Labeled Natural Deduction Systems for a Family of Tense Logics

TL;DR

This paper introduces labeled natural deduction systems for a family of tense logics, proving their soundness, completeness, and normalization properties, and discusses extensions to richer temporal logics like LTL.

Contribution

It presents novel labeled natural deduction systems for tense logics extending Kl, with proofs of soundness, completeness, and normalization, and explores extensions to LTL.

Findings

Systems are sound and complete with respect to Kripke semantics

Derivations reduce to normal forms with subformula property

Extensions to richer temporal logics like LTL are discussed

Abstract

We give labeled natural deduction systems for a family of tense logics extending the basic linear tense logic Kl. We prove that our systems are sound and complete with respect to the usual Kripke semantics, and that they possess a number of useful normalization properties (in particular, derivations reduce to a normal form that enjoys a subformula property). We also discuss how to extend our systems to capture richer logics like (fragments of) LTL.