§1.1.5#2Ὅταν μὲν οὖν ἀντικείμενον ᾖ τὸ καθόλου τῷ κατὰ μέρος, εἴρηται πότʼ ἔσται καὶ πότʼ οὐκ ἔσται συλλογισμός· ὅταν δὲ ὁμοιοσχήμονες ὦσιν αἱ προτάσεις, οἷον ἀμφότεραι στερητεκαὶ ἢ καταφατικαί, οὐδαμῶς ἔσται συλλογισμός.
So when the universal is opposite to the particular, it has been said when there will be and when there will not be a syllogism; but when the premises are of the same form, e.g. both privative or both affirmative, there will in no way be a syllogism.
ἔστωσαν γὰρ πρῶτον στερητικαί, καὶ τὸ καθόλου κείσθω πρὸς τὸ μεῖζον ἄκρον, οἷον τὸ Μ τῷ μέν Ν μηδενὶ τῷ δὲ Ξ τινὶ μὴ ὑπαρχέτω· ἐνδέχεται δὴ καὶ παντὶ καὶ μηδενὶ τῷ Ξ τὸ Ν ὑπάρχειν.
For let them first be privative, and let the universal be placed in relation to the major extreme, e.g. let M belong to no N, but not belong to some Ξ; then it is possible for N to belong both to all and to no Ξ.
ὅροι τοῦ μὲν μὴ ὑπάρχειν μέλαν–χιών–ζῷον· τοῦ δὲ παντὶ. ὑπάρχειν οὐκ ἔστι λαβεῖν, εἰ τὸ Μ τῷ Ξ τινὶ μὲν ὑπάρχει τινὶ δὲ μή.
Terms of belonging to none: black—snow—animal; of belonging to all it is not possible to find, if M belongs to some Ξ and does not belong to some.
εἰ γὰρ παντὶ τῷ Ξ τὸ Ν, τὸ δὲ μηδενὶ τῷ Ν, τὸ Μ οὐδενὶ τῷ Ξ ὑπάρξει· ἀλλʼ ὑπέκειτο τινὶ ὑπάρχειν.
For if N belongs to every Ξ, and M to no N, M will belong to no Ξ; but it was assumed to belong to some.
οὕτω μὲν οὖν οὐκ ἐγχωρεῖλαβεῖν ὅρους, ἐκ δὲ τοῦ ἀδιορίστου δεικτέον·
So in this way it is not possible to find terms, but it must be shown from the indefinite.
ἐπεὶ γὰρ ἀληθεύεται τὸ τινὶ μὴ ὑπάρχειν τὸ Μ τῷ Ξ καὶ εἰ μηδενὶ ὑπάρχει, μηδενὶ δὲ ὑπάρχοντος οὐκ ἦν συλλογισμός, φανερὸν ὅτι οὐδὲ νῦν ἔσται.
For since 'M does not belong to some Ξ' is true also if it belongs to none, and when it belonged to none there was no syllogism, it is clear that neither now will there be one.
πάλιν ἔστωσαν κατηγορικαί, καὶ τὸ καθόλου κείσθω ὁμοίως, οἷον τὸ τῷ μὲν Ν παντὶ τῷ δὲ Ξ τινὶ ὑπαρχέτω.
Again, let them be affirmative, and let the universal be placed similarly, e.g. let M belong to every N, but to some Ξ.
ἐνδέχεται δὴ τὸ Ν τῷ Ξ καὶ παντὶ καὶ μηδενὶ ὑπάρχειν.
Then it is possible for N to belong both to every Ξ and to none.
ὅροι τοῦ μηδενὶ ὑπάρχειν λευκόνκύκνος–λίθος τοῦ δὲ παντὶ οὐκ ἔσται λαβεῖν διὰ τὴν αὐτὴν αἰτίαν ἥνπερ πρότερον, ἀλλʼ ἐκ τοῦ ἀδιορίστου δεικτέον.
