From 2-sequents and Linear Nested Sequents to Natural Deduction for Normal Modal Logics

TL;DR

This paper develops natural deduction systems for normal modal logics from K to S4 by extending linear nested sequents and 2-sequents, featuring simplified rules and proof normalization.

Contribution

It introduces a novel natural deduction framework for modal logics using spatially decorated formulas, eliminating structural rules and explicit accessibility references.

Findings

Systems for modal logics K to S4 are formulated.

Proof normalization is established for intuitionistic versions.

Soundness and consistency are proven for the proposed systems.

Abstract

We extend to natural deduction the approach of Linear Nested Sequents and of 2-sequents. Formulas are decorated with a spatial coordinate, which allows a formulation of formal systems in the original spirit of natural deduction -- only one introduction and one elimination rule per connective, no additional (structural) rule, no explicit reference to the accessibility relation of the intended Kripke models. We give systems for the normal modal logics from K to S4. For the intuitionistic versions of the systems, we define proof reduction, and prove proof normalization, thus obtaining a syntactical proof of consistency. For logics K and K4 we use existence predicates (following Scott) for formulating sound deduction rules.