9780444518910

An Ontological and Epistemological Perspective of Fuzzy Set Theory

by
  • ISBN13:

    9780444518910

  • ISBN10:

    0444518916

  • Format: Hardcover
  • Copyright: 2005-12-01
  • Publisher: Elsevier Science
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Summary

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

Table of Contents

Foreword vii
Preface xi
Foundation
1(54)
A Personal Perspective
2(5)
A Perspective on The Philosophical Grounding of Fuzzy Theories
7(48)
Underlying Philosophical Bases
9(1)
Hierarchy of Levels of Theoretical Inquiry
9(1)
Ontological Level
10(1)
Level 1
10(1)
Level 2
11(1)
General Epistemological Level
11(1)
Level 3
12(1)
Level 4
12(1)
Domain-Specific Epistemological Level
13(1)
Level 5
13(1)
Level 6
13(1)
Application Level (Level 7)
13(1)
Classical vs. Fuzzy Theory
14(1)
A Hierarchical Sketch of the Classical Set and Logic Theory Model
15(8)
A Hierarchical Sketch of Fuzzy Set and Two-valued Logic Theory Model
23(9)
Application of Hierarchical Sketches
32(1)
Explication
33(1)
Assessing Internal Consistency
33(1)
Assessing the Level of Inquiry for an Argument
34(1)
Philosophical Comparison among Different Fuzzy Theories
35(2)
Theory Development
37(1)
Future Work and Conclusions
38(1)
Conclusions
39(16)
Introduction
55(22)
Description and Verity
56(3)
Nature of Truth
59(4)
Fuzzy Truthood
62(1)
Definiteness vs. Indefiniteness
63(3)
Syntax of a Formal Language, a PNL
66(1)
Basic Notations -- Type 1 Theory
67(2)
Basic Notations -- Type 2 Theory
69(4)
Epistemological Concerns
73(4)
Computing With Words
77(12)
Words to Numbers
78(2)
Descriptive Words
78(1)
Veristic Words
79(1)
Descriptive and Veristic Assignments
80(5)
Descriptive Set Assignments
81(2)
Veristic Set Assignments
83(2)
Structure of Sentences
85(4)
Atomic Sentences
85(1)
Complex Sentences
86(3)
Measurement of Membership
89(22)
Interpretations of Grade of Membership
90(7)
The likelihood View
92(2)
Random Set View
94(1)
Similarity View
95(1)
Utility Theory View
96(1)
Measurement Theory View
97(5)
Membership Measurement
98(2)
Ordered Algebraic Structures
100(2)
Membership and Connectives
102(9)
Scale Strength
103(1)
Two or More Subjects
104(2)
Conjoint Measurement
106(1)
Linguistic Concepts and Linguistic Connectives
107(4)
Elicitation Methods
111(12)
Polling Methods
112(1)
Direct Rating Methods
113(2)
Reverse Rating
115(1)
Interval Estimation
115(3)
Membership Exemplification
118(1)
Pair wise Comparison
118(1)
General Remarks on Subjective Methods
119(4)
Fuzzy Clustering Method
123(22)
Fuzzy Clustering Techniques
127(9)
FCM Algorithm
129(1)
Statistical Background
129(3)
Variations of FCM
132(4)
Interval-Valued Fuzzy Sets
136(1)
Type 2 Fuzziness
136(2)
Curve Fitting to Membership Values
138(4)
Neural-fuzzy Technique
142(3)
Classes of Fuzzy Set and Logic Theories
145(26)
Linguistic Expression
146(3)
Meta-Linguistic Expression
149(3)
Propositional Expressions
152(3)
Classes of Fuzzy Sets and Two-Valued Logic
155(2)
Sub-Sub Classes of t-Norms
157(1)
Sub-Sub Classes of t-Conorms
158(1)
Fuzzy-Set Complements
159(1)
De Morgan Triples
159(1)
Parametric t-norms and t-conorms
160(1)
Fundamental Phrases and Clauses
161(10)
Equivalences in Two-Valued Logic
171(16)
Two-Valued Set(Description) and Two-Valued Logic(Verification)
171(1)
(Canonical) Normal Form Derivation
172(2)
(Canonical) Normal Form Derivation Algorithm
172(2)
Discussion
174(1)
Equivalence of Normal Forms
174(2)
Equivalence of DNF and CNF of ``Implication''
174(1)
Equivalence of DNF and CNF of ``OR'' Connective
175(1)
Direct Fuzzification of DNF and CNF Expressions
