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

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”.

Have seen a symbol that looks somewhat like $\forall \!\!\!\! –$ or $\large \vee \!\!\!\!\!\!\! =$ frequently used by my teachers for “for all but finitely many” (equivalently, “for all large enough” in the context of natural numbers). Ergo, used that frequently myself in class. Surprised to find that it is not as standard as I have always taken it to be.

Jaśkowski used $\Pi$ for quantifying over propositions. So he would write

$$\Pi a . \Pi b. a \to b \to a$$

It is useful for formalizing things like “does double negation elimination imply LEM”, which you have to write as

$$(\Pi a. \lnot \lnot a \to a) \to (b \lor \lnot b)$$

IIRC he used $\Sigma$ for existential quantification over propositions.