Elsevier S & T

An Ontological and Epistemological Perspective of Fuzzy Set Theory

Türksen, I. Burhan

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Table of contents
  • Cover
  • CONTENTSxix
  • FOREWORDvii
  • PREFACExi
  • Chapter 0. FOUNDATION1
  • 0.1. A Personal Perspective2
  • 0.2. A Perspective on The Philosophical Grounding of Fuzzy Theories7
  • Chapter 1. INTRODUCTION55
  • 1.1. Description and Verity56
  • 1.2. Nature of Truth59
  • 1.3. Definiteness vs. Indefiniteness63
  • 1.4. Syntax of a Formal Language, a PNL66
  • 1.5. Basic Notations - Type 1 Theory67
  • 1.6. Basic Notations - Type 2 Theory69
  • 1.7. Epistemological Concerns73
  • Chapter 2. COMPUTING WITH WORDS77
  • 2.1. Words to Numbers78
  • 2.2. Descriptive and Veristic Assignments80
  • 2.3. Structure of Sentences85
  • Chapter 3. MEASUREMENT OF MEMBERSHIP89
  • 3.1. Interpretations of Grade of Membership90
  • 3.2. Measurement Theory View97
  • 3.3. Membership and Connectives102
  • Chapter 4. ELICITATION METHODS111
  • 4.1. Polling Methods112
  • 4.2. Direct Rating Methods113
  • 4.3. Reverse Rating115
  • 4.4. Interval Estimation115
  • 4.5. Membership Exemplification118
  • 4.6. Pair wise Comparison118
  • 4.7. General Remarks on Subjective Methods119
  • Chapter 5. FUZZY CLUSTERING METHOD123
  • 5.1. Fuzzy Clustering Techniques127
  • 5.2. Type 2 Fuzziness136
  • 5.3. Curve Fitting to Membership Values138
  • 5.4. Newal-fuzzy Technique142
  • Chapter 6. CLASSES OF FUZZY SET AND LOGIC THEORIES145
  • 6.1. Linguistic Expression146
  • 6.2. Meta-Linguistic Expression149
  • 6.3. Propositional Expression152
  • 6.4. Classes of Fuzzy Sets and Two-Valued Logic155
  • 6.5. Sub-Sub Classes of t-Norms157
  • 6.6. Sub-Sub Classes of t-Conorms158
  • 6.7. Fuzzy-Set Complements159
  • 6.8. De Morgan Triples159
  • 6.9. Parametric t-norms and t-conorms160
  • 6.10. Fundamental Phrases and Clauses161
  • Chapter 7. EQUIVALENCES IN TWO-VALUED LOGIC171
  • 7.1. Two-Valued Set(Description) and Two-Valued Logic(Verification)171
  • 7.2. (Canonical) Normal Form Derivation172
  • 7.3. Equivalence of Normal Forms174
  • 7.4. Direct Fuzzification of DNF and CNF Expression176
  • 7.5. Consequences of D{0,1} V{0,1}178
  • 7.6. Symbols, Proposition and Predicates182
  • Chapter 8. FUZZY-VALUED SET AND TWO-VALUED LOGIC187
  • 8.1. New Construction of Truth Tables187
  • 8.2. Dempster-Pawlak Unification192
  • 8.3. DEMPSTER and PAWLAK Formulations197
  • 8.4. Sets and Logic Constructs201
  • 8.5. Generalization209
  • 8.6. Interval-Valued Type 2 Fuzzy Empty and Universal Sets209
  • Chapter 9. CONTAINMENT OF FDCF IN FCCF219
  • 9.1. Generators of Continuous Archimedean Norms219
  • 9.2. Non Archimedean Triangular Norms and Conorms221
  • 9.3. Ordinal Sums222
  • 9.4. De Morgan Triples223
  • 9.5. Basic Protoforms: FDCF and FCCF224
  • 9.6. Preliminary Observations224
  • 9.7. Containment for continuous Archimedean t-norms228
  • 9.8. Combination of More Than Two Propositions229
  • Chapter 10. CONSEQUENCES OF {D[0,1], V{0,1}}} THEORY241
