Propositional Reasoning that Tracks Probabilistic Reasoning

Bayesians model one’s doxastic state by subjective probabilities. But in traditional epistemology, in logic-based artificial intelligence, and in everyday life, one’s doxastic state is usually expressed in a qualitative, binary way: either one accepts (believes) a proposition or one does not. What is the relationship between qualitative and probabilistic belief? I show that, besides the familiar lottery paradox (Kyburg 1961), there are two new, diachronic paradoxes that are more serious. A solution to the paradoxes, old and new, is provided by means of a new account of the relationship between qualitative and probabilistic belief. I propose that propositional beliefs should crudely but aptly represent one’s probabilistic credences. Aptness should include responses to new information so that propositional belief revision tracks Bayesian conditioning: if belief state B aptly represents degrees of belief p then the revised belief state K∗E should aptly represent the conditional degrees of belief p(·|E). I explain how to characterize synchronic aptness and qualitative belief revision to ensure the tracking property in the sense just defined. I also show that the tracking property is impossible if acceptance is based on thresholds or if qualitative belief revision is based on the familiar AGM belief revision theory of Alchourr ́on, G ̈ardenfors, and Makinson (1985).