The Lottery Paradox
Rational belief and conjunction
First published: H. E. Kyburg, *Probability and the Logic of Rational Belief* (1961).
For each ticket in a million-ticket lottery, you rationally believe it will lose. Conjoin all such beliefs and you believe none will win — contradicting your belief that one will.
Given a fair lottery of one million tickets, for each ticket *t* it is rational to believe that *t* will lose (probability 0.999999). But these beliefs jointly entail that no ticket wins, which contradicts the certain belief that some ticket will. The paradox forces a choice: deny closure of rational belief under conjunction, or deny that high probability suffices for belief, or accept inconsistent beliefs as rational in some attenuated sense. The case is central to debates over the relation between credence and outright belief.
Formulation
Million-ticket fair lottery; rational belief threshold p ≥ 0.99. For each ticket, P(losing) = 0.999999, so believe it loses. Conjunction over all tickets: believe no ticket wins. But P(some ticket wins) = 1.
Dimensions Engaged
Observer
Observer · Knowledge Extent: is rational belief sensitive to conjunction effects?
Information
Bears on Information · Granularity: how do continuous credences relate to discrete belief states?
Responses — How Schools Engage
Affirms / takes the bait 1
A vindication of suspending judgement: rational belief norms over-promise; the lottery makes their inconsistency visible.
Reframes the question 3
Bayesian naturalism: there is no "outright belief" in any deep sense; credences are what matter, and the paradox dissolves once we abandon the threshold-belief framework.
Belief is action-guiding; the paradox arises only when belief is divorced from contexts of action. In any actual decision, the conjunction "no ticket wins" plays no functional role.
The puzzle reveals that ordinary "belief" is not the appropriate doxastic state for science; quantitative probability is.
Holds it inconclusive 1
Live debate: defenders of closure (Williamson) deny that high credence suffices for belief; defenders of high-credence belief (Foley) deny closure. No consensus.
Related Experiments
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Further reading
- Kyburg (1961), op. cit.
- Douven, *The Lottery and Preface Paradoxes* (2014)
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