In this paper, I aim to set out a new theory of vagueness, one that is intended as an alternative to the theories based on supervaluations, or various non-classical logics, or on non-semantic conceptions of vagueness. The theory is based around the concept truer than, a concept that I think is already in common, if implicit, use in a number of other theories. The very brief version of my position is that I abandon classical semantics, that is, the traditional truth tabular semantics for propositional logic and Tarskian semantics for first-order logic, but not classical logic. Traditionally, many theorists have held that vagueness shows we must abandon the bivalent truth tables. The literature on vagueness is replete with descriptions of logics based on truth tables with three values, or continuum-many values, or even more. I think these theorists are fundamentally on the right track – vagueness does show that we must accept intermediate truth values, and that seems to mean we must abandon bivalence. But, this does not mean we must abandon classical logic, for there are non-bivalent semantics for classical logic, and as we shall see vagueness not only poses no threat to their appropriateness, and many even provide some positive reason for adopting them.