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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".

B M
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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. – hmakholm left over Monica 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 00:07
  • 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 00:17

2 Answers2

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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.

Henry
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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
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    No. It is totally unacceptable to 'trust Donald Knuth'!! Consider "$\forall x \in S \exists y \in T ( Q(x,y) )$" ($\forall x \in S \exists y \in T ( Q(x,y) )$). It is extremely poorly spaced. Knuth never anticipated "$\in$" being used in restricted quantifiers. Even the example in your post looks bad to me. – user21820 Jan 25 '20 at 14:49
  • @user21820 - that would not look so bad with some punctuation such as a colon. E.g. $\forall x \in S: \exists y \in T ( Q(x,y) )$ from $\forall x \in S: \exists y \in T ( Q(x,y) )$ – Henry Jan 25 '20 at 14:54
  • It won't look so bad, but my point is simply that LaTeX does not do spacing well at all. Even Knuth himself used lots of manual spacing adjustments! By the way, for my example the nicest spacing to my eye is "$\forall x {\in} S\ \exists y {\in} T\ ( Q(x,y) )$" ($\forall x {\in} S\ \exists y {\in} T\ ( Q(x,y) )$). – user21820 Jan 25 '20 at 14:56
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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.

davidlowryduda
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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? – davidlowryduda Sep 27 '11 at 23:50