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How should I use nouns like proof or evidence in a sentence that requires the noun to be negated when the noun is countable/uncountable? How do I know whether or not to put an article and when to use no or not?

'y is not proof that x exists'

'y is not a proof that x exists'

'y is no proof that x exists'

I found this on another website:

When a noun has an ungradable meaning (it is either something or it is not) we cannot use no + noun:

A potato is not a fruit.

Not: A potato is no fruit.

When a noun has a gradable meaning, no + noun means the same as not a/an + noun:

[a football manager talking about signing a new player]

It’s no secret that we are interested. (= It’s not a secret. A secret is gradable. Something can be more of a secret than something else.)

Now, according to this, it should be ok to say no proof because you can also say not a proof. Am I misinterpreting something here?

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You can safely ignore the "rule" about gradability, which is simply not true. It is, for instance, "violated" by one of the most devastating lines in recent US political history, from the 1988 vice-presidential debate, and still being paraphrased this year. When Dan Quayle compared his experience and readiness to assume the duties of the presidency to that of John Kennedy in 1961, Lloyd Bentsen responded:

Senator, I served with Jack Kennedy. I knew Jack Kennedy. Jack Kennedy was a friend of mine. Senator, you're no Jack Kennedy.

It is difficult to imagine any sense in which Jack Kennedy is "gradable".

The Cambridge Grammar of the English Language (Huddleston and Pullum, 2002), p. 390, gives a better explanation of the difference between no and not:

One place where the no version is semantically distinct is in ascriptive predicative complements. Compare:

[55] a. He isn't a doctor. b. He's no doctor.

Version [a] simply says that he isn't a member of the class of doctors, while [b] says that he doesn't have the properties of a doctor. Similarly, He's no friend of mine implies that I know him and that his behaviour to me is not what one would expect of a friend, while He's not a friend of mine says only that he doesn't belong to the class of friends of mine—it could be that I hardly know him, or indeed that I don't know him at all.

So if you say that Y is not a proof that X exists, you assert only that Y fails to prove the existence of X; but if you say that Y is no proof that X exists, you assert that Y is wholly lacking in the formal qualities required of a proof—it bears no similarity to a proof.

  • When a reference with a name as redolent of authority as The Cambridge Dictionary (complete with coat of arms gules, on a cross ermine between four lions passant guardant) pronounces that "we cannot use no + noun" in these instances (not "we usually don't", or "it is often best to refrain from") how to stop such foolishness from being taken as a ukase? – P. E. Dant Nov 5 '16 at 2:40
  • @StoneyB Thank you so much for the insight! That helps a lot! – Chris Nov 5 '16 at 3:13
  • @P. E. Dant This is exactly what I've been saying (just in a different way, not as eloquent hehe) all the time. There are so many misleading 'rules' on some of the most highly regarded English websites/dictionaries, it's really frustrating at times. – Chris Nov 5 '16 at 3:18
  • @P.E.Dant This is pretty much inevitable in a field that's been evolving with unusual rapidity for sixty years now. Syntax scholar A puts forward a theory to explain Phenomenon X, and it makes just enough sense to people outside the immediate field (lexicographers and teachers, for instance) that it gets picked up and generalized--even though scholars B and C are already jumping in and saying "But wait, what about Phenomena Y and Z--it doesn't work there". – StoneyB Nov 5 '16 at 14:09
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There's a famous scene toward the end of the movie Return of the King where the evil Ringwraith King approaches a warrior and says, "Fool! No man may kill me!" After which (with some help from the hobbit Merry) the warrior pulls off her helmet revealing she is the warrior maiden Eowyn, who declares, "I am no man!"

I mention this because the "A is no B" structure can have a kind of archaic or perhaps refined feeling to it. It's certainly not wrong -- not if Tolkien used it -- but it's kind of like quoting Shakespeare. You have to pick the right context and the right audience for it.

That being said your football manager saying something "is no secret" doesn't have this archaic or refined tone to it. But "A potato is no fruit" does. I suppose there's some subtlety in how you phrase it that makes it sound more or less dramatic.

Anyway, all three of your examples are fine; however "A is no/not proof" and "A is not a proof" require different ways to write "A". With the second, A should be something that is already a proof.

The existence of ever smaller subatomic particles is no/not proof that .. (matter is infinitely divisible? just guessing here)

Einstein's Theory of Relativity is not a proof of ... (curved space-time? the inscrutability of women? you get the idea)

If in doubt, use not -- but pay attention to where native speakers use no, and notice if there are patterns or idioms which you can use.

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    The canonical quotes are "Hinder me? Thou fool. No living man may hinder me!" and "But no living man am I! You look upon a woman." – Jasper Nov 4 '16 at 23:51
  • @Jasper thanks! It's been a while since I read the canon. – Andrew Nov 5 '16 at 0:06

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