A collection of syntactic formulae over which consequence relations corresponding to either syntactic consequence or semantic consequence range. Most frequently, formal languages in logic are recursively defined from a set of atomic formulae with complex formulae generated by repeated applications of construction rules corresponding to connectives, operators, quantifiers, and other logical symbols. For example, the propositional language $ℒ$ of classical logic with a unary negation connective $\neg$ and a binary conjunction connective $\wedge$ is defined by the following scheme:

• • If $\phi$ is an atom, then $\phi \in ℒ$

• • If $\phi ,\psi \in ℒ$, then $\neg \phi \in ℒ$ and $\phi \wedge \psi \in ℒ$

Richer languages are constructed by adding clauses corresponding to other $n$-ary connectives or other logical symbols. For example, a first-order language ${ℒ}_{\sigma }$ in a signature $\sigma$ whose atoms are defined by $\sigma$ might include a clause corresponding to existential and universal quantifiers by appending the following clause to the foregoing definition:

• • If $x$ is a variable and $\phi \in {ℒ}_{\sigma }$, then $\forall x\phi \in {ℒ}_{\sigma }$ and $\exists x\phi \in {ℒ}_{\sigma }$