§2.1.25Ὅτι μὲν οὖν ἡ καθόλου βελτίων τῆς κατὰ μέρος, τοσαῦθʼ ἡμῖν εἰρήσθω· ὅτι δʼ ἡ δεικτικὴ τῆς στερητικῆς, ἐντεῦθεν δῆλον.
Thus, let so much be said by us to show that universal demonstration is better than particular; but that ostensive is better than negative demonstration, is clear from what follows.
ἔστω γὰρ αὕτη ἡ ἀπόδειξις βελτίων τῶν ἄλλων τῶν αὐτῶν ὑπαρχόντων, ἡ ἐξ ἐλαττόνων αἰτημάτων ἢ ὑποθέσεων ἢ προτάσεων.
For let that demonstration be better, other things being the same, which proceeds from fewer postulates or hypotheses or premises.
εἰ γὰρ γνώριμοι ὁμοίως, τὸ θᾶττον γνῶναι διὰ τούτων ὑπάρξει· τοῦτο δʼ αἱρετώτερον.
For if they are equally well known, knowing more quickly will belong to us through these; and this is more desirable.
λόγος δὲ τῆς προτάσεως, ὅτι βελτίων ἡ ἐξ ἐλαττόνων, καθόλου ὅδε· εἰ γὰρ ὁμοίως εἴη τὸ γνώριμα εἶναι τά μέσα, τὰ δὲ πρότερα γνωριμώτερα, ἔστω ἡ μὲν διὰ μέσων ἀπόδειξις τῶν Β Γ Δ ὅτι τὸ Α τῷ Ε ὑπάρχει, ἡ δὲ διὰ τῶν Ζ ὅτι τὸ Α τῷ Ε. ὁμοίως δὴ ἔχει τὸ ὅτι τὸ Α τῷ Δ ὑπάρχει καὶ τὸ Α τῷ Ε. τὸ δʼ ὅτι τὸ Α τῷ Δ πρότερον καὶ γνωριμώτερον ἢ ὅτι τὸ τῷ Ε· διὰ γὰρ τούτου ἐκεῖνο ἀποδείκνυται, πιστότερον δὲ τὸ διʼ οὗ.
The reason for the proposition that the demonstration from fewer is better, is universally as follows: for if the middle terms are known in a similar way, and the prior ones are better known, let one demonstration that A belongs to E be through the middle terms B, C, D, and the other through Z [and H] that A belongs to E. Then, that A belongs to D and that A belongs to E are in a similar position. But that A belongs to D is prior and better known than that [A belongs] to E; for the latter is demonstrated through the former, and that through which something is demonstrated is more trustworthy.
καὶ ἡ διὰ τῶν ἐλαττόνων ἄρα ἀπόδειξις βελτίων τῶν ἄλλων τῶν αὐτῶν ὑπαρχόντων.
Therefore, the demonstration through fewer is also better, other things being the same.
ἀμφότεραι μὲν οὖν διά τε ὅρων τριῶν καὶ προτάσεων δύο δείκνυνται, ἀλλʼ ἡ μὲν εἶναί τι λαμβάνει, ἡ δὲ καὶ εἶναι καὶ μὴ εἶναί τι· διὰ πλειόνων ἄρα, ὥστε χείρων·
Now both are demonstrated through three terms and two premises; but the former assumes that something is, whereas the latter assumes both that something is and that something is not; therefore, it proceeds through more, so that it is inferior.
Ἔτι ἐπειδὴ δέδεικται ὅτι ἀδύνατον ἀμφοτέρων οὐσῶν στερητικῶν τῶν προτάσεων γενέσθαι συλλογισμόν, ἀλλὰ τὴν μὲν δεῖ τοιαύτην εἶναι, τὴν δʼ ὅτι ὑπάρχει, ἔτι πρὸς τούτῳ δεῖ τόδε λαβεῖν.
Furthermore, since it has been shown that it is impossible for a syllogism to come to be when both premises are negative, but one must be such [negative] and the other [must state] that something belongs [affirmative], in addition to this we must grasp the following.
τὰς μὲν γὰρ κατηγορικὰς αὐξανομένης τῆς ἀποδείξεως ἀναγκαῖον γίνεσθαι πλείους, τὰς δὲ στερητικὰς ἀδύνατον πλείους εἶναι μιᾶς ἐν ἅπαντι συλλογισμῷ.
