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.

Answer 2

It is my understanding that this is all about the difference between the words $\textit{some}$ and $\textit{any}$.

$\textit{any}$ means $\forall$, $\textit{some}$ means $\exists$.

Answer 3

According to the dictionary, "any" can mean "one, some, every, all". Often it is obvious from the context which is meant, but not always. Careful writers of mathematics try to choose a more specific word. For example, Halmos never used "any". So there is no answer to your question. Every time you come across "any" you have to try to figure out from the context what the author meant.

Answer 4

This is turning into an English lesson. Here is my take on it as a native UK English speaker: if a bouncer says to a group of students:

"You will all be thrown out of this bar if any of you misbehaves."

The bouncer's "any" is an existential quantifier: one misbehaviour by one student means the group will be thrown out. That is how "A set $X$ is called nice if $P(x)$ holds for any $x$ in $X$" reads to me (one in, all in).

I don't know whether Tate or Halmos ever worked as bouncers, but I would recommend their avoidance of quantifying over "any" (rather than "some" or "all") in a mathematical context unless you are very confident that what you have written is not ambiguous or misleading.

Answer 5

$\newcommand{\eps}{\varepsilon}$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.