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Henry E. Kyburg, Jr.'s Lottery Paradox (1961, p. 197) 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 one ticket will win. Suppose that an event is very likely if the probability of its 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 won't 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 won't win, accepting that ticket 2 won't win, ..., and accepting that ticket 1000 won't 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
- It is rational to accept a proposition that is very likely true,
- It is not rational to accept a proposition that you are aware is inconsistent, and
- If it is rational to accept a proposition A and it is rational to accept another proposition A', then it is rational to accept A & A',
are jointly inconsistent.