The Supervenience of the Vague on the Precise
Our investigation of the combined logic of vagueness and modality is not quite complete. According to a popular slogan, ‘the vague supervenes on the precise’.
For a linguistic account of vagueness, untangling this slogan is a tricky business. Supervenience is a modal notion relating one set of true propositions to another—the A facts supervene on the B facts iff, necessarily, every A proposition is necessitated by the truth of a B proposition. Yet according to linguistic theories of vagueness, the objects of vagueness and precision are not facts or propositions but representations.1312 The condition: (Rxy Λ Sxz) → ∃w(Syw Λ Rzw) is sometimes called the Church-Rosser property. (R î S)xy means that x is an R of an S of y.
13 See chapter 9 ofWilliamson [156] for a good discussion from the perspective of a linguistic theorist. There, things are set up in terms of the supervenience of the vaguely describable facts on the precisely describable facts. It is worth noting, however, that this sort of supervenience may be trivial if all vagueness
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