The Lottery Paradox

The Lottery Paradox, originally noticed by H.E. Kyburg in his 1961 Probability and the Logic of Rational Belief, is studied by epistemologists interested in justification.

The paradox can be simply put. Let's imagine that one wishes to enter a local lottery along with thousands of other participants. However, it's immediately recognizable that the chance of one's ticket losing is so high that one's justified in believing that it won't win. Probability seems to confirm the justification for such a belief. Yet, it's not just one's individually purchased ticket that has such a high probability of losing, but any ticket that's been bought in a fair lottery. Furthermore, since one seems justfied in believing that each individual ticket won't win, one also seems justified in believing that the conjunction of all tickets, or that every ticket won't win. On the other hand, one must also remember that in all lotteries there's the slight probability that a ticket will win. Afterall, there's always one winner. Following this, one doesn't seem justified in believing that the conjuction of all tickets, or every ticket won't win. Therefore, one is paradoxically justified in believing that every ticket won't win, and also not justified in believing that every ticket won't win.

If the paradox had to be put in a few premises, as it was in Peter Klein's Certainty, it would look like this:

Premise 1: There's probabilistic evidence that one is justified in believing that ticket 1 will lose, and justified in believing ticket 2 will lose ... and justified in believing ticket n will lose.

Premise 2: If one is justified in believing that ticket 1 will lose, and justified in believing ticket 2 will lose ... and justified in believing ticket n will lose, then one is justified in believing that ticket 1, and ticket 2 ... and ticket n will lose.

Premise 3: There's probabilistic evidence that one is not justified in believing that ticket 1, and ticket 2 ... and ticket n will lose.

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Conclusion: Therefore, one is justified in believing that ticket 1, and ticket 2 ... and ticket n will lose and not justified in believing that ticket 1, and ticket 2 ... and ticket n will lose.

The Lottery Paradox was also construed slightly differently in David Lewis' "Elusive Knowledge." Let's imagine that one knows how many thousands or millions of tickets there are, and one also knows the number of losing tickets as well as the number of winning tickets, one. Under his interpretation, there's so many tickets and possibilities of losing that no matter how many tickets you know will lose, it's still not great enough to turn your justified belief into knowledge.

External Links and References

• Kyburg, H.E. Probability and the Logic of Rational Belief. Middletown, CT: Wesleyan University Press, 1961.
• Lewis, David. "Elusive Knowledge." Australian Journal of Philosophy, 74, 549-67.
• Klein, Peter. Certainty: a Refutation of Scepticism. Minneapolis, MN: University of Minnesota Press, 1981.
• Links to Jim Hawthorne's paper on the logic of nonmonotonic conditionals (and Lottery Logic)

Paradoxes | Epistemology