 What is less obvious is how to modify the model theory for the language in order to accommodate empty terms. One important choice point is the evaluation of atomic formulas in a model of free quantificational logic with identity. One approach is to declare as false an atomic formula of the form \(P^n_i t_1, \dots, t_n\) or \(t_i = t_j\) if any of the terms involve fail to denote anything in the domain. That would result in a negative model theory for the language. On the other hand, we could make allowance for true atomic formulas even if some of the relevant terms denote nothing in the domain, e.g., we might declare \(t = t\) true even if nothing in the domain of quantification is a denotation for the term in question. That would result in a positive model theory for the language. Finally, we could declare atomic formulas not of the form \(E!t\) to be neither true nor false, which would result in a neutral model theory for the language. The entry on  free logic  explores all these options in detail.