1 APo. A2, 72b9-12, B11, 94a20; Ph. A1, 184 al0-15, B3, 194b17-23; Metaph. A1, 981 a 24-30, A2, 982 b2-4

2 APr. B16, 64b34-36 with APo. A9, 76a19-20; see also APo. A10, 76b23-24.

3 APr. B16, 64b34-36; cf. APo. A2, 72a30-32 and Ph. A1, 184 a12-14.

4 APo. A2, 71b20-22, b29-30, 72a28-39 (esp. 28-29, 31-32, 36-37, 37-38); B19, 100b9-10; cf. A3, 72b5-6, 20-25.

5 APo. A2, 71b9-b16

6 APo. A2, 71b20-21, 72a7-8; A3, 72b18-22 (Also, see A9, 76 a16-17 on theἴδια principles.)

7 APo. 72a7-8

8 See APo. A9, 76a19-20:ἐχ τῶν προτέρων γὰροἶδεν,ὅτανἐχ μὴ αἰτιατῶν εἰδῆ αἰτίων.

9 This is, I take it, the purport of B19, 100b10-11. Note that A3 does not actually commit Aristotle to the claim that there isἐπιστήμη of the first principles. At 72b5 ff., Aristotle is reporting what is thought by some people. In his own response at b18 ff., he accepts the claim that the first principles must be knowable, but as the ’ εἰ ‘ of 20 indicates, he may be accepting it only for purposes of argument.

10 APo. B19, 100b12

11 For instance, see Metaph. Z6, 1031 b20-21.

12 APo. A2, 72 a14-24

13 APo. A10, 76a37ff., esp. 76b14-15; cf. Metaph. B2, 996b26-29. In the case of definitions, we acquire our understanding through a process ofἐπαγωγή that begins with perception of some of the particulars that fall within the scope of the science to which these principles are proper (B19, esp. b4) - provided, of course, that the particulars in question are perceptible.

14 APo. A2, 72a27 (see also A10, 76b14-15); A7, 75a41-42; Metaph. B2, 996b272-29

15 APo. A2, 72a25-29

16 APo. B19, 100b5ff

17 I am here putting aside the well-known problem concerning the axiom of equals. This axiom is not common to all sciences, and it is by no means clear that one must understand it in order to understanding anything at all. (See APo. A10, 76a41; All, 77a30-31.)

18 Metaph.Г3, 1005b19-22. See also B1, 995b-10; B2, 996b29-30; APo. All, 77a10. For present purposes I am conflating the ‘ontological’ with the ‘semantical’ formulations.

19 Metaph. B1, 995b8-10; B2, 996b26-30; cf. APo. A2, 72a27 and A3, 72b21.

20 Metaph.Г3, 100bll-32. Aristotle does not explicitly discuss the connection between inability to disbelieve instances and the necessity to believe them. Furthermore, there is not (here at any rate) a treatment of the relation between instances (and our propositional attitudes with respect to them) and the general proposition (and our propositional attitudes with respect to it). This latter is, I believe, a highly abstract version of the question whether the principles of things are particular or general.

21 Metaph.Г 3, 1005b33-34

22 Metaph.Г3, 1005b16-17

23 As W.D. Ross, Aristotle's Metaphysics, Vol. 1 (Oxford: Clarendon Press 1924), 263, states in his note on 1005b14: ‘ἀνυπόθετον is used quite in the Platonic sense of the word,’ and not with reference to Aristotle's use of ‘ὑπόθεσιζ ’ for one kind of first principle.

24 See note 8.

25 Metaph.Г3, 1005b14-15

26 T.H. Irwin, ‘Aristotle's Discovery of Metaphysics,’ Review of Metaphysics 31 (1977). 210-29

27 Metaph.Г4. 1006a32-33

28 Metaph.Г4, 1007a20-b18

29 APr. B16, 64b36-65al - though here he in fact characterizes question-begging only for alleged proofs of propositions that are not first principles.

