The Lottery Paradox

Trying to find an answer to the question when we can say that we have knowledge I found an article by Baron Reed titled “Fallibilism” (Philosophy Compass 7/9 (2012): 585–596). In a section of this article Reed discusses the Lottery Paradox and its relevance for knowledge as justified true belief. The paradox has been described first by Henry E. Kyburg, Jr, in 1961. I must say that, like the Gettier cases debate, I didn’t follow the discussion that developed since then, so maybe my arguments against it are not new (but I didn’t get the impression when I read Reeds article), but it is interesting to treat the paradox here because of its significance for our concept of knowledge.

I start with Reed’s description of the Lottery Paradox (id, pp. 588-589):

“[The lottery paradox] depends on the following two principles:

1. If it is highly probable that p, then it is rational to believe that p.

2. If it is rational to believe that p and it is rational to believe that q, then it is rational to believe that p & q.”

These two principles are quite plausible, so Reed, and if the “two propositions are each rational to believe, then surely it is rational to believe their conjunction. … Nevertheless, the two principles together yield a counterintuitive result. Suppose there is a fair lottery in which 1000 tickets are sold and in which only one ticket will win. For each ticket, there is thus a .999 probability that it will lose. Principle (1) tells us that it is rational to believe of each ticket that it will lose. So, where proposition p_i is the proposition that ticket t_i will lose, it is rational to believe that p₁, that p₂, …, that p₁₀₀₀. Principle (2) tells us that it is rational to believe the conjunction of all these propositions: that p₁ & p₂ & … & p₁₀₀₀. But, because we know it is a fair lottery, it is also rational for us to believe that some one of the tickets will win – i.e., it is rational for us to believe that either not-p₁, or not-p₂, …, or not-p₁₀₀₀. We know (and rationally believe) that this is equivalent to the proposition that not-(p₁ & p₂ & … & p₁₀₀₀). Using principle (2) again, it is rational to believe that p₁ & p₂ & … & p₁₀₀₀ & not-(p₁ & p₂ & … & p₁₀₀₀). But that proposition, of course, is a contradiction.”

At first glance, the Lottery Paradox may seem plausible, indeed, but at second glance? Premise (1) says that it is rational to believe that p if p is highly probable. But why should it? For example, I work as a security officer on an airport where I check the passengers at the gate. It is highly probable that the next passenger is a decent person and not a terrorist. Nevertheless, for me it is rational to believe that he might be an exception. The only thing that is rational to believe is that p is highly probable and not that it certainly will happen, as premisse (1) seems to state. Otherwise we add information without any argument.

Take now premise (2). What Reed doesn’t make clear is what “&” means. Say I throw the dice. Then there is a probability of .83 that 1-5 will come up. If I throw several times, that doesn’t change for each independent (separate) throw. If we interpret the & for separate throws, the probability for each throw remains .83. However, now we interpret & in the way that the first throw p is 1, 2, 3, 4, or 5 and the next throw q is and the next throw r is etc.. Then after four throws the probability is 0.83x0.83x0.83x0.83=0.48 that this will happen, which is already less than a half! In other words, if p, q etc. are dependent on each other in some way and they have probabilities of less than 1 it is not rational to believe that p & q & etc. (i.e. in my example that each time 1-5 will come up).

This brings me to another flaw in the lottery paradox. Even if the premises are true, the world is wider and it contains also other knowledge, for instance what statisticians say about probabilities. Our rational believe is not based on single isolated facts but it is interwoven with other knowledge, like statistical knowledge. And statisticians can tell us that it is simply not true that we have to analyse the probabilities of the 1000 tickets in the lottery as independent from each other: If we know that one lottery ticket will not win, then the probabilities that the other tickets will win increase. Knowing the probabilities of the separate lottery tickets as such tells us nothing about the probabilities when we put some (or all) tickets together, unless we do the relevant statistical computations. I think that more can be said against the Lottery Paradox, but it is clear that neither premise (1) nor premise (2) is correct. The upshot is that there is no Lottery Paradox.