EQ-algebra; EQ-logic; equational logic; delta connective

Logics based on EQ-algebras are called EQ-logics and they can be considered as special kind of fuzzy logics. After developing propositional and higher-order ones, we address in this contribution the predicate first-order EQ-logic. First, we overview some basic properties of EQ-algebras and the basic propositional EQlogic. Analysis of necessary properties of the fuzzy equality that is in predicate EQ-logic considered not only between truth values (the equivalence) but also between objects revealed that we cannot consider the fuzzy equality in full generality without means enabling us to deal with the classical (crisp) equality. This is possible using the delta-connective. Therefore, we pay a special attention to prelinear EQdelta-algebras and develop the corresponding propositional EQdelta-logic. Finally, we in detail introduce syntax and semantics of the first-order EQ-logics and prove various theorems characterizing its properties including completeness.