The syntax of first-order logic

Now that we are comfortable with the syntax, semantics, and tableau system of zeroth-order logic, we are in a position to consider a more expressive language, the language of first-order logic. Unsurprisingly, we will denote the language of first-order logic by $\mathcal{L}_1$. Just as before, the language has a syntactic aspect, the way its expressions are formed, and a semantic aspect, the way its expressions are given meaning. Clearly, the second depends on the first, so we shall start, again, with the syntax of the language under consideration.

In fact, the language $\mathcal{L}_1$ is very similar to $\mathcal{L}_0$, save for one crucial device; quantification. Accordingly, we still have the same predicates and constants as before, and the same connectives to connect atomic formulas together. However, what we also have are variables, and quantifier symbols that bind these variables.