A modal logic translation of the AGM axioms for belief revision
Giacomo Bonanno
TL;DR
This paper builds a three-modal framework with operators $B$, $>$, and $\square$ to translate the AGM belief revision axioms into modal form. Using Kripke-Lewis frames with a serial belief relation and a Lewis-style selection function, it establishes a precise correspondence between each AGM axiom and a modal axiom or rule, via frame properties $P^*i$ that characterize the dynamics of belief change. The authors define initial beliefs $K_s$ and revised beliefs $K_s \ast \phi$ through $B$-conditionals and show how partial belief-change can extend to full AGM revision; they provide modal axioms that capture $(K^*1)$–$(K^*8)$ and outline proofs in the Appendix. The work yields a rigorous modal-theoretic characterization of belief revision, linking semantic frame properties to syntactic modal rules and enabling reasoning about conditional beliefs within a unified framework. Overall, the approach clarifies how AGM-style belief update can be captured by a compact modal language with clear frame-analytic correspondences, potentially aiding formal analyses of belief dynamics and conditional reasoning.
Abstract
Building on the analysis of Bonanno (Artificial Intelligence, 2025) we introduce a simple modal logic containing three modal operators: a unimodal belief operator, a bimodal conditional operator and the unimodal global operator. For each AGM axiom for belief revision, we provide a corresponding modal axiom. The correspondence is as follows: each AGM axiom is characterized by a property of the Kripke-Lewis frames considered in Bonanno (Artificial Intelligence, 2025) and, in turn, that property characterizes the proposed modal axiom.