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".
 A: 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.
A: 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.
