A first-order language $\mathcal{L}$ consists of:

1. Logical symbols (common to all languages):

   • Variables: $x, y, z, x_1, x_2, \ldots$
   • Logical connectives: $\neg, \land, \lor, \to, \leftrightarrow$
   • Quantifiers: $\forall$ (universal), $\exists$ (existential)
   • Equality: $=$ (optional)

2. Non-logical symbols (specific to $\mathcal{L}$):

   • Constant symbols: $c, d, c_1, c_2, \ldots$
   • Function symbols: $f, g, h, \ldots$ (each with a fixed arity)
   • Predicate symbols: $P, Q, R, \ldots$ (each with a fixed arity)

The set of terms is defined recursively:

1. Every variable is a term
2. Every constant symbol is a term
3. If $f$ is an $n$-ary function symbol and $t_1, \ldots, t_n$ are terms, then $f(t_1, \ldots, t_n)$ is a term
4. Nothing else is a term

Terms represent objects in the domain of discourse.

The set of formulas (or well-formed formulas) is defined recursively:

1. If $P$ is an $n$-ary predicate symbol and $t_1, \ldots, t_n$ are terms, then $P(t_1, \ldots, t_n)$ is a formula (atomic formula)
2. If $t_1$ and $t_2$ are terms, then $t_1 = t_2$ is a formula
3. If $\phi$ is a formula, then $\neg \phi$ is a formula
4. If $\phi$ and $\psi$ are formulas, then $(\phi \land \psi)$, $(\phi \lor \psi)$, $(\phi \to \psi)$, and $(\phi \leftrightarrow \psi)$ are formulas
5. If $\phi$ is a formula and $x$ is a variable, then $\forall x \, \phi$ and $\exists x \, \phi$ are formulas
6. Nothing else is a formula

In a formula, an occurrence of variable $x$ is bound if it appears within the scope of a quantifier $\forall x$ or $\exists x$. Otherwise, it is free. A formula with no free variables is called a sentence or closed formula.