An introduction to the theory of imprecise sets: the mathematics of partial presence

Abstract

In this article, we are going to discuss the mathematics of partial presence of an element in a set. A similar theory well known as the theory of fuzzy sets is already in existence since 1965. However, right at the start, the measure theoretic explanations of fuzziness have taken a wrong turn in the sense that from any given law of fuzziness defined on an interval, workers have since been trying to extract a law of probability giving rise thereby to all sorts of misinterpretations of probability theory. Such developments do not have any classical measure theoretic basis. In fact, not one as popularly believed, but two independent laws of randomness are necessary and sufficient to define a law of fuzziness. Secondly, the existing definition of complement of a fuzzy set is logically incorrect, and hence every result in which that definition had been used is incorrect. As long as we would keep on referring to fuzzy sets, the original definitions that include these two unacceptable points would keep coming up creating an unnecessary confusion thereby. We are therefore going to introduce the theory of imprecise sets in which the two fuzzy set theoretic blunders mentioned above would be absent.