 Since the cardinality of \(\mathcal{V}\) may be greater than 2, the notion of a valuation system provides a natural foundational framework for the very idea of a many-valued logic. The set \(\mathcal{D}\) of designated values is of central importance for the notion of a valuation system. This set can be seen as a generalization of the classical truth value the True in the sense that it determines many central logical notions and thereby generalizes some of the important roles played by Frege’s the True (cf. the introductory remarks about uses of truth values). For example, the set of tautologies (logical laws) is directly specified by the given set of designated truth values: a sentence \(A\) is a tautology in a valuation system \(\mathbf{V}\) iff for every assignment \(a\) relative to \(\mathbf{V}\), \(v_a(A) \in \mathcal{D}\). Another fundamental logical notion—that of an entailment relation—can also be defined by referring to the set \(\mathcal{D}\). For a given valuation system \(\mathbf{V}\) a corresponding entailment relation \((\vDash_V)\) is usually defined by postulating the preservation of designated values from the premises to the conclusion: