So far we have been theorizing in a fairly abstract and informal way about the distinction between vague and precise propositions.
In chapter 4, I argued that there was an important propositional distinction to be studied, and, in chapters 6-10, I formulated some distinctive theses governing the propositional notion of precision and vagueness.
However, I have yet to give anything like a formal and rigorous characterization of this distinction.The formalism of choice for most linguistic theorists is the theory of supervaluations: a formalism with many applications, but which is most saliently associated with a certain semantical apparatus for dealing with vague languages.[157] Supervaluationist semantics is an extremely influential way to make precise the idea that vagueness consists in semantic indecision. A precisification of a vague language, accordingtothisframework,isaway ofmaking each word ofthatlanguage completely precise. A precisification not only tells us what the extension of the predicate ‘bald’ is, and where the cutoff point is in a given sorites sequence, it also tells us what the extension would have been if those people had had different amounts of hair. They are things which, when given a possible world as input, tells us how to assign cutoff points (i.e. complete extensions) to each predicate (and words of other categories) at that world.
Since the rise of supervaluational semantics, however, it has become clear that it is not only theorists who identify vagueness with semantic indecision that can make use of the formalism of precisifications. Epistemicists will appeal to the notion of an interpretation of a language which is not knowable (for distinctive reasons) by the speakers to be incorrect (see Williamson [156]). Inconsistency theorists will talk of ‘acceptable assignments of semantic values’: precise interpretations of the language that come ‘maximally close in satisfying the meaning-constitutive principles for the expressions involved’ (Eklund [40]).
The formalism, in one guise or another, is completely ubiquitous amongst classical approaches to vagueness. Given an appropriate interpretation of‘admissible’ and ‘precisification’, these theorists can all accept the structural claim that a proposition or sentence is borderline iff it is true relative to some but not all admissible precisifications. Many of the things I say about supervaluationism generalize to other approaches that accept this formalism.Although the theory of supervaluations is most commonly associated with linguistic theories of vagueness, it is natural to wonder if it might also be applied to a non-linguistic theory of vagueness. Barnes and Williams [9], for example, develop a metaphysical account of vagueness in which the supervaluationist’s precisifications of a language are replaced by ‘precisifications of the world’.
Supervaluational semantics can indeed be adapted to the kind of non-linguistic theory I have been developing. In addition to the comparatively fine-grained theory of propositions andpropertieswehavebeenworkingwith (which arenot individuated by necessary equivalence), we could also accept the more coarse-grained ideology of possible worlds. A precisification may then be identified with a way of precisifying vague properties and propositions in much the same way that a linguistic theorist would think of one as precisifying predicates and sentences. A precisification in this setting would be a function telling us what the truth value of each vague proposition is at each possible world, telling us what the extension of each vague property is at each world, and so on.
Despite the formal availability of a supervaluationist semantics, I have come to realize that some substantive assumptions are hard-wired into this formalism that do not fit particularly well with the theory of vagueness developed here, and indeed, assumptions that may bring into question its suitability in the modelling of vagueness more generally.
In the following chapters, I’ll elaborate on why the supervaluational semantics is not suitable and explore the ways in which some standard assumptions about vagueness, often guided by the supervaluationist picture, need to be revised.style='font-size:9.0pt;line-height:122%'>I’ll start with the most basic problem for a supervaluational treatment of this theory: while supervaluational semantics is tailor-made to provide analyses of the ‘determinacy’ and ‘borderlineness’ operators, it does not directly give us a characterization of propositional vagueness and precision. In this chapter, I will argue that propositional vagueness and precision are not definable in terms of propositional borderlineness or determinacy. Thus, any theory that takes the latter as primitive (as the supervaluationist does) cannot provide a complete theory of vagueness. By contrast, at the end of the chapter, we will show that one can define propositional borderlineness and determinacy from propositional vagueness and precision.
This fact is particularly relevant to the present theory of vagueness. Recall that the core principles of this theory—Plenitude, Rational Supervenience, and Indifference—are all stated in terms of propositional vagueness and precision. Strikingly, the ubiquitous ‘borderlineness’ and ‘determinacy’ operators are completely absent from these principles; by all accounts they seem to be less theoretically central to the approach. This marks the first important departure from orthodoxy: according to my theory, the notions of precision and vagueness must be taken as primitive.
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