Undefinability of Standard sequent calculi for Paraconsistent three-valued logics

TL;DR

This paper investigates the limitations of standard sequent calculi for three-valued paraconsistent logics, proving their non-existence and proposing a non-standard calculus for one such logic.

Contribution

It demonstrates the non-existence of sound and complete standard sequent calculi for these logics and introduces a non-standard calculus for Paraconsistent Weak Kleene Logic.

Findings

No sound and complete standard sequent calculus exists for these logics.

A non-standard, cut-free sequent calculus is constructed for Paraconsistent Weak Kleene Logic.

The paper advances understanding of proof systems for paraconsistent three-valued logics.

Abstract

In this paper we study the deductive properties of a family of 3-valued paraconsistent logics. We define a notion of standard sequent calculus and prove that there is no sound and complete standard sequent calculus for these logics. Moreover, we provide non-standard sound, complete and cut free sequent calculus for Paraconsistent Weak Kleene Logic using three-sided sequents.