A Kripke-Lewis semantics for belief update and belief revision

Abstract

We provide a new characterization of both belief update and belief revision in terms of a Kripke-Lewis semantics. We consider frames consisting of a set of states, a Kripke belief relation and a Lewis selection function. Adding a valuation to a frame yields a model. Given a model and a state, we identify the initial belief set K with the set of formulas that are believed at that state and we identify either the updated belief set $K\diamond \varphi$ or the revised belief set $K⁎\varphi$ (prompted by the input represented by formula ϕ) as the set of formulas that are the consequent of conditionals that (1) are believed at that state and (2) have ϕ as antecedent. We show that this class of models characterizes both the Katsuno-Mendelzon (KM) belief update functions and the Alchourrón, Gärdenfors and Makinson (AGM) belief revision functions, in the following sense: (1) each model gives rise to a partial belief function that can be completed into a full KM/AGM update/revision function, and (2) for every KM/AGM update/revision function there is a model whose associated belief function coincides with it. The difference between update and revision can be reduced to two semantic properties that appear in a stronger form in revision relative to update, thus confirming the finding by Peppas et al. (1996) [30] that, “for a fixed theory K, revising K is much the same as updating K”. It is argued that the proposed semantic characterization brings into question the common interpretation of belief revision and update as change in beliefs in response to new information.