An Ontological and Epistemological Perspective of Fuzzy Set Theory

by Türksen

• ISBN13: 9780444518910
• ISBN10: 0444518916
• Format: Hardcover
• Copyright: 2005-12-01
• Publisher: Elsevier Science

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