Consistent generalization of classical Boolean two-valued into real-valued theories

Abstract

Consistent Boolean generalization of two-valued into a real-valued theory means preservation of all of its algebraic — value indifferent characteristics: Boolean axioms and theorems. Actually two-valued theories in Boolean frame (classical logic, theory of classical sets, theory of classical relations, etc.) are based on the celebrated two-valued realization of Boolean algebra (BA) and their real-valued consistent generalization should be based on a real-valued realization of BA. The conventional real-valued theories: fuzzy sets, fuzzy logic, fuzzy relations, fuzzy probability, etc., are not in Boolean frame. Interpolative Boolean algebra (IBA) is a real-valued realization of atomic or finite BA. IBA is based on generalized Boolean polynomials (GBP) as a unique figure of every element of finite Boolean algebra. GBP is able to process values from real unit interval so to preserve all algebraic characteristics on a value level as corresponding arithmetic properties (for example: relation ⊆ as ≤). The real-valued realization of atomic or finite BA is adequate for any real problem since gradation offers superior expressiveness in comparison to the black-white outlook. Consistent Boolean generalization is illustrated on representative examples.