1. Aristotle takes children to have debts toward their fathers: "Hence it would appear that a son never ought to disown his father, although a father may disown his son; for a debtor ought to pay what he owes, but nothing a son can do comes up the benefits he has received, so that a son is always in his father's debt. But a creditor may discharge his debtor, and therefore a father may disown his son" (1163b18).

2. H. L. Hart in his The Concept of Law (Oxford: Clarendon Press, 1961), pp. 122-126, points to problems with formulating laws or rules that are similar to the ones Aristotle identifies. But Hart argues that such problems may stem from two different sources. One is the use in formulating rules or laws of general terms that often defy precise definitions. The other, however, is what he calls "our relative indeterminacy of aims." And the indeterminacy of aims results, in turn, in an indeterminacy of our rules. So that the indeterminacy of a law concerning the distribution of land may not necessarily result from the indeterminacy of the term "land" or the concept land , but from the indeterminacy of our aims regarding land.

3. F. Waismann, "Verifiability," in A. Flew (ed.), Logic and Language , First Series (Garden City, N.J.: Anchor Books, 1965). The most detailed examination of the notion of open texture is to be found in J. M. Brennan, The Open-Texture of Moral Concepts (New York: Macmillan, 1977). Brennan makes some important distinctions in his discussion of open texture, but at times he seems to identify open texture with so many and quite different features of our moral concepts that he almost undercuts the usefulness of the notion (see especially pp. 116-134).

4. At Polit. 1326a35 Aristotle remarks, "But there is a due measure of magnitude for a city-state as there also is for all other things—animals, plants, tools; each of these if too small or excessively large will not possess its own proper efficiency, but in the one case will have entirely lost its true nature and in the other will be in a defective condition; for instance, a ship a span long will not be a ship at all, nor will a ship a quarter mile long, and even when it reaches a certain size, in the one case smallness and in the other excessive largeness will make it sail badly."

Aristotle seems to be saying that our concepts provide us with answers even in situations that are similar to those Waismann imagines.

5. G. Frege, "On the Concept of Number," in H. Hermes, F. Kambartel, and F. Kaulbach (eds.), Frege: Posthumous Writings (Chicago: University of Chicago Press, 1979), p. 77.

6. G. Evans, "Can There Be Vague Objects?" Analysis 38 (1978); N. U. Salmon, Reference and Essence (Princeton, N.J.: Princeton University Press, 1981), p. 243ff.; P. Unger, "There Are No Ordinary Things," Synthese 41 (1979), pp. 117-154; S.C. Wheeler, "On That Which Is Not," Synthese 41 (1979), pp. 155-173.

7. Op . cit. , p. 166.

8. P. Geach and M. Black (eds.), Translations from the Philosophical Writings of Gottlob Frege (New York: Rowman and Littlefield, 1952), p. 139. Wittgenstein responds to Frege's claim at Philosophical Investigations I.71 as follows: "Frege compares a concept to an area and says that an area with vague boundaries cannot be called an area at all. This presumably means that we cannot do anything with it.—But is it senseless to say: 'Stand roughly there?' Suppose that I were standing with someone in a city square and said that. As I say it I do not draw any kind of boundary, but perhaps point with my hand—as if I were indicating a particular spot. And this is just how one might explain to someone what a game is" (see also 76, 77).

9. Thus, M. Black, Margins of Precision (Ithaca, N.Y.: Cornell University Press, 1970), writes "So far as I can see all empirical concepts are 'loose' in the sense which I have explained" (p. 6) and "Now if all empirical concepts are loose, as I think they are . . ." (p. 7).

10. For a discussion of the claim that knowledge of the definition of F is necessary for knowing that some x is F and of the Socratic Fallacy in general, see chap. 2 and the references in notes 8 and 29 of that chapter. See also J. L. Austin's discussion of this issue in his "Are There A Priori Concepts?" in his Philosophical Papers (Oxford: Oxford University Press, 1970).

11. This line of argument is developed by G. Ryle in his The Concept of Mind (London: Hutchinson University Library, 1949).

