TITLE: Proper way to read $\forall$ - "for all" or "for every"?
QUESTION [19 upvotes]: I was asked in class the other day by a professor for whom English is a second language why $\forall$ is sometimes read "for all" while other times read "for every." Is there a rule for this? 
I was thinking it might be read "for every" for finite and possibly countably infinite sets, since "every" seems to emphasize the distinctness of unique elements, while "for all" might be used for uncountably infinite sets, but I'm not sure.

REPLY [20 votes]: There are some interesting ambiguities arising from the use of the word "any," for in some situations, this word indicates $\forall$ and in others it indicates $\exists$. 
For example, when one says "any even number larger than two is composite," then the meaning is $\forall$. 
But if I ask, "are there any red balls in the box?" I am clearly not asking whether every red ball is in the box. 
Consider the question, "Is any prime number even?" One answer is "Yes, $2$ is a prime number that is even," and this answer interprets the question as $\exists$. But another answer is, "No, $3$ is a prime number that is odd," interpreting the question as $\forall$.  
Or consider the ambiguous usage, "Is any function satisfying our constraints continuous?"  It isn't clear whether one is asking about a universal claim ($\forall$) or seeking an example ($\exists$).