Terms of belonging to none: white—swan—stone; of belonging to all it will not be possible to find for the same reason as before, but it must be shown from the indefinite.
εἰ δὲ τὸ καθόλου πρὸς τὸ ἔλαττον ἄκρον ἐστί, καὶ τὸ Μ τῷ μὲν Ξ μηδενὶ τῷ δὲ Ν τινὶ μὴ ὑπάρχει, ἐνδέχεται τὸ Ν τῷ Ξ καὶ παντὶ καὶ μηδενὶ ὑπάρχειν.
But if the universal is in relation to the minor extreme, and M belongs to no Ξ, but does not belong to some N, it is possible for N to belong both to all and to no Ξ.
ὅροι τοῦ ὑπάρχειν λευκόν–ζῷονκόραξ, τοῦ μὴ ὑπάρχειν λευκόν–λίθος–κόραξ.
Terms of belonging: white—animal—crow; of not belonging: white—stone—crow.
εἰ δὲ κατηγορικαὶ αἱ προτάσεις, ὅροι τοῦ μὴ ὑπάρχειν λευκόν–ζῷον–χιών, τοῦ ὑπάρχειν λευκόν–ζῷον–κύκνος.
But if the premises are affirmative, terms of not belonging: white—animal—snow; of belonging: white—animal—swan.
φανερὸν οὖν, ὅταν ὁμοιοσχήμονες ὦσιν αἱ προτάσεις καὶ ἡ μὲν καθόλου ἡ δʼ ἐν μέρει, ὅτι οὐδαμῶς γίνεται συλλογισμός.
It is clear, then, that whenever the premises are of the same form and one is universal and the other particular, there is in no way a syllogism.
ἀλλʼ οὐδʼ εἰ τινὶ ἑκατέρῳ ὑπάρχει ἢ μὴ ὑπάρχει, ἢ τῷ μὲν τῷ δὲ μή, ἢ μηδετέρῳ παντί, ἢ ἀδιορίστως.
Nor if it belongs or does not belong to some of each, or to one but not to the other, or to neither of them universally, or indefinitely.
ὅροι δὲ κοινοὶ πάντων λευκόν–ζῷον–ἄνθρωπος, λευκόν–ζῷον–ἀψυχον.
And common terms for all these cases are: white—animal—man; white—animal—inanimate.
Φανερὸν οὖν ἐκ τῶν εἰρημένων ὅτι ἐάν τε οὕτως ἔχωσιν οἱ ὅροι πρὸς ἀλλήλους ὡς ἐλέχθη, γίνεται συλλογισμὸς ἐξ ἀνάγκης, ἄν τʼ συλλογισμός, ἀνάγκη τοὺς ὅρους οὕτως ἔχειν.
It is clear, then, from what has been said, that if the terms are related to one another as was said, a syllogism comes about of necessity, and if there is a syllogism, the terms must be related in this way.
δῆλον δὲ καὶ ὅτι πάντες ἀτελεῖς εἰσὶν οἱ ἐν τούτῳ τῷ σχήματι συλλογισμοί (πάντες γὰρ ἐπιτελοῦνται προσλαμβανομένων τινῶν, ἃ ἢ ἐνυπάρχει τοῖς ὅροις ἐξ ἀνάγκης ἢ τίθενται ὡς ὑποθέσεις, οἷον ὅταν διὰ τοῦ ἀδυνάτου δεικνύωμεν), καὶ ὅτι οὐ γίνεται καταφατικὸς συλλογισμὸς διὰ τούτου τοῦ σχήματος, ἀλλα πάντες στερητικοί, καὶ οἱ καθόλου καὶ οἱ κατὰ μέρος.
And it is also clear that all the syllogisms in this figure are imperfect (for all are completed by taking in addition certain things which either belong to the terms of necessity or are assumed as hypotheses, e.g. when we show through the impossible), and that an affirmative syllogism does not come about through this figure, but all are privative, both the universal and the particular.