Table Of ContentMathematical Principles of Fuzzy Logic
THE KLUWER INTERNATIONAL SERIES
IN ENGINEERING AND COMPUTER SCIENCE
MATHEMATICAL PRINCIPLES OF FUZZY
LOGIC
VILEM NOVĂK
University of Ostrava
Institute for Research and Applications of Fuzzy Modeling
Brafova 7, 701 03 Ostrava 1, Czech Republic
IRINA PERFILIEVA
Moscow State Academy of Instrument Making
Stromynka 20, 107846 Moscow, Russia
and
University of Ostrava
Institute for Research and Applications of Fuzzy Modeling
Brafova 7, 701 03 Ostrava 1, Czech Republic
JIRI MOCKOR
University of Ostrava
Institute for Research and Applications of Fuzzy Modeling
Brafova 7, 701 03 Ostrava 1, Czech Republic
....
"
Springer-Science+Business Media, LLC
Library of Congress Cataloging-in-Publication Data
Novak, Vilem.
Mathematical principles of fuzzy logic I Vilem Novak, Irina
Perfilieva and Jiri Mockor.
p. cm. -- (The Kluwer international series in engineering and
computer science ; secs 517)
Includes bibliographical references and index.
ISBN 978-1-4613-7377-3 ISBN 978-1-4615-5217-8 (eBook)
DOI 10.1007/978-1-4615-5217-8
1. Fuzzy logic. 1. Perfilieva, Irina. 1953- . II. Mockof,
Jin. III. Title. IV. Series.
QA9.64.N68 1999
511.3--dc21 99-37210
CIP
Copyright © 1999 by Springer Science+Business Media New York
Originally published by Kluwer Academic Publishers in 1999
Softcover reprint ofthe hardcover Ist edition 1999
AII rights reserved. No part of this publication may be reproduced, stored in a
retrieval system or transmitted in any form or by any means, mechanical, photo
copying, recording, or otherwise, without the prior written permission of the
publisher, Springer-8cience+Business Media, LLC.
Printed an acid-free paper.
To David and Martin,
to Vitalij,
to Sona, Marek and Katerina,
with love
Contents
Preface xi
1. FUZZY LOGIC: WHAT, WHY, FOR WHICH? 1
1.1 Vagueness and Uncertainty 2
1.2 Vagueness and Fuzzy Sets 6
1.3 What Is Fuzzy Logic 9
1.4 Outline of the Agenda of Fuzzy Logic 12
2. ALGEBRAIC STRUCTURES FOR LOGICAL CALCULI 15
2.1 Algebras for Logics 16
2.1.1 Boolean algebras 16
2.1.2 Residuated lattices and MV-algebras 23
2.2 Filters and Representation Theorems 35
2.3 Elements of the theory of t-norms 41
2.4 Introduction to Topos Theory 52
2.4.1 T opos theory 52
2.4.2 Lattice of subobjects in topos 56
3. LOGICAL CALCULI AND MODEL THEORY 61
3.1 Classical Logic 61
3.1.1 Propositional logic 62
3.1.2 Predicate logic 67
3.1.3 Many-sorted predicate logic 73
3.2 Classical Model Theory 74
3.3 Formal Logical Systems 77
3.4 Model Theory in Categories 83
4. FUZZY LOGIC IN NARROW SENSE 95
4.1 Graded Formal Logical Systems 97
4.2 Truth Values 106
4.2.1 Intuition on truth values and operations with them 106
4.2.2 Truth values as algebras 111
4.2.3 Why Lukasiewicz algebra 112
4.2.4 Enriching structure of truth values 113
4.3 Predicate Fuzzy Logic of First-Order 114
Vlll MATHEMATICAL PRINCIPLES OF FUZZY LOGIC
4.3.1 Syntax and semantics 115
4.3.2 Basic properties of fuzzy theories 120
4.3.3 Consistency of fuzzy theories 129
4.3.4 Extension of fuzzy theories 132
4.3.5 Henkin fuzzy theories 139
4.3.6 Complete fuzzy theories 140
4.3.7 Lindenbaum algebra of formulas and its properties 142
4.3.8 Completeness theorems 147
4.3.9 Sorites fuzzy theories 150
4.3.10 Partial inconsistency and the consistency threshold 151
4.3.11 Additional connectives 153
4.3.12 Disposing irrational logical constants 154
4.4 Fuzzy Theories with Equality and Open Fuzzy Theories 158
4.4.1 Fuzzy theories with equality 159
4.4.2 Consistency theorem in fuzzy logic 160
4.4.3 Herbrand theorem in fuzzy logic 166
4.5 Model Theory in Fuzzy Logic 168
4.5.1 Basic concepts of fuzzy model theory 169
4.5.2 Chains of models 171
4.5.3 Ultraproduct theorem 172
4.6 Recursive Properties of Fuzzy Theories 176
5. FUNCTIONAL SYSTEMS IN FUZZY LOGIC THEORIES 179
5.1 Fuzzy Logic Functions and Their Representation by Formulas 181
5.1.1 Formulas and their relation to fuzzy logic functions 181
5.1.2 Piecewise linear functions and their representation by formulas 185
5.2 Normal Forms for FL-Functions and Formulas of Propositional Fuzzy Logic 189
5.2.1 Normal forms for L-valued FL-functions 190
5.2.2 Normal forms for functions represented by formulas 195
5.3 FL-Relations and Their Connection with Formulas of Predicate Fuzzy Logic 201
5.3.1 FL-Relations and Their Representation by Formulas 201
5.3.2 Normal Forms and Approximation Theorems 204
5.3.3 Representation of FL-relations and Consistency of Fuzzy Theories 208
5.4 Approximation of Continuous Functions by Fuzzy Logic Normal Forms 210
5.4.1 Approximation of continuous functions through FL- relations 210
5.4.2 Approximation of continuous functions determined by a defuzzification
operation 213
5.5 Representation of Continuous Functions by the Conjunctive Normal Form 218
6. FUZZY LOGIC IN BROADER SENSE 223
6.1 Partial Formalization of Natural Language 224
6.1.1 Evaluating and simple conditional linguistic syntagms 224
6.1.2 Translation of evaluating syntagms and predications into fuzzy logic 227
6.1.3 The meaning of simple evaluating syntagms 231
6.2 Formal Scheme of FLb 239
6.3 Special Theories in FLb 242
6.3.1 Independence of formulas 243
6.3.2 Deduction on simple linguistic descriptions 247
6.3.3 Fuzzy approximation based on simple linguistic descriptions 249
Contents IX
7. TOPOl AND CATEGORIES OF FUZZY SETS 259
7.1 Category of O-sets as generalization of fuzzy sets 261
7.2 Category of O-fuzzy sets 275
7.2.1 O-fuzzy sets over MV-algebras 275
7.3 Interpretation of formulas in the category CO - Set 286
8. FEW HISTORICAL AND CONCLUDING REMARKS 299
References 305
Index 315
Preface
This book is an attempt to provide a systematic course of the formal theory
of fuzzy logic. We made a lot of effort to be precise, but at the same time to
explain the motivation and interpretation of all the results and, if possible, to
accompany the theory by examples.
