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Table of Contents
Intro
Foreword
Preface
Contents
Contributors
The Fuzzy Theoretic Turn
Membership Functions
1 Introduction
2 Mathematical Aspects of Membership Functions
2.1 Classical Versus Gradual Properties
2.2 Graduality and Partial Pre-orderings
2.3 Membership Scale: [0, 1] or Not
2.4 Bipolar Scale or Not?
2.5 Beyond Scalar Membership Grades
3 Finding Membership Grades
3.1 Based on Similarity
3.2 Fuzzy Sets as Possibility Distributions
3.3 Fuzzy Sets in Preference Evaluation
4 Concluding Remarks
References
The Evolution of the Notion of Overlap Functions
1 The Origin of the Notion of Overlap Function
2 The Mathematical Definition of Overlap Functions
3 Overlap Functions Versus Triangular Norms
4 n-Dimensional Overlap Functions
5 Applications of Overlap Functions
6 Conclusions
References
Interpolative Reasoning: Valid, Specificity-Gradual and Similarity-Based
1 Introduction
2 Interpolative Reasoning
2.1 Similarity-Based Interpolative Reasoning
3 Similarity Based Axioms for Interpolative Reasoning
3.1 Behaviour When Observation Equals a Premise of a Known Rule
3.2 Homogeneity of Observation-Premise Specificity with Respect to Conclusions
3.3 Monotonicity with Respect to Observation-Conclusion Specificities
3.4 Valid Convex Conclusions
3.5 Graduality with Respect to Rule's Proximity
4 Success and Limitations of Similarity-Based Interpolative Reasoning
4.1 Non-convex Solutions
4.2 Valid and Specificity-Gradual Solution
5 Conclusions
References
A Similarity-Based Three-Valued Modal Logic Approach to Reason with Prototypes and Counterexamples
1 Introduction
2 Prototypes, Counter-Examples and Borderline Cases: A Simple 3-Valued Model for Gradual Properties
2.1 A 3-Valued Approach
2.2 A Refresher on 3-Valued Łukasiewicz Logic Ł3
2.3 Dealing with both Prototypes and Counter-Examples: The Logic Łleq3
3 A Similarity-Based Refined Framework
4 A Multi-modal Approach to Reason About the Similarity-Based Graded Entailments
5 Conclusions and Dedication
References
Analogy
1 Introducing Analogy as One of Two Main Ways of Thinking
2 The Necessity of Fuzzy Logic Computation of Analogical Reasoning and Schemes
3 What Analogy is and What Analogy is Not
4 Models of Solving Analogies and Metaphors for Fuzzy Inference Making
5 Discussion
References
The Role of the Context in Decision and Optimization Problems
1 Introduction
2 The General Decision Problem
3 An Illustrative Example
3.1 Induced Framework
3.2 Ethical Framework
3.3 Sustainability Framework
3.4 Corporate Social Responsibility Framework
4 Conclusions
References
Decision Rules Under Vague and Uncertain Information
1 Introduction
2 Coherent Conditional Decomposable Measures
Foreword
Preface
Contents
Contributors
The Fuzzy Theoretic Turn
Membership Functions
1 Introduction
2 Mathematical Aspects of Membership Functions
2.1 Classical Versus Gradual Properties
2.2 Graduality and Partial Pre-orderings
2.3 Membership Scale: [0, 1] or Not
2.4 Bipolar Scale or Not?
2.5 Beyond Scalar Membership Grades
3 Finding Membership Grades
3.1 Based on Similarity
3.2 Fuzzy Sets as Possibility Distributions
3.3 Fuzzy Sets in Preference Evaluation
4 Concluding Remarks
References
The Evolution of the Notion of Overlap Functions
1 The Origin of the Notion of Overlap Function
2 The Mathematical Definition of Overlap Functions
3 Overlap Functions Versus Triangular Norms
4 n-Dimensional Overlap Functions
5 Applications of Overlap Functions
6 Conclusions
References
Interpolative Reasoning: Valid, Specificity-Gradual and Similarity-Based
1 Introduction
2 Interpolative Reasoning
2.1 Similarity-Based Interpolative Reasoning
3 Similarity Based Axioms for Interpolative Reasoning
3.1 Behaviour When Observation Equals a Premise of a Known Rule
3.2 Homogeneity of Observation-Premise Specificity with Respect to Conclusions
3.3 Monotonicity with Respect to Observation-Conclusion Specificities
3.4 Valid Convex Conclusions
3.5 Graduality with Respect to Rule's Proximity
4 Success and Limitations of Similarity-Based Interpolative Reasoning
4.1 Non-convex Solutions
4.2 Valid and Specificity-Gradual Solution
5 Conclusions
References
A Similarity-Based Three-Valued Modal Logic Approach to Reason with Prototypes and Counterexamples
1 Introduction
2 Prototypes, Counter-Examples and Borderline Cases: A Simple 3-Valued Model for Gradual Properties
2.1 A 3-Valued Approach
2.2 A Refresher on 3-Valued Łukasiewicz Logic Ł3
2.3 Dealing with both Prototypes and Counter-Examples: The Logic Łleq3
3 A Similarity-Based Refined Framework
4 A Multi-modal Approach to Reason About the Similarity-Based Graded Entailments
5 Conclusions and Dedication
References
Analogy
1 Introducing Analogy as One of Two Main Ways of Thinking
2 The Necessity of Fuzzy Logic Computation of Analogical Reasoning and Schemes
3 What Analogy is and What Analogy is Not
4 Models of Solving Analogies and Metaphors for Fuzzy Inference Making
5 Discussion
References
The Role of the Context in Decision and Optimization Problems
1 Introduction
2 The General Decision Problem
3 An Illustrative Example
3.1 Induced Framework
3.2 Ethical Framework
3.3 Sustainability Framework
3.4 Corporate Social Responsibility Framework
4 Conclusions
References
Decision Rules Under Vague and Uncertain Information
1 Introduction
2 Coherent Conditional Decomposable Measures