An Introduction to Many-Valued and Fuzzy Logic : Semantics, Algebras, and Derivation Systems.

By: Bergmann, MerriePublisher: Cambridge : Cambridge University Press, 2007Copyright date: ©2008Description: 1 online resource (343 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9780511376450Subject(s): Fuzzy logicGenre/Form: Electronic books. Additional physical formats: Print version:: An Introduction to Many-Valued and Fuzzy Logic : Semantics, Algebras, and Derivation SystemsDDC classification: 511.313 LOC classification: QA9.64 .B47 2008Online resources: Click to View
Contents:
Cover -- Half-title -- Title -- Copyright -- Dedication -- Contents -- Preface -- 1 Introduction -- 1.1 Issues of Vagueness -- 1.2 Vagueness Defined -- 1.3 The Problem of the Fringe -- 1.4 Preview of the Rest of the Book -- 1.5 History and Scope of Fuzzy Logic -- 1.6 Tall People -- 1.7 Exercises -- 2 Review of Classical Propositional Logic -- 2.1 The Language of Classical Propositional Logic -- 2.2 Semantics of Classical Propositional Logic -- 2.3 Normal Forms -- 2.4 An Axiomatic Derivation System for Classical Propositional Logic -- 2.5 Functional Completeness -- 2.6 Decidability -- 2.7 Exercises -- 3 Review of Classical First-Order Logic -- 3.1 The Language of Classical First-Order Logic -- 3.2 Semantics of Classical First-Order Logic -- 3.3 An Axiomatic Derivation System for Classical First-Order Logic -- 3.4 Exercises -- 4 Alternative Semantics for Truth-Values and Truth-Functions : Numeric Truth-Values and Abstract Algebras -- 4.1 Numeric Truth-Values for Classical Logic -- 4.2 Boolean Algebras and Classical Logic -- 4.3 More Results about Boolean Algebras -- 4.4 Exercises -- 5 5 Three-Valued Propositional Logics: Semantics -- 5.1 Kleene's "Strong" Three-Valued Logic -- 5.2 Lukasiewicz's Three-Valued Logic -- 5.3 Bochvar's Three-Valued Logics -- 5.4 Evaluating Three-Valued Systems -- Quasi-Tautologies and Quasi-Contradictions -- 5.5 Normal Forms -- 5.6 Questions of Interdefinability between the Systems and Functional Completeness -- 5.7 Lukasiewicz's System Expanded -- 5.8 Exercises -- 6 Derivation Systems for Three-Valued Propositional Logic -- 6.1 An Axiomatic System for Tautologies and Validity in Three-Valued Logic -- 6.2 APavelka-Style Derivation System for L3 -- 6.3 Exercises -- 7 Three-Valued First-Order Logics: Semantics -- 7.1 AFirst-Order Generalization of L3 -- 7.2 Quantifiers Based on the Other Three-Valued Systems.
7.3 Tautologies, Validity, and "Quasi-"Semantic Concepts -- 7.4 Exercises -- 8 Derivation Systems for Three-Valued First-Order Logic -- 8.1 An Axiomatic System for Tautologies and Validity in Three-Valued First-Order Logic -- 8.2 APavelka-Style Derivation System for L3 V -- 8.3 Exercises -- 9 Alternative Semantics for Three-Valued Logic -- 9.1 Numeric Truth-Values for Three-Valued Logic -- 9.2 Abstract Algebras for L3,KS3,BI3, and BE3 -- 9.3 MV-Algebras -- 9.4 Exercises -- 10 The Principle of Charity and a New Problem of the Fringe -- 11 Fuzzy Propositional Logics: Semantics -- 11.1 Fuzzy Sets and Degrees of Truth -- 11.2 Lukasiewicz Fuzzy Propositional Logic -- 11.3 Tautologies, Contradictions, and Entailment in Fuzzy Logic -- 11.4 N-Tautologies, Degree-Entailment, and N-Degree-Entailment -- 11.5 Fuzzy Consequence -- 11.6 Fuzzy Generalizations of KS3, BI3, and BE3 -- the Expressive Power of FuzzyL -- 11.7 T-Norms, T-Conorms, and Implication in Fuzzy Logic -- 11.8 Godel Fuzzy Propositional Logic -- 11.9 Product Fuzzy Propositional Logic -- 11.10 Fuzzy External Assertion and Negation -- 11.11 Exercises -- 12 Fuzzy Algebras -- 12.1 More on MV-Algebras -- 12.2 Residuated Lattices and BL-Algebras -- 12.3 Zero and Unit Projections in Algebraic Structures -- 12.4 Exercises -- 13 Derivation Systems for Fuzzy Propositional Logic -- 13.1 An Axiomatic System for Tautologies and Validity in Fuzzy -- 13.2 A Pavelka-Style Derivation System for Fuzzy -- 13.3 An Alternative Axiomatic System for Tautologies and Validity in FuzzyL Based on BL-Algebras -- 13.4 An Axiomatic System for Tautologies and Validity in Fuzzy G -- 13.5 An Axiomatic System for Tautologies and Validity in Fuzzy P -- 13.6 Summary: Comparison of FuzzyL,FuzzyG, and FuzzyP and Their Derivation Systems -- 13.7 External Assertion Axioms -- 13.8 Exercises -- 14 Fuzzy First-Order Logics: Semantics.
