# Punctuation in quantified statement

I am not very experienced in writing longer texts in English and, in particular, I am still very uncertain about English punctuation. A lot of rules seem to me not very well-defined.

In particular, I would like to ask if you should use comma when quantifying or specifying. I am interested in the phrases "For every thing (it holds)(,) (that)..." and "a thing(,) such that" For example:

For any point $$p\in M$$(,) $${\rm d}\Phi_p$$ is injective.

For any vector space $$V$$(,) the set of all invertible linear transformations form a group.

For every $$x\in G$$(,) there is an $$h\in H$$(,) such that $$y=xhx^{-1}$$.

A map $$\Phi\colon X\to Y$$(,) such that for all $$x\in X$$ and $$g\in G$$(,) $$\Phi(gx)=g\Phi(x)$$(,) is called $$G$$-equivariant

Let $$I$$ and $$J$$ be ideals of $$L$$(,) such that $$I\subset J$$.

I will be also grateful if you recommend me a good mathematical writing guide, where such examples are discussed.

P.S. Just to clarify. In sentences like

The kinetic energy is given by $$E={1\over 2}mv^2$$, where $$m$$ is the mass and $$v$$ is the velocity of a given body.

you should always put a comma before "where", do you? (I thought it was a defining clause, since it defines what the variables stand for, but then I googled it and found out, I was probably wrong.)

This isn't a math question per se, but it might be especially relevant to math because of how precise statements must be.

In English, you don't need a comma if the format is "independent clause — dependent clause." However, if you flip it, then you do. For example,

The function is continuous for every point $$x$$. (no comma)

For every point $$x$$, the function is continuous. (always a comma)

That is only one main use of a comma (separating dependent and independent clauses). The last example you gave is one of those cases where you do need a comma.

However, you don't need a comma for the sentence

Let $$I$$ and $$J$$ be ideals such that $$I\subset J$$.

I'm struggling to give a good reason for this, but one way of thinking about it is that you cannot switch the order and say "Such that $$I\subset J$$, let $$I$$ and $$J$$ be ideas." If it is a defendant clause you can always switch the order like I mentioned.

The best way to tell if you need a comma (and realistically what native speakers use) is to put one in wherever there is a pause in speaking. For example, spoken aloud, you say

For every point $$x$$ [pause] the function is continuous.

But there is no pause when you say

The function is continues for every point $$x$$.

Hope that helps some.

• Thank you for your answer, I think I am starting to understand the idea. The confusing thing for me in the first case was, that "for every point x" is not a clause (if I understand the notion properly), because there is no predicate. However, it probably applies for phrases as well. And this is maybe the thing people call "introductory phrases". – Daniel Aug 5 '16 at 16:06
• This is not in fact entirely correct: ‘For every point $x$’ is a prepositional phrase, not a clause, and the comma in your second example is not actually required, though I do think that it’s preferable. – Brian M. Scott Aug 5 '16 at 19:43

I would use the comma in the first and second examples; it’s not absolutely required, but it makes the sentence a little easier to read. There should be no comma in the third example. I would rewrite the fourth example:

A map $\Phi:X\to Y$ is called $G$-equivariant if $\Phi(gx)=g\Phi(x)$ for each $x\in X$ and $g\in G$.

There should be no comma in the fifth example.

Yes, in examples like the one in the postscript you should always place a comma before the where.

The rules are poorly defined; natural languages are not formal languages. Most native speakers don't know any "rules" in any case.

I took the liberty of numbering your examples for reference. My opinion:

(i) You definitely need the comma, just to separate the two "formulas", regardless of whether it's proper English or not. Two bits of mathematical notation separated by nothing look like one formula.

In fact, I disapprove of having two formulas separaed by nothing but a comma; I'd rewrite so they were separated by at least one word.

(ii), (iii) I don't think this is clear-cut. I'd tend to leave out both commas. For some reason I cannot articulate I feel it's close to optional in (ii), while the comma in (iii) strikes me as close to wrong.

(iv) Here it seems to me that using commas or not is ok. Note however that including one of the commas but omitting the other is definitely wrong; that would be "comma fault". (The reason I'm posting this wishy-washy answer is to point that out.)

(v) I'd omit the comma but wouldn't quarrel with someone who included it.

• @BrianM.Scott Reference? – David C. Ullrich Aug 5 '16 at 20:22
• @BrianM.Scott The first reference I found says that not only is the comma not required in the fifth example, it's wrong: cws.illinois.edu/workshop/writers/restrictiveclauses – David C. Ullrich Aug 5 '16 at 20:30
• @BrianM.Scott What? We''re talking about "Let $I$ and $J$ be ideals of $L$ such that $I\subset J$", right? That clause is certainly restrictive! It restricts exactly what ideals we're talking about. Look at the examples of restrictive and nonrestrictive clauses at that link. "Poe, who wrote The Raven.." non-restriictive - we're talking about the same Poe regardless of the second clause. "In 1987, when..." non-restricive - the meaning of "1987" does not requyire the second clause... – David C. Ullrich Aug 5 '16 at 20:38
• @BrianM.Scott "Students who do their homework" restricitve - specifies which students we're talking about. Ideals blah blah such that $I\subset J$: restritive. Specfies which ideals. – David C. Ullrich Aug 5 '16 at 20:38
• My apologies: I remembered which one was last and forgot that it was the sixth. You are of course right about the fifth one, and we’ve been talking about two different sentences all along. – Brian M. Scott Aug 5 '16 at 20:41