"There is to be no drinking beer today" What is the status of "no" and "beer" here?

1. There's no doubting her sincerity.
2. There's no telling what she's done.
3. There's no guessing which way they'll bolt.
4. There's to be no drinking beer today.
5. There's no telling her.

The word no is usually thought of as a determinative. We expect to see it in Determiner position in noun phrases. Of course, we often see determinatives in Determiner function with deverbal nouns:

• The reading of books is prohibited.
• This constituted a breaking of his silence on this topic.

Notice that because the -ing forms above are nouns and not verbs, they cannot take direct objects and the phrases after them are preposition phrases. If we omit the word of the sentences are ungrammatical:

• *The reading books is prohibited. (ungrammatical)
• *This constituted a breaking his silence on this topic. (ungrammatical)

My questions therefore are:

• What part of speech is the word no in examples 1-4?
• How should we regard the syntactic function/grammatical relations of the word no in relation to the phrases it occurs in?
• What is the syntactic function of the strings after the -ing forms. (direct object /complement?)
• What part of speech are these -ing forms?

Solution 1:

I'd say that 1-4 are hybrid constructions where "no" is a determinative functioning as a determiner in construction with a gerund-participial VP head.

In 1. and 4. the post head NPs are objects, but in in 2. and 3. the post-head dependents are not NPs but interrogative complement clauses where 2. means There’s no telling the answer to the question "what has she done”? and 3. means There’s no guessing the answer to the question "which way will they bolt"?

I called them 'hybrid' constructions because the pre-head dependents are characteristic of NP structure while the post-head ones are characteristic of VP structure. They are virtually restricted to the existential construction with “there”.

Solution 2:

These are two idioms involving There-Insertion. (1-3) are examples of one idiom, (4-5) of another.

• There's no Vcog-ing means 'No one can Vcog', where
  Vcog is a cognitive verb (doubt, tell, guess) that is the VP of a gerund complement clause.
  I.e, the NP following no in this idiom (which is an NPI, by the way) is parseable as

  • [np [s [vp Vcog-ing ... vp] s] np]
    which means that the no is an ordinary determiner of the complement NP in this idiom.

• The other idiom is

  • There's no Vvoling, which means 'Vvoling is impossible/not customary/not allowed';
    Vvol is a volitional (hence forbiddable) activity, requiring a sapient agent subject.
    With additional syntactic gymnastics like Neg-Raising the idiom can appear as either
  • There's not to be Vvoling
    or as
  • There's to be no Vvoling
    which both mean 'Vvoling is forbidden' only. As these last two examples demonstrate,
    the no in There's to be no drinking beer is also an NP determiner, and one which is coreferential with the adverbial not of There's not to be drinking beer.

Note that there is at least one modal and one negative involved in each of these constructions, plus obligatory There-Insertion, and idiomaticity all over the place. This produces odd-looking syntax.

As for the specific questions, no is an NP determiner, as noted, modifying the outer NP of the complement clause; the various V-ing are all gerunds, the VPs of gerund complement clauses. Since they're Vs, they can have objects and all the rest, and that's what beer is, for instance.