Focused Labeled Proof Systems for Modal Logic

Focused proofs are sequent calculus proofs that group inference rules into alternating negative and positive phases. These phases can then be used to define macro-level inference rules from Gentzen’s original and tiny introduction and structural rules. We show here that the inference rules of labeled proof systems for modal logics can similarly be described as pairs of such negative and positive phases within the LKF focused proof system (which contains no modal connectives). In particular, we consider the system G3K of Negri for the modal logic K and define a translation from labeled modal formulas into first-order polarized formulas and show a strict correspondence between derivations in the two systems, i.e., each rule application in G3K corresponds to a bipole— a pair of a positive and a negative phases—in LKF. Since, geometric axioms (when properly polarized) induce bipoles, this strong correspondence holds for modal logics whose Kripke frames are characterized by geometric properties. We extend these results to present a focused labeled proof system for this same class of modal logics and show its soundness and completeness. This resulting proof system allows one to define a rich set of normal forms of modal logic proofs. 1998 ACM Subject Classification F.4.1 Mathematical Logic: Modal logic, Proof Theory