176(2)
Conjecture
177(1)
Consequences of D{0,1} V{0,1}
178(4)
Russell' s Paradox
179(1)
Formal Treatment of Russell' s Paradox
179(1)
Barber's Pseudo paradox
179(1)
Catalogue Pseudo paradox
180(1)
Burali-Forti Paradox
180(1)
Cantor's paradox
180(1)
Liar's paradox
181(1)
Symbols, Propositions and Predicates
182(5)
Fuzzy-Valued Set and Two-Valued Logic
187(32)
New Construction of the Truth Tables
187(5)
Dempster-Pawlak Unification
192(5)
Brief Review
193(4)
Dempster and Pawlak Formulations
197(4)
Dempster Construction
198(1)
Pawlak Construction
198(1)
Restatement of Dempster and Pawlak Constructs
199(1)
Pawlak to Dempster Transformation
199(2)
Sets and Logic Constructs
201(8)
Restriction and Modification, T-Formalism
201(2)
Upper and Lower Canonical Forms
203(3)
Five Meta-Linguistic Expression that have ``And'' Composition
206(1)
Five Meta-Linguistic Expression that have ``OR'' Composition
207(1)
Other Six Meta-Linguistic Expressions
208(1)
Results
209(1)
Generalization
209(1)
Interval-Valued Type 2 Fuzzy Empty and Universal Sets
209(10)
Classical Empty Set
210(1)
Type 1 Fuzzy Empty Set
210(1)
Type 2 Fuzzy Sets
211(2)
Combination of Type 1 Fuzzy Sets with Ø and X
213(6)
Containment of FDCF in FCCF
219(22)
Generators of Continuous Archimedean Norms
219(2)
Non Archimedean Triangular Norms and Conorms
221(1)
Ordinal Sums
222(1)
De Morgan Triples
223(1)
Basic Protoforms: FDCF and FCCF
224(1)
Preliminary Observations
224(4)
Containment for continuous Archimedean t-norms
228(1)
Combination of More Than Two Propositions
229(12)
Consequences of {D[0,1], V[0,1]}} Theory
241(26)
Laws of Middle and Contradiction
241(5)
Boolean CNF's and DNF
241(2)
LEM and Contradiction
243(1)
Fuzzy DCF's and CCF's
244(1)
Fuzzy Middle
244(1)
Fuzzy Contradiction
245(1)
Zadehean Fuzzy Middle and Contradiction
246(1)
Fuzzy Middle and Fuzzy Contradiction with t-norms and co-norms
247(4)
Algebraic Product and Sum
248(1)
Bold Intersection and Union
249(2)
Laws of Fuzzy Conservation
251(4)
Laws of Fuzzy Conservation for Zadeh Operators
253(1)
Laws of Fuzzy Conservation for Δ and Combinations
253(1)
Laws of Fuzzy Conservation with Algebraic Product and Sum
253(1)
Laws of Fuzzy Conservation with Bold Intersection and Union
254(1)
Remarks on Laws of Fuzzy Conservation
254(1)
Canonical Forms of Re-Affirmation And Re-Negation
255(4)
Canonical Forms of Re-Affirmation
255(1)
Canonical Forms of Re-Affirmation with ``Or''
255(2)
Canonical Forms of Re-Affirmation with ``And''
257(2)
Canonical Forms of Re-Negation
259(5)
Canonical Forms of Re-Negation with ``Or''
259(3)
Canonical Forms of Re-Negation with ``And''
262(2)
Conclusion
264(3)
Compensatory ``And''
267(22)
Exponential-Compensatory ``And''
268(1)
Containment of FDCF in FCCF of ``And''
269(1)
Compensatory ``Or''
270(2)
Specific Operators
272(7)
Max-Min Operators
272(7)
An Observation
279(2)
``Convex-Linear-Compensatory And''
281(1)
An Observation
282(1)
Conclusion
282(7)
Belief, Plausibility and Probability Measures on Interval-Valued Type 2 Fuzzy Sets
289(24)
Belief and Plausibility over Fuzzy Sets
290(15)
Belief Measures
291(3)
Degree of Evidence
294(2)
Belief on Type 1 Combination of Fuzzy Sets
296(4)
Belief on Type 1 Fuzzy set formed with other De Morgan Triples
300(1)
Algebraic Combination
300(1)
Lucasiewicz Combination
300(1)
Belief on Internal-valued Type 2 Fuzzy Sets
301(2)
Lower Belief Assessments
303(2)
Upper and Lower Probabilities over Interval Valued Type 2 Fuzzy Sets
305(1)
Interval-Valued Type 2 Fuzzy Sets and Fuzzy Beliefs
306(3)
Conclusions
309(4)
