A Kripke-Lewis semantics for belief update and belief revision

We provide a new characterization of both belief update and belief revision in terms of a KripkeLewis 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 circle p or the revised belief set K * p (prompted by the input represented by formula p ) as the set of formulas that are the consequent of conditionals that (1) are believed at that state and (2) have p as antecedent. We show that this class of models characterizes both the Katsuno-Mendelzon (KM) belief update functions and the Alchourron, Gardenfors 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.