1 I wish to thank the Social Sciences and Humanities Research Council of Canada for a Postdoctoral Fellowship, which supported this work. An earlier version of this paper was presented to the Philosophy Department of the University of Western Ontario, and I thank the audience there for discussion, and especially Bill Harper, for pointing out a serious oversight in my initial characterization of Weak Reflection.

2 The discussion to follow of the Surprise Exam Paradox appears also in my ‘Reflection and Truth,’ in Moses, Yoram ed., Theoretical Aspects of Reasoning About Knowledge: Proceedings of the Fourth Conference (San Mateo: Morgan Kaufmann 1992).Google Scholar

3 In an informal presentation to the Philosophy Club at the University of Alberta, Winter 1988.

4 Quine, W.V.O. ‘On a Supposed Antinomy,’ Mind 62 (1953), reprinted in The Ways of Paradox and Other Essays (Cambridge, MA: Harvard University Press 1966), 19-21;Google Scholar and Binkley, Robert ‘The Surprise Exam in Modal Logic,’ Journal of Philosophy 65 (1968) 127-36CrossRefGoogle Scholar

5 It may be objected that the reliability condition R is not appropriately imputed to the students, who, after all, have non-zero probabilities for various events which would prevent the teacher’s intentions from being fulfilled. If we think of N as standing for the proposition that none of these disruptive events takes place (N is a ‘normality’ condition), we can conditionalize R and A_u on N. The inconsistency of A_E A_U, and R (the last two thus conditionalized), still holds, although the proof is not quite so quick. I thank Dave Sharp for an assist with that proof.

6 Sorensen, R. Blindspots (New York: Oxford University Press 1988)Google Scholar

7 The analogue of necessitation is ├ φ ⇒ ├ Baφ, and that of the schema K, ├ Ba(φ →ψ) →. Baφ → Baφ.

8 Koons, Robert C. ‘Doxastic Paradoxes Without Self Reference,’ Australasian Journal of Philosophy 68 (1990) 168-77CrossRefGoogle Scholar

9 In saying ‘no logical relations exist’ I am, of course, excluding ones such as compatibility, and independence, inclusion of which would guarantee that any two propositions are logically related.

10 This fact can also be seen simply by noting that the only principles assumed to govern ‘Ba,’ are shared by some systems of modal logic in which □ (φ &¬ □ φ) is consistent. The model to follow may still be of illustrative value.

11 See, for example, Gaifman, H. ‘A Theory of Higher Order Probabilities,’ in Skyrms, and Harper, eds., Causation, Chance, and Credence, vol. 1, (Dordrecht Kluwer 1988) 191-219CrossRefGoogle Scholar; and Skyrms, B. ‘Higher Order Degrees of Belief,’ in Mellor, D. ed., Prospects for Pragmatism (Cambridge: Cambridge University Press 1986) 109-37.Google Scholar

12 Fraassen, Bas van ‘Belief and the Will,’ Journal of Philosophy 81 (1984) 235-56;CrossRefGoogle Scholar and ‘Belief and the Problem of Ulysses and the Sirens,’ presented to the Philosophy Department of the University of Western Ontario, February 1990. I will henceforth construe the arguments as pertaining specifically to the question of agents’ rationality, as opposed to other sorts of intellectual virtue. Reflection, it may be noted, also goes by the name ‘Miller’s Principle.’

13 David Christensen, following J. Howard Sobel, employs the term ‘Dutch strategy’ to distinguish the kinds of cases presently considered from classical Dutch book. See Christensen, D. ‘Clever Bookies and Coherent Beliefs,’ Philosophical Review 100 (1991) 229-47;CrossRefGoogle Scholar and Sobel, J.H. ‘Self Doubts and Dutch Strategies,’ Australasian Journal of Philosophy 65 (1987) 56-81.CrossRefGoogle Scholar

14 Sobel argues that the ‘ideal intellect’ does not violate Reflection (ibid.). But his conception of the ideal intellect involves virtues that go far beyond simple rationality— e.g., perfect memory and perfect knowledge of one’s beliefs are required of such an agent. Since we cannot, presumably, be faulted for failing to believe what we lack sufficient evidence of, this self-knowledge presupposes (and does not argue) a Cartesian doctrine of accessibility to one’s mental states.

15 It should be noted that van Fraassen is not attempting to assimilate Moore paradoxical beliefs to the logically incoherent ones. His project, particularly as presented in ‘Belief and the Problem of Ulysses and the Sirens,’ is to explore the space between full incoherence and ideal rationality.

16 David Israel makes a similar observation in ‘A Weak Logic of Knowledge and Belief,’ SRI International: Technical Note 359 (1985).