέχεται μηδενὶ τῷ Β, οὐκ ἀνάγκη καὶ τὸ Β ἐνδέχεσθαι μηδενὶ τῷ Α. κείσθω γὰρ τοῦτο, καὶ ἐνδεχέσθω τὸ Β μηδενὶ τῷ Α ὑπάρχειν.
First, then, it must be shown that the privative premise in the case of possibility does not convert; for example, if A is possible to belong to no B, it is not necessary that B is also possible to belong to no A. For let this be assumed, and let B be possible to belong to no A.
οὐκοῦν ἐπεὶ ἀντιστρέφουσιν αἱ ἐν τῷ ἐνδέχεσθαι καταφάσεις ταῖς ἀποφάσεσι, καὶ αἱ ἐναντίαι καὶ αἱ ἀντικείμεναι, τὸ δὲ Β τῷ Α ἐνδέχεται μηδενὶ ὑπάρχειν, φανερὸν ὅτι καὶ παντὶ ἂν ἐνδέχοιτο τῷ Α ὑπάρχειν.
Since, then, affirmations in the case of possibility convert with negations, both the contrary and the contradictory ones, and B is possible to belong to no A, it is clear that it would also be possible to belong to all A.
τοῦτο δὲ ψεῦδος· οὐ γὰρ εἰ τόδε τῷδε παντὶ ἐνδέχεται, καὶ τόδε τῷδε ἀναγκαῖον·
But this is false; for if this is possible to belong to all of that, it is not thereby necessary for this to belong to that.
ὥστʼ οὐκ ἀντιστρέφει τό στερητικόν.
So the privative does not convert.
ἔτι δʼ οὐδὲν κωλύει τὸ μὲν Α τῷ Β ἐνδέχεσθαι μηδενί, τὸ δὲ Β τινὶ τῶν Α ἐξ ἀνάγκης μὴ ὑπάρχειν, οἷον τὸ μὲν λευκὸν παντὶ ἀνθρώπῳ ἐνδέχεται μὴ ὑπάρχειν (καὶ γὰρ ὑπάρχειν), ἄνθρωπον δʼ οὐκ ἀληθὲς εἰπεῖν ὡς ἐνδέχεται μηδενὶ λευκῷ· πολλοῖς γὰρ ἐξ ἀνάγκης οὐχ ὑπάρχει, τὸ δʼ ἀναγκαῖον οὐκ ἦν ἐνδεχόμενον.
Moreover, nothing prevents A from being possible to belong to no B, while B of necessity does not belong to some of the A's. For example, white is possible to belong to no human (for indeed it does belong), but it is not true to say that human is possible to belong to no white; for of necessity it does not belong to many, and the necessary was not the possible.
Ἀλλὰ μὴν οὐδʼ ἐκ τοῦ ἀδυνάτου δειχθήσεται ἀντιστρέφον, οἷον εἴ τις ἀξιώσειεν, ἐπεὶ ψεῦδος τὸ ἐνδέχεσθαι τὸ Β τῷ Α μηδενὶ ὑπάρχειν, ἀληθὲς τὸ μὴ ἐνδέχεσθαι μηδενί (φάσις γὰρ καὶ ἀπόφασις), εἰ δὲ τοῦτʼ, ἀληθὲς ἐξ ἀνάγκης τινὶ τῷ Α ὑπάρχειν· ὥστε καὶ τὸ Α τινὶ τῷ Β·
Furthermore, nor will it be shown to convert through impossibility. For example, if someone should claim that, since it is false that B is possible to belong to no A, it is true that it is not possible to belong to none (for there is assertion and negation); and if this is so, it is true that of necessity it belongs to some of the A's; so that A also belongs to some of the B's; but this is impossible.
τοῦτο δʼ ἀδύνατον. οὐ γὰρ εἰ μὴ ἐνδέχεται μηδενὶ τὸ Β τῷ Α, ἀνάγκη τινὶ ὑπάρχειν.
For if B is not possible to belong to no A, it is not necessary for it to belong to some.
