Strict core fuzzy logics and quasi-witnessed models

In this paper we prove strong completeness of axiomatic extensions of first-order strict core fuzzy logics with the so-called quasi-witnessed axioms with respect to quasi-witnessed models. As a consequence we obtain strong completeness of Product Predicate Logic with respect to quasi-witnessed models, already proven by M.C. Laskowski and S. Malekpour in [19]. Finally we study similar problems for expansions with !, define !-quasi-witnessed axioms and prove that any axiomatic extension of a first-order strict core fuzzy logic, expanded with !, and !-quasiwitnessed axioms are complete with respect to !-quasi-witnessed models.