The semantics of first-order logic

We are now ready to tackle the semantics of first-order logic. In some sense, the semantics of first-order logic is very much like the semantics of zeroth-order logic, with the exception of the newly introduced variables and the quantifiers that bind them. On the other hand, the expressive power of the language $\mathcal{L}_1$ is vastly more powerful than that of $\mathcal{L}_0$. Indeed, for some, first-order classical logic is the logical system, or at any rate, the strongest but still well-behaved logical system. At any rate, this added expressive power comes with added complexity, and everything starts with dealing with the semantic values of our variables.