Indeterminacy and Non-classical Logic

This paper is a response to a challenge set by Timothy Williamson in his Contribution to this volume where he argues that, even leaving aside the difficulties non-classical logicians have if they wish to retain classical pure mathematics, these difficulties become well-nigh insuperable in the case of applied mathematics. I reply, on behalf of my own heterodox non-transitive logic. One novel aspect of the reply is letting assignments take occurrences of expressions as their domain, not expression types. On this basis, after presenting reasons for rejecting Williamson’s epistemicism, I sketch a semantics for vague applied arithmetic including second-order Fregean arithmetic with Hume’s Principle. I then argue that vagueness in at least some scientific contexts can be accommodated by treating it as a phenomenon which occurs ‘inside’ measurement uncertainty and that therefore the application of classical logic in these cases is legitimate, even if one espouses a non-classical theory of vagueness. Finally I argue that the non-classical account is to be preferred to the classical in cases such as the Sorites, particularly the special case of the Slippery Slope Sorites which does crop up in the social sciences.