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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.

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I am also curios should I write "for any" or "for all"? –  Ilya Jul 4 '11 at 9:05
    
Can't every set be given a discrete topology? Are two real numbers not distinct just because there is a continuum many of them? –  Asaf Karagila Jul 4 '11 at 9:18
    
@Asaf: first question: yes. (I'm sure this was rhetorical, although I'm not sure exactly what you have in mind.) Second question: huh? –  Pete L. Clark Jul 4 '11 at 9:21
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@Pete: Both were rhetorical addressing the second paragraph of the question. –  Asaf Karagila Jul 4 '11 at 9:23
    
@Asaf: okay. I don't really get it, but I suppose that's not required. :) –  Pete L. Clark Jul 4 '11 at 9:26

3 Answers 3

up vote 6 down vote accepted

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.

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Perfect, thank you very much. –  jefflovejapan Jul 4 '11 at 9:54
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In view of JDH's answer, let me note that the ambiguous issue with "any" is also indicated on p. 168 of the linked article. Quoting: If you write "If for any $x \in X$, $f(x) > a, \ldots,$" it really is not clear whether you mean "for all $x$" or "for some $x$." –  Shai Covo Jul 4 '11 at 11:24

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 ($\exists$).

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Halmos in How to Write Mathematics suggested to ban any from mathematical writing because every or some are more precise replacements. –  lhf Jul 4 '11 at 11:13
    
I don't find the red ball example illuminating, but the rest of the examples are great. "Any" is often a big headache. –  ShreevatsaR Jul 4 '11 at 11:28
    
The point of the red ball example is that this is a usage of "any" which clearly means $\exists$, whereas many people assert that "any" means $\forall$. The ambiguous phenomenon seems to occur most easily with questions, but it also arises with negated statements. –  JDH Jul 4 '11 at 11:34
    
It also arises in conditionals: "If any student can solve this problem, I will give the whole class a day off". –  Carl Mummert Jul 5 '11 at 12:47

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.

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That's what I figured. I'd never thought about it before he asked me, and had been using them interchangeably myself. –  jefflovejapan Jul 4 '11 at 9:55
    
Why exactly do find it rather charming and insightful? So you think "for all real numbers $x$" is somehow (aesthetically, pedagogically, whatever) better than "for every real number $x$"? –  ShreevatsaR Jul 4 '11 at 11:27
    
I would say for all non-zero real numbers $x$, $x^2$ is positive, while I would say *for every element $g$ of a group there's an element $h$ such that $gh=1$*. My impression is that the use of *every* in the second instance conveys better the idea that $h$ depends on what $g$ you start with. On the other hand, English is not my first language, so I won't push this too much. –  Andrea Mori Jul 4 '11 at 14:29
    
@Andrea: no, I think you are right, but the point is that this is more a matter of linguistics than logic: writing out either statement we use $\forall g \exists h$ and it is the order of the quantifiers which really matters, not the every / all dichotomy. –  Pete L. Clark Jul 4 '11 at 17:44
    
@Shreevatsa: I explained the charm as best as I can: to me, it seemed to come from an unusually close amount of attention paid to English grammar / linguistics. I can't tell you why I find that charming: it's a personal aesthetic reaction. Anyway it is the idea of making some distinction here that I am responding positively to, not the practice. In practice I would not notice that someone used one construction versus the other. There is no pedagogy here whatsoever... –  Pete L. Clark Jul 4 '11 at 17:50

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