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.

Definition in terms of ordinary quantifiers

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

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

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

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

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

See also

• Uniqueness quantification
• Lindström quantifier
• Spectrum of a sentence

References

• Erich Graedel, Martin Otto, and Eric Rosen. "Two-Variable Logic with Counting is Decidable." In Proceedings of 12th IEEE Symposium on Logic in Computer Science LICS `97, Warschau. 1997. Postscript file OCLC 282402933