Lockean Beliefs, Dutch Books, and Scoring Systems

On the Lockean thesis one ought to believe a proposition if and only if one assigns it a credence at or above a threshold (Foley in Am Philos Q 29(2):111–124, 1992). The Lockean thesis, thus, provides a way of characterizing sets of all-or-nothing beliefs. Here we give two independent characterizations of the sets of beliefs satisfying the Lockean thesis. One is in terms of betting dispositions associated with full beliefs and one is in terms of an accuracy scoring system for full beliefs. These characterizations are parallel to, but not merely derivative from, the more familiar Dutch Book (de Finetti in Theory of probability, vol 1, Wiley, London, 1974) and accuracy (Joyce in Philos Sci 65(4):575–603, 1998) arguments for probabilism

In Conjunction With Qualitative Probability

Numerical probabilities (associated with propositions) are eliminated in favor of qualitative notions, with an eye to isolating what it is about probabilities that is essential to judgments of acceptability. A basic choice point is whether the conjunction of two propositions, each (separately) acceptable, must be deemed acceptable. Concepts of acceptability closed under conjunction are analyzed within Keisler's weak logic for generalized quantifiers --- or more specifically, filter quantifiers. In a different direction, the notion of a filter is generalized so as to allow sets with probability non-infinitesimally below 1 to be acceptable