The problem of vagueness is motivated by consideration of a sorites series, and the supervaluationist solution is presented. It is argued that supervaluationism can be regarded as an application of the model-theoretic conception of consequence and should therefore be accepted by all. The limitations of this conception are then explored. In particular, if the theory is to provide a constitutive account of vagueness, an account of vagueness as underspecification is required. University of London Jacobsen Prize Essay 2008