First-Order Nelsonian Paraconsistent Quantum Logic

In this paper, a single-antecedent/succedent sequent calculus NL for first-order Nelsonian paraconsistent quantum logic is investigated. The logic under consideration is regarded as a combination of both Nelson's paraconsistent four-valued logic and Dalla Chiara and Giuntini's paraconsistent quantum logic. The duality and cut-elimination theorems for NL are proved. Decidability, some constructive properties, some constructible falsity properties, and Craig interpolation property are shown for NL. An extend NL with some naive comprehension rules in the naive set theory is also investigated.