The Logical Structure of English Quantifiers

We characterize semantically quantified subjects, type (et,t), in English and show that the Boolean closure of the generalized existential and universal quantifiers is exactly the conservative ones. We prove that all subjects are expressible as Boolean functions of Montagovian individuals and that all mathematically extend to objects, type (eet,et). But quantified objects also include many functions that are not subject extensions, contrary to usual textbook assumptions. This is because two-place predicates (P2s) have more structure than one-place ones (P1s), so quantified objects have more to vary with/depend on. For example, we illustrate how lexical P2s in English can force their models to be infinite; P1s provably cannot.