30 Metaph.Г 4. 1006a16-17

31 Metaph.Г 4, 1006a17-18

32 J. Lukasiewicz attributes to Aristotle the view that ‘an elenchos is an ordinary syllogism differing from a proper proof only in the extrinsic fact that it is used as a means of refutation,’ and hence concludes that the distinction between proof and elenctic proof is ‘entirely empty’ (‘Aristotle on the Law of Contradiction,’ in ]. Barnes, M. Schofield & R. Sorabji, eds., Articles on Aristotle, Vol. 3 [London: Duckworth 1979], 55; originally published as ‘Ueber den Satz des Widerspruchs bei Aristotles,’ in Bulletin International de l'Académie des Sciences de Cracovie [1910]). His criticism is vitiated by the fact that he does not observe that an elenctic proof, unlike a proof proper, need not explain the conclusion. Lukasiewicz also misconstrues the point, at 1006a15-18, about begging the question. He takes Aristotle to be saying that somebody trying to prove PNC is guilty of begging the question, but that if the opponent commits the fallacy ‘an elenchos is possible – and everything is all right.’ The point, however, is that the advocate might seem to beg the question if he attempts a proof proper, but if he can elicit premises from the opponent that can be used to derive PNC, he will then be in a position to give (not a proof proper but) an elenctic proof. Since in an elenctic proof neither party is trying to explain the truth of the conclusion, neither will be committing the fallacy of petitio principii.

33 Metaph.Г 4, 1006a21

34 Metaph.Г 4, 1006b26-28

35 These questions are in the spirit of Metaph. B1, 995b6-10 and B2, 996b26-15.

36 This will be a γένοζ (APo.A10, 76b11-13) or γένοζὑποχείμενον (APo.A7, 75a41-bl, Metaph. B2, 997a6).

37 See APo.B7, 92bl3-14 where it is said that being is not theοὐσία (substance or essence) of anything because it is not a genus. (Also, Metaph. B3, 998b22-27.)

38 Metaph.Г2, 1004b22; cf.?3, 1005b9.

39 These include πάθη (APo.A10, 76bll-15 - see also A7, 75bl and Metaph. B2, 997a6-7) and various kinds of συμβεβηχότα (75bl).

40 See APo.A4 - esp. 73b28-29.

41 APo.A4, 73a34-bl

42 APo.A4, 73bl-4

43 APo.A4, 73b10-16

44 APo.A4, 73b4-5

45 See G.E.L. Owen, ‘Logic and Metaphysics in Some Earlier Works of Aristotle,’ in Aristotle and Plato in the Mid-Fourth Century, ed. I. Düring and G.E.L. Owen (Göteborg: 1960). 163-90.

46 In this sense there is a universal mathematics, distinct from such particular mathematical disciplines as arithmetic, geometry and astronomy (Metaph. El, 1026a25-27; cf. A.Po.A5, 74a17-25). This studies what belongs to numbers, lengths, etc., but not what belongs to them as such, but rather qua some more general term (e.g., quantity).

47 These will be type 2.B.(ii) terms that belong to beings as such.

48 Metaph.Г 2, 1004b5-6

49 Metaph.Г2, l005al0-18

50 See Metaph.Г 3, 1005b26-28 andГ6, 1011b15-22.

51 Metaph.Г3, 1005b32-4

52 Metaph.Г3, 1005b22ff

53 Metaph.Г4, 1006a3-5

54 See Jonathan Lear, Aristotle and Logical Theory (Cambridge: 1980), Chapter 6, ‘Proof by refutation,’ 99, 113-14. My thinking on the matters discussed in this paper was greatly assisted by Lear's book.

55 This paper was delivered to the Pacific Division of the American Philosophical Association in March 1984. Montgomery Furth and S. Marc Cohen were the commentators, and their replies are published in this issue. Substantial portions of previous drafts were presented to both the Philosophy and Classics Departments of the University of Texas, Austin, and to the Logic Seminar of the Section on Foundations, Department of Mathematics, University of California, Berkeley. In addition to the members of those audiences, I would like to thank Allan Silverman and Gregory Vlastos for comments and advice. I owe a special debt to H.P. Grice. The central topic of this paper (‘Which Science Investigates the Principle of Non-Contradiction?’) and the strategy for dealing with it both arose from extended discussion with him concerning the characterization of metaphysics.