allows for both a collective reading, where the children as a group ate a single pizza, and a distributive reading, where each of the children ate a pizza (presumably a different one!)  One way of dealing with such ambiguities in practice is to treat plural NPs as ambiguous between a collection-denoting reading and an “each member of the collection” reading. For example the children in (3.9) would be treated as ambiguous between the collection of children (which is the basic sense of the phrase) and each of the children. This entails that a reading of type each of the people  should also be available in (3.8) — but we can assume that this is ruled out  because (intransitive) gather requires a collective argument.  In a sentence such as