 The earliest and most extensive applications of free logic have been to the theory of definite descriptions. A definite description is a phrase that may be expressed in the form “the \(x\) such that \(A\),” where \(A\) is an open formula with only \(x\) free. Formally, this is written using a special logical operator, the definite description operator ‘\(\iota\)’, as \(\iota xA\). Contra Russell, free logic treats definite descriptions not as merely apparent singular terms in formulas whose logical form is obtainable only by elaborate contextual definitions, but as genuine singular terms. Thus, like an individual constant, \(\iota xA\) may be attached to predicates and (under appropriate conditions) substituted for variables. For any object \(d\) in the domain \(\bD, \iota xA\) denotes \(d\) if and only if among all objects in \(\bD\), \(d\) and only \(d\) satisfies \(A\). If in \(\bD\) there is more than one object satisfying \(A\), or none, \(\iota xA\) is empty. The description operator therefore satisfies: