Sequent Calculi for Semi-De Morgan and De Morgan Algebras

TL;DR

This paper introduces two new sequent calculi for semi-De Morgan and De Morgan algebras, demonstrating their properties and embeddings into classical and intuitionistic logic frameworks.

Contribution

It develops contraction-free, cut-free sequent calculi for semi-De Morgan and De Morgan algebras, with proven decidability and interpolation, and establishes their embeddings into classical and intuitionistic logic.

Findings

Cut rule is admissible in both calculi.

Both calculi are decidable and enjoy Craig interpolation.

G3DM embeds into G3SDM and classical logic; G3SDM embeds into G3ip.

Abstract

A contraction-free and cut-free sequent calculus $\msf{G3SDM}$ for semi-De Morgan algebras, and a structural-rule-free and single-succedent sequent calculus $\msf{G3DM}$ for De Morgan algebras are developed. The cut rule is admissible in both sequent calculi. Both calculi enjoy the decidability and Craig interpolation. The sequent calculi are applied to prove some embedding theorems: $\msf{G3DM}$ is embedded into $\msf{G3SDM}$ via G\"odel-Gentzen translation. $\msf{G3DM}$ is embedded into a sequent calculus for classical propositional logic. $\msf{G3SDM}$ is embedded into the sequent calculus $\msf{G3ip}$ for intuitionistic propositional logic.