You buy a lottery ticket, for no good reason. Indeed, you know that the chance that your ticket will win is at least 10 million to one, since at least 10 million tickets have been sold, as you learn later on the evening news, before the drawing (assume that the lottery is fair and that a winning ticket exists).
So you are rationally justified in believing that your ticket will lose—in fact, you’d be crazy to believe that your ticket will win. Likewise, you are justified in believing that your friend Jane’s ticket will lose, that your uncle Harvey’s ticket will lose, that your dog Ralph’s ticket will lose, that the ticket bought by the guy ahead of you in line at the convenience store will lose, and so on for each ticket bought by anyone you know or don’t know. In general, for each ticket sold in the lottery, you are justified in believing: “That ticket will lose.”
It follows that you are justified in believing that all tickets will lose, or (equivalently) that no ticket will win. But, of course, you know that one ticket will win. So you’re justified in believing what you know to be false (that no ticket will win). How can that be?
The lottery constitutes an apparent counterexample to one version of a principle known as the deductive closure of justification:
If one is justified in believing P and justified in believing Q, then one is justified in believing any proposition that follows deductively (necessarily) from P and Q.
For example, if I am justified in believing that my lottery ticket is in the envelope (because I put it there), and if I am justified in believing that the envelope is in the paper shredder (because I put it there), then I am justified in believing that my lottery ticket is in the paper shredder.
Since its introduction in the early 1960s, the lottery paradox has provoked much discussion of possible alternatives to the closure principle, as well as new theories of knowledge and belief that would retain the principle while avoiding its paradoxical consequences.