  Semantically we can model definite descriptions by partial intensions. We say the term  \(\atoi y\phi (y)\)  designates at state \(\Gamma\) of a partial FOIL model  \(\cM\) with respect to valuation  \(v\) if   \(\cM, \Gamma \vDash_{w} \phi (y)\) for exactly one \(y\)-variant \(w\) of \(v\). Then the conditions from section 3.5.1 are extended as follows.