14.1 Fuzzy Interpretations -- 14.2 Lukasiewicz Fuzzy First-Order Logic -- 14.3 Tautologies and Other Semantic Concepts -- 14.4 Lukasiewicz Fuzzy Logic and the Problems of Vagueness -- 14.5 Godel Fuzzy First-Order Logic -- 14.6 Product Fuzzy First-Order Logic -- 14.7 The Sorites Paradox: Comparison of FuzzyL…,FuzzyG…, and FuzzyP… -- 14.8 Exercises -- 15 Derivation Systems for Fuzzy First-Order Logic -- 15.1 Axiomatic Systems for Fuzzy First-Order Logic: Overview -- 15.2 APavelka-Style Derivation System for Fuzzy -- 15.3 An Axiomatic Derivation System for Fuzzy -- 15.4 Combining Fuzzy First-Order Logical Systems -- External Assertion -- 15.5 Exercises -- 16 Extensions of Fuzziness -- 16.1 Fuzzy Qualifiers: Hedges -- 16.2 Fuzzy "Linguistic" Truth-Values -- 16.3 Other Fuzzy Extensions of Fuzzy Logic -- 16.4 Exercises -- 17 Fuzzy Membership Functions -- 17.1 Defining Membership Functions -- 17.2 Empirical Construction of Membership Functions -- 17.3 Logical Relevance? -- 17.4 Exercises -- APPENDIX: Basics of Countability and Uncountability -- Bibliography -- Index.
Summary: An accessible introduction to many-valued and fuzzy logic designed for undergraduate and graduate students.
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Cover -- Half-title -- Title -- Copyright -- Dedication -- Contents -- Preface -- 1 Introduction -- 1.1 Issues of Vagueness -- 1.2 Vagueness Defined -- 1.3 The Problem of the Fringe -- 1.4 Preview of the Rest of the Book -- 1.5 History and Scope of Fuzzy Logic -- 1.6 Tall People -- 1.7 Exercises -- 2 Review of Classical Propositional Logic -- 2.1 The Language of Classical Propositional Logic -- 2.2 Semantics of Classical Propositional Logic -- 2.3 Normal Forms -- 2.4 An Axiomatic Derivation System for Classical Propositional Logic -- 2.5 Functional Completeness -- 2.6 Decidability -- 2.7 Exercises -- 3 Review of Classical First-Order Logic -- 3.1 The Language of Classical First-Order Logic -- 3.2 Semantics of Classical First-Order Logic -- 3.3 An Axiomatic Derivation System for Classical First-Order Logic -- 3.4 Exercises -- 4 Alternative Semantics for Truth-Values and Truth-Functions : Numeric Truth-Values and Abstract Algebras -- 4.1 Numeric Truth-Values for Classical Logic -- 4.2 Boolean Algebras and Classical Logic -- 4.3 More Results about Boolean Algebras -- 4.4 Exercises -- 5 5 Three-Valued Propositional Logics: Semantics -- 5.1 Kleene's "Strong" Three-Valued Logic -- 5.2 Lukasiewicz's Three-Valued Logic -- 5.3 Bochvar's Three-Valued Logics -- 5.4 Evaluating Three-Valued Systems -- Quasi-Tautologies and Quasi-Contradictions -- 5.5 Normal Forms -- 5.6 Questions of Interdefinability between the Systems and Functional Completeness -- 5.7 Lukasiewicz's System Expanded -- 5.8 Exercises -- 6 Derivation Systems for Three-Valued Propositional Logic -- 6.1 An Axiomatic System for Tautologies and Validity in Three-Valued Logic -- 6.2 APavelka-Style Derivation System for L3 -- 6.3 Exercises -- 7 Three-Valued First-Order Logics: Semantics -- 7.1 AFirst-Order Generalization of L3 -- 7.2 Quantifiers Based on the Other Three-Valued Systems.

