A. Brasoveanu 2010Modified Numerals as Post-suppositions

The paper provides a compositional account of cumulative readings with non-increasing modified numerals (aka van Benthem's puzzle), e.g., Exactly three boys saw exactly five movies. The main proposal is that modified numerals make two kinds of semantic contributions. Their asserted/at-issue contribution is a maximization operator that introduces the maximal set of entities that satisfies their restrictor and nuclear scope. The second contribution is a post-supposition, i.e., a cardinality constraint that needs to be satisfied relative to the context that results after the at-issue meaning is evaluated. We take contexts to be sets of variable assignments relative to which quantificational expressions are interpreted and which are updated as a result of their interpretation. The main difference between this account and Krifka (1999) is conceptual: we take modified numerals to constrain quantificational -- and not focus -- alternatives, where a quantificational alternative is one of the sets of assignments satisfying a quantificational expression.