  • 10.1. Laws of Middle and Contradiction241
  • 10.2. Zadehean Fuzzy Middle and Contradiction246
  • 10.3. Fuzzy Middle and Fuzzy Contradiction with t-norms and co-norms247
  • 10.4. Laws of Fuzzy Conservation251
  • 10.5. Canonical Forms of Re-Affirmation And Re-Negation255
  • 10.6. Canonical Forms of Re-Negation259
  • 10.7. Conclusion264
  • Chapter 11. COMPENSATORY "AND"267
  • 11.1. Exponential-Compensatory "AND"268
  • 11.2. Containment of FDCF in FCCF of "AND"269
  • 11.3. Compensatory "OR"270
  • 11.4. Specific Operators272
  • 11.5. An Observation279
  • 11.6. "Convex-Linear-Compensatory AND"281
  • 11.7. An Observation282
  • 11.8. Conclusion.282
  • Chapter 12. BELIEF, PLAUSIBILITY AND PROBABILITY MEASURES ON INTERVAL-VALUED TYPE 2 FUZZY SETS289
  • 12.1. Belief and Plausibility over Fuzzy Sets290
  • 12.2. Upper and Lower Probabilities over Interval Valued Type 2 Fuzzy Sets305
  • 12.3. Interval-Valued Type 2 Fuzzy Sets and Fuzzy Beliefs306
  • 12.4. Conclusions309
  • Chapter 13. VERISTIC FUZZY SETS OF TRUTHOODS313
  • 13.1. Modal Logic314
  • 13.2. Meta-Theory Based On Modal Logic315
  • 13.3. "AND", "OR" and "COMPLEMENT"323
  • 13.4. Canonical Forms for the Synchronous Case338
  • 13.5. Canonical Forms for the Asynchronous Case347
  • 13.6. Soft computing example348
  • 13.7. Conclusion349
  • Chapter 14. APPROXIMATE REASONING*353
  • 14.1. Classical Reasoning Methods354
  • 14.2. Classical Modus Ponens354
  • 14.3. Generalized Modus Ponens359
  • 14.4. Type 1 Fuzzy Rules361
  • 14.5. Type 1 Fuzzy Inference: Single Antecedent GMP365
  • 14.6. Information Gap in Type 1 GMP367
  • 14.7. Type 1 Fuzzy Inference: Two Antecedent GMP368
  • 14.8. Decomposition369
  • 14.9. Computational Complexity372
  • 14.10. Implementation with Type 1 Reasoning373
  • 14.11. Type 1 Fuzzy System Modeling375
  • 14.12. Case studies381
  • Chapter 15. INTERVAL-VALUED TYPE 2 GMP387
  • 15.1 . Interval-Valued Type 2 Fuzzy Rules387
  • 15.2. Some Properties Interval-Valued Type 2 Implication389
  • 15.3. Information Gap in Interval-Valued Type 2 GMP391
  • 15.4. Implementations of Interval-Valued Type 2 Reasoning395
  • 15.5. Interval-Valued Type 2 System Modeling396
  • 15.6. Application to Case Studies with Interval Valued Type 2 Reasoning401
  • Chapter 16. A THEORETICAL APPLICATION OF INTERVAL-VALUED TYPE 2 REPRESENTATION423
  • 16.1. Background425
  • 16.2. Strict Preference429
  • 16.3. Conclusion441
  • Chapter 17. A FOUNDATION FOR COMPUTING WITH WORDS: META-LINGUISTIC AXIOMS445
  • 17.1. Introduction445
  • 17.2. Meta-Linguistic Axioms447
  • 17.3. Consequences of the Proposed Meta-Linguistic Axioms450
  • 17.4. Meta-Linguistic Reasoning481
  • 17.5. Conclusion483
  • EPILOGUE487
  • REFERENCES489
  • INDEX505
  • AUTHOR INDEX513
Book details
  • Vendor Elsevier S & T
  • SKU 9780444518910
  • ISBN-13 9780080525716
  • Author Türksen, I. Burhan
  • Category Mathematics
  • Subject Logic

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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