Preface | p. VII |
Notations and Some Preliminaries | p. XI |
An Introduction to Fuzzy Implications | p. 1 |
Definition and Basic Examples | p. 1 |
Continuity of Fuzzy Implications | p. 7 |
Basic Properties of Fuzzy Implications | p. 9 |
Negations from Fuzzy Implications | p. 13 |
Fuzzy Negations | p. 13 |
Natural Negations of Fuzzy Implications | p. 17 |
Laws of Contraposition | p. 20 |
Reciprocal Fuzzy Implications | p. 28 |
Bibliographical Remarks | p. 32 |
Analytical Study of Fuzzy Implications | |
Fuzzy Implications from Fuzzy Logic Operations | p. 41 |
Fuzzy Conjunctions: Triangular Norms | p. 41 |
Fuzzy Disjunctions: Triangular Conorms | p. 45 |
Relationships between Negations, T-Norms and T-Conorms | p. 49 |
Natural Negations of T-Norms and T-Conorms | p. 49 |
Laws of Excluded Middle and Contradiction | p. 52 |
De Morgan Triples | p. 55 |
(S,N)-Implications and S-Implications | p. 57 |
Motivation, Definition and Examples | p. 57 |
Characterizations of (S,N)-Implications | p. 59 |
(S,N)-Implications and the Identity Principle | p. 64 |
(S,N)-Implications and the Ordering Property | p. 64 |
Intersections between Subfamilies of (S,N)-Implications | p. 67 |
R-Implications | p. 68 |
Motivation, Definition and Examples | p. 68 |
Properties of R-Implications | p. 70 |
Characterizations and Representations of R-Implications | p. 75 |
R-Implications and Laws of Contraposition | p. 85 |
Intersections between Subfamilies of R-Implications | p. 89 |
QL-Implications | p. 90 |
Definition, Examples and Basic Properties | p. 91 |
QL-Implications and the Exchange Principle | p. 99 |
QL-Implications and the Identity Principle | p. 100 |
QL-Implications and the Ordering Property | p. 101 |
QL-Implications and the Law of Contraposition | p. 103 |
Bibliographical Remarks | p. 105 |
Fuzzy Implications from Generator Functions | p. 109 |
f-Generated Implications | p. 109 |
Definition and Examples | p. 109 |
Properties of f-Implications | p. 111 |
g-Generated Implications | p. 115 |
Definition and Examples | p. 116 |
Properties of g-Implications | p. 118 |
Bibliographical Remarks | p. 124 |
Intersections between Families of Fuzzy Implications | p. 127 |
Intersections between (S,N)- and R-Implications | p. 127 |
Intersections between (S,N)- and QL-Implications | p. 131 |
Intersections between R- and QL-Implications | p. 135 |
Intersections between Yager's f- and g-Implications | p. 137 |
Intersections between Yager's and (S,N)-Implications | p. 139 |
Intersections between Yager's and R-Implications | p. 141 |
Intersections between Yager's and QL-Implications | p. 141 |
Bibliographical Remarks | p. 142 |
Fuzzy Implications from Uninorms | p. 145 |
Uninorms | p. 145 |
Definitions and Examples | p. 145 |
Pseudo-continuous Uninorms | p. 146 |
Idempotent Uninorms | p. 147 |
Representable Uninorms | p. 149 |
Natural Negations of Fuzzy Implications - Revisited | p. 150 |
(U,N)-Implications | p. 151 |
Definition and Basic Properties | p. 152 |
Characterizations of (U,N)-Implications | p. 155 |
RU-Implications | p. 160 |
Definition and Basic Properties | p. 160 |
RU-Implications from Pseudo-continuous Uninorms | p. 162 |
RU-Implications from Representable Uninorms | p. 166 |
RU-Implications from Idempotent Uninorms | p. 168 |
Intersections between (U,N)- and RU-Implications | p. 172 |
