An Ontological and Epistemological Perspective of Fuzzy Set Theory

I. Burhan Türksen

• Publisher: Elsevier Science
• Year: 2005
• Print ISBN: 9780444518910
• eBook ISBN: 9780080525716
• Topic: Computer Science
• Subtopic: Artificial Intelligence (AI) & Semantics
• Index: Computer Science

Fuzzy set and logic theory suggest that all natural language linguistic expressions are imprecise and must be assessed as a matter of degree. But in general membership degree is an imprecise notion which requires that Type 2 membership degrees be considered in most applications related to human decision making schemas. Even if the membership functions are restricted to be Type1, their combinations generate an interval – valued Type 2 membership. This is part of the general result that Classical equivalences breakdown in Fuzzy theory. Thus all classical formulas must be reassessed with an upper and lower expression that are generated by the breakdown of classical formulas.Key features:- Ontological grounding- Epistemological justification- Measurement of Membership- Breakdown of equivalences- FDCF is not equivalent to FCCF- Fuzzy Beliefs- Meta-Linguistic axioms- Ontological grounding- Epistemological justification- Measurement of Membership- Breakdown of equivalences- FDCF is not equivalent to FCCF- Fuzzy Beliefs- Meta-Linguistic axioms

Chapter 1

INTRODUCTION

The ontological and epistemological foundations of fuzzy sets and fuzzy logics are reassessed from the perspective of equivalences of the classical theory which breaks down in fuzzy theory. In turn, it is shown that there are additional new formulae and laws of conservation which demonstrate in part the richness of fuzzy theory beyond the classical theory. In particular, combinations of linguistic variables with linguistic connectives generate part of this richness within the framework of CWW and CWP proposed by L.A. Zadeh.

Another unique perspective of this work is to emphasize the fact that theories of sets and logics are similar but separate even though they are used together in deriving the formulas of combined concepts within the scope of computing with words and information granules for the construction of “expressions”. First, it should be noted that there are two separate disciplines in most universities under the headings of “set theories and their courses” which are generally included in a mathematics department and “logic theories and their courses” which are generally included in a philosophy department.

Formalization of logic theories has its origins in Aristolalian works and the formalization of set theory have its origins in Cantor’s work. With Zadeh’s (1965) seminal work, fuzzy set theory was introduced into the set theoretic studies. Later Zadeh (1973, 1975) introduced fuzzy logic as a foundation for approximate reasoning. Naturally Zadeh’s contributions demonstrate the convergence and integration of many earlier investigations and contributions by many world renowned scientists such as Max Black and J. Lukasiewicz, etc.

It is found that in most of the current works in fuzzy theory, fuzzy set membership values and fuzzy logic values are assumed to be the same at times implicitly (in most works). At other times, this assumption is made more explicit. This causes a great deal of confusion in understanding and in restructuring the foundations of fuzzy theory for the construction of expression in CWW. In this regard, Zadeh (1997, Prague IFSA Congress) has pointed out that there are four aspects to fuzziness: (1) fuzzy sets, (2) fuzzy logics, (3) fuzzy relations, and (4) fuzzy semantics. Each of these aspects need to be studied and investigated on their own as well as in relation to others. More recently, Zadeh pointed out the necessity to construct “Precisiated Natural Language, PNL, expressions,” for the implementation of CWW. In this work, we present some of the foundational concepts related to the ontological and epistemological aspects of fuzzy theories in general, and in particular to set and logic aspects, and their impact on the construction of “PNL expressions” for CWW. At the end, we propose a re-structuring the basic axioms of set and logic theory by expressing them linguistically in order to provide a foundation for CWW.

It appears that there has been an ongoing discussion amongst the mathematicians and logicians who attempt to distinguish the notions of set and logic theories and their essential normative foundations. For example, there is a response to Quine’s (1970) statement that “second-order logic (with standard semantics) is set theory in sheep’s clothing”. In “Metaphysical Myths, Mathematical Practice”, Azzouni (1994) points out that “second order logic with standard semantics is hardly set theory, if by that phrase is meant first-order set theory. They simply don ‘t have the same models.” The approach in this work is from a more practical point of view. That is the extraction and acquisition of membership values and functions of descriptive assignments from experts and/or from data-mining and knowledge discovery of input-output data vectors in order to capture concept definitions, i.e., semantic representation of concepts, and then their verification by independent observers. These are two separate but related assignments of membership values, i.e., descriptive and veristic assignments which address the epistemological foundation of fuzzy theories. They are used in the combination of concepts for the derivation of PNL expressions in order to represent our knowledge both qualitatively and quantitatively. Furthermore, such combination of concepts must allow one-to-many representation of linguistic connectives in order to capture our knowledge properly and to expose the non-linearity and the second order imprecision and its associated uncertainty that can be represented in fuzzy theories.

1.1 Description and Verity

In an attempt to identify the assignment of membership functions within the framework of “Computing With Words”, (Zadeh, IFSA 1997 and IFSA 1995 Conferences) it needs to be pointed out that either there is an assignment of words to numbers i.e., membership values and hence functions, or there is an assignment of numbers, i.e., memberships values and hence functions, to words. But in either case after one defines or describes the characteristics of an object (entity or event) by these two types of assignments that have been indicated above, one has to affirm or deny, i.e., verify, in the sense of ...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright
• Dedication
• FOREWORD
• PREFACE
• Chapter 0: FOUNDATION
• Chapter 1: INTRODUCTION
• Chapter 2: COMPUTING WITH WORDS
• Chapter 3: MEASUREMENT OF MEMBERSHIP
• Chapter 4: ELICITATION METHODS
• Chapter 5: FUZZY CLUSTERING METHOD
• Chapter 6: CLASSES OF FUZZY SET AND LOGIC THEORIES
• Chapter 7: EQUIVALENCES IN TWO-VALUED LOGIC
• Chapter 8: FUZZY-VALUED SET AND TWO-VALUED LOGIC
• Chapter 9: CONTAINMENT OF FDCF IN FCCF
• Chapter 10: CONSEQUENCES OF {D[0,1], V{0,1}}} THEORY
• Chapter 11: COMPENSATORY “AND”
• Chapter 12: BELIEF, PLAUSIBILITY AND PROBABILITY MEASURES ON INTERVAL-VALUED TYPE 2 FUZZY SETS
• Chapter 13: VERISTIC FUZZY SETS OF TRUTHOODS
• Chapter 14: APPROXIMATE REASONING
• Chapter 15: INTERVAL-VALUED TYPE 2 GMP
• Chapter 16: A THEORETICAL APPLICATION OF INTERVAL-VALUED TYPE 2 REPRESENTATION
• Chapter 17: A FOUNDATION FOR COMPUTING WITH WORDS: META-LINGUISTIC AXIOMS
• EPILOGUE
• REFERENCES
• INDEX
• AUTHOR INDEX