 The emphasis on (bivalent) valuations throughout our discussion may suggest some limitation of scope: are we not excluding ‘many-valued logic’ (or better: matrix semantics)? And what about Kripke style semantics in which truth is doubly relativized not only to a model but to a point therein? No, there is no limitation. Every (generalized) consequence relation is determined by a class of valuations. The matrix assignments of values to formulas are homomorphisms from the algebra L of formulas to the algebra A—presumed to be of the same similarity type of L—of a matrix (A, D), where D ⊆ A; D is called the set of designated elements of the matrix. Thus the compositional semantics is done in terms of such homomorphisms, any one of which, h, say, gives rise to a bivalent valuation vh defined by setting vh(φ) = T iff h(φ) ∈ D, and a Set-Fmla or Set-Set sequent is valid in the matrix just in case it holds—in the usual bivalent sense—on the valuation vh for every such homomorphism h, the (generalized) consequence relation determined by the matrix is the (generalized) consequence relation determined by the class of all such vh,  etc.[56]  Similarly for classes of matrices, with the quantification now over all vh where h is a homomorphism from the language to any matrix in the class. Likewise, in the Kripke-semantical case, any model M and point x therein gives rise to the valuation vMx defined by vMx(φ) = T iff M⊨x φ, and the (generalized) consequence relation determined by a class of models is that determined by all such vMx with M in the class. (This is the so-called local (generalized) consequence relation; for the global variant, use valuations vM with vM(φ) = T iff for all points x in the model, M⊨x φ.) There is an obvious further adaptation of these definitions if one wants a formulation in terms of determination by (“soundness and completeness with respect to”) a class of frames rather than models.