Intersection between I[subscript U,N] and I[subscript U subscript M] | p. 172 |
Intersection between I[subscript U,N] and I[subscript U subscript R] | p. 172 |
Intersection between I[subscript U,N] and I[subscript U subscript I] | p. 173 |
Intersection between I[subscript U subscript M] and I[subscript U subscript R] | p. 174 |
Intersection between I[subscript U subscript M] and I[subscript U subscript I] | p. 174 |
Intersection between I[subscript U subscript R] and I[subscript U subscript I] | p. 175 |
Bibliographical Remarks | p. 175 |
Algebraic Study of Fuzzy Implications | |
Algebraic Structures of Fuzzy Implications | p. 183 |
Lattice of Fuzzy Implications | p. 183 |
Convex Classes of Fuzzy Implications | p. 188 |
Conjugacy Classes of Fuzzy Implications | p. 190 |
Semigroups of Fuzzy Implications | p. 197 |
Composition of Fuzzy Implications | p. 197 |
Semigroups of Fuzzy Implications | p. 201 |
Bibliographical Remarks | p. 205 |
Fuzzy Implications and Some Functional Equations | p. 207 |
Contrapositive Symmetrization of Fuzzy Implications | p. 207 |
Upper and Lower Contrapositivisations | p. 208 |
Medium Contrapositivisation | p. 212 |
Distributivity of Fuzzy Implications | p. 214 |
On the Equation I(S(x, y), z) = T(I(x, z), I(y, z)) | p. 215 |
On the Equation I(T(x, y), z) = S(I(x, z), I(y, z)) | p. 217 |
On the Equation I(x, T[subscript 1](y, z)) = T[subscript 2](I(x, y), I(x, z)) | p. 218 |
On the Equation I(x, S[subscript 1](y, z)) = S[subscript 2](I(x, y), I(x, z)) | p. 221 |
The Law of Importation | p. 223 |
(S,N)-Implications and the Law of Importation | p. 224 |
R-Implications and the Law of Importation | p. 225 |
QL-Implications and the Law of Importation | p. 226 |
f- and g-Implications and the Law of Importation | p. 227 |
Fuzzy Implications and T-Conditionality | p. 229 |
(S,N)-Implications and T-Conditionality | p. 231 |
R-Implications and T-Conditionality | p. 232 |
QL-Implications and T-Conditionality | p. 233 |
Characterization through Functional Equations | p. 233 |
Bibliographical Remarks | p. 237 |
Applicational Study of Fuzzy Implications | |
Fuzzy Implications in Approximate Reasoning | p. 245 |
Approximate Reasoning | p. 246 |
Classical Implication in Inference Schemas | p. 246 |
Fuzzy Implication in Inference Schemas | p. 246 |
Approximate Reasoning | p. 247 |
Fuzzy IF-THEN Rules | p. 247 |
Possibility Distribution | p. 247 |
Fuzzy Statements | p. 247 |
Fuzzy IF-THEN Rules | p. 248 |
Inference Schemes in Approximate Reasoning | p. 249 |
Generalized Modus Ponens (GMP) | p. 249 |
Compositional Rule of Inference (CRI) | p. 250 |
Inference in CRI with Multiple Rules | p. 253 |
Similarity Based Reasoning (SBR) | p. 254 |
Effectiveness of Inference Schemes in AR | p. 257 |
GMP Rules and AR | p. 258 |
Function Approximation and AR | p. 261 |
Efficiency of Inference Schemes in AR | p. 263 |
Modification of the CRI Inference Algorithm | p. 264 |
Transformation of the Structure of the Rules | p. 268 |
Bibliographical Remarks | p. 275 |
Appendix | |
Some Results on Real Functions | p. 281 |
References | p. 289 |
List of Figures | p. 303 |
List of Tables | p. 305 |
Index | p. 307 |
Table of Contents provided by Ingram. All Rights Reserved. |
The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.
The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.