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Modal Boolean Connexive Logics: Semantics and Tableau Approach
Bulletin of the Section of Logic, Tome 48 (2019) no. 3, pp. 213-243.

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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.
Keywords: Boolean connexive logics, connexive logic, modal Boolean connexive logics, modal logics, normal modal logics, possible worlds semantics, relatedness, relating logic, relating semantics, tableau methods 
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     author = {Jarmu\.zek, Tomasz and Malinowski, Jacek},
     title = {Modal {Boolean} {Connexive} {Logics:} {Semantics} and {Tableau} {Approach}},
     journal = {Bulletin of the Section of Logic},
     pages = {213--243},
     publisher = {mathdoc},
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     number = {3},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BSL_2019_48_3_a4/}
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Jarmużek, Tomasz; Malinowski, Jacek. Modal Boolean Connexive Logics: Semantics and Tableau Approach. Bulletin of the Section of Logic, Tome 48 (2019) no. 3, pp. 213-243. http://geodesic.mathdoc.fr/item/BSL_2019_48_3_a4/