Van Benthem's problem, exhaustivity, and distributivity
We discuss the problem of deriving upper-bounded meanings of few, fewer than, and related expressions, in treatments where they are taken to denote predicates of individuals. In such analyses, the determiner-like uses of these expressions are derived by existentially- closing their predicate denotations, but this is known to give rise to problems (van Benthem, 1986). We show that the needed upper bound can be derived by applying an exhaustification operator above existential closure. Crucially, this exhaustification operator is insensitive to the distributive properties of the predicates in the sentence, an assumption that we see as consistent with recent work supporting the blindness view of implicatures (Fox and Hackl, 2006; Magri, 2009). We also discuss some similarities and differences between our analysis and Buccola and Spector’s (2016) maximality-based approach.