Mixed-polarity pluralities: A solution to van Benthem’s problem

In this paper, I suggest a new way of resolving a number of problems with an orthodox treatment of modified numerals as existential quantifiers over pluralities. It’s well known that existential quantification renders upper bounds inert—an issue known as ‘van Benthem’s problem’. This can be addressed by building a notion of maximality into the semantics of modified numerals, but this fails to capture attested cumulative readings due to how maximality takes scope with respect to other operators. Existing work addressing this compositional stalemate exploits powerful mechanisms for side-stepping scopal interactions, such as post-suppositions (e.g., Brasoveanu 2013). Here, I maintain a simple treatment of modified numerals as existential quantifiers over pluralities, while significantly enriching the ontology of pluralities themselves. Concretely, I explore the idea that pluralities can encode both positive and negative information, building on recent work by Bledin (2024).