 The right way to think of the model-theoretic definition is that we have sentences whose truth value varies according to the situation where they are used. So the nonlogical constants are not variables; they are definite descriptions whose reference depends on the context. Likewise the quantifiers have this indexical feature, that the domain over which they range depends on the context of use. In this spirit one can add other kinds of indexing. For example a Kripke structure is an indexed family of structures, with a relation on the index set; these structures and their close relatives are fundamental for the semantics of modal,  temporal  and  intuitionist  logic.