Common Nouns as Types

When modern type theories are employed for formal semantics, common nouns (CNs) are interpreted as types, not as predicates. Although this brings about some technical advantages, it is worthwhile to ask: what is special about CNs that merits them to be interpreted as types? We discuss the observation made by Geach that, unlike other lexical categories, CNs have criteria of identity, a component of meaning that makes it legitimate to compare, count and quantify. This is closely related to the notion of set (type) in constructive mathematics, where a set (type) is not given solely by specifying its objects, but together with an equality between its objects, and explains and justifies to some extent why types are used to interpret CNs in modern type theories. It is shown that, in order to faithfully interpret modified CNs as Σ-types so that the associated criteria of identity can be captured correctly, it is important to assume proof irrelevance in type theory. We shall also briefly discuss a proposal to interpret mass noun phrases as types in a uniform approach to the semantics of CNs.