A Duplication and Loop Checking Free Proof System for S4

Most of the sequent/tableau based proof systems for the modal logic S4 need to duplicate formulas and thus are required to adopt some method of loop checking [7, 13, 10]. In what follows we present a tableau-like proof system for S4, based on D’Agostino and Mondadori’s classical KE [3], which is free of duplication and loop checking. The key feature of this system (let us call it KES4) consists in its use of (i) a label formalism which models the semantics of the modal operators according to the usual conditions for S4; and (ii) a label unification scheme which tells us when two labels “denote” the same world in the S4-model(s) generated in the course of proof search. Moreover, it uses special closure conditions to check models for putative contradictions.