Cut elimination for systems of transparent truth with restricted initial sequents

TL;DR

This paper explores systems of disquotational truth with restricted initial sequents, demonstrating cut elimination, proof-theoretic properties, and connections to fixed-point semantics without additional premiss restrictions.

Contribution

It introduces a new class of truth systems with simple proof-theoretic properties and establishes cut elimination and fixed-point semantics without premiss restrictions.

Findings

Cut is eliminable in the systems.

Cut remains eliminable with added arithmetical axioms.

Links to fixed-point semantics are established without premiss restrictions.

Abstract

The paper studies a cluster of systems for fully disquotational truth based on the restriction of initial sequents. Unlike well-known alternative approaches, such systems display both a simple and intuitive model theory and remarkable proof-theoretic properties. We start by showing that, due to a strong form of invertibility of the truth rules, cut is eliminable in the systems via a standard strategy supplemented by a suitable measure of the number of applications of truth rules to formulas in derivations. Next, we notice that cut remains eliminable when suitable arithmetical axioms are added to the system. Finally, we establish a direct link between cut-free derivability in infinitary formulations of the systems considered and fixed-point semantics. Noticeably, unlike what happens with other background logics, such links are established without imposing any restriction to the premisses of the truth rules.