Probability, rational belief and belief change

A simple model of rational belief holds that: (i) an instantaneous snapshot of an ideally rational belief system corresponds to a probability distribution; and (ii) rational belief change occurs by Bayesian conditionalization. But a priori probability distributions of the Kolmogorov sort cannot distinguish between propositions that are simply true from propositions that are necessarily true. Further, propositions accepted by Bayesian conditionalization become necessary truths of the updated distribution. Thus on this model, once you have accepted a proposition, it is impossible to change your mind. These problems are not avoided by Jeffrey conditionalization nor by adopting infinitesimal probability values. In contrast, conditional probability distributions are able to distinguish propositions that are simply true from propositions that are necessarily true. However, Bayesian conditionalization as a model of belief change still makes the newly accepted proposition necessarily true, and hence immune to future revision. In this paper we develop a different revision scheme for conditional probability distributions that does permit one to accept a proposition with probability 1, but to subsequently change one’s mind. keywords: logic, probability, belief representation, belief change, belief revision I. Belief Revision Using Kolmogorov and Bayes Our most detailed theory about reasoning under conditions of uncertainty is formal probability theory. A simple model of rational belief holds that: (i) an instantaneous snapshot of an ideally rational belief system corresponds to a probability distribution; and (ii) rational belief change occurs by Bayesian conditionalization. The most well known account of the classical theory of elementary probability is due to Kolmogorov in [5]. The theory consists of a set of constraints which any function P must satisfy in order to be a probability function. KP1 P is defined on a F-field of sets.