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An Ontological and Epistemological Perspective of Fuzzy Set Theory

Book Description

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

Table of Contents

  1. Cover image
  2. Title page
  3. Table of Contents
  4. Copyright
  5. Dedication
  6. FOREWORD
  7. PREFACE
  8. Chapter 0: FOUNDATION
    1. 0.1 A Personal Perspective
    2. 0.2 A Perspective on The Philosophical Grounding of Fuzzy Theories
    3. Appendix 1 Scientific and Philosophical Contribution of L.A. ZADEH
    4. Appendix 2 Fuzzy Systems Applications in Operations Research, Management Science and Decision Support Systems
  9. Chapter 1: INTRODUCTION
    1. 1.1 Description and Verity
    2. 1.2 Nature of Truth
    3. 1.3 Definiteness vs. Indefiniteness
    4. 1.4 Syntax of a Formal Language, a PNL
    5. 1.5 Basic Notations – Type 1 Theory
    6. 1.6 Basic Notations – Type 2 Theory
    7. 1.7 Epistemological Concerns
  10. Chapter 2: COMPUTING WITH WORDS
    1. 2.1 Words to Numbers
    2. 2.2 Descriptive and Veristic Assignments
    3. 2.3 Structure of Sentences
  11. Chapter 3: MEASUREMENT OF MEMBERSHIP
    1. 3.1 Interpretations of Grade of Membership
    2. 3.2 Measurement Theory View
    3. 3.3 Membership and Connectives
  12. Chapter 4: ELICITATION METHODS
    1. 4.1 Polling Methods
    2. 4.2 Direct Rating Methods
    3. 4.3 Reverse Rating
    4. 4.4 Interval Estimation
    5. 4.5 Membership Exemplification
    6. 4.6 Pair wise Comparison
    7. 4.7 General Remarks on Subjective Methods
  13. Chapter 5: FUZZY CLUSTERING METHOD
    1. 5.1 Fuzzy Clustering Techniques
    2. 5.2 Type 2 Fuzziness
    3. 5.3 Curve Fitting to Membership Values
    4. 5.4 Neural-fuzzy Technique
  14. Chapter 6: CLASSES OF FUZZY SET AND LOGIC THEORIES
    1. 6.1 Linguistic Expression
    2. 6.2 Meta-Linguistic Expression
    3. 6.3 Propositional Expressions
    4. 6.4 Classes of Fuzzy Sets and Two-Valued Logic
    5. 6.5 Sub-Sub Classes of t-Norms
    6. 6.6 Sub-Sub Classes of t-Conorms
    7. 6.7 Fuzzy-Set Complements
    8. 6.8 De Morgan Triples
    9. 6.9 Parametric t-norms and t-conorms
    10. 6.10 Fundamental Phrases and Clauses
    11. Appendix 6.1 Basic Properties of negations and t-norms
  15. Chapter 7: EQUIVALENCES IN TWO-VALUED LOGIC
    1. 7.1 Two-Valued Set(Description) and Two-Valued Logic(Verification)
    2. 7.2 (Canonical) Normal Form Derivation
    3. 7.3 Equivalence of Normal Forms
    4. 7.4 Direct Fuzzification of DNF and CNF Expressions
    5. 7.5 Consequences of D{0,1} V{0,1}
    6. 7.6 Symbols, Propositions and Predicates
  16. Chapter 8: FUZZY-VALUED SET AND TWO-VALUED LOGIC
    1. 8.1 New Construction of the Truth Tables
    2. 8.2 Dempster-Pawlak Unification
    3. 8.3 DEMPSTER and PAWLAK Formulations
    4. 8.4 Sets and Logic Constructs
    5. 8.5 Generalization
    6. 8.6 Interval-Valued Type 2 Fuzzy Empty and Universal Sets
    7. Appendix 8.1 Fuzzy Canonical Form Derivation Algorithm
    8. Appendix 8.2 Canonical Form Derivation with Set Membership Consideration
  17. Chapter 9: CONTAINMENT OF FDCF IN FCCF
    1. 9.1 Generators of Continuous Archimedean Norms
    2. 9.2 Non Archimedean Triangular Norms and Conorms
    3. 9.3 Ordinal Sums
    4. 9.4 De Morgan Triples
    5. 9.5 Basic Protoforms: FDCF and FCCF
