On the quantified version of the Belnap-Dunn modal logic

Abstract

We develop a quantified version of the propositional modal logic $\mathsf{BK}$ from an article by Odintsov and Wansing, which is based on the (non-modal) Belnap-Dunn system; we denote this version by $\mathsf{QBK}$. First, by using the canonical model method we prove that $\mathsf{QBK}$, as well as some important extensions of it, is strongly complete with respect to a suitable possible world semantics. Then we define translations (in the spirit of Gödel-McKinsey-Tarski) that faithfully embed the quantified versions of Nelson's constructive logics into suitable extensions of $\mathsf{QBK}$. In conclusion, we discuss interpolation properties for $\mathsf{QBK}$-extensions. Bibliography: 21 titles.