Neighbourhood Semantics for FDE-Based Modal Logics

Abstract

We investigate some non-normal variants of well-studied paraconsistent and paracomplete modal logics that are based on N. Belnap’s and M. Dunn’s four-valued logic. Our basic non-normal modal logics are characterized by a weak extensionality rule, which reflects the four-valued nature of underlying logics. Aside from introducing our basic framework of bi-neighbourhood semantics, we develop a correspondence theory in order to prove completeness results with respect to our neighbourhood semantics for non-normal variants of $$\mathsf {BK}$$ , $$\mathsf {BK^{FS}}$$ and $$\mathsf {MBL}$$ .