Generalizing the Lottery Paradox

This paper is concerned with formal solutions to the lottery paradox on which high probability defeasibly warrants acceptance. It considers some recently proposed solutions of this type and presents an argument showing that these solutions are trivial in that they boil down to the claim that perfect probability is sufficient for rational acceptability. The argument is then generalized, showing that a broad class of similar solutions faces the same problem. Over the past decades, there has been a steadily growing interest in utilizing probability theory to elucidate, or even analyze, concepts central to traditional epistemology. Special attention in this regard has been given to the notion of rational acceptability. Many have found the following thesis at least prima facie a promising starting point for a probabilistic elucidation of that notion: Sufficiency Thesis (ST) A proposition φ is rationally acceptable if Pr(φ) > t, where Pr is a probability distribution over propositions and t is a threshold value close to 1. Another plausible constraint is that when some propositions are rationally acceptable, so is their conjunction: Conjunction Principle (CP) If each of the propositions φ and ψ is rationally acceptable, so is φ ∧ψ. From CP we can easily derive its generalization to any finite number of conjuncts, by mathematical induction. Of course, one can think of readings of ‘rationally acceptable’ on which CP fails. Suppose that we have generalized the consequence relation beyond deduction to include the results of good abductive, inductive, statistical and probabilistic reasoning too. Call the generalized relation between premise sets and conclusions ‘general 1We think of the Pr-function as representing the probability of the various propositions on the relevant evidence. We are neutral as to whether such evidential probabilities should be conceived as the degrees of belief of a rational agent or more objectively, for example in the manner of Williamson ([2000], pp. 209–37). In any case, we assume that they satisfy the standard axioms of probability theory.