Abstract Several authors proposed to devise logical structures for Natural Language (NL) semantics in which noun phrases yield referential terms rather than standard Generalized Quantifiers. In this view, two main problems arise: the need to refer to the maximal sets of entities involved in the predications and the need to cope with Independent Set (IS) readings, where two or more sets of entities are introduced in parallel. The article illustrates these problems and their consequences, then presents an extension of the proposal made in Sher (J Philos Logic 26:1–43, 1997) in order to properly represent the meaning of IS readings involving NL quantifiers. The solution proposed here allows to uniformly deal with both standard linear and IS readings, regardless of their actual existence in NL or quantifiers’ monotonicity. Sentences featuring nested quantifications are particularly problematic. By avoiding parallel nested quantification in the formulae, the proper true values are achieved.
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