Sequent Calculi for Semi-De Morgan and De Morgan Algebras

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.