# “at all the vertices”, what does this mean?

My professor has written a statement like this:

function is non-negative at all the vertices of the structure S and positive at some vertex

for a publication. It is a peer-reviewed publication so it should not have mistakes but I find at least two different meanings for this statement:

1. Does it mean "function is non-negative at all of the vertices in the structure S and positive at some vertex"?

2. Or does it mean "the function is not non-negative at all in the vertices of the structure S"?

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Older thread with some decent discussion and references. No time to follow-up with a references check, so here's the link. forum.wordreference.com/showthread.php?t=605984 –  dotsamuelswan Apr 14 '13 at 19:56
@dotsamuelswan it is not quite the same -- here I would use "of" to make sure the reader does not even by accident misunderstand the phrase at all. It could be understood in two different ways: "function is not non-negative" and "function is non-negative". This is the issue I would like to understand, how to make sure what my pro meant? –  hhh Apr 14 '13 at 20:06
It means #1. I don't think the of is necessary. For that matter, I don't think the the is required, either: function is non-negative at all vertices of the structure S.... –  J.R. Apr 14 '13 at 22:08
Quite simply, "non-negative at all" CANNOT mean "not negative at all." By using "non-negative" ("non" instead of "not," and hyphenating it) the writer unquestionably links the negation to the word "negative" AND NOT to anything else. If you have to rewrite the sentence so drastically to show the other meaning you THINK might be applicable, you probably are imagining things. (Your rewrite, possibility 2, adds an additional negative, changes a preposition, and drops an entire substantive phrase!! If the writer meant something THAT different, he probably would have written it that way.) –  John M. Landsberg Apr 15 '13 at 9:39

It means "[The] function is non-negative at all of the vertices in the structure S and positive at some vertex". The of isn't necessary though it may help to clarify the meaning.

There may be some confusion between the phrases not negative at all and non-negative at all. Some examples may help to clarify what I mean. Note that these sentences do not have the same meaning as your original sentence.

This sentence asserts that the function is never negative.

The function is not negative at all.

It could also be written as

The function is not at all negative.

The phrase at all serves to emphasize not.

However in your sentence the phrase at all refers to the set of vertices. It could be written as

The function is not negative at all the vertices ...

(Though, as jwpat points out, there are better ways to say this.)

If it were reordered it would not make sense. Thus at all is not serving to emphasize not.

*The function is not at all negative the vertices ...

This question discusses the difference between at all and not at all.

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I read “The function is not negative at all the vertices” as implying it is negative at most or many of them, and zero or positive at some. –  jwpat7 Apr 14 '13 at 21:11
@jwpat makes a good point. I was attempting to clarify the difference between not negative and non-negative. Hopefully the edited version helps. –  Wayne Johnston Apr 14 '13 at 21:30
I upvoted your answer as useful, but you could improve it by changing “The function is not negative at all the vertices” to eg “At all the vertices, the function is not negative” to get the meaning you say it has. Also, use a > instead of 4 spaces at beginnings of inset lines, and asterisks rather than backticks around words or phrases that should be italicized –  jwpat7 Apr 14 '13 at 21:48
Thanks jwpat for the suggestions. –  Wayne Johnston Apr 14 '13 at 23:17
Downvotes are OK. They tell you that you need to do better. As David Brin says "criticism is the only known antidote to error". I have a better answer now as a result. –  Wayne Johnston Apr 14 '13 at 23:19

[The] function is non-negative at all the vertices of the structure S and positive at some vertex...

That statement has a specific meaning, and it is a proper and precise way of stating that meaning:

The function is not negative at any vertex of S (that is, there is no vertex of S where the function is negative). In addition, there exists at least one vertex of S where the function is positive.

Note that zero is neither negative nor positive; to say that a function is non-negative means that its value is zero or positive. To say a function is positive means that it has a value greater than zero.

Most native English speakers will not notice any difference due to of being present or absent in the example sentence. If you are concerned about it, change all to every:

The function is non-negative at every vertex of the structure S and is positive at some vertex...

Edit: onomatomaniak commented,

I wonder, though, if “some vertex” means not at least one vertex, as you indicate, but rather [precisely] one vertex. If I wanted to indicate at least one, my inclination would be to write some vertices, not some vertex.

Because phrasing like “positive at one vertex of S and zero at all others” is the obvious way to express a single-vertex-positive condition, I think it would be perverse or misguided for someone to write “non-negative at all the vertices of S and positive at some vertex” to mean positive at one and only one vertex.

Anyhow, the usual intent of phrasing like “non-negative at all the vertices of the structure S and positive at some vertex” is to describe a function that never is negative, and by dint of going positive somewhere, is non-trivial.

Regarding “some vertices” vs “some vertex”: (1) In mathematical writing one desires to make premises only as strong as necessary for a proof to go through. If we take “some vertices” as implying or suggesting multiple vertices, and “some vertex” as one or more, then the latter condition is weaker, hence desired. (2) In a proof, “some vertex” is likely to be part of a phrase like “some vertex, say v...” – a construction where “some vertices” would not work.

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I am always worried about using "at all" because it has its special meanings such as "in the slightest degree or in any respect" -- look for example this: "function is non-negative at all in the structure S having the vertices X Y Z and positive at some vertex" or "function is not negative at all in the structure S having the vertices X Y Z and positive at some vertex"-- now this "at all" has totally different meaning. Is there any concern to mix this "at all" to its different meaning like the earlier examples illustrated? –  hhh Apr 14 '13 at 21:57
@hhh, I'm not concerned about it and would expect any and all native speakers to have no trouble if “at all” is correctly used. But neither example in your comment is properly formed, so both are problematic. –  jwpat7 Apr 14 '13 at 22:19
+1. I wonder, though, if "some vertex" means not at least one vertex, as you indicate, but rather [precisely] one vertex. If I wanted to indicate at least one, my inclination would be to write some vertices, not some vertex. Thoughts? Is this usage specific to mathematical writing? –  onomatomaniak Apr 15 '13 at 9:17
@onomatomaniak "some vertex" does not mean "precisely one vertex", it can be just one vertex but there can be more than one. If you want to stress the uniqueness, you add words such as unique, precisely one etc. –  hhh Apr 15 '13 at 13:47
@onomatomaniak, also see edit –  jwpat7 Apr 15 '13 at 14:19

"At all" has its special meaning only in a phrase with a Negative Polarity trigger such as no, none, never, or a yes-no question. Words with the prefix "non-", despite their negative meaning, are not grammatically Negative Polarity words and so the phrase at all cannot occur in this sentence and that particular parse is impossible.

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+1 thanks for clarification, easy to overlook different negations. –  hhh Apr 14 '13 at 23:46