We observed in section 4.4 that an adequate linguistic account of vagueness must provide more than an account of sentential vagueness.
One can also have vague predicates, like ‘is bald, and further machinery beyond the notion of sentential vagueness is needed to analyse predicative vagueness.
Indeed, vagueness can occur at almost every grammatical category: the quantifier ‘some bald person', the operator ‘it is known that', the determiner ‘most', the name ‘Mt Kilimanjaro' (and so on) are all vague—it is hard to think of a grammatical category that does not contain vague expressions. It is natural, then, to wonder what analogous kinds of vagueness arise on a non-linguistic theory of vagueness. Of course, in the non-linguistic setting we are not concerned with linguistic items belonging to certain grammatical categories, but with the sorts of things the linguistic items in those grammatical categories express. Reasoning about these sorts of things receives a more precise treatment in the setting of a typed higher-order logic: a theory that allows one to quantify into positions other than the position of a singular term (see, for example, Williamson [165]).Central to our discussion of vagueness at other types will be the status of the non- linguistic analogue of vague names: vague objects. Indeed, we shall see that extending the theory of vagueness to objects, in addition to propositions, will be the key to understanding vagueness elsewhere in the type-theoretic hierarchy. The first question we must get clear on is therefore: what is a vague object? After outlining the general framework in section 16.1, I discuss some of the dominant approaches to this question in section 16.2 and 16.3, and in sections 16.4, I develop my own test for being a vague object.
This chapter offers less by way of substantive conclusions than the preceding chapters. One particular question—one might think the central question concerning vague objects—is left open: whether there are any vague objects. The non-linguistic theory of vagueness defended in this book provides a friendly environment for the discussion of vague objects whereas, by contrast, a linguistic theory of vagueness does not. However, this theory of vagueness does not straightforwardly commit us to the existence of vague objects either.
16.1