 Van Fraassen’s aim was to develop a semantics for sentences containing non-denoting singular terms. Even if one grants atomic sentences containing non-denoting singular terms and that some attributions of vague predicates are neither true nor false, it nevertheless seems natural not to preclude that compound sentences of a certain shape containing non-denoting terms or vague predications are either true or false, e.g., sentences of the form ‘If \(A\), then \(A\)’. Supervaluational semantics provides a solution to this problem. A three-valued assignment \(a\) into \(\{T, I, F\}\) may assign a truth-value gap (or rather the value \(I)\) to the vague sentence ‘Picasso’s Guernica is beautiful’. Any classical assignment \(a'\) that agrees with \(a\) whenever \(a\) assigns \(T\) or \(F\) may be seen as a precisification (or superassignment) of \(a\). A sentence may than be said to be supertrue under assignment \(a\) if it is true under every precisification \(a'\) of \(a\). Thus, if \(a\) is a three-valued assignment into \(\{T, I, F\}\) and \(a'\) is a two-valued assignment into \(\{T, F\}\) such that \(a(p) = a'(p)\) if \(a(p) \in \{T, F\}\), then \(a'\) is said to be a superassignment of \(a\). It turns out that if \(a\) is an assignment extended to a valuation function \(v_a\) for the Kleene matrix \(\mathbf{K}_3\), then for every formula \(A\) in the language of \(\mathbf{K}_3\), \(v_a (A) = v_{a'}(A)\) if \(v_a (A) \in \{T, F\}\). Therefore, the function \(v_{a'}\) may be called a supervaluation of \(v_a\). A formula is then said to be supertrue under a valuation function \(v_a\) for \(\mathbf{K}_3\) if it is true under every supervaluation \(v_{a'}\) of \(v_a\), i.e., if \(v_{a'}(A) = T\) for every supervaluation \(v_{a'}\) of \(v_a\). The property of being superfalse is defined analogously.