Non-commutative first-order EQ-logics

EQ-algebras are meet semilattices endowed with two additional binary operations: fuzzy equality and multiplication. These algebras serve as a basic algebraic structure of truth values for many-valued logics based on (fuzzy) equality instead of implication. Therefore, they generalize residuated lattices in the sense that each residuated lattice is an EQ-algebra but not vice-versa.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 paper the predicate first-order EQ-logic. First, we overview some basic properties of EQ-algebras and the basic propositional EQ-logic.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 Δ-connective. Therefore, we pay a special attention to prelinear EQΔEQΔ-algebras and develop the corresponding propositional EQΔEQΔ-logic. Finally, we in detail introduce syntax and semantics of the first-order EQ-logics and prove various theorems characterizing its properties including completeness.