Modal weak Kleene logics: axiomatizations and relational semantics

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 [4] and Halldén [30], 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 $\textsf{B}_{\text{e}}^{\square }$ and $\textsf{PWK}^{\Box }_{\text{e}} $, 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 $\textsf{B}_{\text{e}}$ and $\textsf{PWK}_{\text{e}}$, which we provide using the recent algebraization results about these two logics. In particular, we prove the algebraizability of $\textsf{PWK}_{\text{e}}$. Finally some standard extensions of the basic modal systems are provided with their completeness results w.r.t. special classes of frames.