The Ockham Society (Friday - Week 2, HT20)

According to supervaluationists, a sentence in a vague language is true iff true on all reasonable ways of making it precise, false iff false on all reasonable ways of making it precise, and neither true nor false otherwise. Supervaluationist treatments of vagueness face difficulties in accounting for higher-order vagueness. In particular, an argument from Graff Fara (‘Gap principles, penumbral consequence, and infinitely higher-order vagueness’, 2008) shows that the standard supervaluationist semantics is inconsistent with the possibility of higher-order vagueness. Cobreros’s region-valuationism (‘Supervaluationism and Fara’s Argument Concerning Higher- Order Vagueness’, 2011) adjusts the supervaluationist semantics to avoid this problem. This approach stresses the similarities between the standard supervaluationist semantics and the possible-worlds semantics for the modal logic S5, and avoids Graff Fara’s paradox by restricting the accessibility relation to reflect the fact that whether or not a particular precisification of the language is reasonable is itself a vague matter. However, as Graff Fara argues (‘Truth in a region’, 2011), region-valuationism leaves us without a defensible notion of truth in the intended model.

In this presentation, I outline an alternative approach. Second-order vagueness, reflected in the vagueness of the metalinguistic predicate ‘x is a reasonable precisification’, ought to be captured by providing many different candidate accessibility relations and supervaluating over the resulting models. We might call the resulting model a supersupervalutation. This process may be iterated to account for third- and higher-order vagueness.

Keefe (Theories of Vagueness, 2000) suggests that higher-order vagueness ought to be accounted for by ascending up a hierarchy of metalanguages. My approach differs from hers in that we continue to discuss a single object language, by using a single metalanguage. Rather than ascending up a hierarchy of languages, higher-order vagueness is captured by ascending up the levels of a single hierarchical supermodel for the object language. The ‘definitely’ and ‘borderline’ operator are understood as devices for ascending up the levels of this hierarchy.

I discuss some concerns with this approach; in particular, I ask whether this semantics can furnish us with satisfactory notions of truth and validity. I tentatively suggest that they can. I suggest that my semantics can explain our ordinary conceptions of truth and validity as norms that admit of degree. However, we can also define notions of absolute truth and validity, should we so wish.