Facts and figures on fuzzified normal forms

Abstract

The idea behind normal forms is to provide a standard representation or approximation of various kinds of functions. In Boolean logic, for instance, this amounts to expressing a given well-formed formula (WFF) in terms of the disjunction (respectively, conjunction) of some conjunctions (respectively, disjunctions) of several elementary ones. Interpreting these well-known identical disjunctive and conjunctive normal forms in a poorer structure such as a Kleene or De Morgan algebra, leads to a lower and upper approximation only of the corresponding (identical) normal forms (and, hence, of the given WFF) in that poorer structure. In this paper, we address the question whether a similar interpretation of these normal forms in a BL-algebra still provides lower and upper approximations of a given WFF. This question falls apart in two subquestions: First, are these interpretations comparable, and second, if so, is the WFF located in between. The first question can be answered positively for certain BL-algebras, while the second one is answered negatively.