Preface |
Notations and Some Preliminaries |
An Introduction to Fuzzy Implications / 1: |
Definition and Basic Examples / 1.1: |
Continuity of Fuzzy Implications / 1.2: |
Basic Properties of Fuzzy Implications / 1.3: |
Negations from Fuzzy Implications / 1.4: |
Fuzzy Negations / 1.4.1: |
Natural Negations of Fuzzy Implications / 1.4.2: |
Laws of Contraposition / 1.5: |
Reciprocal Fuzzy Implications / 1.6: |
Bibliographical Remarks / 1.7: |
Analytical Study of Fuzzy Implications / Part I: |
Fuzzy Implications from Fuzzy Logic Operations / 2: |
Fuzzy Conjunctions: Triangular Norms / 2.1: |
Fuzzy Disjunctions: Triangular Conorms / 2.2: |
Relationships between Negations, T-Norms and T-Conorms / 2.3: |
Natural Negations of T-Norms and T-Conorms / 2.3.1: |
Laws of Excluded Middle and Contradiction / 2.3.2: |
De Morgan Triples / 2.3.3: |
(S,N)-Implications and S-Implications / 2.4: |
Motivation, Definition and Examples / 2.4.1: |
Characterizations of (S,N)-Implications / 2.4.2: |
(S,N)-Implications and the Identity Principle / 2.4.3: |
(S,N)-Implications and the Ordering Property / 2.4.4: |
Intersections between Subfamilies of (S,N)-Implications / 2.4.5: |
R-Implications / 2.5: |
Properties of R-Implications / 2.5.1: |
Characterizations and Representations of R-Implications / 2.5.3: |
R-Implications and Laws of Contraposition / 2.5.4: |
Intersections between Subfamilies of R-Implications / 2.5.5: |
QL-Implications / 2.6: |
Definition, Examples and Basic Properties / 2.6.1: |
QL-Implications and the Exchange Principle / 2.6.2: |
QL-Implications and the Identity Principle / 2.6.3: |
QL-Implications and the Ordering Property / 2.6.4: |
QL-Implications and the Law of Contraposition / 2.6.5: |
Fuzzy Implications from Generator Functions / 2.7: |
f-Generated Implications / 3.1: |
Definition and Examples / 3.1.1: |
Properties of f-Implications / 3.1.2: |
g-Generated Implications / 3.2: |
Properties of g-Implications / 3.2.1: |
Intersections between Families of Fuzzy Implications / 3.3: |
Intersections between (S,N)- and R-Implications / 4.1: |
Intersections between (S,N)- and QL-Implications / 4.2: |
Intersections between R- and QL-Implications / 4.3: |
Intersections between Yager's f- and g-Implications / 4.4: |
Intersections between Yager's and (S,N)-Implications / 4.5: |
Intersections between Yager's and R-Implications / 4.6: |
Intersections between Yager's and QL-Implications / 4.7: |
Fuzzy Implications from Uninorms / 4.8: |
Uninorms / 5.1: |
Definitions and Examples / 5.1.1: |
Pseudo-continuous Uninorms / 5.1.2: |
Idempotent Uninorms / 5.1.3: |
Representable Uninorms / 5.1.4: |
Natural Negations of Fuzzy Implications - Revisited / 5.2: |
(U,N)-Implications / 5.3: |
Definition and Basic Properties / 5.3.1: |
Characterizations of (U,N)-Implications / 5.3.2: |
RU-Implications / 5.4: |
RU-Implications from Pseudo-continuous Uninorms / 5.4.1: |
RU-Implications from Representable Uninorms / 5.4.3: |
RU-Implications from Idempotent Uninorms / 5.4.4: |
Intersections between (U,N)- and RU-Implications / 5.5: |
Intersection between I[subscript U,N] and I[subscript U subscript M] / 5.5.1: |
Intersection between I[subscript U,N] and I[subscript U subscript R] / 5.5.2: |
Intersection between I[subscript U,N] and I[subscript U subscript I] / 5.5.3: |
Intersection between I[subscript U subscript M] and I[subscript U subscript R] / 5.5.4: |
Intersection between I[subscript U subscript M] and I[subscript U subscript I] / 5.5.5: |
Intersection between I[subscript U subscript R] and I[subscript U subscript I] / 5.5.6: |
Algebraic Study of Fuzzy Implications / 5.6: |
Algebraic Structures of Fuzzy Implications / 6: |
Lattice of Fuzzy Implications / 6.1: |
Convex Classes of Fuzzy Implications / 6.2: |
Conjugacy Classes of Fuzzy Implications / 6.3: |
Semigroups of Fuzzy Implications / 6.4: |
Composition of Fuzzy Implications / 6.4.1: |
Fuzzy Implications and Some Functional Equations / 6.4.2: |
Contrapositive Symmetrization of Fuzzy Implications / 7.1: |
Upper and Lower Contrapositivisations / 7.1.1: |
Medium Contrapositivisation / 7.1.2: |
Distributivity of Fuzzy Implications / 7.2: |
On the Equation I(S(x, y), z) = T(I(x, z), I(y, z)) / 7.2.1: |
On the Equation I(T(x, y), z) = S(I(x, z), I(y, z)) / 7.2.2: |
On the Equation I(x, T[subscript 1](y, z)) = T[subscript 2](I(x, y), I(x, z)) / 7.2.3: |
On the Equation I(x, S[subscript 1](y, z)) = S[subscript 2](I(x, y), I(x, z)) / 7.2.4: |
The Law of Importation / 7.3: |
(S,N)-Implications and the Law of Importation / 7.3.1: |
R-Implications and the Law of Importation / 7.3.2: |
QL-Implications and the Law of Importation / 7.3.3: |
f- and g-Implications and the Law of Importation / 7.3.4: |
Fuzzy Implications and T-Conditionality / 7.4: |
(S,N)-Implications and T-Conditionality / 7.4.1: |
R-Implications and T-Conditionality / 7.4.2: |
QL-Implications and T-Conditionality / 7.4.3: |
Characterization through Functional Equations / 7.5: |
Applicational Study of Fuzzy Implications / 7.6: |
Fuzzy Implications in Approximate Reasoning / 8: |
Approximate Reasoning / 8.1: |
Classical Implication in Inference Schemas / 8.1.1: |
Fuzzy Implication in Inference Schemas / 8.1.2: |
Fuzzy IF-THEN Rules / 8.1.3: |
Possibility Distribution / 8.2.1: |
Fuzzy Statements / 8.2.2: |
Inference Schemes in Approximate Reasoning / 8.2.3: |
Generalized Modus Ponens (GMP) / 8.3.1: |
Compositional Rule of Inference (CRI) / 8.3.2: |
Inference in CRI with Multiple Rules / 8.3.3: |
Similarity Based Reasoning (SBR) / 8.3.4: |
Effectiveness of Inference Schemes in AR / 8.4: |
GMP Rules and AR / 8.4.1: |
Function Approximation and AR / 8.4.2: |
Efficiency of Inference Schemes in AR / 8.5: |
Modification of the CRI Inference Algorithm / 8.5.1: |
Transformation of the Structure of the Rules / 8.5.2: |
Appendix / 8.6: |
Some Results on Real Functions / A: |
References |
List of Figures |
List of Tables |
Index |
Preface |
Notations and Some Preliminaries |
An Introduction to Fuzzy Implications / 1: |