Modal Boolean Connexive Logics: Semantics and Tableau Approach

Abstract

<jats:p>In this paper we investigate Boolean connexive logics in a language with modal operators: □, ◊. In such logics, negation, conjunction, and disjunction behave in a classical, Boolean way. Only implication is non-classical. We construct these logics by mixing relating semantics with possible worlds. This way, we obtain connexive counterparts of basic normal modal logics. However, most of their traditional axioms formulated in terms of modalities and implication do not hold anymore without additional constraints, since our implication is weaker than the material one. In the final section, we present a tableau approach to the discussed modal logics.</jats:p>

Axiomatization of Some Basic and Modal Boolean Connexive Logics

“…Moreover, Kapsner's conditions of weakly and strongly connexive logics presented in [10] as well as the condition of properly connexive logic (the nonsymmetry of connexive implication) given by Jarmużek and Malinowski in [8] are satisfied. Some further steps in the analysis of Boolean connexive logics were made in [9] where some modal systems were introduced (cf. [15]).…”

“…Furthermore well-known relational properties of the accessibility relation are not enough to validate modal laws such as (T), (D), (B), ( 4) and (5). Two methods of making such laws valid were presented in [9]. In this article we are going to focus on the first one and only briefly consider the second one.…”

“…The main goal of our article is to axiomatize some of the logics considered in [8,9]. More specifically, we will consider axiomatic systems for the least Boolean connexive logic, its non-modal extension generated by models based on relations that are closed under negation and some modal extensions (generated by models that validate well-known modal schemata).…”

“…In order to validate Aritotle's and Botheius' theses we consider the following relational conditions discussed in [8,9]:…”

“…In [9] two ways were presented in which we can validate the modal laws. The first one assumes some restrictions on modal and relating frames.…”