Semisimple Algebras of Infinite Valued Logic and Bold Fuzzy Set Theory

Abstract

In classical two-valued logic there is a three way relationship among formal systems, Boolean algebras and set theory. In the case of infinite-valued logic we have a similar relationship among formal systems, MV-algebras and what is called Bold fuzzy set theory. The relationship, in the latter case, between formal systems and MV-algebras has been known for many years while the relationship between MV-algebras and fuzzy set theory has hardly been studied. This is not surprising. MV-algebras were invented by C. C. Chang [1] in order to provide an algebraic proof of the completeness theorem of the infinitevalued logic of Lukasiewicz and Tarski. Having served this purpose (see [2]), the study of these algebras has been minimal, see for example [6], [7]. Fuzzy set theory was also being born around the same time and only in recent years has its connection with infinite-valued logic been made, see e.g. [3], [4], [5]. It seems appropriate then, to take a further look at the structure of MV-algebras and their relation to fuzzy set theory.