^{}University of Ostrava, NSC IT4Innovations, 30. dubna 22, 702 00 Ostrava, Czech Republic

Abstract

In this paper we continue development of formal theory of a special class of fuzzy logics, called EQ-logics. Unlike fuzzy logics being extensions of the MTL-logic in which the basic connective is implication, the basic connective in EQ-logics is equivalence. Therefore, a new algebra of truth values called EQ-algebra was developed. This is a lower semilattice with top element endowed with two binary operations of fuzzy equality and multiplication. EQ-algebra generalizes residuated lattices, namely, every residuated lattice is an EQ-algebra but not vice-versa. In this paper, we introduce additional connective $logdelta$ in EQ-logics (analogous to Baaz delta connective in MTL-algebra based fuzzy logics) and demonstrate that the resulting logic has again reasonable properties including completeness. Introducing $Delta$ in EQ-logic makes it possible to prove also generalized deduction theorem which otherwise does not hold in EQ-logics weaker than MTL-logic.

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