Classical Modal Display Logic in the Calculus of Structures and Minimal Cut-free Deep Inference Calculi for S5

Abstract

We begin by showing how to faithfully encode the Classical Modal Display Logic (CMDL) of Wansing into the Calculus of Structures (CoS) of Guglielmi. Since every CMDL calculus enjoys cut-elimination, we obtain a cut-elimination theorem for all corresponding CoS calculi. We then show how our result leads to a minimal cut-free CoS calculus for modal logic S5. No other existing CoS calculi for S5 enjoy both these properties simultaneously.