EQ-algebra; EQ-logic; equational logic; delta connective; generalized deduction theorem

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 by 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.