Modal weak Kleene logics: axiomatizations and relational semantics

Abstract: Weak Kleene logics are three-valued logics characterized by the presence of an infectious truth-value. In their external versions, as they were originally introduced by Bochvar and Hallden, these systems are equipped with an additional connective capable of expressing whether a formula is classically true. In this paper we further expand the expressive power of external weak Kleen logics by modalizing them with a unary operator. The addition of an alethic modality gives rise to the two systems $\B^{\square}$ and $MPWK$, which have two different readings of the modal operator. We provide these logics with a complete and decidable Hilbert-style axiomatization w.r.t. a three-valued possible worlds semantics. The starting point of these calculi are new axiomatizations for the non-modal bases B and PWKe, which we provide using the recent algebraization results about these two logics. In particular, we prove the algebraizability of $\PWKe$. Finally some standard extensions of the basic modal systems are provided with their completeness results w.r.t. special classes of frames.