Sentence

Definition

Let $\sigma$ be a signature. The set $\operatorname{FV}(\varphi)$ of free variables of a formula $\varphi$ over $\sigma$ is defined by recursion on $\varphi$:

1. for an atomic formula, $\operatorname{FV}(\varphi)$ is the set of variables that occur in its terms;
2. $\operatorname{FV}(\neg \varphi) = \operatorname{FV}(\varphi)$, and $\operatorname{FV}(\varphi \ast \psi) = \operatorname{FV}(\varphi) \cup \operatorname{FV}(\psi)$ for each binary connective $\ast$;
3. $\operatorname{FV}(\forall x \, \varphi) = \operatorname{FV}(\exists x \, \varphi) = \operatorname{FV}(\varphi) \setminus \set{x}$.

A sentence over $\sigma$ is a formula $\varphi$ with $\operatorname{FV}(\varphi) = \emptyset$.