Analytic Cut in Epistemic Logics with Distributed Knowledge

2026-06-30

AI Summary

This work addresses the absence of cut elimination in existing sequent calculi for distributed knowledge in epistemic logic, which hinders proof-theoretic analysis. Focusing on the modal logics K45, KD45, and S5, the paper constructs sequent systems that enjoy the analytic cut property by restricting cut formulas to subsets of the conclusion’s subformulas. It establishes, for the first time, analytic cut elimination for epistemic logics with distributed knowledge and extends the result to cases involving empty groups—where distributed knowledge collapses to a global modality. Building on Takano’s (2018) restricted-cut strategy and integrating multi-agent semantics with sequent calculus techniques, the study proves that all three logics satisfy the analytic cut property and derives Craig interpolation as a corollary. These results remain valid under an interpretation incorporating global modalities.

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.

Problem

• distributed knowledge
• epistemic logic
• sequent calculus
• analytic cut
• cut elimination

Innovation

• analytic cut
• distributed knowledge
• sequent calculus
• epistemic logic
• Craig interpolation