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.

Belief Functions on MV-Algebras of Fuzzy Sets: An Overview

“…X is clearly an MV-algebra, which we will henceforth call MV-algebra of fuzzy sets in order to point out that every fuzzy subset of X is indeed included into A. Every MV-subalgebra of [0,1] X is called an MV-clan or simply a clan (cf. [4,46]).…”

“…[4,46]). Notice that, for a finite non-empty set X, the Boolean skeleton of the MV-algebra of fuzzy sets [0,1] X coincides with the power set 2 X of X.…”

“…This is, in particular, a key point of distinction between MV and Boolean algebras. Remember, in fact, that every Boolean algebra is semisimple, and that all MV-algebras of fuzzy sets A = [0, 1] X are semisimple as well.…”

“…Fuzzy sets were introduced by Zadeh [57] as an extension of classical sets: given a referential set X, a fuzzy subset of X can be identified with a function f from X into the real unit interval [0,1]. Given a fuzzy sets f , the idea is to interpret, for every x ∈ X, the value f (x) ∈ [0, 1] as the degree of membership of x to f .…”

“…Moreover there are several different ways to generalize Boolean algebras to algebras of fuzzy sets. Usually the generalization of belief functions to this frame, is done in the algebraic context of the so called De Morgan triples (or Zadeh-algebras) over classes of fuzzy sets, and where intersection, union, and complementation, are replaced in [0,1] X by the pointwise extensions of the operations in [0, 1] of min, max, and standard negation ¬ : x → 1 − x respectively (see for instance [60, §2.1…”