There are a lot of other books on various aspects of fuzzy logic. Our book
is more specific from the point of view of several aspects. First, it is based
on logical formalism demonstrating that fuzzy logic is a well developed logical
theory. Second, it includes the theory of functional systems in fuzzy logic,
which provide explanation of what, and how can be represented by formulas of
fuzzy logic calculi. Third, except for the generalization of the classical way of
interpretation within the environment of fuzzy sets constructed over classical
sets, it also presents much 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. Last but not least, the leading
philosophical point of view is presentation of fuzzy logic especially as the theory
of vagueness as well as the theory of the common-sense human reasoning, which
is based on the use of natural language, the distinguished feature of which is
vagueness of its semantics.
We expect the book to be read by people interested in fuzzy logic and related
areas, and also by logicians, mathematicians and computer scientists interested
in mathematical aspects of fuzzy logic. It can be used in special courses of
fuzzy logic, artificial and computational intelligence, in master and post-gradual
university studies, in advanced courses on various applications including fuzzy
control, decision-making and others.
The book is divided into eight chapters. The first chapter is introductory
and it provides motivation for the development of fuzzy logic, describes its
structure and outlines its potential for applications. Fuzzy logic is there di
vided into that in narrow sense (FLn), which is a special many-valued logic
aiming at description of the vagueness phenomenon and that in broader sense
(FLb), whose aim is to provide a formal theory for modeling of natural human
deduction based on the use of natural language. It is argued that characteriza
tion of the vagueness phenomenon is fundamental for further development of
fuzzy logic as well as its applications.
xii MATHEMATICAL PRINCIPLES OF FUZZY LOGIC
The second chapter is an overview of the basic algebraic concepts necessary
for characterization of the structure of truth values and for understanding to
the subsequent chapters.
The third chapter briefly reminds basic concepts of classical logic, and also
the notation which is employed further.
The main chapters of the book are fourth to seventh. The fourth chapter con
tains explanation of fuzzy logic in narrow sense. We confine ourselves mainly to
fuzzy logic based on the Lukasiewicz algebra of the truth values. Many reasons
have been given arguing that this requirement is a necessary consequence of the
assumptions, which seem to be natural and convincing. Among them, essential
are continuity of the connectives (this follows from the conviction that vague
ness phenomenon requires continuity) and completeness (balance between syn
tax and semantics). We prove lemmas and theorems characterizing behaviour
of FLn, including deduction, contradiction, and others. The main result is the
completeness theorem. Besides this, we also outline fundamentals of the model
theory and discuss questions concerning computability.
The fifth chapter describes functional systems related to propositional and
predicate calculi of fuzzy logic. A notable role here is played by the well known
McNaughton theorem. We provide a constructive proof of it and then, deduce a
canonical representation of formulas of propositional fuzzy logic as well as some
special representation of formulas of predicate fuzzy logic. A special attention
has been paid to the generalization of disjunctive and conjunctive normal forms
into fuzzy logic. Among the main properties of them, the ability to represent
approximately continuous functions including evaluation of the quality of the
approximation has been investigated.
The sixth chapter extends the previous theory to obtain a theory of parts
of natural language to constitute fuzzy logic in broader sense. We present a
formalization of the concepts of intension and extension, evaluating linguistic
syntagms and linguistic description (a set of IF-THEN rules), and demonstrate
some theorems characterizing their behaviour.
The seventh chapter is more abstract and deals with fuzzy sets and fuzzy
logic within category theory. First, we present properties of three possible
categories of fuzzy sets. The subsequent sections provide the reader with a
general picture of fuzzy logic interpreted in top os and its place in topos logic.
We tried to show two possible approaches. The first one is based on top os
theory and Heyting algebra structures defined on the set of subobjects. The
second one is based on the MV -algebra structures defined on these sets. In this
book, we present some of the results and consequences, which could be useful
from the point of view of fuzzy set theory. In particular, we tried to show
explicitly similarities between the obtained categories and topoi, and to show
how the internal logic in these categories could be directly developed.
The book is concluded by a short, eighth chapter giving a brief overview of
the history of fuzzy logic and outlining some of its actual problems to be solved
in the future.
Sections 2.4, 3.4 and Chapter 7 have been written by J. Mockor. Chapter 5
has been written by 1. Perfilieva and Chapters 4 and 6 by V. Novak. The rest
has has been written by by the latter two authors together.