τὸ γὰρ μὴ ἐνδέχεσθαι μηδενὶ διχῶς λέγεται, τὸ μὲν εἰ ἐξ ἀνάγκης τινὶ ὑπάρχει, τὸ δʼ εἰ ἐξ ἀνάγκης τινὶ μὴ ὑπάρχει·
For not to be possible to belong to none is said in two ways: one, if of necessity it belongs to some, and the other, if of necessity it does not belong to some.
τὸ γὰρ ἐξ ἀνάγκης τινὶ τῶν Α μὴ ὑπάρχον οὐκ ἀληθὲς εἰπεῖν ὡς παντὶ ἐνδέχεται μὴ ὑπάρχειν, ὥσπερ οὐδὲ τὸ τινὶ ὑπάρχον ἐξ ἀνάγκης ὅτι παντὶ ἐνδέχεται ὑπάρχειν.
For of that which of necessity does not belong to some of the A's, it is not true to say that it is possible to belong to all; just as of that which of necessity belongs to some, it is not true to say that it is possible to belong to all.
εἰ οὖν τις ἀξιοίη, ἐπεὶ οὐκ ἐνδέχεται τὸ Γ τῷ Δ παντὶ ὑπάρχειν, ἐξ ἀνάγκης τινὶ μὴ ὑπάρχειν αὐτό, ψεῦδος ἂν λαμβάνοι·
If, then, someone should claim that, since it is not possible for Γ to belong to all Δ, of necessity it does not belong to some of them, he would assume what is false.
παντὶ γὰρ ὑπάρχει, ἀλλʼ ὅτι ἐνίοις ἐξ ἀνάγκης ὑπάρχει, διὰ τοῦτό φαμεν οὐ παντὶ ἐνδέχεσθαι.
For it belongs to all, but because it belongs to some of necessity, we say on this account that it is not possible to belong to all.
ὥστε τῷ ἐνδέχεσθαι παντὶ ὑπάρχειν τό τʼ ἐξ ἀνάγκης τινὶ ὑπάρχειν ἀντίκειται καὶ τὸ ἐξ ἀνάγκης τινὶ μὴ ὑπάρχειν.
So to being possible to belong to all, both belonging to some of necessity and not belonging to some of necessity are opposed.
ὁμοίως δὲ καὶ τῷ ἐνδέχεσθαι μηδενί.
And similarly also to being possible to belong to none.
δῆλον οὖν ὅτι πρὸς τὸ οὕτως ἐνδεχόμενον καὶ μὴ ἐνδεχόμενον ὡς ἐν ἀρχῇ διωρίσαμεν οὐ τὸ ἐξ ἀνάγκης τινὶ ὑπάρχειν ἀλλὰ τὸ ἐξ ἀνάγκης τινὶ μὴ ὑπάρχειν ληπτέον.
It is clear, then, that in relation to the possible and not possible in the sense we defined at the beginning, we must assume not that of necessity it belongs to some, but rather that of necessity it does not belong to some.
τούτου δὲ ληφθέντος οὐδὲν συμβαίνει ἀδύνατον, ὥστʼ οὐ γίνεται συλλογισμός.
But when this is assumed, no impossible consequence occurs, so that no syllogism comes about.
φανερὸν οὖν ἐκ τῶν εἰρημένων ὅτι οὐκ ἀντιστρέφει τὸ στερητικόν.
It is clear, then, from what has been said that the privative does not convert.
Τούτου δὲ δειχθέντος κείσθω τὸ Α τῷ μὲν Β ἐνδέχεσθαι μηδενί, τῷ δὲ Γ παντί.
This having been shown, let A be assumed to be possible to belong to no B, and to all Γ.
διὰ μὲν οὖν τῆς ἀντιστροφῆς οὐκ ἔσται συλλογισμός· εἴρηται γὰρ ὅτι οὐκ ἀντιστρέφει ἡ τοιαύτη πρότασις.
Through conversion, then, there will be no syllogism; for it has been said that such a premise does not convert.
ἀλλʼ οὐδὲ διὰ τοῦ ἀδυνάτου τεθέντος γὰρ τοῦ Β ⟨μὴ⟩ παντὶ τῷ Γ ἐνδέχεσθαι ⟨μὴ⟩ ὑπάρχειν οὐδὲν συμβαίνει ψεῦδος· ἐνδέχοιτο γὰρ ἂν τὸ τῷ Γ καὶ παντὶ καὶ μηδενὶ ὑπάρχειν.
