Vagueness: where degree-based approaches are useful, and where we can do without

Abstract

The vagueness of a property becomes apparent when more than one level of granularity is addressed in a discourse. For instance, the adjective "tall" can be used to distinguish just two kinds of persons--those who are tall as opposed to those who are not. This coarse distinction contrasts with any finer one, in particular with the finest possible one, on which we specify sizes, beyond all limits of precision, by real numbers. We understand vagueness as a relative notion, causing a problem when information on a coarse level is to be transferred to a fine level. Under this viewpoint there is no problem to accept that, depending on the application, reasoning under vagueness may require different formal frameworks. In this paper, we consider the same kind of vague properties in two different contexts. We first discuss a version of the sorites paradox. In this case, it is necessary to combine reasoning on two levels of granularity in a single formalism. Following established practice, we choose an approach based on numerical degrees. Second, we consider generalised Aristotelian syllogisms. In this case, degree-based solutions are unnecessarily rich in structure and are not found adequate. We give preference to a formalism that stays entirely on the coarse level of argumentation. We conclude that there is no reason to call for a uniform formalism to cope with the problem of vagueness. In particular, degree-based approaches are usually applicable, but there can be simpler alternatives. Different problems may call for different solutions, and choosing diversity does not mean that we approach the problem of vagueness incoherently.