Ontological and Epistemological Perspective of Fuzzy Set Theory (eBook)

Front Cover 1
AN ONTOLOGICAL AND EPISTEMOLOGICAL PERSPECTIVE OF FUZZY SET THEORY 4
Copyright Page 5
CONTENTS 20
FOREWORD 8
PREFACE 12
Chapter 0. FOUNDATION 30
0.1. A Personal Perspective 31
0.2. A Perspective on The Philosophical Grounding of Fuzzy Theories 36
Chapter 1. INTRODUCTION 84
1.1. Description and Verity 85
1.2. Nature of Truth 88
1.3. Definiteness vs. Indefiniteness 92
1.4. Syntax of a Formal Language, a PNL 95
1.5. Basic Notations - Type 1 Theory 96
1.6. Basic Notations - Type 2 Theory 98
1.7. Epistemological Concerns 102
Chapter 2. COMPUTING WITH WORDS 106
2.1. Words to Numbers 107
2.2. Descriptive and Veristic Assignments 109
2.3. Structure of Sentences 114
Chapter 3. MEASUREMENT OF MEMBERSHIP 118
3.1. Interpretations of Grade of Membership 119
3.2. Measurement Theory View 126
3.3. Membership and Connectives 131
Chapter 4. ELICITATION METHODS 140
4.1. Polling Methods 141
4.2. Direct Rating Methods 142
4.3. Reverse Rating 144
4.4. Interval Estimation 144
4.5. Membership Exemplification 147
4.6. Pair wise Comparison 147
4.7. General Remarks on Subjective Methods 148
Chapter 5. FUZZY CLUSTERING METHOD 152
5.1. Fuzzy Clustering Techniques 156
5.2. Type 2 Fuzziness 165
5.3. Curve Fitting to Membership Values 167
5.4. Newal-fuzzy Technique 171
Chapter 6. CLASSES OF FUZZY SET AND LOGIC THEORIES 174
6.1. Linguistic Expression 175
6.2. Meta-Linguistic Expression 178
6.3. Propositional Expression 181
6.4. Classes of Fuzzy Sets and Two-Valued Logic 184
6.5. Sub-Sub Classes of t-Norms 186
6.6. Sub-Sub Classes of t-Conorms 187
6.7. Fuzzy-Set Complements 188
6.8. De Morgan Triples 188
6.9. Parametric t-norms and t-conorms 189
6.10. Fundamental Phrases and Clauses 190
Chapter 7. EQUIVALENCES IN TWO-VALUED LOGIC 200
7.1. Two-Valued Set(Description) and Two-Valued Logic(Verification) 200
7.2. (Canonical) Normal Form Derivation 201
7.3. Equivalence of Normal Forms 203
7.4. Direct Fuzzification of DNF and CNF Expression 205
7.5. Consequences of D{0,1} V{0,1} 207
7.6. Symbols, Proposition and Predicates 211
Chapter 8. FUZZY-VALUED SET AND TWO-VALUED LOGIC 216
8.1. New Construction of Truth Tables 216
8.2. Dempster-Pawlak Unification 221
8.3. DEMPSTER and PAWLAK Formulations 226
8.4. Sets and Logic Constructs 230
8.5. Generalization 238
8.6. Interval-Valued Type 2 Fuzzy Empty and Universal Sets 238
Chapter 9. CONTAINMENT OF FDCF IN FCCF 248
9.1. Generators of Continuous Archimedean Norms 248
9.2. Non Archimedean Triangular Norms and Conorms 250
9.3. Ordinal Sums 251
9.4. De Morgan Triples 252
9.5. Basic Protoforms: FDCF and FCCF 253
9.6. Preliminary Observations 253
9.7. Containment for continuous Archimedean t-norms 257
9.8. Combination of More Than Two Propositions 258
Chapter 10. CONSEQUENCES OF {D[0,1], V{0,1}}} THEORY 270
10.1. Laws of Middle and Contradiction 270
10.2. Zadehean Fuzzy Middle and Contradiction 275
10.3. Fuzzy Middle and Fuzzy Contradiction with t-norms and co-norms 276
10.4. Laws of Fuzzy Conservation 280
10.5. Canonical Forms of Re-Affirmation And Re-Negation 284
10.6. Canonical Forms of Re-Negation 288
10.7. Conclusion 293
Chapter 11. COMPENSATORY "AND" 296
11.1. Exponential-Compensatory "AND" 297
11.2. Containment of FDCF in FCCF of "AND" 298
11.3. Compensatory "OR" 299
11.4. Specific Operators 301
11.5. An Observation 308
11.6. "Convex-Linear-Compensatory AND" 310
11.7. An Observation 311
11.8. Conclusion. 311
Chapter 12. BELIEF, PLAUSIBILITY AND PROBABILITY MEASURES ON INTERVAL-VALUED TYPE 2 FUZZY SETS 318
12.1. Belief and Plausibility over Fuzzy Sets 319
12.2. Upper and Lower Probabilities over Interval Valued Type 2 Fuzzy Sets 334
12.3. Interval-Valued Type 2 Fuzzy Sets and Fuzzy Beliefs 335
12.4. Conclusions 338
Chapter 13. VERISTIC FUZZY SETS OF TRUTHOODS 342
13.1. Modal Logic 343
13.2. Meta-Theory Based On Modal Logic 344
13.3. "AND", "OR" and "COMPLEMENT" 352
13.4. Canonical Forms for the Synchronous Case 367
13.5. Canonical Forms for the Asynchronous Case 376
13.6. Soft computing example 377
13.7. Conclusion 378
Chapter 14. APPROXIMATE REASONING* 382
14.1. Classical Reasoning Methods 383
14.2. Classical Modus Ponens 383
14.3. Generalized Modus Ponens 388
14.4. Type 1 Fuzzy Rules 390
14.5. Type 1 Fuzzy Inference: Single Antecedent GMP 394
14.6. Information Gap in Type 1 GMP 396
14.7. Type 1 Fuzzy Inference: Two Antecedent GMP 397
14.8. Decomposition 398
14.9. Computational Complexity 401
14.10. Implementation with Type 1 Reasoning 402
14.11. Type 1 Fuzzy System Modeling 404
14.12. Case studies 410
Chapter 15. INTERVAL-VALUED TYPE 2 GMP 416
15.1 . Interval-Valued Type 2 Fuzzy Rules 416
15.2. Some Properties Interval-Valued Type 2 Implication 418
15.3. Information Gap in Interval-Valued Type 2 GMP 420
15.4. Implementations of Interval-Valued Type 2 Reasoning 424
15.5. Interval-Valued Type 2 System Modeling 425
15.6. Application to Case Studies with Interval Valued Type 2 Reasoning 430
Chapter 16. A THEORETICAL APPLICATION OF INTERVAL-VALUED TYPE 2 REPRESENTATION 452
16.1. Background 454
16.2. Strict Preference 458
16.3. Conclusion 470
Chapter 17. A FOUNDATION FOR COMPUTING WITH WORDS: META-LINGUISTIC AXIOMS 474
17.1. Introduction 474
17.2. Meta-Linguistic Axioms 476
17.3. Consequences of the Proposed Meta-Linguistic Axioms 479
17.4. Meta-Linguistic Reasoning 510
17.5. Conclusion 512
EPILOGUE 516
REFERENCES 518
INDEX 534
AUTHOR INDEX 542