 There are two ways of introducing first-order counterparts for other propositional t-norm based fuzzy logics. On the one hand, the axiomatization of MTL\(\forall\) can be extended by adding the first-order instances of propositional axioms like those seen in the previous sections. In this manner one obtains syntactic presentations of first-order variants of, e.g., Łukasiewicz logic, Gödel–Dummett logic, Product logic, or Hájek’s basic fuzzy logic BL. On the other hand, the semantic definition of MTL\(\forall\) can be easily modified by defining the consequence relations given by the corresponding (sets of) t-norms. Then, the natural question is whether, in each case, these two routes lead to the same logic (as it happened for MTL). For soundness there is no problem, as the axiomatic systems are easily checked to be sound with respect to their corresponding classes of structures. As for completeness, there is no general answer and the problem has to be considered case by case.