TITLE: Is the word "any" a $\forall$ or an $\exists$?
QUESTION [18 upvotes]: 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.

REPLY [3 votes]: $\newcommand{\eps}{\varepsilon}$Good writing facilitates understanding.
In my experience, the greatest risk of confusion comes from a predicate of the form "if for any...":

A function $f$ is continuous at $a$ if for any $\eps > 0$, there is a $\delta > 0$ such that if $|x - a| < \delta$, then $|f(x) - f(a)| < \eps$. (Here, "any" means "every".)
A function $f$ is discontinuous on a set $A$ if for any $a$ in $A$, $f$ is discontinuous at $a$. (Here, "any" means "some".)

In each case, the intended meaning is far from obvious until the definition of continuity has been absorbed. Even then, "if for any" makes even a fluent reader stop and re-read, breaks the train of thought.
In other words, the phrase obstructs learning and hampers communication. It belongs only in manuals of expository sabotage.