A simple modal logic for belief revision: Extended Abstract

The notions of static belief and of belief revision have been extensively studied in the literature. However, there is a surprising lack of uniformity in the two approaches. In the philosophy and logic literature the notion of static belief has been studied mainly within the context of modal logic. On the syntactic side a belief operator B is introduced, with the intended interpretation of B as “the individual believes that ”. Various properties of beliefs are then expressed by means of axioms, such as the positive introspection axiom B ! BB , which says that if the individual believes then she believes that she believes . On the semantic side Kripke structures are used, consisting of a set of states (or possible worlds) together with a binary relation B on , with the interpretation of B as “at state the individual considers state possible”. The connection between syntax and semantics is then obtained by means of a valuation V which associates with every atomic sentence p the set of states where p is true. The pair h ;Bi is called a frame and the addition of a valuation V to a frame yields a model. Rules are given for determining the truth of an arbitrary formula at every state of a model; in particular, the formula B is true at state if and only if is true at every such that B , that is, if is true at every state that the individual considers possible at . A property of the accessibility relation B is said to correspond to an axiom if every instance of the axiom is true at every state of every model based on a frame that satis…es the property and vice versa. The AGM theory of belief revision, on the other hand, has followed a di¤erent path. In this literature beliefs are modeled as sets of formulas in a given syntactic language and the problem that has been studied is how a belief set ought to be modi…ed when new information, represented by a formula , becomes available. With a few exceptions, the tools of modal logic have not been explicitly employed in the analysis of belief revision. In the economics and game theory literature, it is standard to represent