Vagueness Throughout the Type Hierarchy
Let us begin by investigating the nature of vagueness in other semantic categories. The natural background setting in which to frame such a discussion is type theory.
In what follows we shall assume two basic types: e, representing the type of individuals—the semantic values of singular terms—and type t representing the type of propositions— the semantic values of sentences.1 Complex types can be constructed from these in a recursive fashion: if σ and τ are types then so is σ → τ.In the usual set theoretic models for type theory, the domain of the latter sort of type consists of the set of all functions from the domain σ to the domain of τ, and I shall often refer to these things as functions for convenience. However, these models should not be taken too seriously: strictly speaking, sets, and thus functions, are individuals, and in the intended model they will all belong to the basic type e. Quantification at higher types cannot be straightforwardly thought about in terms of quantification over entities at all, set theoretic or otherwise. (Some of these issues are fraught; for a defence of the intelligibility of this sort of quantification, see Prior [112] and Williamson [161]).In this setting, the semantic values of a variety of expressions can be represented quite easily. A predicate—‘is red' for example—determines a mapping from individuals to propositions, in this case the function that maps x to the proposition that x is red. So, predicates belong to the type e → t.[180] [181] An operator expresses a function from propositions to propositions, t → t,a quantifier takes a predicate to a sentence and so expresses a function in (e → t) → t,a determiner takes a predicate to a quantifier and so expresses things of type (e → t) → ((e → t) → t), and so on.
The logical expressions also fit into this hierarchy: conjunction (written ∧) of type t → (t → t), negation (written —) of type t → t and, for each type σ, a universal quantifier (written ∀σ) over functions of that type, which has the type (σ → t) → t.I began by noting that each of these grammatical types are occupied by vague expressions. On the sort of theory I am espousing, this vagueness results from vagueness in the semantic values of these expressions—of the functions belonging to the types t → t, (e → t) → t, and so on. However, although I have given a treatment of vagueness in type t (i.e. an account of vague propositions) I have said very little to connect this with vagueness at other types.
The standard line is to define a predicate F as vague if it has a borderline case: ∃xV Fx. Clearly, such a definition at best captures an extensional notion of vagueness for predicates.[182] But even setting that aside, it is subject to an objection, under the assumption that there are vague objects. The property of being exactly 19,341 feet is intuitively precise, but assuming Mt Kilimanjaro is a vague object with an indeterminate height, it has a borderline case; i.e. there is some x (Mt Kilimanjaro) such that it is borderline whether x is exactly 19,341 feet tall.
A natural way to fix this is by taking the notion of precision for both of the basic types as primitive. That is, we will begin by taking for granted both the notion of a precise proposition and a precise object. With the notion of precision at type e and type t given, we can extend the definition up the type hierarchy recursively.[183] [184] Suppose that we have already defined precision at types σ and τ:
Precision at Higher Types: An entity F of type σ → τ is precise if and only if F maps every precise thing of type σ to a precise thing of type τ.5
The informal statement hides the fact that, to state, it we need to employ higher-order quantification.
To spell out the idea in a bit more precision, suppose that we have introduced higher-order predicates flσ and Qτ of type σ → t and τ → t respectively. Then flσ → τ := λX(Vy(βσy → QτXy). Let me emphasize that here we are not theorizing with the extensional notion of precision. This has an important conceptual advantage: although the extensional notion of precision for propositions—determinacy—seems clear enough (and can be defined in terms of precision), it's not entirely clear what the extensional analogue of being a precise object is. Indeed, the way we defined determinacy from precision in the propositional case relied on features that were distinctive of propositions: that they had a logical ordering given by entailment (recall that we defined a proposition to be determinate if it is entailed by a precise truth). There is no straightforward analogue of entailment in the objectual case, and thus no obvious analogue to our notion of propositional extensional determinacy for objects.A compelling principle governing linguistic expressions states that a complex expression is vague only if one of its constituents is vague or, conversely, that if the constituents are precise, so is the composite. Our definition of precision at higher types entails a similar principle in the non-linguistic setting: combining precise things can only deliver precise things. However, this definition makes sense even given a relatively coarse-grained theory of content. The notion of a ‘constituent’ is absent from the definition—it is phrased in terms of functional application—and so we do not need to assume that propositions have constituents in the same way that sentences do.
In summary, to generalize the propositional notion of vagueness up the functional type hierarchy we need to take the notion of a vague object and vague proposition as primitive. Although we have paid considerable attention to the latter notion, we have said relatively little about the former. We will now turn to some popular accounts of the distinction.
16.2