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This paper deals with one class of propositional fuzzy logics. Recently, fuzzy logic systems have been introduced as logics being complete with respect to linearly ordered algebras, in particular, algebras on the unit interval [0,1]. One of important trends in this logic is to introduce logic systems having more general structures. As one work of this trend, we introduce implicational tonoid fuzzy logics as fuzzy logics with tonic properties. For this, we first define implicational tonoid fuzzy logics in general. We then introduce their corresponding ternary relational semantics, called Routley–Meyer–style semantics. Routley–Meyer semantics was first introduced as semantics for relevance logics and then has been generalized to semantics for other non-classical logics. Finally, we prove that implicational tonoid fuzzy logics are sound and complete with respect to their corresponding Routley–Meyer–style semantics.
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