1 This line of argument is popular among logical positivists. See, for example, Ayer, A.J. Language, Truth and Logic (New York: Dover 1952)Google Scholar, Ch. 4; and Hempel, Carl ‘On the Nature of Mathematical Truth,’ reprinted in Benacerraf, P. and Putnam, H. eds., Philosophy of Mathematics (Englewood Cliffs, NJ: Prentice-Hall 1964)Google Scholar. A more recent version of the argument can be found in Field, Hartry Science Without Numbers (Princeton: Princeton University Press 1980)Google Scholar, Ch. 1.

2 Both Ayer and Hempel offer such a defense of premise (1) in the works cited in n. 1.

3 It is important to recognize that the antecedent countings which establish that there are two pairs of objects present are observationally independent of the third counting which establishes that there are four objects present. For, as an anonymous referee has correctly stressed, if the truth of the claim that there are two pairs of objects present and the truth of the claim that there are four objects present could not be observationally established independently of one another, then it would be plausible to maintain that it is analytically true that if one pair of objects is added to a second pair of objects then the total number of objects is four.

4 Hilary Putnam has stressed that there are two different ways in which experience can disconfirm a proposition: (1) it can provide evidence for the negation of the proposition; or (2) it can provide evidence that some of the concepts involved in the proposition should be given up. See, for example, his paper, ‘There Is At Least One A Priori Truth,’ Erkettnis 13 (1978), 153–70Google Scholar.

5 This position has been forcefully defended by Putnam, Hilary in ‘Philosophy of Logic’ and ‘What Is Mathematical Truth?’ both of which are reprinted in his Mathematics, Matter and Method: Philosophical Papers, Vol. 1, 2nd ed. (Cambridge: Cambridge University Press 1979).CrossRefGoogle Scholar

6 This response to the argument of the previous paragraph was made by an anonymous referee.

7 This point is stressed by Hempel, Carl in ‘The Nature of Mathematical Truth,’ and in ‘Geometry and Empirical Science,’ reprinted in Feigl, H. and Sellars, W. eds., Readings in Philosophical Analysis (New York: Appleton-Century-Crofts 1949)Google Scholar.

8 This point was first argued by W.V. Quine in his classic paper ‘Truth by Convention,’ reprinted in Feigl and Sellars, eds., Readings in Philosophical Analysis.

9 The best known statement of this argument is found in Russell, Bertrand The Problems of Philosophy (Oxford: Oxford University Press 1971), Ch. 7.Google Scholar It has been resurrected recently by Kim, Jaegwon in ‘The Role of Perception in A Priori Knowledge: Some Remarks,’ Philosophical Studies 40 (1981), 339–54CrossRefGoogle Scholar. Perhaps Kant had this argument in mind when he claimed that ‘strict universability’ is a mark of the a priori in the Introduction to the Critique of Pure Reason, trans. Smith, Norman Kemp (New York: St. Martin's Press 1965), Sect. 2Google Scholar.

10 R.M. Chisholm maintains that demonstrative a priori knowledge is not certain. He also maintains that the conclusions of long demonstrations which involve reliance upon memory are not known a priori. See Theory of Knowledge, 2nd ed. (Englewood Cliffs, NJ: Prentice-Hall 1977), 44. Kitcher, Philip endorses both of these points in The Nature of Mathematical Knowledge (Oxford: Oxford University Press 1983), Ch. 2Google Scholar.

11 Roderick Firth, ‘The Anatomy of Certainty,’ reprinted in Chisholm, R. M. and Swartz, R. J. eds., Empirical Knowledge (Englewood Cliffs, NJ: Prentice-Hall 1973).Google Scholar

12 See Pappas, George ‘Incorrigibility, Knowledge and Justification, Philosophical Studies 25 (1974), 219–25CrossRefGoogle Scholar; and Lehrer, Keith Knowledge (Oxford: Oxford University Press 1974), Ch. 4Google Scholar.

13 This sense of certainty is suggested by both Kitcher, P. The Nature of Mathematical Knowledge, Ch. 1Google Scholar; and Kroon, F. ‘Contingency and the A Posteriori,’ Australasian Journal of Philosophy 60 (1982), 40–54.CrossRefGoogle Scholar

14 Chisholm, Theory of Knowledge, 10Google Scholar

15 Ibid., 12-15

16 See definitions 3.1 and 3.2 in Theory of Knowledge, 42.

17 Firth offers this definition in ‘The Anatomy of Certainty’ (Empirical Knowledge, 215) as an account of Malcolm's use of the term in ‘Knowledge and Belief,’ reprinted in Malcolm's Knowledge and Certainty (Englewood Cliffs, NJ: Prentice-Hall 1963).

18 This point was raised by an anonymous referee.

19 The locus classicus of this argument is the Introduction to Immanuel Kant's Critique of Pure Reason, Parts II and V. It is echoed by Russell in Ch. 7 of The Problems of Philosophy and by Chisholm in Ch. 3 of Theory of Knowledge. Kim appears to reject the argument in ‘The Role of Perception in A Priori Knowledge: Some Remarks’ but does so by rejecting premise (1).

20 Kant, Critique of Pure Reason, 43Google Scholar

21 Ibid., 52

22 Kripke, Saul ‘Identity and Necessity,’ in Munitz, M. K. ed., Identity and Individuation (New York: New York University Press 1971)Google Scholar and Naming and Necessity (Cambridge, MA: Harvard University Press 1980)

23 See, for example, Fitch, G.W. ‘Are There Necessary A Posteriori Truths?’ Philosophical Studies 30 (1976), 243–7CrossRefGoogle Scholar and Plantiga, Alvin The Nature of Necessity (Oxford: Oxford University Press 1974), 81–7Google Scholar.

24 Chisholm suggests such an approach to testimonial evidence in Theory of Knowledge, 47. For a discussion of the epistemic significance of computer results, see Tymoczko, T. ‘The Four-Color Problem and its Philosophical Significance,’ Journal of Philosophy 76 (1979), 57–83CrossRefGoogle Scholar; Teller, P. ‘Computer Proof,’ and Detlefsen, M. and Lukes, M. ‘The Four-Color Theorem and Mathematical Proof,’ both in Journal of Philosophy 77 (1980), 797–820CrossRefGoogle Scholar.

25 See, for example, Chisholm, R.M. Theory of Knowledge, 37Google Scholar; and McGinn, C. ‘A Priori and A Posteriori Knowledge,’ Proceedings of the Aristotelian Society 76 (1975-76), 195–208CrossRefGoogle Scholar. Kitcher, Philip ‘Apriority and Necessity’ (Australasian Journal of Philosophy 58 (1980), 100–1)CrossRefGoogle Scholar, also maintains that the plausibility of the Modal version of the Argument from Necessity depends on this claim. He goes on to reject the argument for reasons different from mine.

26 I want to thank Robert Audi and Philip Hugly for their comments on earlier versions of this paper. The reports of two anonymous referees for the Canadian Journal of Philosophy were of invaluable assistance in improving the paper.