Lotteries and justification

The lottery paradox shows that the following three individually highly plausible theses are jointly incompatible: (i) highly probable propositions are justifiably believable, (ii) justified believability is closed under conjunction introduction, (iii) known contradictions are not justifiably believable. This paper argues that a satisfactory solution to the lottery paradox must reject (i) as versions of the paradox can be generated without appeal to either (ii) or (iii) and proposes a new solution to the paradox in terms of a novel account of justified believability. 1. Consider the following three theses: Sufficiency Thesis (ST). If the probability of p on one’s evidence is very high, then p is justifiably believable for one. Conjunction Closure (CC). If p is justifiably believable for one and q is justifiably believable for one, then their conjunction, p and q, is justifiably believable for one. No Contradictions (NC). No proposition one knows to be a contradiction is justifiably believable for one. While individually plausible, it turns out that ST, CC and NC are jointly inconsistent. To see this, notice that no matter how high we set the standards for satisfaction of the predicate ‘very likely’, there will be some fair lottery with exactly one winner such that it is very likely on my evidence that each ticket will lose. So suppose that a ticket will very likely lose if the chance that it will lose is at least (n−1)/n and let l be a fair lottery I know to have n tickets and exactly one winner. By ST, for each ticket i ∈ l, it is justifiably believable for me that i will lose. By CC, it is justifiably believable for me that all tickets in l will lose. Since I also know that l has exactly one winner, by a further application of CC, it is justifiably believable for me that all tickets in l will lose and that exactly one ticket in l will win. However, I know that this is a contradiction and so, by NC, it is not justifiably believable for me. This is Kyburg’s [1961, 1970] famous lottery paradox. ∗Centre for Logic and Analytic Philosophy, KU Leuven, Kardinaal Mercierplein 2, BE–3000 Leuven, Email: [email protected]