1st Edition

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

By Mircea S. Reghis, Eugene Roventa Copyright 1998
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:

• Propositional and predicate logic
• Logic networks
• Logic programming
• Proof of correctness
• Semantics
• Syntax
• Completenesss
• 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.
• 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

### Biography

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

"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