# Quantifier

The general name for a logical operation that constructs from a predicate $P(x)$ a statement characterizing the domain of validity of $P(x)$. In mathematical logic, the most widely used quantifiers are the universal quantifier $\forall$ and the existential quantifier $\exists$. The statement $\forall x . P(x)$ means that the domain of validity of $P(x)$ is the same as the domain of values of the variable $x$. The statement $\exists x . P(x)$ means that the domain of validity of $P(x)$ is non-empty. If one is interested in the behaviour of the predicate $P(x)$ not on the whole domain of values of $x$, but only on the part singled out by a predicate $R(x)$, then one often uses the restricted quantifiers $\forall_{R(x)}$ and $\exists_{R(x)}$. In this case, the statement $\exists_{R(x)} x . P(x)$ means the same as $\exists x . R(x) \wedge P(x)$, while $\forall_{R(x)} x . P(x)$ has the same meaning as $\forall x . R(x) \rightarrow P(x)$, where $\wedge$ is the conjunction sign and $\rightarrow$ is the implication sign.