Some time ago, a friend of mine asked me how to put a symbol *below* a quantifier—in other words, how to typeset e.g. instead of (this is quite common in Polish typography).

That’s easy, I said. You just put `\limits`

right after the `\forall`

. Well, easy as it was, it didn’t work. The reason is obvious (at least, after a reading session with *The TeXbook*): `\limits`

(and it’s twin, `\nolimits`

) work only after so-called *operators*. For example, you can get when you say

\int\limits_\Omega \int\nolimits_0^1

to (La)TeX. (Note that LaTeX, and particularly AMS-LaTeX, has some defaults, different for, e.g., sums and integrals. Note also that you may change them globally: either by using some options of AMS-LaTeX, or by saying something like

\let\originalint=\int \renewcommand{\int}{\originalint\limits}

in the preamble of the LaTeX file; plain TeX or ConTeXt users would use `\def\int{...}`

instead of `\newcommand{\int}{...}`

.)

Since by default, `\forall`

is an *atom* of type *Ord* (as in “*Ord*inary symbol”; for discussion of atoms and their types, see Chapter 17 of *The TeXbook*), it can’t take `\limits`

(neither `\nolimits`

). It is easy, however, to convince TeX to treat it as an *Op* (”*Op*erator”) atom: you just have to say `\mathop\forall`

instead of just `\forall`

. This way, you may type

```
$\mathop\forall\limits_x x>0$
```

to get the result shown above.

Note that this has also an interesting side-effect: *Op* atoms, unlike *Ord* ones, are vertically centered. I don’t think this is a problem, but if you really don’t like it, here’s how to avoid it:

$\mathop{\hbox{$\forall$}}\limits_x x>0$;

don’t ask me why this works, just look at Rule 13 of Appendix G in *The TeXbook*.

But this is not the end of the story. The quantifier symbol, even with `\mathop`

, does not really behave like an operator: it has a fixed, rather small, size; we would sometimes like to have it a bit bigger. (In fact, it would be best if its size adapted to different circumstances—but I guess it can be safely assumed that no-one puts quantifiers into sub- or superscripts, so I don’t want to go into this kind of details. If you *really* want to know how to do this, look up `\mathchoice`

in *The TeXbook*‘s index.) An easy solution would be to say

```
$\mathop{\large\forall}\limits_x x>0$
```

or something similar; again, this does not work. The reason (sorry to say that again, but this is explained in detail… well, you know where) is that TeX chooses fonts for a math formula on the basis of current settings *at the end of the formula*; maybe because of that, if you try to use `\large`

in a math formula, you get a `LaTeX Font Warning: Command \large invalid in math mode...`

This way, we have at least two choices. One is to redefine some (La)TeX internals and start to mess up with font families, *mathcodes* etc. I didn’t want to do this, since the thing is complicated even in plain TeX. The easier solution might be to try the `\hbox`

trick again (notice the two occurences of `\mathop`

):

$\mathop{\hbox{\Large$\mathop\forall$}}\limits_x x>0$,

which yields . However, this is again not very good—the “for all” symbol seems shifted a bit up. This is true: although it is centered around the so-called *vertical axis*, the axis height is taken from the `\Large`

size, i.e., it is higher than it should be here. How to overcome this new problem? Again, it is (of course!) possible, and in fact not that difficult: you just write

$\mathop{\vcenter{\hbox{\Large$\forall$}}}\limits_x x>0$

and finally get ! (Note that the `\hbox`

is necessary here, since `\vcenter`

takes *vertical material* as its argument; try removing the `\hbox`

and see (and explain) what happens! Also, the second `\mathop`

is now made obsolete by `\vcenter`

.)

Obviously, if you need such a construct, it is best to use a macro so that you don’t have to repeat the above code every time you need a quantifier; for example, in LaTeX you write

\let\originalforall=\forall \renewcommand{\forall}{\mathop{\vcenter{\hbox{\Large$\originalforall$}}}}

and use `\forall`

as usual (with or even without subscripts).

Two final remarks are:

- Yes, I know that you want to say that the example formula is wrong. It isn’t; I just forgot to mention that it should be interpreted in the universe .
- Even the last thing is not really (typographically) optimal: the best solution would be to use a font with a predesigned large “for all” symbol (which is absent from Computer Modern fonts; also the
*Comprehensive LaTeX Symbol List*does not mention any font containing it…). The used way (with`\Large`

and similar commands) may result in incosistent line widths etc. In case of , this seems to be tolerable: you get thicker lines, which make the symbol visually heavier, but still the lines are thinner than in, e.g., (`\bigvee`

). But I’m not a font designer, and thus can’t help this problem…

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