But neither through impossibility; for if B is assumed not to be possible to belong to all Γ, no falsehood occurs; for it could be possible to belong to all and to none of the Γ's.
ὅλως δʼ εἰ ἔστι συλλογισμός, δῆλον ὅτι τοῦ ἐνδέχεσθαι ἂν εἴη διὰ τὸ μηδετέραν τῶν προτάσεων εἰλῆφθαι ἐν τῷ ὑπάρχειν, καὶ οὗτος ἢ καταφατικὸς ἢ στερητικός·
And in general, if there is a syllogism, it is clear that it would be of the possible, because neither of the premises has been assumed in the sense of belonging, and this either affirmative or privative.
οὐδετέρως δʼ ἐγχωρεῖ.
But in neither way is it possible.
καταφατικοῦ μὲν γὰρ τεθέντος δειχθήσεται διὰ τῶν ὅρων ὅτι οὐκ ἐνδέχεται ὑπάρχειν, στερητικοῦ δέ, ὅτι τὸ συμπέρασμα οὐκ ἐνδεχόμενον ἀλλʼ ἀναγκαῖόν ἐστιν.
For if an affirmative is assumed, it will be shown through the terms that it is not possible to belong; but if a privative, that the conclusion is not possible but necessary.
ἔστω γὰρ τὸ μὲν Α λευκόν, τὸ δὲ Β ἄνθρωπος, ἐφʼ ᾧ δὲ Γ ἵππος.
For let A be white, B human, and Γ horse.
τὸ δὴ Α, τὸ λευκόν, ἐνδέχεται τῷ μὲν παντὶ τῷ δὲ μηδενὶ ὑπάρχειν.
A, then, namely white, is possible to belong to all of the one and to none of the other.
ἀλλὰ τὸ Β τῷ Γ οὔτε ὑπάρχειν ἐνδέχεται οὔτε μὴ ὑπάρχειν.
But B is neither possible to belong nor not possible to belong to Γ.
ὅτι μὲν οὖν ὑπάρχειν οὐκ ἐγχωρεῖ, φανερόν· οὐδεὶς γὰρ ἵππος ἄνθρωπος.
That it is not possible to belong is clear; for no horse is human.
ἀλλʼ οὐδʼ ἐνδέχεσθαι μὴ ὑπάρχειν· ἀνάγκη γὰρ μηδένα ἵππον ἄνθρωπον εἶναι, τὸ δʼ ἀναγκαῖον οὐκ ἦν ἐνδεχόμενον.
But neither is it possible not to belong; for of necessity no horse is human, and the necessary was not the possible.
οὐκ ἄρα γίνεται συλλογισμός.
Therefore, no syllogism comes about.
ὁμοίως δὲ δειχθήσεται καὶ ἂν ἀνάπαλιν τεθῇ τὸ στερητικόν, κἂν ἀμφότεραι καταφατικαὶ ληφθῶσιν ἢ στερητικαί (διὰ γὰρ τῶν αὐτῶν ὅρων ἔσται ἡ ἀπόδειξις)· καὶ ὅταν ἡ μὲν καθόλου ἡ δʼ ἐν μέρει, ἢ ἀμφότεραι κατὰ μέρος ἢ ἀδιόριστοι, ἢ ὁσαχῶς ἄλλως ἐνδέχεται μεταλαβεῖν τὰς προτάσεις· ἀεὶ γὰρ ἔσται διὰ τῶν αὐτῶν ὅρων ἡ ἀπόδειξις.
And similarly it will be shown also if the privative is assumed the other way around, and if both are assumed as affirmative or as privative (for the proof will be through the same terms); and when one is universal and the other particular, or both as particular or indefinite, or in whatever other way it is possible to vary the premises; for the proof will always be through the same terms.
φανερὸν οὖν ὅτι ἀμφοτέρων τῶν προτάσεων κατὰ τὸ ἐνδέχεσθαι τιθεμένων οὐδείς γίνεται συλλογισμός.
It is clear, then, that when both premises are assumed in relation to the possible, no syllogism comes about.