A Combined System for Update Logic and Belief Revision

In this paper we propose a logical system combining the update logic of A. Baltag, L. Moss and S. Solecki (to which we will refer to by the generic term BMS, [BMS04]) with the belief revision theory as conceived by C. Alchouròn, P. Gärdenfors and D. Mackinson (that we will call the AGM theory, [GardRott95]) viewed from the point of view of W. Spohn ([Spohn90,Spohn88]). We also give a proof system and a comparison with the AGM postulates. Introduction and Motivation: Update logic is a modal logic trying to model epistemic situations involving several agents, and changes that can occur in these situations due to incoming information or more generally incoming action. Belief revision theory typically deals with changes (revisions) that a database representing a belief state of a unique agent must undergo after adding conflicting information to the database. Roughly speaking, these two theories thus deal with the same kind of phenomenon. However, there are some dissimilarities. On the one hand, belief revision theory is not a logic and it deals with a single agent, unlike update logic. On the other hand, belief revision theory deals with revision (and expansion) of information unlike update logic which deals only with expansion of information. Far from being in contradiction, it seems then that these theories have a lot to give each other. So it makes sense to look for a way in which they can be merged. In Sect. 1, we will set out the BMS theory and the AGM theory viewed from the point of view of W. Spohn. In Sect. 2 we will propose a system combining these two theories. In Sect. 3, we will give an axiomatization of it with a soundness and completeness proof. In Sect. 4, we will show that it fulfills the 8 AGM postulates. 1 Update Logic and Belief Revision Theory