It is not true that any...

[Seiichi Myoga]

I'd like to prove that "no" in (1) is really a sentence modifier but not something that modifies the word "problem."

Do you agree that (2) is a paraphrase of (1)?

(1) No problem is so big that you can't find a solution to it.
(2) It is not [true / the case] that any problem is so big that you can't find a solution to it.

Thank you in advance
Seiichi MYOGA

[Barb_D]

No, it modifies "problem."

All problems, no matter how large, have solutions.
There are no problems, no matter how large, that lack a solution.
It is not the case that a problem exists that does not have a corresponding solution.

[MrPedantic]

It's an interesting approach. I would agree with Barb, though; if we accept "no" as a sentence modifier, in that example, we must also accept other qualifying items in the same position as sentence modifiers, e.g.

1. Some problems are so big that you can't find a solution to it.
2. Your problem is so big that you won't find a solution to it.
3. Mike's problem is so big that we are extremely unlikely to find a solution to it.

etc.

All the best,

MrP

[Jamie (K)]

Seiichi, you're on the wrong track. There is a restriction on double negation within the same clause, but there is no restriction on using negation in two independent clauses in the same sentence. If there were, "Nobody said he didn't want cake," would be ungrammatical, but, in fact, it is perfectly grammatical.

The problem with sentence (2) is that in its underlying structure it has a null subject position that is filled with an expletive subject.

(2a) [ø] is not true that any problem is so big that you can't find a solution to it.

The portion of the sentence after "true" is the logical subject of the sentence, and it can even be put into that position:

(2b) [That any problem is so big that you can't find a solution to it] is not true.

It's a very awkward sentence, but it's grammatically correct.

The "no" in your sentence (1) isn't a sentence modifier, but a noun phrase modifier, and it is actually the "any" in your second sentence. It is not the "not" in your sentence (2).

The reason that positive and negative polarity applies across clause boundaries in sentence (2) is that the second and third clause are the logical subject of the sentence, and they are therefore subject to the polarity rules as if they were in the same clause with "not".

[Seiichi Myoga]

Dear Barb_D, MrPedantic and Jamie (K)

I thank each and every one of you for their help and detailed comments.

Seiichi MYOGA