Vagueness at every order: the prospects of denying B

A number of arguments purport to show that vague properties determine sharp boundaries at higher orders. That is, although we may countenance vagueness concerning the location of boundaries for vague predicates, every predicate can instead be associated with precise knowable cut-off points deriving from precision in their higher order boundaries. I argue that this conclusion is indeed paradoxical, and identify the assumption responsible for the paradox as the Brouwerian principle B for vagueness: that if p then it's determinate that it's not determinate that not p. Other paradoxes which do not appear to turn on B turn instead on some subtle issues concerning the relation between assertion, belief and higher order vagueness. In this paper a theory of assertion, knowledge and logic is outlined which allows one to avoid any kind of higher order precision. A class of realistic models containing counterexamples to B and a number of weakenings of B are introduced and its logic is shown to support vagueness at every order.
Keywords Higher order vagueness  vagueness  Brouwers principle
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