Tolerance and higher-order vagueness

The idea of higher-order vagueness is usually associated with conceptions of vagueness that focus on the existence of borderline cases. What sense can be made of it within a conception of vagueness that focuses on tolerance instead? A proposal is offered here. It involves understanding ‘definitely’ not as a sentence operator but as a predicate modifier, and more precisely as an intensifier, that is, an operator that shifts the predicate extension along a scale. This idea is combined with the author’s earlier approach to the semantics of vague expressions, which builds on the idea of a central gap associated with a predicate. The central gap approach is generalized to handle arbitrarily many iterations of ‘definitely’.