The Lottery Paradox

Henry Kyburg was a philosopher who developed a theory of probability called statistical (or epistemological or sometimes just Kyburgian) probability, in contrast with Bayesian probability and logical probability. He's famous for coming up with the Lottery Paradox. Say there's a raffle with one million tickets, so one of the tickets between 1 and 1,000,000 will be picked. Since ticket 1 only has a one in a million chance of being selected, rationality requires us to believe that ticket 1 won't win. What is the probability that ticket 2 will win? Well, the same, one in a million. So rationality requires us to believe that ticket 2 won't win either. In fact, since every individual ticket only has a one in a million chance of winning, we should believe that every individual ticket will lose. But of course, we also know that one of those tickets will win. So rationality requires us to believe that ticket 1 will lose, ticket 2 will lose, etc. all the way up to ticket 1,000,000, but also that one of those tickets will win. This is a contradiction, there is no possible world where all the beliefs are true, yet it is irrational to deny any of them.

Kyburg was pointing out that there are three rational principles that lead to contradiction, and so we must reject one of them. First, if a proposition is probably true, it is rational to accept it. Second, if rationality requires us to accept proposition X and rationality also requires us to accept proposition Y, then rationality requires us to believe X and Y. Third, it is not rational to accept an inconsistent proposition. The Lottery Paradox shows that the second principle would mean we should accept that each ticket will lose and that one of them will win. This contradicts the third principle. To avoid this, we need to reject one of the three principles. Kyburg advocated rejecting the second: just because it is rational to believe ticket 1 will lose, ticket 2 will lose, etc., it doesn't mean it's rational to believe every ticket between 1 and 1,000,000 will lose. Many logicians, however, argue that we should reject the first principle: we shouldn't believe that ticket 1 will lose, we should believe it is very improbable that ticket 1 will win, but it might. Another possibility is to just throw up your hands and accept that an ideal form of rationality would still be imperfect and could entail contradictions. This is a rejection of the third principle. But accepting all three is not an option.

I'm inclined to reject the first premise -- or rephrase it so that instead of saying rationality requires us to believe a low probability entails falsehood, we say rationality requires us to believe a low probability entails ... wait for it ... a low probability. But regardless, the Lottery Paradox shows that logic, rationality, and probability are not as simple as they may appear.