Boolean Connexive Logics: Semantics and tableau approach

Abstract

In this paper we define a new type of connexive logics which we call Boolean connexive logics. In such logics negation, conjunction and disjunction behave in the classical, Boolean way. We determine these logics through application of the relating semantics. In the final section we present a tableau approach to the discussed logics.

Modal Boolean Connexive Logics: Semantics and Tableau Approach

“…In the present paper, we continue the investigations initiated in [6] by generalizing its results towards the area of modal logics. In a natural way, by modal Boolean connexive logics we mean a logic formulated in the sentential language with implication, classical negation, classical disjunction, classical conjunction, necessity and possibility operators satisfying the Aristotle's and Boethius' laws.…”

“…In order to simplify the notation, we define the language of Boolean connexive logics as identical to For CPL . Although in [6] we used the language generated with: variables Var = {p, q, r, p 1 , q 1 , r 1 , . .…”

“…On the other hand, it is a natural idea to keep as close as possible to the classical logic while investigating the connexive logics by interpreting the Aristotelian and Boethian laws. This very idea guided us in our research published in paper [6]. There, we interpreted negation, conjunction, and disjunction in the classical Boolean way, leaving at the same time aside a broad spectrum of possible interpretations of implication by application of relating semantics to it.…”

“…In [6] we constructed 32 logics by determining respective classes of relations R. In two of them Aristotelian and Boethian laws hold and at the same time the negation, conjunction, and disjunction preserve the Boolean meaning. That is why we proposed the name Boolean connexive logics for such logics.…”

“…A motivation for considering this particular combination of two semantics is natural. Since in [6] we considered the connexive logics without possible worlds, here we extend our ideas onto the possible worlds framework.…”