 Given a language and a structure, both of them sharing the same signature, each closed term of the language will denote an element in the structure and each atomic sentence is either true or false in the structure. Nevertheless, the scope of our definition is widened so that each term will receive a denotation and each formula a truth value. To achieve that, we define  assignments[5]  from the set of variables to the domains of the structure, arguments and values should be of the same sort. For any individual \(\mathbf{x}\in \mathbf{A}_{i}\) and variable \(x^{i}\) of sort \(i\), we use \(s(\mathbf{x}/x^{i})\) to denote the assignment which is like assignment \(s\) except that the value at \(x^{i}\) has to be \(\mathbf{x}\).