Interpolative Realization of Boolean Algebra as a Consistent Frame for Gradation and/or Fuzziness

Abstract

L. Zadeh has ingeniously recognized importance and necessity of gradation in relations generally (theory of sets – fuzzy sets, logic – fuzzy logic, relations – fuzzy relations) for real applications. Common for all known approaches for treatment gradation is the fact that either they are not complete (from the logical point of view) or they are not in the Boolean frame. Here is given Interpolative Boolean algebra (IBA) as a consistent MV realization of finite (atomic) Boolean algebra. Since, axioms and lows of Boolean algebra are actually meter of value independent structure of IBA elements, all axioms and all laws of Boolean algebra are preserved in any type of value realization (two-valued, three-valued,..., [0, 1]). To every element of IBA corresponds generalized Boolean polynomial with ability to process all values of primary variables from real unit interval [0, 1]. The possibility of new approach is illustrated on two examples: generalized preference structure and interpolative sets as consistent realization of idea of fuzzy sets.