 Suppose \(\mathfrak{L}\) is a syntactically defined logical system \(\mathfrak{L}\) with a consequence relation \(\vdash_{ \mathfrak{L} }\), specified as a relation between the power-set of \(\mathcal{L}\) and \(\mathcal{L}\). Then a valuational system \(\mathbf{V}\) is said to be strictly characteristic for \(\mathfrak{L}\) just in case \(Δ\vDash_V A \textrm{ iff } Δ\vdash_{ \mathfrak{L} }A\) (see Dummett 1981: 431). Conversely, one says that \(\mathfrak{L}\) is characterized by \(\mathbf{V}\). Thus, if a valuation system is said to determine a logic, the valuation system by itself is, properly speaking, not a logic, but only serves as a semantic basis for some logical system. Valuation systems are often referred to as (logical) matrices. Note that in Urquhart 1986, the set \(\mathcal{D}\) of designated elements of a matrix is required to be non-empty, and in Dunn &amp; Hardegree 2001, \(\mathcal{D}\) is required to be a non-empty proper subset of \(\mathbf{V}\). With a view on semantically defining a many-valued logic, these restrictions are very natural and have been taken up in Shramko &amp; Wansing 2011 and elsewhere. For the characterization of consequence relations (see the supplementary document  Suszko’s Thesis), however, the restrictions do not apply.  