§1.1.25#1Δῆλον δὲ καὶ ὅτι πᾶσα ἀπόδειξις ἔσται διὰ τριῶν ὅρων καὶ οὐ πλειόνων, ἐὰν μὴ διʼ ἄλλων καὶ ἄλλων τὸ αὐτὸ συμπέρασμα γίνηται, οἷον τὸ διά τε τῶν Β καὶ διὰ τῶν Γ Δ, ἢ διὰ τῶν Α Β καὶ Α Γ Δ· πλείω γὰρ μέσα τῶν αὐτῶν οὐδὲν εἶναι κωλύει.
It is clear also that every proof will be through three terms and no more, unless the same conclusion comes about through different sets of terms, as for example [the conclusion] through B and through C, D, or through A, B and A, C, D; for nothing prevents there being several middle terms for the same [conclusions].
τούτων δʼ ὄντων οὐχ εἷς ἀλλὰ πλείους εἰσὶν οἱ συλλογισμοί.
But when this is the case, the syllogisms are not one but several.
ἢ πάλιν ὅταν ἑκάτερον τῶν Α Β διὰ συλλογισμοῦ ληφθῇ (οἷον τὸ Α διὰ τῶν Δ Ε καὶ πάλιν τὸ Β διὰ τῶν Θ), ἢ τὸ μὲν ἐπαγωγῇ, τὸ δὲ συλλογισμῷ.
Or again when each of A, B is assumed through a syllogism (for example A through D, E and again B through Θ), or one by induction and the other by syllogism.
ἀλλὰ καὶ οὕτως πλείους οἱ συλλογισμοί· πλείω γὰρ τὰ συμπεράσματα ἐστιν, οἷον τό τε Α καὶ τὸ Β καὶ τὸ Γ.
Εἰ δʼ οὖν μὴ πλείους ἀλλʼ εἷς, οὕτω μὲν ἐνδέχεται γενέσθαι διὰ πλειόνων τὸ αὐτὸ συμπέρασμα, ὡς δὲ τὸ διὰ τῶν Α Β, ἀδύνατον.
But even so the syllogisms are several; for the conclusions are several, as for example A, B, and Γ. If, then, there are not several but one, in the former way indeed it is possible for the same conclusion to come about through several terms, but in the way of that through A, B, it is impossible.
ἔστω γὰρ τὸ Ε συμπεπερασμένον ἐκ τῶν Α Β Γ Δ. οὐκοῦν ἀνάγκη τι αὐτῶν ἄλλο πρὸς ἄλλο εἰλῆφθαι, τὸ μὲν ὡς ὅλον τὸ δʼ ὡς μέρος·
For let E be concluded from A, B, Γ, Δ.
τοῦτο γὰρ δεδεικται πρότερον, ὅτι ὄντος συλλογισμοῦ ἀναγκαῖον οὕτως τινὰς ἔχειν τῶν ὅρων.
It is necessary, therefore, that some of them be assumed in relation to others, one as a whole and the other as a part; for this has been shown before, that if there is a syllogism, some of the terms must necessarily be related in this way.
ἐχέτω οὖν τὸ οὕτως πρὸς τὸ Β. ἔστιν ἄρα τι ἐξ αὐτῶν συμπέρασμα.
Let, then, [A] be thus related to B. There is, therefore, some conclusion from them.
οὐκοῦν ἤτοι τὸ ἢ τῶν Γ Δ θάτερον ἢ ἄλλο τι παρὰ ταῦτα.
Therefore it is either E, or one of Γ, Δ, or something else besides these.
καὶ εἰ μὲν τὸ Ε, ἐκ τῶν Α Β μόνον ἂν εἴη ὁ συλλογισμός.
And if indeed it is E, the syllogism would be from A, B alone.
τὰ δὲ Γ Δ εἰ μὲν ἔχει οὕτως ὥστʼ εἶναι τὸ μὲν ὡς ὅλον τὸ δʼ ὡς μέρος, ἔσται τι καὶ ἐξ ἐκείνων, καὶ ἤτοι τὸ Ε ἢ τῶν Α Β θάτερον ἢ ἄλλο τι παρὰ ταῦτα.
And if Γ, Δ are related in such a way that one is as a whole and the other as a part, there will be some conclusion from them too, and it will be either E, or one of A, B, or something else besides these.
καὶ εἰ μὲν τὸ Ε ἢ τῶν Α Β θάτερον, ἢ πλείους ἔσονται οἱ συλλογισμοί, ἢ ὡς ἐνεδέχετο ταὐτὸ διὰ πλειόνων ὅρων περαίνεσθαι συμβαίνει· εἰ δʼ ἄλλο τι παρὰ ταῦτα, πλείους ἔσονται καὶ ἀσύναπτοι οἱ συλλογισμοὶ πρὸς ἀλλήλους.
And if indeed it is E or one of A, B, either the syllogisms will be several, or it happens as was possible for the same thing to be concluded through several terms; but if it is something else besides these, the syllogisms will be several and disconnected from one another.
εἰ δὲ μὴ οὕτως ἔχοι τὸ Γ πρὸς τὸ Δ ὥστε ποιεῖν συλλογισμόν, μάτην ἔσται εἰλημμένα, εἰ μὴ ἐπαγωγῆς ἢ κρύψεως ἤ τινος ἄλλου τῶν τοιούτων χάριν.
And if Γ is not related to Δ in such a way as to produce a syllogism, they will have been assumed in vain, unless for the sake of induction, or concealment, or some other such purpose.
Εἰ δʼ ἐκ τῶν Α Β μὴ τὸ Ε ἀλλʼ ἄλλο τι γίγνεται συμπέρασμα, ἐκ δὲ τῶν Γ Δ ἢ τούτων θάτερον ἢ ἄλλο παρὰ ταῦτα, πλείους τε οἱ συλλογισμοὶ γίνονται καὶ οὐ τοῦ ὑποκειμένου· ὑπέκειτο γὰρ εἶναι τοῦ Ε τὸν συλλογισμόν.
But if from A, B some conclusion other than E arises, and from Γ, Δ either one of these [A, B] or something else besides these arises, the syllogisms will be several and not concerning the proposed question; for it was assumed that the syllogism is of E.