Page:Mind (New Series) Volume 15.djvu/234

 220 ISAAC HUSIK: Callias (KaAAt'ai' el /cat /AT) KaXAt'av), still a? long as man is animal and not not-animal, it would follow that Callias is animal and not not-animal. The reason for this is, he goes on to say, that the major term is more extensive than the middle, and applies to not- man as well as to man, and the middle term is more extensive than the minor and applies to not-Callias as well as to Callias ; and therefore even if Callias is both man and not-man (et TO pea-ay KOL avro Ian Kol p.r) avro), this does not prevent the major term animal (and not not-animal) from applying to it. Similarly even if the minor term is both Callias and not-Callias, the major term still applies to it through the middle. I should therefore translate the passage, with more precision than elegance, as follows : " That it is impossible at the same time both to affirm and deny, no demonstration assumes, unless it be necessary to demonstrate the conclusion in this manner [i.e., excluding the opposite]. [In the latter case] the demonstration is effected by assuming that it is true to affirm the major term of the middle, and not true to deny it. But as to the middle term, it makes no difference if we assume that it both is and is not ; and similarly with the minor term. For if we grant [sc. in the major premiss], that that whereof it is true to predicate man (even if it be true also to predicate not- man) as long, at any rate, as we grant that man is animal and not not-animal, it will be true to say that Callias (even if he be at the same time not-Callias) is still animal and not not-animal. The reason is that the major term is predicated not only of the middle but also of something else, because it is more extensive [than the middle] ; and therefore even if the middle is both itself and not itself, it will make no difference to the conclusion. But that everything must be either affirmed or denied is assumed by the demonstration known as reductio ad absurdum. ..." That my interpretation of the first sentence is correct is also proved from the last sentence, which refers to the principle of excluded middle (Waitz confuses the two). That Aristotle thought of them together is shown in a number of places in the Meta- physics where they are mentioned together as the bases of thought. Here he tells us that, whereas the method of proof known as reductio ad absurdum cannot be effected unless we assume the principle of excluded middle, the direct proof does not require the principle of contradiction, unless we want the conclusion to exclude its opposite. And in the latter case, too, it is sufficient to assume the law of contradiction in the major premiss ; it is not necessary even then in the minor. This being the true meaning of Aristotle in the above passage, it is next in order to see whether the view here expressed is consonant with views held elsewhere by Aristotle. In the begin- ning of the Prior Analytics (24 b, 28-30) Aristotle gives a defini- tion of the dictum de omni et nullo as follows : Aeyo/iev 8e TO Kara TTUVTOS Ka-nqyopelcrOai, orav /J.rj8ev 77 Xafieiv rCtv TOV ov, /ca$' ov Oarepov ov )(6if](rfTai, /cat TO /u^Sei/