On Revocable and Irrevocable Belief Revision

Abstract

Krister Segerberg proposed irrevocable belief revision, to be contrasted with ‘standard’ belief revision, in a setting wherein belief of propositional formulas is modelled explicitly. In standard belief revision one can unmake (‘revoke’) belief in any formula, given yet further information that contradicts it. But irrevocable formulas remain believed forever. We compare traditional AGM belief revision with Segerberg’s dynamic doxastic logic, and with dynamic epistemic logical approaches to belief revision. Our work falls in the latter category. In that context with explicit belief operators and dynamic modal operators \([* \varphi ]\) for belief revision with \(\varphi \), we define revocable belief revision as belief revision satisfying that \(\psi \leftrightarrow [* \varphi ] [* \lnot \varphi ] \psi \) is valid; such that irrevocable means not revocable. Segerberg’s irrevocable belief revision is indeed irrevocable in that sense. We give semantic constraints (on multi-agent Kripke models) for revocable belief revision. In order for belief revision to be revocable: (i) the agents should consider the same states possible before and after revision, (ii) states that are non-bisimilar before revision may not be bisimilar after revision (if states are non-bisimilar, they can be distinguished from one another in the logical language), and (iii) it should be possible that states that are not equally plausible before revision become equally plausible after revision. We reformulate four well-known belief revision operators (hard update, soft update, conservative revision, severe revision) as qualitative dynamic belief revision operators. They are irrevocable in the (strong) sense above, because they violate one or more of these three requirements. However, single-agent severe revision is revocable in a weaker sense that following a revision \(*\varphi \) there is a sequence of further revisions recovering the initial state of belief. The work may be relevant for restricted-memory or other bounded rationality approaches to belief revision, e.g., when only a finite number of plausibility distinctions may be stored in memory. Therefore, it may be relevant for the study of logic and cognition.