Counting quantification
{{Short description|Mathematical logical term}}
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 denote "there exist exactly ". Then
:
\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 (P y \land y \neq x))
\end{align}
Let denote "there exist at least ". Then
:
\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 (P y \land y \neq x))
\end{align}
See also
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. [http://www-mgi.informatik.rwth-aachen.de/Publications/pub/graedel/gorc2.ps Postscript file] {{oclc|282402933}}
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