forthcoming in José L. Falguera and Concha Martínez-Vidal (eds.), Abstract Objects: For and Against, Cham: Springer (Synthese Library).
The main features of typed object theory are: (1) it is formalized in a relational type theory in which relations are primitive and are not reconstructed as functions, (2) it contains a formal theory of relations, with identity conditions that allow for the hyperintensionality of relations, (3) the domain of each type t divides up into ordinary entities of type t and abstract entities of type t, and (4) the denotation and the sense of a natural language term of type t are assigned objects of the very same type, though the sense of a term of type t is always an abstract object of that type.
We show how typed object theory compares with other intensional type theories. For example, typed object theory doesn’t require a primitive type for truth-values, doesn’t require a primitive type for possible worlds, doesn’t collapse the types for individuals and propositions (as suggested by Partee, Liefke, and Liefke & Werning); and uses only one set of types for both its syntax and semantics (unlike Williamson’s logic. Finally, we analyze as number of natural language contexts without requiring the technique of ‘type-raising’.