[Math] Is the word “any” a $\forall$ or an $\exists$

Question

I was wondering how should the word "any" be used in mathematical context. Is it a "for all" or an "it exists"?

For example, assume I stated something like

A set $X$ is called nice if $P(x)$ holds for any $x\in X$.

Would that mean that $X$ is nice only if all of its elements satisfy $P$, or that $X$ is nice as long as one of its elements satisfies $P$?

Personally, I always assumed the second case, but English is not my mother tongue, and I have seen the word being used both ways.

Answer 1

Here is the opinion of P. Halmos, extracted from his highly recommended article How to write mathematics, p. 142:

The point is that in everyday English "any" is an ambiguous word; depending on context it may hint at an existential quantifier ("have you any wool ?", "if anyone can do it, he can") or a universal one ("any number can play"). Conclusion: never use "any" in mathematical writing. Replace it by "each" or "every", or recast the whole sentence.