EQ-logics: Non-commutative fuzzy logics based on fuzzy equality

In this paper, we develop a specific formal logic in which the basic connective is fuzzy equality and the implication is derived from the latter. Moreover, the fusion connective (strong conjunction) is non-commutative. We call this logic EQ-logic. First, we formulate the basic EQ-logic which is rich enough to enjoy the completeness property. Furthermore, we introduce two extensions which seem to us interesting. The first one is IEQ-logic which is EQ-logic with double negation. The second one adds prelinearity that enables us to prove a stronger variant of the completeness property. Finally, we extend the latter logic by three more axioms including the residuation one (importation-exportation law) and prove that the resulting logic is equivalent with MTL-logic. Formal proofs in this paper proceed mostly in an equational style.