Lottery Paradox

Henry E. Kyburg, Jr.'s lottery paradox arises from considering a fair 1000-ticket lottery that has exactly one winning ticket. If this much is known about the execution of the lottery it is therefore rational to accept that some ticket will win. Suppose that an event is very likely only if the probability of it occurring is greater than 0.99. On these grounds it is presumed rational to accept the proposition that ticket 1 of the lottery will not win. Since the lottery is fair, it is rational to accept that ticket 2 will not win either—indeed, it is rational to accept for any individual ticket i of the lottery that ticket i will not win. However, accepting that ticket 1 will not win, accepting that ticket 2 will not win, and so on until accepting that ticket 1000 will not win entails that it is rational to accept that no ticket will win, which entails that it is rational to accept the contradictory proposition that one ticket wins and no ticket wins.

The lottery paradox was designed to demonstrate that three attractive principles governing rational acceptance lead to contradiction, namely that

The paradox remains of continuing interest because it raises several issues at the foundations of knowledge representation and uncertain reasoning: the relationships between fallibility, corrigible belief and logical consequence; the roles that consistency, statistical evidence and probability play in belief fixation; the precise normative force that logical and probabilistic consistency have on rational belief.

Although the first published statement of the lottery paradox appears in Kyburg's 1961 Probability and the Logic of Rational Belief, the first formulation of the paradox appears in his "Probability and Randomness", a paper delivered at the 1959 meeting of the Association for Symbolic Logic, and the 1960 International Congress for the History and Philosophy of Science, but published in the journal Theoria in 1963. This paper is reprinted in Kyburg (1987).

Raymond Smullyan presents the following variation on the lottery paradox: One is either inconsistent or conceited. Since the human brain is finite, there are a finite number of propositions p
1…p
n that one believes. But unless you are conceited, you know that you sometimes make mistakes, and that not everything you believe is true. Therefore, if you are not conceited, you know that at least some of the p
i are false. Yet you believe each of the p
i individually. This is an inconsistency.(Smullyan 1978, p. 206)

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