  A first order model is a structure \(\langle \bG, \bR, \bD_{O}, \bI\rangle\) where \(\langle \bG, \bR\rangle\) is a frame, as in Section 3.1, \(\bD_{O}\) is a non-empty object domain, and \(\bI\) is an interpretation that assigns to each \(n\)-place relation symbol \(P\) a mapping, \(\bI(P)\) from \(\bG\) to subsets of \(\bD_{O}^{n}\). We’ll write \(\bI(P, \Gamma )\) as an easier-to-read version of \(\bI(P)(\Gamma )\). It is required that \(\bI(=, \Gamma )\) is the equality relation on \(\bD_{O}\), for every state \(\Gamma\), and \(\bI(E, \Gamma )\) is non-empty, for every \(\Gamma\). A first-order valuation in a model is a mapping \(v\) that assigns a member of \(\bD_{O}\) to each variable. Note that first order valuations are not state-dependent in the way that interpretations are. A first order valuation \(w\) is an \(x\)-variant of valuation \(v\) if \(v\) and \(w\) agree on all variables except possibly for \(x\). Truth, at a state \(\Gamma\) of a model \(\cM = \langle \bG, \bR, \bD_{O}, \bI\rangle\), with respect to a first order valuation \(v\), is characterized as follows, where \(P(x_{1}, \ldots ,x_{n})\) is an atomic formula: