Sequent Calculi for the classical fragment of Bochvar and Halld\'en's Nonsense Logics

TL;DR

This paper develops sequent calculi for the classical fragment of Bochvar and Halldén's nonsense logics, establishing their soundness, completeness, and cut elimination, highlighting their semantic relationships.

Contribution

It introduces new sequent calculi for the classical fragment of Bochvar and Halldén's nonsense logics with proven soundness, completeness, and cut elimination.

Findings

Sequent calculi for B3 and H3 are constructed.

Soundness and completeness theorems are established.

Cut elimination theorems are proved.

Abstract

In this paper sequent calculi for the classical fragment (that is, the conjunction-disjunction-implication-negation fragment) of the nonsense logics B3, introduced by Bochvar, and H3, introduced by Halld\'en, are presented. These calculi are obtained by restricting in an appropriate way the application of the rules of a sequent calculus for classical propositional logic CPL. The nice symmetry between the provisos in the rules reveal the semantical relationship between these logics. The Soundness and Completeness theorems for both calculi are obtained, as well as the respective Cut elimination theorems.