Symmetric Paraconsistent Quantum Logic

In this study, a new logic called symmetric paraconsistent quantum logic (SPQL) is introduced as a dual monosequent system, which is a restricted and indexed sequent calculus. The logic SPQL is regarded as an extension and generalization of Dalla Chiara and Giuntini's paraconsistent quantum logic (PQL). Theorems for syntactically and semantically embedding SPQL into PQL and vice versa are proved. Additionally, cut-elimination, symmetry-elimination, contraposition-elimination, and algebraic-completeness theorems are proved for SPQL.