Weak Belnapian modal logic

Abstract In this paper, a new modal logic extending the first-degree entailment (FDE) system is introduced. This system, called weak Belnapian modal logic (WBK), is closely related to two other modal logics: modal bilattice logic (MBL) and Belnapian modal logic (BK), both of which have some claims towards inducing an (in some sense) minimal semantic framework. In fact, two versions of WBK are provided: one in the language of BK and one in the language of MBL. Calling them versions of the same system is justified by the fact that both of them share the same modal axioms and rules and that the latter is a conservative extension of the former by some non-modal connectives. In this paper, we show that WBK gives rise to a semantic framework that is more general than those of MBL and BK while still retaining an intuitive interpretation. We provide an axiom system and semantics for our logic and prove a number of correspondence results, mainly showing how semantics of BK and MBL can be recovered from that of WBK. Finally, we make some remarks on the expressive power of the version of WBK over the language of MBL.