# Questions about notation $\forall$ and $\exists$

I had a few questions about notation $\forall$ and $\exists$. I am posing each of the question for the former, but I have the corresponding question for the latter as well:

1) If I have only one object to quantify, is it more common to write $\forall a$ or $\forall$ $a$ (with or without space).

2) If I have more than one object to quantify, is it more common to write $\forall a,b,c$ or $\forall$ $a,b,c$ (with or without space).

3) Is using these symbols in papers on subjects other than logic, considered a bad habit? Most papers I have read outside of logic, actually spell out "For all" and "there exists".

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I would always write them without a space, but that's just my habit, and shouldn't be taken as a definitive answer. I don't think it's a bad habit to use these symbols outside of a paper on logic. I think most mathematicians would know what they mean. However, if you think it is clearer to write 'for all $x$' instead of $\forall x$ and it doesn't get in the way of your exposition, then feel free to spell it out in longhand instead. –  Chris Taylor Sep 27 '11 at 23:33
@Chris: Thanks for the comment. After seeing the answers I have decided to leave my paper unchanged, since the way I typed it seems OK to others. –  B M Sep 27 '11 at 23:46
And for a contrasting view, I find it somewhat jarring when one mixes symbols and English text. One can embed a symbolic formula in English text, of course, but not the other way around except in informal or very special situations. So if you use a symbol for the quantifier, you should let the entire scope of the quantified variable be a symbolic expression. Otherwise it just looks like textspeak. –  Henning Makholm Sep 27 '11 at 23:58
I think that spelling out "for all" and "there exists" is the thing to do if it appears as part of an English sentence. As part of a displayed formula, $\forall$ and $\exists$ are fine, but I'd still prefer spelling it out unless there is a specific reason for doing otherwise. –  George Lowther Sep 28 '11 at 0:07
3, yes. Use these symbols only when doing symbolic logic, or when doing quick abbreviations for your own use. Ordinarily (especially in papers) write things out in words. –  GEdgar Sep 28 '11 at 0:17

My view is that you can leave it to $\TeX$ to sort out the spacing, trusting Donald Knuth, so the first of each of your two examples.

I would say that it is acceptable to use in mathematics (which I do not see as a subset of logic) as in $$\forall n \in \mathbb{N}: \sum_{i=0}^n i = \frac{n(n+1)}{2}$$ but rare outside logic and mathematics.

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Thanks for the answer, Henry. By outside of logic, I did imply the universal set of Mathematics. I was mainly concerned when there was more than one quantity involved since TEX does typeset a space after the comma. –  B M Sep 27 '11 at 23:44

For the first 2, I don'r really even notice. I happen to not use a space, so that long statements are clumped and are (in my opinion) easier to grasp. For example, $\forall f: \mathbb{R} \to \mathbb{R}\; \text{strictly increasing}, \; \forall n \in \mathbb{N}, \; \exists m \in \mathbb{N} \; \mathrm{s.t.} \; f(m) > f(n)$ versus $\forall \;f: \mathbb{R} \to \mathbb{R}\; \text{strictly increasing}, \; \forall \;n \in \mathbb{N}, \; \exists \; m \in \mathbb{N} \; \mathrm{s.t.} \; f(m) > f(n)$

But even looking at them now, I barely notice. Perhaps I don't use the space because I'm lazy.

With respect to your last question, it's absolutely fine to use 'mathspeak' in real papers. I have read many number theory papers with loads of mathspeak.

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Thanks for the answer, mixed math. I was about to revise my entire paper, but it seems the general consensus is to prefer not having a space, which is the way I typed it, so I will leave it as is. –  B M Sep 27 '11 at 23:45
@BM I would be curious to see your paper. Have you uploaded a preprint or anything? –  mixedmath Sep 27 '11 at 23:50