  To distinguish the models described here from those in Section 3.2.2, these will be referred to as FOIL models, standing for first order intensional logic. They are discussed more fully in (Fitting 2004). A FOIL model is a structure  \(\cM =  \langle \bG, \bR, \bD_{O}, \bD_{i}, \bI\rangle\) where \(\langle \bG, \bR, \bD_{O}, \bI\rangle\) meets the conditions of Section 3.2.2, and in addition, \(\bD_{i}\) is a non-empty set of functions from \(\bG\) to \(\bD_{O}\); it is the intension domain.