§1.1.6#2Ἐὰν δʼ ὁ μὲν ἦ κατηγορικὸς ὁ δὲ στερητικός, καθόλου δὲ ὁ κατηγορικός, ὅταν μὲν ὁ ἐλάττων ἦ κατηγορικός, ἔσται συλλογισμός.
And if one is affirmative and the other privative, the affirmative being universal, when the minor is affirmative, there will be a syllogism.
εἰ γὰρ τὸ παντὶ τῷ Σ, τὸ δὲ τινὶ μὴ ὑπάρχει, ἀνάγκη τὸ τινὶ τῷ Ρ μὴ ὑπάρχειν.
For if Π belongs to every Σ, and Ρ does not belong to some Σ, Π must of necessity not belong to some Ρ.
εἰ γὰρ παντί, καὶ τὸ Ρ παντὶ τῷ Σ, καὶ τὸ Π παντὶ τῷ ὑπάρξει· ἀλλʼ οὐχ ὑπῆρχεν.
For if it belongs to all, since Ρ belongs to every Σ, Π will also belong to every Σ; but it did not belong.
δείκνυται δὲ καὶ ἄνευ τῆς ἀπαγωγῆς, ἐὰν ληφθῇ τι τῶν Σ ᾧ τὸ μὴ ὑπάρχει.
And it is also demonstrated without the reduction to the impossible, if some Σ is taken to which Π does not belong.
ὅταν δʼ ὁ μείζων ᾖ κατηγορικός, οὐκ ἔσται συλλογισμός, οἷον εἰ τὸ μὲν παντὶ τῷ Σ, τὸ δὲ Ρ τινὶ τῷ Σ μὴ ὑπάρχει.
But when the major is affirmative, there will not be a syllogism, as for instance if Π belongs to every Σ, and Ρ does not belong to some Σ.
ὅροι τοῦ παντὶ ὑπάρχειν ἔμψυχον—ἄνθρωπος—ζῷον.
Terms for belonging to all: animate—man—animal.
τοῦ δὲ μηδενὶ οὐκ ἔστι λαβεῖν ὅρους, εἰ τινὶ μὲν ὑπάρχει τῷ Σ τὸ Ρ, τινὶ δὲ μή· εἰ γὰρ παντὶ τὸ Π τῷ ὑπάρχει, τὸ δὲ Ρ τινὶ τῷ Σ, καὶ τὸ τινὶ τῷ Ρ ὑπάρξει· ὑπέκειτο δὲ μηδενὶ ὑπάρχειν.
And it is not possible to take terms for belonging to none, if Ρ belongs to some Σ and does not belong to some; for if Π belongs to every Σ, and Ρ to some Σ, Π will also belong to some Ρ; but it was assumed to belong to none.
ἀλλʼ ὥσπερ ἐν τοῖς πρότερον λητπτέον· ἀδιορίστου γὰρ ὄντος τοῦ τινὶ μὴ ὑπάρχειν καὶ τὸ μηδενὶ ὑπάρχον ἀληθὲς εἰπεῖν τινὶ μὴ ὑπάρχειν· μτηδενὶ δὲ ὑπάρχοντος οὐκ ἦν συλλογισμός.
But as in the previous cases, it must be taken so; for since 'not belonging to some' is indefinite, it is also true to say of that which belongs to none that it 'does not belong to some'; and when it belonged to none, there was no syllogism.
φανερὸν οὖν ὅτι οὐκ ἔσται συλλογισμός.
It is clear, then, that there will not be a syllogism.
ἐὰν δʼ ὁ στερητικὸς ᾖ καθόλου τῶν ὅρων, ὅταν μὲν ὁ μείζων ᾖ στερητικὸς ὁ δὲ ἐλάττων κατηγορικός, ἔσται συλλογισμός.
And if the privative is universal of the terms, when the major is privative and the minor is affirmative, there will be a syllogism.
εἰ γὰρ τὸ μηδενὶ τῷ Σ, τὸ δὲ Ρ τινὶ ὑπάρχει τῷ Σ, τὸ Π τινὶ τῷ Ρ οὐχ ὑπάρξει· πάλιν γὰρ ἔσται τὸ πρῶτον σχῆμα τῆς Ρ Σ προτάσεως ἀντιστραφείσης.
For if Π belongs to no Σ, and Ρ belongs to some Σ, Π will not belong to some Ρ; for there will be the first figure again when the Ρ-Σ premise is converted.
ὅταν δὲ ὁ ἐλάττων ᾖ στερητικός, οὐκ ἔσται συλλογισμός.
But when the minor is privative, there will not be a syllogism.
ὅροι τοῦ ὑπάρχειν ζῷον—ἄνθρωτπος—ἄγριον, τοῦ μὴ ὑπάρχειν ζῷον—ἐπιστήμη—ἄγριονν· μέσον ἐν ἀμφοῖν τὸ ἄγριον.
Terms for belonging: animal—man—wild; for not belonging: animal—knowledge—wild; middle in both: wild.
οὐδʼ ὅταν ἀμφότεροι στερητικοὶ τεθῶσιν, ᾖ δʼ ὁ μὲν καθόλου ὁ δʼ ἐν μέρει.
Nor when both are assumed privative, and one is universal and the other particular.
ὅροι ὅταν ὁ ἐλάττων ᾖ καθόλου πρὸς τὸ μέσον, ζῷον—ἐπιστήμη—ἄγριον, ζῷον—ἄνθρωπος—ἄγριον· ὅταν δʼ ὁ μείζων, τοῦ μὲν μὴ ὑτπάρχειν κόραξ—χιών—λευκόν.
Terms when the minor is universal in relation to the middle, for belonging: animal—knowledge—wild; for not belonging: animal—man—wild; and when the major is universal, for not belonging: raven—snow—white.
τοῦ δʼ ὑπάρχειν οὐκ ἔστι λαβεῖν, εἰ τὸ Ρ τινὶ μὲν ὑπάρχει τῷ Σ, τινὶ δὲ μὴ ὑπάρχει.
And it is not possible to take terms for belonging, if Ρ belongs to some Σ and does not belong to some.
εἰ γὰρ τὸ παντὶ τῷ Ρ, τὸ δὲ Ρ τινὶ τῷ Σ, καὶ τὸ τινὶ τῷ Σ· ὑπέκειτο δὲ μηδενί.
For if Π belongs to every Ρ, and Ρ to some Σ, Π will also belong to some Σ; but it was assumed to belong to none.
ἀλλʼ ἐκ τοῦ ἀδιορίστου δεικτέον.
But it must be shown from the indefinite.
Οὐδ’ ἂν ἑκάτερος τινὶ τῷ μέσῳ ὑπάρχῃ ἢ μὴ ὑπάρχῃ, ἢ ὁ μὲν ὑπάρχῃ ὁ δὲ μὴ ὑπάρχῃ, ἢ ὁ μὲν τινὶ ὁ δὲ μὴ παντί, ἢ ἀδιορίστως, οὐκ ἔσται συλλογισμὸς οὐδαμῶς.
Nor if each belongs or does not belong to some of the middle, or one belongs and the other does not, or one to some and the other not to all, or indefinitely, will there be a syllogism in any way.
ὅροι δὲ κοινοὶ πάντων ζῷον-ἄνθρωπος—λευκόν, ζῷον—ἄψυχον—λευκόν.
Common terms for all: animal—man—white; animal—inanimate—white.
Φανερὸν οὖν καὶ ἐν τούτῳ τῷ σχήματι πότʼ ἔσται καὶ πότʼ οὐκ ἔσται συλλογισμός, καὶ ὅτι ἐχόντων τε τῶν ὅρων ὡς ἐλέχθη γίνεται συλλογισμός ἐξ ἀνάγκης, ἄν τʼ ᾖ συλλογισμός, ἀνάγκη τοὺς ὅρους οὕτως ἔχειν.
It is clear, then, in this figure too, when there will and when there will not be a syllogism, and that when the terms are related as has been said, a syllogism comes about of necessity, and if there is a syllogism, the terms must be so related.
φανερὸν δὲ καὶ ὅτι πάντες ἀτελεῖς εἰσὶν οἱ ἐν τούτῳ τῷ σχήματι συλλογισμοί (πάντες γάρ τελειοῦνται προσλαμβανομένων τινῶν) καὶ ὅτι συλλογίσασθαι τὸ καθόλου διὰ τούτου τοῦ σχήματος οὐκ ἔσται, οὔτε στερητικόν οὔτε καταφατικόν.
It is also clear that all the syllogisms in this figure are imperfect (for they are all completed by taking additional propositions) and that it is not possible to syllogize a universal conclusion through this figure, whether privative or affirmative.