Analytic Cut in Epistemic Logics with Distributed Knowledge

Abstract

Distributed knowledge is a notion of group knowledge studied in multi-agent epistemic logic. Semantically, the distributed knowledge of a group is interpreted via an accessibility relation given by the intersection of the epistemic accessibility relations of the agents in that group. This paper investigates sequent calculi for epistemic logics of distributed knowledge based on K45, KD45, and S5. While cut elimination holds in existing sequent calculi for modal logics K45 and KD45, it fails in all the systems mentioned above. Instead, we establish the analytic cut property for all three systems by adapting Takano' s (2018) strategy, which restricts the cut formulas to the set of subformulas of the conclusion of the cut rule. As a corollary, the Craig interpolation theorem holds for all logics considered. We also show that all proof-theoretic results remain valid when the empty group is allowed for the distributed-knowledge operator, in which case the distributed knowledge for the empty group is interpreted as the global modality.