In this paper we introduce and study certain L-fuzzifications of the notions of valuation of a formal language and (logical) matrix of a propositional language. Specifying, according to the principles of L-fuzzy logic, validity of sequents of a (propositional) language under fuzzy valuations (matrices), we define the concept of the sequential consequence of a class of fuzzy valuations (matrices). The main result of our paper is that a (propositional) sequential consequence satisfies all the structural rules iff it is the sequential consequence of a class of fuzzy valuations (matrices). We also prove that the consequence of a (propositional) sequential calculus with derivable structural rules is the sequential consequence of a class of crisp valuations (matrices) iff the calculus is, in a sense, disjunctive. Our general study is exemplified by considering the propositional sequential calculi LK and LJ (more precisely, its multiple-conclusion version) for the classical and the intuitionistic propositional logics, respectively.