How Vagueness Could Cut out at any order

Timothy Williamson (1999) presents, without endorsing, an argument that for any n > 1, any sentence that is nth order vague is (n + 1)th order vague. The setting for the argument is a standard Kripke-style model theory for a propositional language enriched with an operator ∆ for ‘Definitely’. A model is a triple 〈W,≺, [[]]〉, where W is a domain of “points”, ≺ is an “accessibility” relation on W, and [[]] a function from the sentences of the language to subsets of W, subject to the standard recursive rules: [[¬α]] = W \ [[α]]; [[α∨ γ]] = [[α]]∪ [[γ]]; [[∆α]] = {x : ∀y(x ≺ y⊃ y ∈ [[α]])}. A sentence α is precise at x ∈W iff x ∈ [[∆α∨∆¬α]]; otherwise, α is vague at x. α is precise in a model iff α is precise at every point in the model, otherwise α is vague in the model.1 Williamson proposes the following as the most natural definition of higher-order vagueness and precision. For any set X of sentences, CX is the closure of X under the truth-functional operators (¬ and ∨). CnX is defined inductively: C1X = CX Cn+1X = C{∆α : α ∈ CnX}