After defining what truth values are, we will next explain, in which context it makes sense to assign a truth value to a string of a given formal language. Imagine, we want to assign a truth value to the following English sentence:
"The equation $x+4 = 2$ has always a solution."
Remember, that we are now at the model level of our study and this sentence is up to now a string of some formal language without any meaning. If we want to construct a logical calculus, in which we can assign a truth value to this string, we have to pay attention to the context, in which we do so. Otherwise, our assignment might be ambiguous. For instance, if the context is integers, and our logical calculus is laying out the corresponding theory, then our logical calculus should be able to assign the value true to this string because the integer \(x = -2\) is a solution. But if the context is natural numbers, then the string is false, since there is no such natural number $x$, for which $x+4=2$.
The context, in which we are studying the logic of given strings is known as the domain of discourse. Now, we will define it formally:
A domain of discourse (also called universe of discourse or just universe) is a non-empty universal set $U$, in which we study a given formal language \(L\subseteq (\Sigma^*,\cdot) \) over an alphabet \(\Sigma\).