# Symbols for Quantifiers Other Than $\forall$ and $\exists$

The symbols $\forall$ and $\exists$ denote "for all" and "there exists" quantifiers. In some papers, I saw the (not so common) quantifiers $Я$ and $\exists^+$, denoting "for a randomly chosen element of" and "for most elements in", respectively.

Are there other symbols for quantifiers?

I'm specially interested in quantifiers for:

• for all but finitely many elements of...
• for infinitely many elements of...

Edit: After seeing some of the comments, I found the list of logic symbols and the table of mathematical symbols, which I could be useful for others.

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$\exists !$: There exist exactly one – Sune Jakobsen Feb 10 '11 at 15:57
$\nexists$: Does not exist – kennytm Feb 10 '11 at 16:01
I've sometimes seen $\forall\forall$ used to mean "for all but finitely many elements", but I don't know if this is a common convention or not. – user1728 Feb 10 '11 at 16:05
@Chris: For ${\mathcal U}$ an ultrafilter, I have seen ${\mathcal U}$ itself used as a quantifier, so ${\mathcal U}x\phi(x)$ means that the set of $x$ such that $\phi(x)$ is in ${\mathcal U}$. This is convenient in a context where one may be using several different ultrafilters or where the role of a specific ${\mathcal U}$ wants to be emphasized. – Andrés E. Caicedo Feb 10 '11 at 17:34
@Andres: that notation could subsume the "for all but finitely many" case once you introduce a fixed symbol for the Fréchet filter. – Chris Eagle Feb 10 '11 at 18:20

The most commonly used symbols to express "for all but finitely many" and "there are infinitely many" are $\forall^\infty$ and $\exists^\infty$, respectively.

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Other common quantifiers are $\forall^*$ and $\exists^*$, used in the context of a Polish space (such as ${\mathbb R}$). The first means "for all but a meager set of points", the second "there is a non-meager set of points". – Andrés E. Caicedo Feb 10 '11 at 17:32
This makes a lot of sense. Thanks! – M.S. Dousti Feb 10 '11 at 18:46
For sequences (and sort of for nets—but uglier since cofinality comes into the picture) I've also seen the terms "eventually" and "frequently" for "for all but finitely many" and "for infinitely many" respectively. E.g. "$n^2 \gt 2n$ eventually" ($\exists N : n \geq N \implies n^2 \gt 2n$) and "$\sin n \lt 0$ frequently" ($\forall N \exists n : n \geq N \land \sin n \lt 0$) – kahen Feb 10 '11 at 19:56
An extension of this would be models of languages with added quantifiers that express "there exist uncountably many" or "there exist $\kappa$ many, for $\kappa > \aleph_{1}$". The first case I've seen explored when the language is our standard first-order logic, but I've not seen treatments of the latter. – JakeR Jan 24 '12 at 5:18

Not quite a symbol per se, but of course there is "a.e." ("almost everywhere"): "for all but a set of measure zero". Probabilists call it "a.s." ("almost surely"). They are also in the habit of writing "a.a." ("almost all"): "for all but finitely many" and "i.o." ("infinitely often"): "for infinitely many".

In potential theory, one also sees "q.e." ("quasi-everywhere"): "for all but a set of capacity zero".

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