The Lottery Paradox
If high probability is enough for knowledge, then you know your lottery ticket will lose. But you also know every other ticket will lose. And one of them will win. So you know something false.
Henry Kyburg introduced this paradox in 1961 to expose a tension in probabilistic accounts of knowledge. The case shows that if we allow sufficiently high probability to generate knowledge, we end up committed to knowing a contradiction.
Kyburg, H. (1961). Probability and the Logic of Rational Belief. Wesleyan University Press.
The paradox
You hold one of 1,000 lottery tickets. The odds are 999-to-1 that your ticket loses. That's a 99.9% chance. If high enough probability is sufficient for knowledge, it seems you know your ticket will lose.
But the same reasoning applies to every other ticket. For each ticket in the lottery, there's a 99.9% chance it loses. So you know ticket 1 will lose, and you know ticket 2 will lose, and so on for all 1,000 tickets. But one ticket will win. You now know something false.
The problem with the threshold account
The threshold account of knowledge holds that a belief counts as knowledge when the probability is above some threshold, say 0.99 or 0.999. The lottery paradox shows this leads to trouble.
If knowledge is closed under conjunction (if you know P and you know Q, you know P and Q), then from knowing each individual ticket loses, you can derive that you know all tickets lose. But you also know that exactly one ticket wins. These together are a contradiction you know.
Lowering the threshold doesn't help. Whatever threshold you set, a large enough lottery will break it.
The available exits
Three responses dominate the literature. First, deny that high probability ever yields knowledge. Knowledge requires something more than probability, perhaps certainty or safety or sensitivity. On this view, you never knew your ticket would lose, even at 999-to-1 odds.
Second, accept the lottery case and reject the closure principle. Maybe knowing each conjunct doesn't always get you knowledge of the conjunction. This preserves the probability account but at significant cost to the logic of knowledge.
Third, distinguish between types of justification. You have strong statistical reason to believe your ticket loses, but this kind of evidence may be the wrong kind to ground knowledge. Some philosophers argue that knowledge requires individuating evidence, something about this ticket specifically, not just base rates.
None of these exits is free. Each abandons something that seemed obvious before the paradox arose.
Discussion questions
- Is it rational to believe each ticket will lose while also believing some ticket will win?
- Can you think of a case in real life where you hold beliefs that are individually reasonable but collectively inconsistent?
- Does the lottery paradox show something is wrong with our concept of rational belief?
Take it to the dinner table.
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