G3-style sequent calculi are introduced for a family of logics with strong negation: Gurevich logic, Nelson logic, intuitionistic propositional logic, Avron logic, De-Omori logic, and classical propositional logic. Structural properties including cut elimination are established for these calculi. In addition, a Glivenko theorem for embedding classical propositional logic into Gurevich logic is shown.

In this study, an extended paradefinite logic with classical negation (EPLC), which has the connectives of conflation, paraconsistent negation, classical negation, and classical implication, is introduced as a Gentzen-type sequent calculus. The logic EPLC is regarded as a modification of Arieli, Avron, and Zamansky’s ideal four-valued paradefinite logic (4CC) and as an extension of De and Omori’s extended Belnap–Dunn logic with classical negation (BD+) and Avron’s self-extensional four-valued paradefinite logic (SE4). The completeness, cut-elimination, and decidability theorems for EPLC are proved (...) and EPLC is shown to be embeddable into classical logic. The strong equivalence substitution property and the admissibilities of the rules of negative symmetry, contraposition, and involution are shown for EPLC. Some alternative simple Gentzen-type sequent calculi, which are theorem-equivalent to EPLC, are obtained via these characteristic properties. (shrink)