Higher order vagueness and rough sets

Abstract

It is well-known the naive approach to consider the main flaw to vague terms is that there are no sharp boundaries between positive and negative extensions, i.e., some borderline cases exist if the predicates are vague. However, Z. Pawlak has proposed a prominent approach to vagueness based on rough set theory but it seemed to be implausible due to the boundary region as the theory constructed must be precise (or “crisp”) but not vague in some sense. On the basis of Pawlak's creative idea and efforts, A. Skowron and R. Swiniarski proceeded to examine some further problems, one of them is the problem of higher order vagueness which is exactly to say that there is not only no sharp boundary between positive and negative extensions but also no sharp boundaries between positive extension and borderline cases or between borderline cases and negative extension, etc. And the main idea of the theory was within adaptive learning framework. Contrast to their approach, I aim to provide some supplied principles in this paper to show that the problem of higher order vagueness ipso facto can be assimilated by the revised rough set theory from a philosophical point of view.