Veristic Fuzzy Sets of Truthoods
313(40)
Modal Logic
314(1)
Kripke Modal Framework
314(1)
Meta-Theory Based On Modal Logic
315(8)
Consonance and Dissonance of Propositions
319(1)
Dissonant Propositions
320(1)
Consonant Propositions
320(3)
``And'', ``Or'' and ``Complement''
323(15)
``And'', ``Or'' Operations
323(5)
Worlds and Synchronisation
328(1)
Conjunction and synchronisation
328(2)
Disjunction and synchronisation
330(1)
Complement and synchronisation
331(4)
The Effect of Transformation Γ of the worlds
335(2)
Transformation and the absurd condition
337(1)
Canonical Forms for the Synchronous Case
338(9)
Equivalence or Containment
341(1)
Equivalence of FDNF and FCNF
341(2)
Containment of FDNF and FCNF
343(4)
Canonical Forms for the Asynchronous Case
347(1)
Soft computing example
348(1)
Conclusion
349(4)
Approximate Reasoning*
353(34)
Classical Reasoning Methods
354(1)
Classical Modus Ponens
354(5)
Generalized Modus Ponens
359(2)
Type 1 Fuzzy Rules
361(4)
Implication Functions
361(1)
S-Implications
362(2)
R-Implications
364(1)
Type 1 Fuzzy Inference: Single Antecedent GMP
365(2)
Information Gap in Type 1 GMP
367(1)
Type 1 Fuzzy Inference: Two Antecedent GMP
368(1)
Decomposition
369(3)
Inference with rule decomposition
370(1)
Inference with operation decomposition
371(1)
Computational Complexity
372(1)
Implementation with Type 1 Reasoning
373(2)
Type 1 Fuzzy System Modeling
375(6)
FATI and FITA Representations
376(1)
Crisp connectives of fuzzy theory
377(1)
Type 1 Implication and Aggregation
378(1)
Type 1 inference with a rule set
379(2)
Defuzzification
381(1)
Parameter optimization
381(1)
Case studies
381(6)
Case 1 - An Industrial Process
382(1)
Case 2 - A nonlinear system
382(1)
Case 3 - Gas furnace model
383(4)
Interval-Valued Type 2 GMP
387(36)
Interval-Valued Type 2 Fuzzy Rules
387(2)
Some Properties Interval-Valued Type 2 Implication
389(2)
Information Gap in Interval-Valued Type 2 GMP
391(4)
Implementations of Interval-Valued Type 2 Reasoning
395(1)
Interval-Valued Type 2 System Modeling
396(5)
Type 1 Representation and Type 2 Reasoning
396(2)
Type 2 Representation and Type 2 Reasoning
398(3)
Application to Case Studies with Interval Valued Type 2 Reasoning
401(22)
Comparison of reasoning methods
405(1)
Conclusion
406(17)
A Theoretical Application of Interval-Valued Type 2 Representation
423(22)
Background
425(4)
Relations and Fuzzy Relations
425(3)
Interval Valued Type 2 Fuzzy Sets
428(1)
Strict Preference
429(12)
Transitivity of the Order Relation
432(3)
Numerical Example
435(1)
Threshold Relations
436(3)
The Case where the Interval Vanishes
439(2)
Non-unfuzzy Non-dominated Elements
441(1)
Conclusion
441(4)
A Foundation for Computing With Words: Metalinguistic Axioms
445(42)
Introduction
445(2)
Meta-Linguistic Axioms
447(3)
Consequences of the Proposed Meta-Linguistic Axioms
450(31)
Classical Axioms
451(1)
Investigation of Meta-Linguistic Axioms
452(1)
Fuzzy Set Theoretic Axioms for CWW
452(1)
Fuzzy Involution
452(1)
Fuzzy Commutativity
453(2)
Fuzzy Associativity
455(5)
Fuzzy Distributivity
460(7)
Fuzzy Idempotency
467(1)
General Fuzzy Absorption
468(2)
Fuzzy Absorption
470(2)
Fuzzy Identity
472(1)
Law of Fuzzy Contradiction ``A And Not(A) Ø''
473(1)
Law of Fuzzy Middle ``A Or Not(A)X''
474(1)
Fuzzy De Morgan Law in Form (1) ``Not(A And B) vs. Not(A) Or Not(B)''
475(3)
Fuzzy De Morgan Law in Form (2) ``Not(A Or B) vs. Not(A) And Not(B)''
478(3)
Meta-Linguistic Reasoning
481(2)
Conclusion
483(4)
Epilogue 487(2)
References 489(16)
Index 505(8)
Author Index 513

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