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.
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.
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