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 [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.

In this paper we propose a semantic analysis of Lewis' counterfactuals. By exploiting the structural properties of the recently introduced boolean algebras of conditionals, we show that counterfactuals can be expressed as formal combinations of a conditional object and a normal necessity modal operator. Specifically, we introduce a class of algebras that serve as modal expansions of boolean algebras of conditionals, together with their dual relational structures. Moreover, we show that Lewis' semantics based on sphere models can be reconstructed in this framework. As a consequence, we establish the soundness and completeness of a slightly stronger variant of Lewis' logic for counterfactuals with respect to our algebraic models. In the second part of the paper, we present a novel approach to the probability of counterfactuals showing that it aligns with the uncertainty degree assigned by a belief function, as per Dempster-Shafer theory, to its associated conditional formula. Furthermore, we characterize the probability of a counterfactual in terms of Gärdenfors' imaging rule for the probabilistic update.

Abstract The containment companion of a logic $\vdash $ consists of the consequence relation $\vdash ^{r}$ which satisfies all the inferences of $\vdash $, where the variables of the conclusion are contained into those of the set of premises, in case this is not inconsistent. Following the algebraic analysis started in Bonzio and Pra Baldi (2021, Studia Logica, 109, 969–994), this paper characterizes the algebraic counterpart of a finitary containment logic $\vdash ^{r}$ and investigates the structure of the Leibniz and Suszko reduced models. The analysis is carried within the framework of abstract algebraic logic.Mathematics Subject Classification: Primary: 03G27. Secondary: 03G25

The paper introduces the notion of state for involutive bisemilattices, a variety which plays the role of algebraic counterpart of weak Kleene logics and whose elements are represented as Płonka sums of Boolean algebras. We investigate the relations between states over an involutive bisemilattice and probability measures over the (Boolean) algebras in the Płonka sum representation and, the direct limit of these algebras. Moreover, we study the metric completion of involutive bisemilattices, as pseudometric spaces, and the topology induced by the pseudometric.