What Is the Function of the Determinative in the Construction 'They were all visiting their families' according to CGEL?

Question

In Cambridge Grammar of the English Language (CGEL), Huddleston and Pullum talk about 'universal determinatives' (pp.374–378), 'both' and 'all.' They also talk about the 'distributive determiner' 'each' (pp.378–379).

On page 377, under the sub-heading 'Other functions,' they give examples of 'all' being used as a modifier and specify that it can modify PPs, AdjPs, AdvPs, and NPs. Here are some examples:

She did it [all by herself].

I'm [all wet].

Having flipped through pages that I think are relevant, I can't seem to find any mention of the following construction, which is strikingly similar:

They were all/both/each visiting their families.'

As this appears to fall within the VP, I would imagine that if there is no reference in the above-mentioned sections, discussion of these constructions should be located under Chapter 8 ('The clause: adjuncts'). However, I cannot find anything.

For those familiar with CGEL, could you describe what this construction is called? I assume that it is a type of predicative adjunct in the clause structure—perhaps depictive.

Answer 1

9.2 Fusion of determiner and head (p 412)

Partitives (p 413)

    (c) Quantificational adjuncts (p 413)

Subsection (c) touches on the subject:

One use of fused-head partitives that merits separate mention here is that where they function as quantificational adjunct in clause structure:

[13] i Her parents both felt she had been exploited.
ii They had none of them intended to cause so much ill will.

The quantificational adjuncts serve to quantify the subject. Example [i] is thus equivalent to Both her parents felt she had been exploited and [ii] to None of them had intended to cause so much ill will. Universal all, both, and each can occur as implicit partitives, while the set of determinatives found in explicit partitives is considerably larger: these three together with existential some, any, none, cardinal numerals, multal many and much, paucal few, a few, and several. The implicit partitives can occur in preverbal position, as in [i]; explicit partitives are somewhat questionable in this position, strongly preferring post-auxiliary position. Compare ?Her parents both of them felt she had been exploited and Her parents had both of them felt she had been exploited.