Thinking in terms of probabilities can give us a valuable lens on uses of vague language. In particular, it holds out the promise of bringing formally-tractable theories closer to empirical observations about how speakers actually use vague language. However, most existing applications of probability to vague language assume a classical approach to probability. This may be fine as a first approximation, but does not deal well with certain observed phenomena that have been used to motivate formal treatments of vague language based on nonclassical logics. Here I have in mind some phenomena around what have been called "borderline contradictions". In particular, speakers seem relatively happy to agree, of a borderline case of "tall" (for example), that they are both tall and not tall; but speakers are also resistant, in such cases, to agree that such a person is tall, or that they are not tall. These phenomena have been used to motivate three-valued non-probabilistic theories of vague language. In this talk, try to bring these approaches together, in a way that hopefully achieves some of the virtues of both probabilistic and nonclassical approaches. I give an outline of some of the reasons probabilistic approaches to vague language seem promising and enlightening. Then I turn to borderline contradictions, arguing that classical probabilities are not well-suited for understanding this phenomenon. Finally, I make steps towards a theory of nonclassical probabilities that (I hope) can achieve the goods of existing probabilistic theories of vagueness, while fitting with a plausible approach to borderline contradictions.
