 Before we modify the definition of formula for the expanded language, we may give a recursive definition of a term of the language of the classical theory of quantification and identity. An expression \(t\) is a term if, and only if, (i) \(t\) is an individual variable, or (ii) \(t\) is an individual constant, or (iii) \(t\) is an expression of the form \(f^{n}_{i}t_{1}, \ldots, t_{n}\), where \(f^{n}_{i}\) an \(n\)-place function symbol and \(t_1\), …, \(t_n\) are themselves terms. Given the definition of term, we can reformulate the definition of atomic formula for the expanded language to consist of all and only expressions \(A\) of the form \(P^n_i t_1, \ldots, t_n\), where \(P^n_i\) is an \(n\)-place predicate and \(t_1\), …, \(t_n\) are each terms or \(t_i = t_j\) in which \(t_i\) and \(t_j\) are terms. At this point, we may adapt the usual recursive definition of formula for the expanded language.