Classical and Fuzzy Concepts in Mathematical Logic and Applications, Professional Version: 1st Edition (Hardback) book cover

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

1st Edition

By Mircea S. Reghis, Eugene Roventa

CRC Press

384 pages

Purchasing Options:$ = USD
Hardback: 9780849331978
pub: 1998-05-20
$170.00
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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.

  • 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

    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

    Subject Categories

    BISAC Subject Codes/Headings:
    COM000000
    COMPUTERS / General
    MAT003000
    MATHEMATICS / Applied
    MAT028000
    MATHEMATICS / Set Theory