    6. 9.6 Preliminary Observations
    7. 9.7 Containment for continuous Archimedean t-norms
    8. 9.8 Combination of More Than Two Propositions
    9. Appendix 9.1 (Bilgic, 1995)
    10. Appendix 9.2 (Bilgic, 1995)
  18. Chapter 10: CONSEQUENCES OF {D[0,1], V{0,1}}} THEORY
    1. 10.1 Laws of Middle and Contradiction
    2. 10.2 Zadehean Fuzzy Middle and Contradiction
    3. 10.3 Fuzzy Middle and Fuzzy Contradiction with t-norms and co-norms
    4. 10.4 Laws of Fuzzy Conservation
    5. 10.5 Canonical Forms of Re-Affirmation And Re-Negation
    6. 10.6 Canonical Forms of Re-Negation
    7. 10.7 Conclusion
  19. Chapter 11: COMPENSATORY “AND”
    1. 11.1 Exponential-Compensatory “AND”
    2. 11.2 Containment of FDCF in FCCF of “AND”
    3. 11.3 Compensatory “OR”
    4. 11.4 Specific Operators
    5. 11.6 “Convex-Linear-Compensatory AND”
    6. 11.7 An Observation
    7. 11.8 Conclusion
    8. Appendix 11.1
    9. Appendix 11.2
    10. Appendix 11.3
  20. Chapter 12: BELIEF, PLAUSIBILITY AND PROBABILITY MEASURES ON INTERVAL-VALUED TYPE 2 FUZZY SETS
    1. 12.1 Belief and Plausibility over Fuzzy Sets
    2. 12.2 Upper and Lower Probabilities over Interval Valued Type 2 Fuzzy Sets
    3. 12.3 Interval-Valued Type 2 Fuzzy Sets and Fuzzy Beliefs
    4. 12.4 Conclusions
    5. Appendix 12.1
  21. Chapter 13: VERISTIC FUZZY SETS OF TRUTHOODS
    1. 13.1 Modal Logic
    2. 13.2 Meta-Theory Based On Modal Logic
    3. 13.3 “AND”, “OR” and “COMPLEMENT”
    4. 13.4 Canonical Forms for the Synchronous Case
    5. 13.5 Canonical Forms for the Asynchronous Case
    6. 13.6 Soft computing example
    7. 13.7 Conclusion
  22. Chapter 14: APPROXIMATE REASONING
    1. 14.1 Classical Reasoning Methods
    2. 14.2 Classical Modus Ponens
    3. 14.3 Generalized Modus Ponens
    4. 14.4 Type 1 Fuzzy Rules
    5. 14.5 Type 1 Fuzzy Inference: Single Antecedent GMP
    6. 14.6 Information Gap in Type 1 GMP
    7. 14.7 Type 1 Fuzzy Inference: Two Antecedent GMP
    8. 14.8 Decomposition
    9. 14.9 Computational Complexity
    10. 14.10 Implementation with Type 1 Reasoning
    11. 14.11 Type 1 Fuzzy System Modeling
    12. 14.12 Case studies
    13. Appendix 14.1 Brief explanation of the proposed stages of fuzzy system modeling
  23. Chapter 15: INTERVAL-VALUED TYPE 2 GMP
    1. 15.1 Interval-Valued Type 2 Fuzzy Rules
    2. 15.2 Some Properties Interval-Valued Type 2 Implication
    3. 15.3 Information Gap in Interval-Valued Type 2 GMP
    4. 15.4 Implementations of Interval-Valued Type 2 Reasoning
    5. 15.5 Interval-Valued Type 2 System Modeling
    6. 15.6 Application to Case Studies with Interval Valued Type 2 Reasoning
    7. 15.6.2 Conclusion
    8. Appendix 15.1
    9. Appendix 15.1 Representation of rules
    10. Appendix 15.3 Actual output data vs the results of reasoning methods
  24. Chapter 16: A THEORETICAL APPLICATION OF INTERVAL-VALUED TYPE 2 REPRESENTATION
    1. 16.1 Background
    2. 16.2 Strict Preference
    3. 16.3 Conclusion
  25. Chapter 17: A FOUNDATION FOR COMPUTING WITH WORDS: META-LINGUISTIC AXIOMS
    1. 17.1 Introduction
    2. 17.2 Meta-Linguistic Axioms
    3. 17.3 Consequences of the Proposed Meta-Linguistic Axioms
    4. 17.4 Meta-Linguistic Reasoning
    5. 17.5 Conclusion
  26. EPILOGUE
  27. REFERENCES
  28. INDEX
  29. AUTHOR INDEX