The (topo)logic of vagueness
Abstract
Zeno's dichotomy paradox of the runner and the sorites paradox exhibit certain interesting similarities. Both of them involve a long series of steps, each of which seems legitimate, but which, taken together, apparently lead to an unacceptable conclusion. In this article, a particular interpretation of a common reply to Zeno's paradox is presented, which recognises that to defuse the paradox, it is necessary to assert that the number of stages that the runner has completed on Zeno's in nite sequence of times is not an appropriate measure of whether he nishes the race or not. Applying this style of reply to the sorites argument, one would reject the argument on the grounds of the inappropriateness of the number of hairs for reasoning about baldness. Such an attitude to the sorites argument implies a certain conception of the problem posed by vague terms, according to which the problem is to understand such relationships between terms as the appropriateness of one for reasoning about the other. Consequently, it poses a certain set of challenges to prospective theories of vagueness.