1 Here and throughout I intend ‘know’ to be taken in the propositional sense, and I assume that the traditional account of knowledge as Justified true belief provides a necessary condition for knowledge in this sense.

2 Lehrer, Keith Knowledge (Oxford: Clarendon Press 1974).Google Scholar Parenthetical page references to this text are prefixed by ‘K’.

3 Lehrer calls this an incorrigible belief according to the following definition: ‘S has an incorrigible belief that p if and only if (i) it is contingent that p and (ii) it is logically impossible that S believes that p and it is false that p’;. (K, 83) Lehrer argues that too few beliefs would qualify as basic, much less incorrigible, to constitute a foundation for knowledge.

4 Harman, Gilbert Thought (Princeton: Princeton U. P. 1973) 119.Google Scholar Harman attributes the suggestion to Robert Nozick but gives no further reference. Keith Lehrer takes up the suggestion himself in ‘Reason and Consistency’ in Keith Lehrer ed. Analysis and Metaphysics, Essays in Honor of R.M. Chisholm (Dordrecht: D. Reidel 1975). Raymond Smullyan, who claims to have thought of the idea some thirty years ago, mentions it approvingly in his What Is the Name of This Book?, The Riddle of Dracula and other Logical Puzzles (Englewood Cliffs; Prentice Hall 1978) 206. In the June 1979 meetings of the Canadian Philosophical Association, there were two papers on this topic: ‘Doublethink’ (unpublished) by Michael Stack, and an earlier version of the present paper.

5 If the set of beliefs is infinite there is a complication. For it is impossible to prove that a contradiction can be deduced from the set in question. (See Quine, W.V.O. Set Theory and Its Logic (Cambridge, Mass:, Harvard U. P. 1963) 304f.Google Scholar) There would still be an inconsistency in some sense, however, since not all the beliefs could be true. Moreover, none of the points below will be affected if we restrict ourselves to finite sets of beliefs.

6 Lehrer, op. cit.; parenthetical page references to this text are prefixed by ‘RC’.

7 Harman, op. cit., 119.

8 Another version of this puzzle is mentioned by Lehrer (RC, 60) and attributed to Henry E. Kyburg, Jr., ‘Conjunctivitis,’ in Swain, Marshall ed. Induction, Acceptance, and Rational Belief (Dordrecht: D. Reidel 1971) 77.Google Scholar

9 I beg the reader's indulgence. It is unlikely that anyone reading this paper would be confused on this point. Yet the present objection keeps cropping up, and unless I have completely misunderstood it, the objection rests partly on this mistake.

10 We can prove this result formally be deducing a contradiction from the set. The steps would involve DeMorgan's Rule and repeated applications of Disjunctive Syllogism. It is important to note that we need not assume that the set itself is closed under conjunction.

11 SEP and CC are analogous to the modal rule RR [A → B ⇒ □ A → □ B] and the axiom schema K [□ A & □ B → □ (A & B) ]. See Krister Segerberg, An Essay in Classical Modal Logic (Stockholm, 1971) 12. It is easy to demonstrate that RR and K are equivalent to a rule of closure under consequence, i.e., for Г =﹛AI□A﹜, Г⊢ A ⇒ A ϵ Г. The reasoning is essentially the same to show that SEP and CC are equivalent to JEP. There is a slight disanalogy, since SEP and JEP stipulate that the proposition that q is known to be entailed singly or Jointly, not merely that it is entailed. Without this qualification, SEP would not be plausible in my view; but the equivalence holds in any case.

12 There are other reasons for challenging CC. A classic paper on this topic is Henry E. Kyburg, Jr., ‘Conjunctivitis,’ in Swain, op. cit.

13 Such an assumption is sometimes thought to be responsible for the lottery paradox. On this see: Marshall Swain, ‘The Consistency of Rational Belief’ in Swain, op. cit., 118·120. Keith Lehrer believes that one natural defense of the position that I am advocating depends on a closely related assumption: if P is more probable than Q, and it is reasonable to accept Q, then it is reasonable to accept P (RC, 66-69).

14 It might be asked whether we should figure in the value of the other beliefs in making the choice. For example, to believe truly that our other beliefs are true entails having other true beliefs and shouldn't this fact affect the value of the first belief? Lehrer's method of calculating expected utility implies a negative answer, and rightly so. The other beliefs are already held at the point of decision and thus their value as true or false beliefs is a constant factor whatever we choose to believe about them.

15 Intuitively ‘P & Q & R’ is the strongest competitor to ‘~P v ~ Q v ~ R’ and this intuition agrees with Lehrer's technical definitions: h * is a strongest competitor of h for S if and only if h * competes with h for S and for any k, if k competes with h for S then p (h *) is at least as great as p(k); and r competes with h for S if and only if r has strong negative relevance to h within the corrected doxastic system of S, i.e., p(h,r) is less than p(h) and the disjunction d which is logically equivalent to h and contains as disjuncts members of the epistemic partition of h for S is such that no disjunction d’ of any of those members can be formed where p(h, d’) = p (h). Now the epistemic partition of ‘~ P v ~ Q v ~ R’ consists of the eight possible three member conjunctions containing P or ~P, Q or ~ Q, and R or ~ R; consequently ‘P & Q & R’ has strong negative relevance to ‘~ P v ~ Q v ~ R’ and indeed is its only competitor. (K, 192-197, 201, 207)

16 On this view the content of tautologies will be minimum, while that of contradictions will be maximum. Moreover, it cannot be the case that a more informative statement be more probable than a less informative one. Lehrer attempts to develop a more reasonable conception of informative content in ‘Belief and Error’ in M.S. Gram and E.D. Klemke, (eds.), The Ontological Turn (Iowa City: University of Iowa Press 1974). But that view implies that vt (S2)= p (~ S2) I p(S2) and vf (S2)= -1 so that the expected utility of accepting S2 is not negative.

17 In fairness Lehrer concedes that a truth-seeker might not endorse the ideal of maxivericity. Lehrer believes that this endorsement is a reasonable option but not logically required: ‘If someone finds the ideal maxivericity so unrealistic as to be worthy of benign neglect, and considers even the smallest gain in information to be of greater value than clinging to an almost miniscule change of avoiding error totally, we do not pretend to have refuted him. Here we reach the absolutely bottom rock of epistemic preferences’ (RC, 71). What Lehrer fails to recognize is that perfection seeking of the sort that he advocates is not a possible option for someone aiming to maximize true belief and minimize false belief.

18 Lehrer does in fact make precisely this assumption in Chapter 6 of Knowledge, op. cit., 146-150, although he takes a different position, as we have seen, in Chapter8.

19 Considerations of this sort are mentioned by Lehrer, RC, 72.

20 I make the Kantian assumption that it cannot be impossible to fulfill our real obligations. For a contrary view see P. K. Schotch and R.E. Jennings, ‘Non-Kripkean Deontic Logic’ forthcoming in Hilpinen, R. ed. Readings in Deontic Logic (Dordrecht: D. Reidel).Google Scholar

21 I am indebted to Nathan Brett, my colleagues at Dalhousie, and the referee of this Journal for helpful discussions of earlier drafts of this essay.