During my PhD thesis (but don't waste your time reading it :-) If you really want to know more, read this), I worked on representing the meaning of NL quantifiers in logics, specifically in underspecified logics. I propose a new logical framework, called Dependency Tree Semantics to handle quantifier scope ambiguities via Skolem-like dependencies built on dependency trees. These structures may be then algorithmically translated into formulae in second order logic involving Generalized Quantifiers.
The interesting part of the thesis are the second order logic formulae obtained at the end of the process. Those are flat formulae able to represent (what I termed as) Independent Set readings, i.e. cumulative, collective, and cover readings, which appeared to be highly problematic in any other frameworks for quantifier scope ambiguities proposed in the literature at that time.
In the literature, Independent Set readings are usually known as Scopeless readings, but I prefer to avoid this term because it recalls the concept of "scope" (of quantifiers, on subformulae). And, in my view, embedding subformulae within other quantifiers prevents the proper representation of these readings. On the other hand, the second order logic formulae I proposed are flat, in the sense that quantifiers are never embedded in the scope of other quantifiers. For this reason, I prefer to use the term "Independent Set reading", in order to emphasize the fact that the desired interpretations are conceptually distinct from the formal instrument used to represent them. Possible dependencies among quantifiers, needed to represent standard distributive readings, are still allowed, and they are modeled in terms of Skolem-like functional dependencies.
A similar proposal in Natural Language Semantics is the logical formalism proposed by Jerry R. Hobbs, whose insights are summarized here. However, Jerry R. Hobbs focuses on predications, while mostly neglects quantifiers. The two formalisms are perfectly integrable.
The second order formulae are presented in some journal papers I authored after my thesis. [Robaldo, 2010a], [Robaldo, 2010a], [Robaldo, 2011] present the main insights (read them in this order :-)). In [Robaldo and Di Carlo, 2014] I proposed an expressively complete version of Dependency Tree Semantics allowing flexible disambiguation, while in [Robaldo, 2013], I argue that the well-known property of Conservativity, must be a necessary property to assert in the formulae, in order to properly handle the truth values of IS readings.
The last journal paper I published on the topic is [Robaldo, Szymanik, and Meijering, 2014], co-authored with Jakub Szymanik and Ben Meijering, which reports the results of an online questionnaire showing that the interpretation of quantifiers appears to have a strong cognitive dependence on pragmatic factors, and that the second order formulae proposed in [Robaldo, 2011] are able to formally take into account these factors.
After [Robaldo et al, 2014], I stopped working on NL quantifiers, and I really think I will not work on that again as now I am focused on Computational Law. On the other hand, Jakub was later awarded by an ERC Starting Grant on related topics (developing cognitive semantics of generalized quantifiers). Congrats Jakub!