Proper way to read $\forall$ - "for all" or "for every"? 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.
 A: I think your question is in fact most appropriate for an English language website.
If I were to pretend that this were such a website, I might answer as follows: for every emphasizes the individuality of the objects in the collection and is technically a singular ("for every element x..."), whereas for all emphasizes the collection itself and is technically a plural ("for all elements x...") 
Based on this I have to say that I find your idea of using for every when referring to "discrete" or countable sets and for all when referring to "continuous" or uncountable sets rather charming and insightful.  However, as a matter of mathematical practice, in all my experience the terms are used absolutely interchangeably.  Many people say one, many say the other, and many say both.  Given that the universal quantifier $\forall$ is written as an upside-down $A$ and is latexed as "\forall", probably a lot of people are trained to say "for all" more often than "for every" and maybe especially in more formal contexts.  But really either usage is absolutely permissible -- indeed, should go completely unnoticed (except by your student the professor?) -- in practice no distinction is made.
A: Universal quantifications can be written as "for all", "for any","for every", and "for each"; see the beginning of p. 168 here and http://en.wikipedia.org/wiki/List_of_mathematical_symbols.
A: 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 ($\exists$).
