  The language stays the same, but intension variables are now interpreted by partial functions on the set of states—functions whose domains may be proper subsets of the set of states. Thus   \(\cM = \langle \bG, \bR, \bD_{O}, \bD_{i}, \bI\rangle\) is a partial FOIL model if it is as in Section 3.4 except that members of \(\bD_{i}\) are partial functions from \(\bG\) to \(\bD_{O}\). Given a partial FOIL model  \(\cM\)  and a valuation \(v\) in it, an intension variable \(f\) designates at state \(\Gamma\) of this model with respect to \(v\) if \(\Gamma\) is in the domain of \(v(f)\).