# Bounded quantifier

In the study of formal theories in mathematical logic, bounded quantifiers (a.k.a. restricted quantifiers) are often included in a formal language in addition to the standard quantifiers "∀" and "∃". Bounded quantifiers differ from "∀" and "∃" in that bounded quantifiers restrict the range of the quantified variable. The study of bounded quantifiers is motivated by the fact that determining whether a sentence with only bounded quantifiers is true is often not as difficult as determining whether an arbitrary sentence is true.

## Examples

Examples of bounded quantifiers in the context of real analysis include:
* $\forall x>0$ - for all x where x is larger than 0
* $\exists y<0$ - there exists a y where y is less than 0
* $\forall x\in \mathbb {R}$ - for all x where x is a real number
* $\forall x>0\quad \exists y<0\quad (x=y^{2})$ - every positive number is the square of a negative number

## Bounded quantifiers in arithmetic

Suppose that L is the language of Peano arithmetic (the language of second-order arithmetic or arithmetic in all finite types would work as well). There are two types of bounded quantifiers: $\forall n<t$ and $\exists n<t$. These quantifiers bind the number variable n using a numeric term t not containing n but which may have other free variables. ("Numeric terms" here means terms such as "1 + 1", "2", "2 × 3", "m + 3", etc.)
These quantifiers are defined by the following rules ($\phi$ denotes formulas):
    $\exists n<t\,\phi \Leftrightarrow \exists n(n<t\land \phi )$
    $\forall n<t\,\phi \Leftrightarrow \forall n(n<t\rightarrow \phi )$
There are several motivations for these quantifiers.
* In applications of the language to recursion theory, such as the arithmetical hierarchy, bounded quantifiers add no complexity. If $\phi$ is a decidable predicate then $\exists n<t\,\phi$ and $\forall n<t\,\phi$ are decidable as well.
* In applications to the study of Peano arithmetic, the fact that a particular set can be defined with only bounded quantifiers can have consequences for the computability of the set. For example, there is a definition of primality using only bounded quantifiers: a number n is prime if and only if there are not two numbers strictly less than n whose product is n. There is no quantifier-free definition of primality in the language $\langle 0,1,+,\times ,<,=\rangle$, however. The fact that there is a bounded quantifier formula defining primality shows that the primality of each number can be computably decided.
In general, a relation on natural numbers is definable by a bounded formula if and only if it is computable in the linear-time hierarchy, which is defined similarly to the polynomial hierarchy, but with linear time bounds instead of polynomial. Consequently, all predicates definable by a bounded formula are Kalmár elementary, context-sensitive, and primitive recursive.
In the arithmetical hierarchy, an arithmetical formula that contains only bounded quantifiers is called $\Sigma _{0}^{0}$, $\Delta _{0}^{0}$, and $\Pi _{0}^{0}$. The superscript 0 is sometimes omitted.

## Bounded quantifiers in set theory

Suppose that L is the language $\langle \in ,\ldots ,=\rangle$ of the Zermelo–Fraenkel set theory, where the ellipsis may be replaced by term-forming operations such as a symbol for the powerset operation. There are two bounded quantifiers: $\forall x\in t$ and $\exists x\in t$. These quantifiers bind the set variable x and contain a term t which may not mention x but which may have other free variables.
The semantics of these quantifiers is determined by the following rules:
    $\exists x\in t\ (\phi )\Leftrightarrow \exists x(x\in t\land \phi )$
    $\forall x\in t\ (\phi )\Leftrightarrow \forall x(x\in t\rightarrow \phi )$
A ZF formula that contains only bounded quantifiers is called $\Sigma _{0}$, $\Delta _{0}$, and $\Pi _{0}$. This forms the basis of the Lévy hierarchy, which is defined analogously with the arithmetical hierarchy.
Bounded quantifiers are important in Kripke–Platek set theory and constructive set theory, where only Δ$_0$ separation is included. That is, it includes separation for formulas with only bounded quantifiers, but not separation for other formulas. In KP the motivation is the fact that whether a set x satisfies a bounded quantifier formula only depends on the collection of sets that are close in rank to x (as the powerset operation can only be applied finitely many times to form a term). In constructive set theory, it is motivated on predicative grounds.