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{align}

\exists^{= 0} x P x &\leftrightarrow \neg \exists x P x \\ \exists^{= k+1} x P x &\leftrightarrow \exists x (P x \land \exists^{= k} y (y \neq x \land P y)) \end{align}$$ Let $$\exists^{\geq k}$$ denote "there exist at least $$k$$". Then
 * $$\begin{align}

\exists^{\geq 0} x P x &\leftrightarrow \top \\ \exists^{\geq k+1} x P x &\leftrightarrow \exists x (P x \land \exists^{\geq k} y (y \neq x \land P y)) \end{align}$$