Page:Mind (New Series) Volume 15.djvu/235

" We call that de omni predication, where it is not possible to take any individual denoted by the subject of which the other [i.e. the attribute denoted by the predicate] is not predicated; and de nullo predication is to be defined likewise." Farther on (25 632- 40), a definition and illustration are given of the syllogism in the first figure ; and the syllogism of the first mode is based directly upon the preceding definition of de omni predication: —

ὅταν οὖν ὅροι τρεῖς οὕτως ἔχουσι πρὸς ἀλλήλους ὥστε τὸν ἔσχατον ἐν ὅλῳ εἶναι τῷ μέσῳ καὶ τὸν μέσον ἐν ὅλῳ τῷ πρώτῳ ἢ εἶναι ἢ μὴ εἶναι, ἀνάγκη τῶν ἄκρων εἶναι συλλογισμὸν τέλειον … εἰ γὰρ τὸ Α κατὰ παντὸς τοῦ Β καὶ τὸ Β κατὰ παντὸς τοῦ Γ, ἀνάγκη τὸ Α κατὰ παντὸς τοῦ Γ κατηγορεῖσθαι· πρότερον γὰρ εἴρηται πῶς τὸ κατὰ παντὸς λέγομεν.

"When, therefore, three terms are so related that the minor in its whole extent is the subject of the middle taken affirmatively, and the middle in its whole extent the subject of the major taken affirmatively or negatively, there must be a perfect syllogism of the two extreme terms. . . . For if A is predicated of all B and B of all C, A must be predicated of all C, for we have defined above what is meant by de omni predication."

It is thus seen that Aristotle bases the syllogism upon the dictum de omni et nullo, which is a definition and has nothing to say about the compatibility or incompatibility of the positive and its negative. A still clearer case of Aristotle's reducing the syllogism to -a definition of de omni and de nullo predication can be made out by a reference to his treatment of the syllogism in the contingent mode (32 b 38-33 a 5):—

ὅταν οὖν τὸ Α παντὶ τῷ Β ἐνδέχηται καὶ τὸ Β παντὶ τῷ Γ, συλλογισμὸς ἔσται τέλειος ὅτι τὸ Α παντὶ τῷ Γ ἐνδέχεται ὑπάρχειν. τοῦτο δὲ φανερὸν ἐκ τοῦ ὁρισμοῦ. τὸ γὰρ ἐνδέχεσθαι παντὶ ὑπάρχειν οὕτως ἐλέγομεν. ὁμοίως δὲ καὶ εἰ τὸ μὲν Α ἐνδέχεται μηδενι τῷ Β, τὸ δὲ Β παντὶ τῷ Γ, ὅτι τὸ Α ἐνδέχεται μηδενὶ τῷ Γ· τὸ γὰρ καθ' οὗ τὸ Β ἐνδέχεται, τὸ Α μὴ ἐνδέχεσθαι τοῦτ᾽ ἦν τὸ μηδὲν ἀπολείπειν τῶν ὑπὸ τὸ Β ἐνδεχομένων.

"When, therefore, A may belong to all B and B to all C, there will be a perfect syllogism that A may belong to all C. This is evident from the definition, For this is what we defined de omni contingent predication to mean. Similarly if A may belong to no B and B to all C, it follows that A may belong to no C. For the premiss, 'A may not belong to whatever B may belong to,' has been defined to mean that nothing should be excluded which may come under B."

Similarly in the Posterior Analytics (73 a 28 sq.) Aristotle defines the dictum de omni, and as an illustration gives a syllogism (οἷον εἰ κατὰ παντὸς ἀνθρώπου ζῷον, εἰ ἀληθὲς τόνδ᾽ εἰπεῖν ἄνθρωπον, ἀληθὲς καὶ ζώον…).

But it would be wrong to infer from all this that Aristotle assigns no importance to the principles of contradiction and ex- cluded middle. Quite the contrary. He devotes the entire second part of the fourth (Γ), and a few chapters of the eleventh (Κ) book of the Metaphysics to show that we cannot get along without them, and they are the basis of all ἀπόδειξις and συλλογισμός