A Logical Representation of Belief Update Based on Modal Logics of Ordered Arrows

Abstract

We propose modal logics of ordered arrows to represent belief update as logical calculus of modal logic. Belief update, proposed by Katsuno and Mendelzon, is a theoretical formulation of the way we change our beliefs by getting or losing information based on changes of the world. Modal logics of ordered arrows are the extension of modal logics of arrows proposed by Vakarelov. We propose ordered arrow frames (OA frames, for short) that represent structures of state transition diagrams and relative plausibility of state transitions. OA frames provide Kripke semantics for modal logics of ordered arrows (OA models, for short). We also propose an axiomization of the logic of all OA frames and call this logic OAL(ordered arrow logic). OAL contains all axioms and inference rules of BAL(basic arrow logics). We prove that OAL is sound and complete for the class of all OA models. Moreover, we represent belief update operations and postulates as formulae in OAL, which means that belief update is characterized as logical calculus of OAL.