Propositional Reasoning that Tracks Probabilistic Reasoning

Abstract

This paper concerns the extent to which uncertain propositional reasoning can track probabilistic reasoning, and addresses kinematic problems that extend the familiar Lottery paradox. An acceptance rule assigns to each Bayesian credal state p a propositional belief revision method ${\sf B}_{p}$, which specifies an initial belief state ${\sf B}_{p}(\top)$ that is revised to the new propositional belief state ${\sf B}(E)$ upon receipt of information E. An acceptance rule tracks Bayesian conditioning when ${\sf B}_{p}(E) = {\sf B}_{p|_{E}}(\top)$, for every E such that p(E) > 0; namely, when acceptance by propositional belief revision equals Bayesian conditioning followed by acceptance. Standard proposals for uncertain acceptance and belief revision do not track Bayesian conditioning. The “Lockean” rule that accepts propositions above a probability threshold is subject to the familiar lottery paradox (Kyburg 1961), and we show that it is also subject to new and more stubborn paradoxes when the tracking property is taken into account. Moreover, we show that the familiar AGM approach to belief revision (Harper, Synthese 30(1–2):221–262, 1975; Alchourrón et al., J Symb Log 50:510–530, 1985) cannot be realized in a sensible way by any uncertain acceptance rule that tracks Bayesian conditioning. Finally, we present a plausible, alternative approach that tracks Bayesian conditioning and avoids all of the paradoxes. It combines an odds-based acceptance rule proposed originally by Levi (1996) with a non-AGM belief revision method proposed originally by Shoham (1987).