# 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.

More generally, the model-theoretic interpretation of an arbitrary "quantifier" $Q$ (with the same syntactic behaviour as $\forall$ and $\exists$) can (according to A. Mostowski) be given by a mapping associating with each model $(A,\dots)$ a class $\tilde Q$ of subsets of $A$. Then one stipulates as a truth-definition for $Q$ that, e.g., a sentence $Qx\ \Phi(x)$ holds in $(A,\dots)$ if and only if the set $\{a\in A:\Phi(a)\text{ holds in }(A,\dots)\}$ is in $\tilde Q$. Thus, with the existential quantifier $\exists$ is associated the class of non-empty subsets of $A$ and with the universal quantifier $\forall$ is associated the class $\{A\}$. However, there are many more possible quantifiers, e.g. given by $\{B\subset A:B\text{ finite}\}$, $\{B\subset A:B\text{ has the same cardinality as }A\}$ (the Chang quantifier), $\{B\subset A:B\text{ is uncountable}\}$, etc.
This set-up can be generalized to polyadic "quantifiers" binding more than one variable occurring in more than one formula (example: the equi-cardinality quantifier $Q$ binding two variables $x$ and $y$ in two formulas $\Phi(x)$ and $\Psi(y)$, yielding the formula $Qxy(\Phi(x),\Psi(y))$, which is interpreted by $\{(B,C):B\text{ and }C\text{ have the same cardinality}\}$). Even more general is the Lindström quantifier. And each quantifier has its own logic.