Implicational tonoid logics and their extensions with abstract Galois properties have been introduced by Yang and Dunn. They introduced matrix semantics for the implicational tonoid logics but did not do for the extensions. Here we provide such semantics for implicational partial gaggle logics as one sort of such extensions. To this end, first we discuss implicational partial gaggle logics in Hilbert-style. We next introduce one kind of matrix semantics based on Lindenbaum– Tarski matrices for the logics and show that those (...) logics are complete with respect to the matrix semantics. Finally, we further introduce a slightly different kind of matrix semantics based on reduced models for the logics and show that those logics are complete with respect to this matrix semantics. (shrink)

This paper addresses a set-theoretic completeness based on a relational semantics for fuzzy extensions of two versions Rt and R T of R (Relevance logic). To this end, two fuzzy logics FRt and FRT as extensions of Rt and R T, respectively, and the relational semantics, so called Routley-Meyer semantics, for them are first recalled. Next, on the semantics completeness results are provided for them using a set-theoretic way.

Here algebraic Routley-Meyer semantics is addressed for two fuzzy versions of the logic of relevant implication R. To this end, two versions R t and R T of R and their fuzzy extensions FRt and FRT , respectively, are first discussed together with their algebraic semantics. Next algebraic Routley-Meyer semantics for these two fuzzy extensions is introduced. Finally, it is verified that these logics are sound and complete over the semantics.