b16, 10054 21, > 6, 11, and the same narrowing down of the scope of pirocopia is implied in the distinction drawn between it and mathematics in A. 992733, A. 1073? 4.
Philosophy distinguished from mathematics and physics (ch. 4).
1061» 17. Since even the mathematician uses the axioms only in a special form, it is the task of first philosophy to study them. That if equals be taken from equals equals remain is common to all quanti- ties, but mathematics studies the several parts of its subject-matter (lines, angles, numbers, &c.), not gva being but gua continuous; while philosophy studies particular things only in so far as they have being.
27. Physics is like mathematics; it studies the properties and principles of things gua moving, not gua being, while the first science studies the attributes of being as being. Hence physics and mathe- matics are only parts of wisdom.
This chapter corresponds to IT. 1005*19-»2, and answers the question raised in 1059? 23-26.
1061” 18. tots Kowois, i.e. the aéimpara, cf. B. 997% 10.
TouTuy, i.e, Tov pabnpatiKOv.
20. kody pev éotiy éml mavtwy tav woody, i.e. this proposition is neither a common principle of all sciences nor peculiar to one; it is common to all the sciences of guantity. It thus belongs to a type not recognized in Aristotle’s classification of dpxai, An. Post. 76% 38. This axiom is similarly treated as a xowov, though really common only to the sciences of quantity, in Aw. Post. 774 30, 31.
21. drodkaBodoa, ‘cutting a part off for separate consideration’. Cf. Poet. 1459% 35 and the use of drorepvec$a, I. 1003% 24.
25. 4 3é prdogodia, cf. 1. 5 n.
26-27. mept To dy . . . Oewpet. The construction seems to be, ‘but speculates about that which is, in so far as each particular thing is’ (7 bv trav TowtTwy exactoy is opposed to 7 ovvexés airy ExacTov, 1, 24)—which is a brachylogy for ‘but speculates about that which is, and about particular things only in so far as each of them is’.
32. tavTny apparently refers to 7 vou; the clause answers to I, 1005) 1 éori d€ codia tis kal 7 pvoixy. THy démpaerny ... erepov 7. must therefore be treated as parenthetical. The point of connexion between 8.6, &c., and what precedes is not very clear. Alexander supplies a reason for the conclusion that physics and mathematics are branches of wisdom, viz. érevd) dodekvvovor Kal 6 pabnpatiKds Kal 6 pvorkos, kal ovdérote Wetdovrat, but this cannot properly be read into the text. It seems more likely that pépy is to be stressed. Because mathematics and physics do not study their objects gva being but gua continuous or gua moving, they are merely branches of wisdom; wisdom proper is
K. 4. 1ro61> 18—33 315
the more comprehensive science which studies being as such. This answers to the corresponding statement inT. 1005» 1 gor 5¢ copia Tis Kat 4 pvouxy, GAN od zpern.
33. For this wide sense of codéa in which it includes mathematics — and physics as well as philosophy cf. A. 981227, T. roo5b1, and EN. vi. 7, where it includes the study of é€ dy 6 xécpos ovvéornker, the constituents of the physical universe (11419 1),
Defence of the law of contradiction (ch. 5).
1061> 34. There is a principle about which it is impossible to be deceived—that the same thing cannot at the same time be and not be. Such truths cannot be proved absolutely because there is no more certain premise from which they can be inferred.
10628 5. But they can be proved to a particular person, viz. to any- one who makes contradictory assertions; the method is to assume something that is identical with the truth in question but does not seem so to our opponent.
11. People who are to discuss together must to some extent under-
stand each other; therefore each of their words must mean something, and ifit is ambiguous the intended meaning must be indicated. Now (1) he who says ‘A is and is not B’ is saying that the word B does not mean what it does mean, and since this is impossible the law of contradiction is proved. - 1g. (2) If the word means something and this meaning is truly asserted of a subject, the subject must necessarily have this character, and therefore can never not have it; so that the opposite statements cannot be true of the same subject.
23. (3) If the affirmation is no more true than the negation, ‘Ais a man’ is no truer than ‘A is ‘not a man’, and therefore a fortior? no truer than ‘A is not a horse’, and therefore (if contradictions are both true) no truer than ‘ A is a horse’; so that the same object is a man, a horse, &c.
gt. If one had argued thus with Heraclitus he might have abandoned his view ; he adopted it without realizing what he was saying. If his dictum were true, even it itself would not be true. For if‘ A is B ’ is no truer than“ A is not B’, ‘ A is both B and not B’ is no truer than ‘A is neither B nor not B’.
by, Further, if nothing can be truly affirmed it is false to say that no affirmation is true; while if something can be truly affirmed the dictum of these destroyers of all discussion is upset.
316 Commentary
The chapter covers the ground of IT. 3. 1005» 8-end and of TI. 4, with the exception of
(1) the argument that if the law of contradiction be denied, substance is denied and everything reduced to accident (10078 20-» 18),
(2) the group of arguments in 10084 7-b 42,~
(3) the argument that the opponents of the law of contradiction themselves act upon it (1008 r2—1009%5). For this cf. K. 1063? 28-35.
1061» 341062? 2 answers to 1005” 8-34.
36—106222. Two kinds of contradictory proposition are here referred to, (1) ‘A is’ and ‘ A is not’, (2) ‘A is B’ and ‘A is not B’ (réAXa. Ta TOdTOV avToIs avTiKe(“eva TOV Tporov). Of these the second is the more general, since the first may be exhibited as a species of it, ‘ A is existent’ and ‘A is not existent’.
