Extending paraconsistent quantum logic: a single-antecedent/succedent system approach

In this study, some conservative extensions of paraconsistent quantum logic, such as Nelsonian, modal, infinitary and temporal, are investigated by extending a single-antecedent/succedent sequent calculus PQL for paraconsistent quantum logic. A sequent calculus NQL, which is obtained from PQL by adding implication and co-implication, is introduced as a variant of Nelson's paraconsistent four-valued logic. Sequent calculi MPQL, IPQL and TPQL are introduced, respectively, as modal, infinitary and temporal extensions of PQL. The cut-elimination and duality theorems for these calculi are proved, and some extended calculi including NQL and MPQL, as well as their fragments, are shown to be decidable. A theorem for embedding NQL into its negation-free fragment and a theorem for embedding TPQL into IPQL are proved.