Counting quantification

A counting quantifier is a mathematical term for a quantifier of the form "there exists at least k elements that satisfy property X". In first-order logic with equality, counting quantifiers can be defined in terms of ordinary quantifiers, so in this context they are a notational shorthand. However, they are interesting in the context of logics such as two-variable logic with counting that restrict the number of variables in formulas. Also, generalized counting quantifiers that say "there exists infinitely many" are not expressible using a finite number of formulas in first-order logic.

Counting quantifiers can be defined recursively in terms of ordinary quantifiers.

Let ${\displaystyle \exists _{=k}}$ denote "there exist exactly ${\displaystyle k}$". Then

${\displaystyle {\begin{aligned}\exists _{=0}xPx&\leftrightarrow \neg \exists xPx\\\exists _{=k+1}xPx&\leftrightarrow \exists x(Px\land \exists _{=k}y(Py\land y\neq x))\end{aligned}}}$

Let ${\displaystyle \exists _{\geq k}}$ denote "there exist at least ${\displaystyle k}$". Then

${\displaystyle {\begin{aligned}\exists _{\geq 0}xPx&\leftrightarrow \top \\\exists _{\geq k+1}xPx&\leftrightarrow \exists x(Px\land \exists _{\geq k}y(Py\land y\neq x))\end{aligned}}}$