Reflecting on finite additivity

Leendert Huisman

pp. 1785-1797

in: Synthese 192 (6), 2015.

Abstract

An infinite lottery experiment seems to indicate that Bayesian conditionalization may be inconsistent when the prior credence function is finitely additive because, in that experiment, it conflicts with the principle of reflection. I will show that any other form of updating credences would produce the same conflict, and, furthermore, that the conflict is not between conditionalization and reflection but, instead, between finite additivity and reflection. A correct treatment of the infinite lottery experiment requires a careful treatment of finite additivity. I will show that the results of the experiment, paradoxical though they may be, are not inconsistent, but that they conflict with one particular version of reflection (special reflection). I will reject that version and I will propose a slight modification of a different version of reflection (general reflection) such that the modified version does maintain the mutual consistency of finite additivity, reflection and conditionalization. As a result, I will strengthen the case for finite additivity by showing that Bayesian conditionalization is fully consistent with it.