ORTHODOX AND NON-ORTHODOX SETS - SOME PHILOSOPHICAL REMARKS

摘要

We outline the relationship between classical (orthodox) sets from one side, and fuzzy and rough (non-orthodox) sets from another side. The classical concept of a set used in mathematics leads to antinomies, i.e., it is contradictory. This deficiency has, however, rather philosophical than practical meaning. Antinomies are associated with very "artificial" sets constructed in logic but not found in sets used in mathematics. That is why one can use mathematics safely. Fuzzy set and rough set theory are two different approaches to vagueness and are not remedy for classical set theory difficulties. Fuzzy set theory addresses gradualness of knowledge, expressed by the fuzzy membership, whereas rough set theory addresses granularity of knowledge, expressed by the indiscernibility relation. From practical point of view both theories are not competing but are rather complementary.