1st Edition

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

378 Pages

by
CRC Press

378 Pages

by
CRC Press

**Also available as eBook on:**

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:

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.

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

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

### Biography

Mircea S. Reghis (Author) , Eugene Roventa (York University) (Author)

--Quan Lei, Zentralblatt MATH, Vol. 944