Concepts in Fuzzy Scaling Theory: Order and Granularity

Karl Erich Wolff

Fachbereich Mathematik und Naturwissenschaften, Fachhochschule Darmstadt,

Schöfferstr. 3, D-64295 Darmstadt, Germany

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PT 04

ABSTRACT:

This paper explains mainly by examples the mathematical foundation of Fuzzy Scaling Theory described in Wolff (1998). It illustrates the solution of fundamental problems in Fuzzy Theory, especially the problem of the pragmatic meaning of ordered sets as logics, the problem of the meaning of Fuzzy implications and the problem of the construction of direct products of linguistic variables.

The key to the solution of these problems is the description of membership functions by cut contexts and of linguistic variables by conceptual scales. This renders the translation of Conceptual Scaling Theory into Fuzzy Scaling Theory and the generalization of the latter to L-Fuzzy Scaling Theory for an arbitrary ordered set L as logic.

This paper bridges the gap between two scientific communities both using "graded concepts" as tools to handle granularity: it is shown that L-Fuzzy Scaling Theory is equivalent to Conceptual Scaling Theory.

KEYWORDS:

Fuzzy Scaling Theory, linguistic variables, Formal Concept Analysis, Fuzzy implications