 To make sense of interpretations over a universal domain, one may conceive of an interpretation as a value of a second-order predicate variable, \(I\), which applies to ordered pairs of a certain sort. An interpretation, \(I\), applies to ordered pairs of the form \(\langle \forall, o \rangle\), when \(o\) is in the domain of the interpretation, and \(I\) applies to ordered pairs of the form \(\langle P^{n}_{i}, \langle a_1, \ldots, a_n\rangle \rangle\), when \(P^{n}_{i}\) is taken to apply to the objects in \(\langle a_1, \ldots, a_n\rangle\). The usual definition of an assignment is then replaced by one on which an assignment is given by the value of a second-order variable \(S\), which applies to an ordered pair of the form \(\langle x, o\rangle\), where \(o\) is an object, for each first-order variable in the language, and to ordered pairs of the form \(\langle X, o\rangle\), where \(o\) is an object, for each second-order variable in the language. An assignment \(S\) is required to pair each first-order variable with exactly one object, but it is allowed to pair a second-order variable with any objects whatever. In the presence of these definitions, we can generalize Tarski’s definition satisfaction in an interpretation by an assignment, truth, and  validity.[12]