In this paper we extend the fuzzy-set-theoretic approach to semantics of sequent calculi, developed in (Fuzzy Sets Systems, 121 (2001) 397), by allowing the truth lattices of fuzzy sets involved to be not necessarily distributive. The main result of our paper is that a (propositional) sequent consequence (in Tarski's sense) satisfies weak structural rules, that is, structural rules with Cut splitted into two weaker rules typical of sequent calculi related to orthologic, iff it is the sequent consequence of a class of not necessarily distributive fuzzy valuations (matrices). Our general study is exemplified by considering two propositional sequent calculi for orthologic and its combination with Belnap's four-valued logic.