Can vagueness cut out at any order?

Australasian Journal of Philosophy 86 (3):499 – 508 (2008)
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.
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