2010. Modified 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.