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Bulletin of the Section of Logic Volume 37:3/4 (2008), pp. 185–196 Krystyna Mruczek-Nasieniewska Marek Nasieniewski PARACONSITENT LOGICS OBTAINED BY ´ J.-Y. BEZIAU’S METHOD BY MEANS OF SOME NON-NORMAL MODAL LOGICS Abstract The paper presents a formulation of some propositional logics. In [2] J.-Y. B´eziau formulated a logic called Z. J.-Y. B´eziau’s idea was generalized independently in [7] and [8]. The present paper (based on results from [8]) is a contribution to a further generalization for those frames in which non-normal worlds are allowed. Introduction The main point of the logic Z was to understand negation as “it is possible that not” [2]. While defining this logic, B´eziau used modal logic S5. In the present paper we are using B´eziau’s negation in the case of modal log- ics expressible by Kripke semantics where non-normal worlds are allowed. Considered logics turn out to be paraconsistent. 1. Class R Definition 1. Let For be the set of all propositional formulae in the language with connectives {∼, ∧, ∨, →, ↔} and the set of propositional variables Var. 186 Krystyna Mruczek-Nasieniewska and Marek Nasieniewski Definition 2. Let R be the class of all logics being any non-trivial subset of For, containing the full positive classical logic in the language {∧, ∨, →, ↔}, including (dM1→) ∼(p ∧ q) → (∼p ∨ ∼ q), (dM1→) and closed under Modus Ponens (MP), any substitution, and (CONTR) A → B . (CONTR) ∼ B → ∼ A Since the classical logic fulfills the above conditions we conclude that R = ∅. It is easy to see that for any L ∈ R, L contains the fol