Well-formed formula

Definition

Let $\sigma$ be a signature and $V$ a set of variables. The set of first-order formulas over $\sigma$ is the smallest set of strings such that:

1. if $t_1$ and $t_2$ are terms, then $t_1 = t_2$ is a formula;
2. if $R$ is a relation symbol of arity $n$ and $t_1, \dots, t_n$ are terms, then $R(t_1, \dots, t_n)$ is a formula;
3. if $\varphi$ and $\psi$ are formulas, then $\neg \varphi$, $(\varphi \land \psi)$, $(\varphi \lor \psi)$, $(\varphi \to \psi)$ and $(\varphi \leftrightarrow \psi)$ are formulas;
4. if $\varphi$ is a formula and $x \in V$, then $\forall x \, \varphi$ and $\exists x \, \varphi$ are formulas.

The formulas from clauses 1 and 2 are the atomic formulas.