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?