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.