 Kripke defines truth for sentences of \(\mathcal{L}^{\Sigma}\) recursively in terms of truth in \(\mathcal{L}\). If \(A\) is an atomic sentence of \(\mathcal{L}^{\Sigma}\)—which may well be a complex sentence of \(\mathcal{L}\), then \(A\) is true in \(\mathcal{L}^{\Sigma}\) if, and only if, \(A\) is true in \(\mathcal{L}\). If \(A\) is a sentence of the form \(\lnot B\), then \(A\) is true in \(\mathcal{L}^{\Sigma}\) if, and only if, \(B\) is not true in \(\mathcal{L}^{\Sigma}\). If \(A\) is of the form \((B \rightarrow C)\), then \(A\) is true in \(\mathcal{L}^{\Sigma}\) if, and only if, \(B\) is not true or \(C\) is true in \(\mathcal{L}^{\Sigma}\). Finally, and more crucially, if \(A\) is of the form \(\Sigma \alpha \ B\), then \(A\) is true in \(\mathcal{L}^{\Sigma}\) if and only if \(B(\epsilon/\alpha)\) is true in \(\mathcal{L}^{\Sigma}\) for some \(\epsilon\) in the substitution class \(C\) associated with \(\Sigma\).