Classical and Fuzzy Concepts in Mathematical Logic and Applications, Professional Version  book cover
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Classical and Fuzzy Concepts in Mathematical Logic and Applications, Professional Version



ISBN 9780849331978
Published May 20, 1998 by CRC Press
378 Pages

 
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Book Description

Classical and Fuzzy Concepts in Mathematical Logic and Applications provides a broad, thorough coverage of the fundamentals of two-valued logic, multivalued logic, and fuzzy logic.
Exploring the parallels between classical and fuzzy mathematical logic, the book examines the use of logic in computer science, addresses questions in automatic deduction, and describes efficient computer implementation of proof techniques.
Specific issues discussed include:

  • Propositional and predicate logic
  • Logic networks
  • Logic programming
  • Proof of correctness
  • Semantics
  • Syntax
  • Completenesss
  • Non-contradiction
  • Theorems of Herbrand and Kalman
    The authors consider that the teaching of logic for computer science is biased by the absence of motivations, comments, relevant and convincing examples, graphic aids, and the use of color to distinguish language and metalanguage. Classical and Fuzzy Concepts in Mathematical Logic and Applications discusses how the presence of these facts trigger a stirring, decisive insight into the understanding process. This view shapes this work, reflecting the authors' subjective balance between the scientific and pedagogic components of the textbook.
    Usually, problems in logic lack relevance, creating a gap between classroom learning and applications to real-life problems. The book includes a variety of application-oriented problems at the end of almost every section, including programming problems in PROLOG III. With the possibility of carrying out proofs with PROLOG III and other software packages, readers will gain a first-hand experience and thus a deeper understanding of the idea of formal proof.
  • Table of Contents

    Preliminaries of "Naive" Mathematical Logic
    PART I. Propositional Logic
    The Formal Language of Propositional Logic
    The Formal Language Lo of Propositional Logic Using Parentheses
    The Formal Language Lo of Propositional Logic without Parentheses (Polish Notation)
    The Truth Structure on Lo in Semantic Version
    Boolean Interpretations of the Language Lo
    Semantic Deduction
    The Semantic Lindenbaum Algebra of Lo
    The Truth Structure of Lo in the Syntactic Version
    The System of Hilbert H: Axioms, Inference, Theorems
    Metatheorems
    The Syntactic Lindenbaum Algebra of Lo. Normal Formulas
    Connections between the Truth Structures on Lo in Semantic and Syntactic Versions
    All Theorems Are Tautologies (Soundness of Prepositional Logic)
    All Tautologies Are Theorems (Completeness of Propositional Logic)
    Another Proof of the Completeness Metatheorem
    Other Syntactic Versions of the Truth Structure on Lo
    The Systems L and M; Their Equivalence to the System H
    Some Remarks about the Independence of Axioms
    The System C of Lukasiewicz and Tarski
    Elements of Fuzzy Propositional Logic
    Some Elementary Notions about Fuzzy Sets
    The Language of Fuzzy Propositional Logic
    The Semantic Truth Structure of Fuzzy Propositional Logic
    Elements of Fuzzy Propositional Logic in Syntactic Version
    Applications of Propostional Logic in Computer Science
    Recall about Lindenbaum Algebra of the Language Lo
    Some Connections of Lo with Programming Languages
    Karnaugh Maps
    Switching Networks
    Logical Networks
    Exercises for Part I
    PART II. Predicate Logic
    Introductory Considerations
    The Formal Language of Predicate Logic
    The Formal Alphabet of Predicate Logic; Formal Words
    Terms and Formulas
    The Semantic Truth Structure on the Language L of Predicate Logic
    The Notion of Interpretation of the Language L
    Semantic Deduction in Predicate Logic
    The Syntactic Truth Structure on the Language L of Predicate Logic
    Axioms, Theorems
    Some Remarkable Metatheorems
    Completeness of Predicate Logic
    Elements of Fuzzy Predicate Logic
    The Language of Fuzzy Predicate Logic
    The Semantic Truth Structure of Fuzzy Predicate Logic
    The Syntactic Truth Structure of Fuzzy Predicate Logic
    Further Applications of Logic in Computer Science
    Elements of the Theory of Resolution
    Elements of Logical Foundations of Prolog
    Elements of Approximate Reasoning for Expert Systems Design
    Exercises for Part II
    A. Boolean Algebras
    B. MV-Algebras
    C. General Considerations about Fuzzy Sets
    Index
    References

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    Reviews

    "This textbook is useful for students at the advanced undergraduate level in mathematics, computer science and engineering; it could be helpful for university teachers, engineers and any person interested in learning and applying logical concepts."
    --Quan Lei, Zentralblatt MATH, Vol. 944