12. The most important criticisms of the claim that any definition of F is by itself sufficient for determining whether something is an F are those developed by Wittgenstein in the Philosophical Investigations . Wittgenstein, by examining the conditions that need to be met for applying a rule correctly, concludes that no rule is by itself sufficient for determining anything. A whole system of beliefs and practices is, according to him, presupposed or required in applying any rule.

13. See Aristotle's discussion of this matter in Post. Anal. II.xix and Anim. II and the perceptive comments of M. Wedin in his "Aristotle on the Mechanics of Thought," Ancient Philosophy 9 (1989), especially pp. 80-81.

14. See the discussion by M. Black, op. cit. , pp. 11-12.

15. For the problem of the identity of vague objects, see G. Evans and N. U. Salmon, op. cit. The problem with the transitivity of identity of such objects has been discussed by S. Kripke, Naming and Necessity (Oxford: Oxford University Press, 1980), p. 51.

16. Thus, Frege writes that "The law of the excluded middle is really just another

form of the requirement that the concept should have a sharp boundary," op. cit. , p. 139.

17. B. Russell, "Vagueness," Australasian Journal of Psychology and Philosophy I (1923), p. 88. See Wittgenstein's criticisms of this view in Philosophical Investigations I, 104, 105, 106, 107.

18. This point has been made by J. Lear, Aristotle and Logical Theory (Cambridge: Cambridge University Press, 1980), pp. 110-111.

19. Black, who agrees with Aristotle that all empirical concepts are "loose" or that "nothing perceptible can be precisely defined," points to the unwelcome consequence of denying the possibility of valid reasoning with imprecise or vague concepts: "Now if all empirical concepts are loose, as I think they are, the policy becomes one of abstention from any reasoning from empirical premises. If this is a cure, it is one that kills the patient" ( op. cit. , p. 7). Black goes on to argue along lines that Aristotle would probably have found congenial: "In using a loose concept, I must know that there are instances that are indisputably 'clear' and must be able to recognize such cases; and I must also be able to recognize 'borderline cases'. I must understand that the use of the customary logical principles presupposes an ad hoc demarcation, somewhere in the region of cases that are not indisputably clear; and, finally, I must understand that there are no rules for drawing such lines. Provided I understand all this, I may properly use a loose concept and reason with it, even at times about cases which are not indisputably clear" ( op. cit. , p. 12).

20. See Post. Anal. I.xxvii and Met. A.3, as well as the discussion in chap. 4 above.

21. It may be objected in this connection that Aristotle takes the moral virtue (vice) of courage (cowardice) not to be simply a natural disposition, since, according to him, it cannot exist without practical wisdom ( N. E. VI.xiii). But we cannot conclude from this that no causal-explanatory account of the moral virtue (vice) of courage (cowardice) can be given. For, among other reasons, it is not obvious that no causal-explanatory account cannot be given in the case of practical wisdom itself.

22. I do not, of course, mean that there are no causal explanations purporting to explain why cities come into being or why they grow the way they do. I am only questioning whether the causal explanation of these kinds of phenomena about cities is relevant to the question concerning the vagueness or indefiniteness of "city" or of the concept city .

23. A. Grant, The Ethics of Aristotle (New York: Arno Press, 1973), p. 488, and Eustratius' commentary in G. Heylbut (ed.), Eustratii et Michaelis et Anonyma in Ethica Nicomachea Commentaria (Berlin: Reimer, 1892), p. 126.

24. Aristotle himself defines art (craft) in a few lines at N.E. 1140a20 while he devotes most of the Poet. to defining tragedy. Concerning the problems with defining tragedy, see the discussion by B. Vickers, Towards Greek Tragedy (London: Longman, 1973), especially chs. 1 and 2.

25. For a discussion of the problem of trying to make ethical concepts more exact by assigning to them numerical values, see K. Baier, The Moral Point of View (Ithaca, N.Y.: Cornell University Press, 1958), pp. 66-72.

26. I have in mind here Aristotle's argument from function at N.E. I.vii where he concludes that the "human good turns out to be activity of soul in conformity with excellence [or virtue]" (1098a15).

27. J. Rawls, A Theory of Justice (Cambridge: Harvard University Press, 1972), p. 399.

28. W. D. Ross, The Right and the Good (Oxford: Oxford University Press, 1930), p. 67.