7.3 Tautologies, Validity, and "Quasi-"Semantic Concepts -- 7.4 Exercises -- 8 Derivation Systems for Three-Valued First-Order Logic -- 8.1 An Axiomatic System for Tautologies and Validity in Three-Valued First-Order Logic -- 8.2 APavelka-Style Derivation System for L3 V -- 8.3 Exercises -- 9 Alternative Semantics for Three-Valued Logic -- 9.1 Numeric Truth-Values for Three-Valued Logic -- 9.2 Abstract Algebras for L3,KS3,BI3, and BE3 -- 9.3 MV-Algebras -- 9.4 Exercises -- 10 The Principle of Charity and a New Problem of the Fringe -- 11 Fuzzy Propositional Logics: Semantics -- 11.1 Fuzzy Sets and Degrees of Truth -- 11.2 Lukasiewicz Fuzzy Propositional Logic -- 11.3 Tautologies, Contradictions, and Entailment in Fuzzy Logic -- 11.4 N-Tautologies, Degree-Entailment, and N-Degree-Entailment -- 11.5 Fuzzy Consequence -- 11.6 Fuzzy Generalizations of KS3, BI3, and BE3 -- the Expressive Power of FuzzyL -- 11.7 T-Norms, T-Conorms, and Implication in Fuzzy Logic -- 11.8 Godel Fuzzy Propositional Logic -- 11.9 Product Fuzzy Propositional Logic -- 11.10 Fuzzy External Assertion and Negation -- 11.11 Exercises -- 12 Fuzzy Algebras -- 12.1 More on MV-Algebras -- 12.2 Residuated Lattices and BL-Algebras -- 12.3 Zero and Unit Projections in Algebraic Structures -- 12.4 Exercises -- 13 Derivation Systems for Fuzzy Propositional Logic -- 13.1 An Axiomatic System for Tautologies and Validity in Fuzzy -- 13.2 A Pavelka-Style Derivation System for Fuzzy -- 13.3 An Alternative Axiomatic System for Tautologies and Validity in FuzzyL Based on BL-Algebras -- 13.4 An Axiomatic System for Tautologies and Validity in Fuzzy G -- 13.5 An Axiomatic System for Tautologies and Validity in Fuzzy P -- 13.6 Summary: Comparison of FuzzyL,FuzzyG, and FuzzyP and Their Derivation Systems -- 13.7 External Assertion Axioms -- 13.8 Exercises -- 14 Fuzzy First-Order Logics: Semantics.

14.1 Fuzzy Interpretations -- 14.2 Lukasiewicz Fuzzy First-Order Logic -- 14.3 Tautologies and Other Semantic Concepts -- 14.4 Lukasiewicz Fuzzy Logic and the Problems of Vagueness -- 14.5 Godel Fuzzy First-Order Logic -- 14.6 Product Fuzzy First-Order Logic -- 14.7 The Sorites Paradox: Comparison of FuzzyL…,FuzzyG…, and FuzzyP… -- 14.8 Exercises -- 15 Derivation Systems for Fuzzy First-Order Logic -- 15.1 Axiomatic Systems for Fuzzy First-Order Logic: Overview -- 15.2 APavelka-Style Derivation System for Fuzzy -- 15.3 An Axiomatic Derivation System for Fuzzy -- 15.4 Combining Fuzzy First-Order Logical Systems -- External Assertion -- 15.5 Exercises -- 16 Extensions of Fuzziness -- 16.1 Fuzzy Qualifiers: Hedges -- 16.2 Fuzzy "Linguistic" Truth-Values -- 16.3 Other Fuzzy Extensions of Fuzzy Logic -- 16.4 Exercises -- 17 Fuzzy Membership Functions -- 17.1 Defining Membership Functions -- 17.2 Empirical Construction of Membership Functions -- 17.3 Logical Relevance? -- 17.4 Exercises -- APPENDIX: Basics of Countability and Uncountability -- Bibliography -- Index.

An accessible introduction to many-valued and fuzzy logic designed for undergraduate and graduate students.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2019. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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