Logical Features

What are the main logical differences between this approach to propositional vague­ness and its rivals? As may be expected, the central notions of this sort of theory are most aptly captured using operator locutions (as in Fine [56], for example) as opposed to metalinguistic predicates of sentences (as in, e.g.

Other differences, however, require more emphasis. Throughout this book we draw a sharp distinction between being borderline and being vague. The distinction is best dramatized by examples. Suppose that we have a sorites sequence consisting of individuals of gradually increasing heights. Every proposition stating that one of these individuals is tall is vague, for they all ascribe the vague property of being tall. On the other hand, not all of these propositions are borderline—only the propositions ascribing tallness to individuals towards the middle of the sequence are borderline, the rest are determinate!/ true or determinate!/ false. On the intended understanding of vagueness, a sufficient condition for a proposition to be vague (but not an analysis) is for it to be possibly borderline, and each individual is possibly borderline tall even if they’re in fact not borderline tall.

The majority of work on the logic of vagueness has focused on the notion of being borderline, and the cognate notion of being determinate (where p is determinate if and only if p and it’s not borderline whether p). Indeed, the main formalisms for dealing with vagueness—such as the theory of precisifications associated with supervaluationism (see chapter 12)—are all geared towards providing analyses for determinacy and borderlineness operators.

The usual strategy is to try and reduce vagueness to borderlineness by means of some kind of modal notion—perhaps a proposition is vague if it could have been borderline. Although I think there are general problems with this reduction, we will see in chapter 12 that the theory of propositional vagueness I outline above requires a primitive notion of precision and vagueness which cannot be defined in terms of propositional borderlineness and metaphysical modality. On this sort of view vagueness must instead be taken as primitive, and borderlineness must be defined in terms of it.

This raises a wider question about the suitability of the usual formalism—the theory of precisifications—for thinking about vagueness in connection with this view. With­out the modal reduction of vagueness to borderlineness this formalism has nothing to say about the vague/precise distinction. To model these notions a new formalism is needed, and in chapter 13, I argue that the theory of symmetries advertised in section 3.3 fits the bill. The resulting theory does away with the traditional formalism for dealing with vagueness involving possible worlds and precisifications, and uses an alternative formalism formulated in terms of a set of propositions and a certain group of automorphisms, the style='font-style:italic'>symmetries.

As noted, on this theory the central notions are formalized using operators, by analogy with the operators used to formalize metaphysical modalities. Given the formal similarity, many natural questions about the interaction ofvagueness-theoretic operators and modal operators arise. One such question is the status of a certain supervenience principle:

Supervenience.

Here the notion of supervenience is to be explicated in terms of metaphysical modality. This principle is formulated precisely and adopted in chapter 15.

One consequence of Supervenience for our theory of propositional vagueness is that the theory of propositions must be a hyperintensional one: there must be necessarily equivalent but distinct propositions (indeed, given Boolean Precision, Supervenience guarantees that every vague proposition is necessarily equivalent to a precise one). A broadly supervaluationist take on vague propositions might identify them with certain sets of ordered pairs consisting of worlds and precisifications. It will be seen that this sort of theory of propositions secures the supervenience of the vague on the precise in a particularly simple way. It is also clearly a hyperintensional theory, since propositions are more fine-grained than sets of worlds.

However, there are a number of substantive assumptions hard-wired into this broadly supervaluationist account of vague propositions, some of which are highly contentious and ought to be brought into the open. One of these is the commitment to certain principles concerning the interaction of the modal and determinacy operators. For example:

It's determinately necessary that A if and only if it's necessarily determinate that A.

It’s possibly determinate that A only if it’s determinately possible that A.

In chapter 15, I show how these principles fall out of a supervaluationist semantics for determinacy and modality, and present some difficulties for these consequences. Another issue is that, given a certain resolute attitude towards the use of possible worlds in supervaluationist semantics, the theory is subject to paradoxes of higher- order vagueness (see chapter 7).

Although my main objective in this book is to spell out a theory of propositional vagueness, the more general perspective of this project is that vagueness is a property of non-linguistic entities; propositions being merely the non-linguistic analogue of a sentence. But there are also non-linguistic analogues of other linguistic items: objects correspond to names, properties to predicates, and so on. Given that there can be vagueness in names, predicates, and so on, it is worth wondering how to generalize a theory of propositional vagueness to other types. I explore this topic more systematically, in the context of type theory, in chapter 16.

3.3