Natural Deduction for Connexive Paraconsistent Quantum Logic

Abstract

In this study, a new logic called the connexive paraconsistent quantum logic is introduced as a common denominator of a paraconsistent logic and a quantum logic. A natural deduction system for this logic is introduced, and the weak normalization theorem for this system is shown. A typed lambda calculus for the implication-negation fragment of this logic is developed on the basis of the Curry-Howard correspondence. The strong normalization theorem for this calculus is proved.