Structural Functionality as a Fundamental Property of Boolean Algebra and Base for Its Real-Valued Realizations

Abstract

The value of the complex Boolean function can be calculated directly on the basis of its components value. It is a principle known as the truth functionality. Properties of the Boolean algebra have indifferent values. The truth functional principle is taken as a valid principle in general case in the conventional generalization: multi-valued and/or real-valued realizations (fuzzy logic in the broad sense). This paper presents that truth functionality is not valued indifferent property of the Boolean algebra and it is valid only in two-valued realization, and thus it cannot be the basic of the value generalization. The value generalization (real-valued realizations) enables incomparably more descriptiveness than the two-valued classical Boolean algebra, so that the finite Boolean algebra is enough for any real application. Each finite Boolean algebra is atomic. Every Boolean function (the element of the analyzed finite Boolean algebra) can be presented uniquely as disjunction of the relevant atoms - disjunctive canonical form. Which atoms are and which are not included in the analyzed Boolean function is defined by its structure: 0-1 vector which dimension matches the number of atoms (in the case of n independent variables, the number of atoms is 2n). Atom corresponds uniquely to each vector structure position and value 0 means that the adequate atom is not included in the analyzed function, and 1 means that it is included. The principle of the structural functionality is: the structure of the complex Boolean function is defined directly on the basis of its components structure. The truth functionality is a value image of the structural functionality only in the case of two-valued realization. Each insisting on the truth functionality, such as in the case of conventional multi-valued logic and fuzzy logic in general sense, is unjustified from the point of the Boolean consistency. © Springer International Publishing Switzerland 2014.