Can Vagueness Cut Out at Any Order?

Abstract

Could a sentence be, say, 3rd order vague, but 4th order precise? In Williamson 1999 we find an argument that seems to show that this is impossible: every sentence is either 1st order precise, 2nd order precise, or infinitely vague. The argument for this claim is unpersuasive, however, and this paper explains why.

Notes

1 An interpretation of a language is ‘sharp’ iff under that interpretation, bivalence holds.

2 The supervaluationist may be able to give a response to Williamson similar to the response I recommend in this paper to the epistemicist. The supervaluationist may be able to reject Williamson's argument—as (I show) the epistemicist can—provided that the supervaluationist can coherently claim that ‘definitely’ is a vague term on her account.

3 Williamson introduces the concept of the ‘standpoint’ of a given point 1997: 262]. I take it that ‘from the standpoint of i’ means something like ‘if the interpretation at i were the correct interpretation of the language’.

4 By ‘the sorts of models we are interested in’, I mean ‘generated’ models. A generated model contains a point that can reach every other point through a series of accessibility relations. Williamson states that ‘formally we lose nothing by restricting our attention to generated models' [Williamson 1999: 131].

5 Each model corresponds to a fixed precise state of the world. A ‘class’ of models contains, for every possible precise state of the world, a model corresponding to that state. Williamson claims that ⊨α is true just in case α is true at every point in every model in the relevant class 1999: 129]. Following Williamson, however, we can restrict our attention to sentences that are necessarily true or necessarily false. For example, rather than considering sentences like, ‘Bruce Willis is bald’, we can focus on sentences like ‘any possible person with 5,383 hairs is bald’. If such a sentence is true at every point in a given model, it will be true at every point in every model in the relevant class. Thus, for our purposes, ⊨α is true just in case α is true at every point in a given model.

6 Williamson suggests that this would also be a natural assumption for the supervaluationist [Williamson 1999: 130]. In this paper I focus only on how a follower of Williamson's brand of epistemicism should understand the framework.

7 To see this, suppose that sentence α is nth order precise, and so that every sentence in C_n{α} is precise. Take some sentence β from C_n{α}. Given that β is precise, it follows that Δβ is precise. For (on the sort of models that we are interested in—see note 3), if β is precise then either β is true at every point (in which case Δβ is also true at every point), or β is false at every point (in which case Δβ is false at every point). Δβ is thus either true at every point or false at every point—and so precise. Thus for any sentence β in C_n{α}, the definitization of that sentence (i.e. Δβ) is precise. C_n+1{α} consists of definitizations of the sentences in C_n{α}, combined with truth-functors. As vague sentences cannot be created by combining precise sentences with truth-functors (Rule Z above [Williamson 1999: 132]), C_n+1{α} consists of precise sentences, and so α is n+1th order precise. In general, if a sentence is precise at the nth order, then it is precise at the n+1th order: a sentence cannot be lower-order precise and higher-order vague [Williamson 1999: 134].

8 This is where the condition that n≥1 becomes relevant. If n is 0 or less, then there will be no classification C_n{α}.

9 I would like to thank an anonymous reviewer for suggesting this way of defining the distance between two points.