  Then we conclude, classically, that the sentence Ga is true in M. It will be useful to have some notation for the classical truth value of a sentence S in a classical model M. We will write ValM(S). In this case, ValM(Ga) = t. In Example 3.1, we did not start with a classical model of the whole language L, but only a classical model of the T-free fragment of L. But then we added a hypothesis, in order to get a classical model of all of L. Let's use the notation M + h for the classical model of all of L that you get when you extend M by assigning T an extension via the hypothesis h. Once you have assigned an extension to the predicate T, you can calculate the truth values of the various sentences of L. That is, for each sentence S of L, we can calculate