Prolegomena to Dynamic Logic for Belief Revision

Abstract

In ‘belief revision’ a theory${\cal K}$ is revised with a formula φ resulting in a revised theory ${\cal K}\ast\varphi$. Typically, $\neg\varphi$ is in ${\cal K}$, one has to give up belief in $\neg\varphi$ by a process of retraction, and φ is in ${\cal K}\ast\varphi$. We propose to model belief revision in a dynamic epistemic logic. In this setting, we typically have an information state (pointed Kripke model) for the theory ${\cal K}$ wherein the agent believes the negation of the revision formula, i.e., wherein $B\neg\varphi$ is true. The revision with φ is a program *φ that transforms this information state into a new information state. The transformation is described by a dynamic modal operator [*φ], that is interpreted as a binary relation [ [*φ] ] between information states. The next information state is computed from the current information state and the belief revision formula. If the revision is successful, the agent believes φ in the resulting state, i.e., Bφ is then true. To make this work, as information states we propose ‘doxastic epistemic models’ that represent both knowledge and degrees of belief. These are multi-modal and multi-agent Kripke models. They are constructed from preference relations for agents, and they satisfy various characterizable multi-agent frame properties. Iterated, revocable, and higher-order belief revision are all quite natural in this setting. We present, for an example, five different ways of such dynamic belief revision. One can also see that as a non-deterministic epistemic action with two alternatives, where one is preferred over the other, and there is a natural generalization to general epistemic actions with preferences.