Rigged lotteries: a diachronic problem for reducing belief to credence

Lin and Kelly (J Philos Log 41(6):957–981, 2012) and Leitgeb (Ann Pure Appl Log 164(12):1338–1389, 2013, Philos Rev 123(2):131–171, 2014), offer similar solutions to the Lottery Paradox, defining acceptance rules which determine a rational agent’s beliefs in terms of broader features of her credal state than just her isolated credences in individual propositions. I express each proposal as a method for obtaining an ordering over a partition from a credence function, and then a belief set from the ordering. Although these proposals avoid the original Lottery Paradox, I raise a diachronic case which illustrates that neither satisfies both (i) Lin and Kelly’s constraint that the update on orderings track the update on credence functions, and (ii) the intuitive constraint that credence of at least 0.5 is necessary for rational belief. I conclude by suggesting that we reformulate these proposals in terms of orderings over entire algebras based on partitions rather than orderings just over the partitions themselves. Reformulating both rules in this way yields acceptance rules which avoid the Lottery Paradox while satisfying both the tracking and likeliness constraints.