Lottery paradox
From Wikipedia, the free encyclopedia
The Lottery Paradox is a paradox studied by epistemologists interested in justification. It was originally noticed by H. E. Kyburg Probability and the Logic of Rational Belief (1961). In short, the Lottery Paradox is that one seems to be both justified and not justified in believing the truth of a proposition regarding the outcome of a fair lottery, in which one ticket is a winner in each drawing.
[edit] The paradox
Imagine that one wishes to enter a local lottery along with thousands of other participants. It is immediately recognizable that the chance of one's ticket losing is so high that one is justified in believing that it will not win. Probability seems to confirm the justification for such a belief. Yet, it is not just one's individually purchased ticket that has such a high probability of losing, but any ticket that has been bought in a fair lottery. Furthermore, since one seems justified in believing that each individual ticket will not win, one also seems justified in believing that the conjunction of all tickets, or that every ticket, will not win. Yet at the same time, one must also remember that in all lotteries there is the slight probability that a ticket will win. After all, there is often one winner. How, therefore, can one be justified in believing that every ticket will not win, but that one ticket will win?
Peter Klein put the paradox in a few premises[1]:
- 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.
- 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's Elusive Knowledge (1996). Let us 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 are so many tickets and possibilities of losing that no matter how many tickets you know will lose, it is still not great enough to turn your justified belief into knowledge.
One possible way to use logic to get around the paradox is to match math with the human instinct to "assume." Let a number between 0 and 1 be the probability of a ticket winning in a fair lottery. If there are 1000 tickets, then the probability of ticket 1 winning is 0.001. With the human assumption coming into play now, the assumption of winning is zero. Simply rounded to the nearest whole number, 0.001 becomes zero. When you multiply 0 by the number of tickets, 1000, you still get zero. This is not true, because the human assumption is not equal to the actual probability. If you multiply the actual probability, 0.001, times 1000, you get one. The outcome of premise 3, if you buy all of the tickets, is that you win. The problem with the paradox is that the human assumption is not equal to the actual probability, so premise 2 is flawed.
[edit] Notes
- ^ Klein, P., Certainty (1981)
[edit] References and further reading
- Hawthorne, John. (2004). Knowledge and Lotteries, New York: Oxford University Press.
- Klein, Peter. (1981). Certainty: a Refutation of Scepticism, Minneapolis, MN: University of Minnesota Press.
- Kyburg, H.E. (1961). Probability and the Logic of Rational Belief, Middletown, CT: Wesleyan University Press.
- Lewis, David. (1996). "Elusive Knowledge", Australian Journal of Philosophy, 74, pp. 549-67.