For when the demonstration is expanded, it is necessary that the affirmative premises become more, but it is impossible for the negative premises to be more than one in any syllogism.
ἔστω γὰρ μηδενὶ ὑπάρχον τὸ Α ἐφʼ ὅσων τὸ Β, τῷ δὲ Γ ὑπάρχον·
For let A belong to nothing of which B is predicated, and let B belong to all C.
παντὶ τὸ Β. ἂν δὴ δέῃ πάλιν αὔξειν ἀμφοτέρας τὰς προτάσεις, μέσον ἐμβλητέον.
Now if it is necessary to expand both premises again, a middle term must be inserted.
τοῦ μὲν Β ἔστω τὸ Δ, τοῦ δὲ Β τὸ Ε. τὸ μὲν δὴ Ε φανερὸν ὅτι κατηγορικόν, τὸ δὲ Δ τοῦ μὲν Β κατηγορικόν, πρὸς δὲ τὸ στερητικὸν κεῖται.
Let D be the middle of [the premise concerning] B, and E of [the other concerning] B. Now it is clear that E is affirmative, while D is affirmative of B, but is placed in relation to the negative premise.
τὸ μὲν γὰρ Δ παντὸς τοῦ Β, τὸ δὲ οὐδενὶ δεῖ τῶν Δ ὑπάρχειν.
For D must belong to all B, and [A] must belong to none of D.
γίνεται οὖν μία στερητικὴ πρότασις ἡ τὸ Α Δ. ὁ δʼ αὐτὸς τρόπος καὶ ἐπὶ τῶν ἑτέρων συλλογισμῶν.
Thus there is only one negative premise, namely A-D. And the same mode applies also to the other syllogisms.
ἀεὶ γὰρ τὸ μέσον τῶν κατηγορικῶν ὅρων κατηγορικὸν ἐπʼ ἀμφότερα· τοῦ δὲ στερητικοῦ ἐπὶ θάτερα στερητικὸν ἀναγκαῖον εἶναι, ὥστε αὕτη μία τοιαύτη γίνεται πρότασις, αἱ δʼ ἄλλαι κατηγορικαί.
For the middle of the affirmative terms is always affirmative in both directions; but of the negative, it must be negative in one of the directions, so that this becomes only one such premise, and the others are affirmative.
εἰ δὴ γνωριμώτερον διʼ οὗ δείκνυται καὶ πιστότερον, δείκνυται δʼ ἡ μὲν στερητικὴ διὰ τῆς κατηγορικῆς, αὕτη δὲ διʼ ἐκείνης οὐ δείκνυται, προτέρα καὶ γνωριμωτέρα οὖσα καὶ πιστοτέρα βελτίων ἂν εἴη.
If, then, that through which something is demonstrated is better known and more trustworthy, and the negative is demonstrated through the affirmative, while the latter is not demonstrated through the former, then being prior and better known and more trustworthy, the affirmative would be better.
ἔτι εἰ ἀρχὴ συλλογισμοῦ ἡ καθόλου πρότασις ἄμεσος, ἔστι δʼ ἐν μὲν τῇ δεικτικῇ καταφατικὴ ἐν δὲ τῇ στερητικῇ ἀποφατικὴ ἡ καθόλου πρότασις, ἡ δὲ καταφατικὴ τῆς ἀποφατικῆς προτέρα καὶ γνωριμωτέρα (διὰ γὰρ τὴν κατάφασιν ἡ ἀπόφασις γνώριμος, καὶ προτέρα ἡ κατάφασις, ὥσπερ καὶ τὸ εἶναι τοῦ μὴ εἶναι)· ὥστε βελτίων ἡ ἀρχὴ τῆς δεικτικῆς ἢ τῆς στερητικῆς· ἡ δὲ βελτίοσιν ἀρχαῖς χρωμένη βελτίων.
Furthermore, if the principle of a syllogism is a universal immediate premise, and the universal premise in ostensive demonstration is affirmative, while in negative demonstration it is negative, and the affirmative is prior to and better known than the negative (for the negation is known through the affirmation, and the affirmation is prior, just as being is prior to non-being); so that the principle of ostensive demonstration is better than that of negative demonstration; and that which uses better principles is better.
ἔτι ἀρχοειδεστέρα· ἄνευ γὰρ τῆς δεικνυούσης οὐκ ἔστιν ἡ στερητική.
Furthermore, it is more like a principle; for without ostensive demonstration there is no negative demonstration.