# New book on Mathematical Principles of Fuzzy Logic

Vilem Novak (novakv@prf1.osu.cz)
Thu, 2 Sep 1999 17:26:33 +0200 (MET DST)

Dear Colleagues,

let me announce that a new book has just been published:

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Vilem Novak, Irina Perfilieva, Jiri Mockor

MATHEMATICAL PRINCIPLES OF FUZZY LOGIC

Kluwer Academic Publishers, Boston/Dordrecht /London 1999.
ISBN 0-7923-8595-0

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MATHEMATICAL PRINCIPLES OF FUZZY LOGIC provides a systematic
study of the formal theory of fuzzy logic. The book is based on
logical formalism demonstrating that fuzzy logic is a well-developed
logical theory. It includes the theory of functional systems in fuzzy
logic, providing an explanation of what, and how it can be represented
by formulas of fuzzy logic calculi. It also presents a more general
interpretation of fuzzy logic within the environment of other proper
categories of fuzzy sets stemming either from the topos theory, or even
generalizing the latter.

This book presents fuzzy logic as the mathematical theory of vagueness
as well as the theory of the common-sense human reasoning, which
is based on the use of natural language, the distinguishing feature
is vagueness of its semantics.

MATHEMATICAL PRINCIPLES OF FUZZY LOGIC will be of interest to all
researchers of fuzzy logic, including mathematicians and computer
scientists interested in the mathematical aspects of fuzzy logic. It
may be used as a text in advanced level courses on various fuzzy logic
applications, artificial and computational intelligence,
decision-making and more.

Preface

1. Fuzzy logic: what, why, for which?
1.1 Vagueness and Uncertainty
1.2 Vagueness and Fuzzy Sets
1.3 What Is Fuzzy Logic
1.4 Outline of the Agenda of Fuzzy Logic

2. Algebraic structures for logical calculi
2.1 Algebras for Logics
2.2 Filters and Representation Theorems
2.3 Elements of the theory of t-norms
2.4 Introduction to Topos Theory

3. Logical calculi and model theory
3.1 Classical Logic
3.2 Classical Model Theory
3.3 Formal Logical Systems
3.4 Model Theory in Categories

4. Fuzzy logic in narrow sense
4.2 Truth Values
4.3 Predicate Fuzzy Logic of First-Order
4.4 Fuzzy Theories with Equality and Open Fuzzy Theories
4.5 Model Theory in Fuzzy Logic
4.6 Recursive Properties of Fuzzy Theories

5. Functional Systems in Fuzzy Logic Theories
5.1 Fuzzy Logic Functions and Their Representation by Formulas
5.2 Normal Forms for FL-Functions and Formulas of Propositional Fuzzy
Logic
5.3 FL-Relations and Their Connection with Formulas of Predicate
Fuzzy Logic
5.4 Approximation of Continuous Functions by Fuzzy Logic Normal Forms
5.5 Representation of Continuous Functions by the Conjunctive Normal
Form

6. Fuzzy Logic in Broader Sense
6.1 Partial Formalization of Natural Language
6.2 Formal Scheme of FLb
6.3 Special Theories in FLb

7. Topoi and categories of fuzzy sets
7.1 Category of Omega-sets as generalization of fuzzy sets
7.2 Category of Omega-fuzzy sets
7.3 Interpretation of formulas in the category C\Omega-Set

8. Few historical and concluding remarks

References
Index

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Vilem Novak, DSc., Associate Professor
University of Ostrava
IRAFM (Institute for Research and Applications of Fuzzy Modeling)
Brafova 7
701 03 Ostrava 1
Czech Republic

tel: +420-69-6160 234
fax: +420-69-6120 478
mob: +420-602-576 477
e-mail: Vilem.Novak@osu.cz, novakv@osu.cz
WEB: http://www1.osu.cz/irafm
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