 The interpretation of the language of pure quantificational logic requires one to specify both a domain for the variables to range over and an extension for each non-logical predicate of the language. Alfred Tarski developed what is now known as a model theory for a wide range of formal languages. We recall some definitions from the entry on  model theory.  A model for the language of pure quantificational logic is an ordered pair, \(\langle D, I\rangle\), where \(D\) is a non-empty set, and \(I\) is an interpretation function, which maps each \(n\)-place predicate of the language into a set of ordered \(n\)-tuples of objects in the set \(D\). This set provides the domain of discourse for the quantifier, and \(I\) provides each predicate of the language with an extension: one-place predicates are mapped into subsets of \(D\), two-place predicates are mapped into sets of ordered pairs of members of \(D\), and more generally, \(n\)-place predicates are mapped into sets of \(n\)-tuples of members of \(D\).