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 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 ^ 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. Finally some standard extensions of the basic modal systems are provided with their completeness results w. r. t. special classes of frames.