1062? 2-5 answers to 1006? 5-18.
5-I9 answers to 10064 18—1007? 20.
12, adtav = dhAnAwv. Cf. L. and S. s. v. m1.
16-19. In view of 1061> 36—10624 2 (where see n.) the meaning seems to be ‘ He, then, who says that A is B and that it is not B de- nies what he asserts, so that what the word B means he says it does not mean ; but this is impossible, so that if ‘‘being B” means something ’ (or ‘if any word means “ being something” ’), ‘the contradictory of it cannot be truly asserted of the same subject’.
17. ToUvopa, Sc. TO elvar, Says Bz., but the reference is to any predicate, cf. tév é6vopatwv exacrov, 1, 13. In 1006 30, to which Bz. refers, it is not clear whether 70 civai 7} a) etvar is as he supposes explicative of 76 dvopa, OF 70 €lvat 7 pa) eivar Todi is the object of oypaiver, so that 7o dvowa would be quite general.
18. elwep onpaiver tr 7d elvor Té8e. 7. may be either subject or object of onpatver.
Ig-23 answers to 1006) 28-34. The meaning is expressed more clearly in I. dvdyxn roivuv, «i te €otw GAnGes ciety O71 GvOpwros, Gov civar dirovy (rodro yap jv 6 éojpawe 70 dvOpwros) <i 3 dvayKy TOUTO, OdK evdexeTaL py Elvat (TOTE) TO adTo LGov dirow.
23-30 answers to 1007) 18—1008? 2,
31-35 answers to 1005) 23-26.
36-7 answers to 1008%4-7. The point is, however, different. Aristotle is there showing that if the law of contradiction is denied, the law of excluded middle must be denied too (ovx dvayxy 7 paver 7 aropavat). Ifit is true that Aisa man and not a man, it will (or may) also be true that he is neither a man nor not a man. Here the point which is proved is that if the law of contradiction is denied, the denial of it must be denied too. If it is true that A is a man and not a man, then this proposition itself is no more true than its opposite.
°6. otf pa\ov * dxddacis 3} Td Sdov ds ev Katagpace TiHEwevor GAnGevcerat gives just the opposite of the right sense. The simplest emendation is to transpose # to before , where it would easily have dropped out. A less probable alternative is to read jrrov for uadAor. 7-9 answers to ror2? 13-18.
Lnadequacy of the grounds for denying the law of contradiction (ch. 6).
1062> 12, Protagoras’ saying, ‘man is measure of all things’, is similar to the views we have been discussing. He means that what seems to each man is so; then since things seem different to different people, the same thing will be and not be.
20. The problem will be solved if we consider the origin of this belief. (1) With some, it arose from the doctrine of the physicists ; (2) with others, from observing that people have different impressions about the same thing.
24. (1) That nothing can come into being from not-being but everything from being is a view common to most of the physicists. Therefore, since a thing does not become white if it was perfectly white, the white must come from what is not white ; so that according to them it must come from not-being, unless the same thing was white and not white. The difficulty is easily removed; we have stated in the P&ysics in what sense things come to be from being and in what sense from not-being.
38. (2) (@) It is foolish to attend equally to opposite opinions. The same thing never seems sweet and sour to two people unless the sense-organ of the one has been injured; in that case only the other man is a measure. So too with good and bad, &c. We might just as well claim that the object which seems to be two when you press your eye must de two.
1063* 10. (4) We ought to judge of the nature of things not from the changes in the things around us but from the changelessness of the heavenly bodies.
17. (c) Again, if there is motion, there is a moving thing, which moves from something and into something ; and when it is in the one it is not in the other,
as. (¢) Even if things in this world were always changing in quantity—and the observation of such changes is one of the chief motives of the theory—they need not be always changing in quality, and essence depends on quality, which is determinate, not on quantity, which is indeterminate.
318 Commentary
28. (e) Again, the fact that people act on the doctor’s orders shows that they think things have a determinate nature which is not always changing.
35. (/) If we are always changing, the changes in our perceptions do not show the objects to be changing ; if we-are mo¢ changing, then there is something that is at rest.
b7. It is not easy to meet those who feel these difficulties on dialectical grounds, for if they will not posit something and no longer demand a reason for it, they make discussion impossible; those, on the other hand, who are puzzled by the traditional difficulties may be answered as we have shown.
15. Contradictory statements, then, cannot both be true; nor can contraries, because contrariety involves privation, as may be seen by analysing the definitions of contraries.
1g. Nor can an intermediate be predicated of the same subject of which one of the extremes is asserted. If A is white we cannot truly say it is neither black nor white, for we shall be saying that it is both white and not white.
24. Therefore neither the view of Heraclitus is right, nor that of Anaxagoras that there is a portion of everything in everything; if everything is present acfually in everything, contraries are true of the same thing.
go. Similarly all'statements cannot be false, nor all true—for this reason in addition to others, that if all are false this statement itself is false, and if all are true it will be true that all are false.
This chapter covers almost the whole ground covered in T. 5-8, the main sections omitted being (1) the references to earlier views in 1009 38—1010% 22, (2) the arguments in 1010 26—1011% 2, 1orT® 17-12,
1062 12-24 answers to 10092 6-16, 22-30.
21-24. Of the two reasons for the Protagorean theory, the first is discussed in 24-33, the second in 33—1063? 7.
24-33 answers to 1009? 30-36, though not closely.