 More formally, a dual-domain model for a language \(\bL\) of the sort defined in  Section 2  is a triple \(\langle \bD,\bD_o,\bI\rangle\), where \(\bD\) is a possibly empty inner domain, \(\bD_o\) is a nonempty outer domain such that \(\bD \subseteq \bD_o\), and \(\bI\) is an interpretation function such that for every individual constant \(t\) of \(\bL, \bI(t) \in \bD_o\), and for every \(n\)-place predicate \(P\) of \(\bL, \bI(P) \subseteq \bD_o^n\). Given a model \(\langle \bD,\bD_o,\bI\rangle\), the valuation function \(\bV\) assigns truth values to atomic and quantified formulas as follows: