# Restricted quantifier

A quantifier applied to predicates not with respect to the whole range of a given object variable, but with respect to a part of it defined by a predicate \$R(x)\$. When used in this restricted sense, the universal quantifier \$(\forall x)\$ and the existential quantifier \$(\exists x)\$ are usually denoted by \$(\forall x)_R(x)\$ and \$(\exists x)_R(x)\$ (or \$\forall x\colon R(x)\$ and \$\exists x\colon R(x)\$, respectively). If \$P(x)\$ is a predicate, then \$(\forall x)_R(x)P(x)\$ means

\$\$\forall x(R(x)\supset P(x)),\$\$

that is, the predicate \$P(x)\$ is true for all values of the variable \$x\$ satisfying the predicate \$R(x)\$. The proposition \$(\exists x)_R(X)P(x)\$ means

\$\$\exists x(R(x)\&P(x)),\$\$

that is, the intersection of the truth domains of the predicates \$R(x)\$ and \$P(x)\$ is non-empty.

Restricted quantifiers of the form \$(\forall x)_{x<t}\$ and \$(\exists x)_{x<t}\$ (more commonly called bounded quantifiers) play an important role in formal arithmetic (cf. Arithmetic, formal), where \$t\$ is a term not containing \$x\$. When these quantifiers are applied to a decidable predicate, the result is a decidable predicate.

How to Cite This Entry:
Restricted quantifier. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Restricted_quantifier&oldid=33093
This article was adapted from an original article by V.E. Plisko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article