Table Of ContentFuzzy Sets, Logics and Reasoning about Knowledge
APPLIED LOGIC SERIES
VOLUME 15
Managing Editor
Dov M. Gabbay, Department of Computer Science, King s College, Londen, U.K.
Co-Editor
John Barwise, Department of Philosophy, Indiana University, Bloomington, IN,
U.S.A.
Editorial Assistant
s
Jane Spurr, Department of Computer Science, King College, London, U.K.
seOPE OF THE SERIES
Logic is applied in an increasingly wide variety of disciplines, from the traditional
subjects of philosophy and rnathematics to the more recent disciplines of cognitive
science, computer science, artificial intelligence, and linguistics, leading to new
vigor in this ancient subject. Kluwer, through its Applied Logic Series, seeks to
provide ahorne for outstanding books and research monographs in applied logic,
and in doing so demonstrates the underlying unity and applicability of logic.
The titZes published in this series are listed at the end of this voZume.
Fuzzy Sets, Logics and
Reasoning about
Knowledge
edited by
DIDIER DUBOIS
I.R.I.T., C.N.R.S.,
University ofToulouse-lll,
France
HENRI PRADE
I.R.I.T., C.N.R.S.,
University of Toulouse-lll,
France
and
ERICH PETER KLEMENT
Johannes Kepler Universität,
Linz, Austria
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P.·Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-5324-4 ISBN 978-94-017-1652-9 (eBook)
DOI 10.1007/978-94-017-1652-9
Logo design by L. Rivin
Printed on acid-free paper
All Rights Reserved
© 1999 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1999
Softcover reprint of the hardcover 1s t edition 1999
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner
CONTENTS
Editorial Preface ix
Dov M. Gabbay
Introduction: Bridging the Gap between Multiple-valued 1
Logics, Fuzzy Logic, Uncertain Reasoning and Reasoning
about Knowledge
Didier Dubois, Henri Prade and Erich Peter
Klement
Part I: Advances in Mutiple-valued Logics
The Poincare Paradox and Non-classical Logics 7
Ulrich Höhle
Propositional Fuzzy Logics based on Frank t-norms: A 17
comparison
Erich Peter Klement and Mirko Navara
AResolution-based Axiomatisation of 'Bold' Propositional 39
Fuzzy Logic
Stephan Lehmke
How to Make Your Logic Fuzzy: Fibred Semantics and The 51
Weaving of Logics
Dov M. Gabbay
Introducing Grade to Some Metalogical Notions 85
Mihir K. Chakraborty and Sanjukta Basu
Closure Operators, Fuzzy Logic and Constraints 101
Giangiacomo Gerla
vi
Part 11: Aigebraic Aspects of Multiple-valued Logics
Ulam Game, the Logic of MaxSat, and Many-valued 121
Partitions
Daniele Mundici
A Many-valued Generalisation of the Ultrapower 139
Construction
Costas A. Drossos
Gabriel Filters and the Spectrum of an MV-Algebra 151
Panagis Karazeris
Conditional States in Finite-valued Logics 161
Antonio Di Nola, George Georgescu and Ada
Lettieri
Conditioning on MV-algebras and Additive 175
Measures-further results
Siegfried Weber
Part 111: Advances in Approximate Reasoning
Toward Adequacy Conditions for Inference Schemata in 201
Approximate Reasoning:
The Case of the Rule of Syllogism
Siegfried Gottwald
Formal Theories in Fuzzy Logic 213
Vilem Novcik
A Note on Fuzzy Inference as Deduction 237
Lluis Godo and Petr Hajek
The Role of Similarity in Fuzzy Reasoning 243
Frank Klawonn
T-indistinguishability Operators and Approximate 255
Reasoning via CR!
Dionis Boixader and Joan Jacas
About Similarity-based Logical Systems 269
Francesco Esteva, Pere Garcia and Lluis Godo
CONTENTS vii
On Similarity-based Fuzzy Clusterings 289
Helmut Thiele
Part IV: Reasoning about Information and
Knowledge
Informational Representability: Abstract Models versus 301
Concrete Models
Stephane Demri and Ewa Orlowska
From Possibilistic Information to Kleene's Strong 315
Multi-valued Logics
Gert De Cooman
A Roadmap of Qualitative Independence 325
D. Dubois, L. Fariiias dei Cerro, A. Herzig,
and H. Prade .
Truth Functionality and Measure-based Logics 351
Luca Boldrin and Claudio Sossai
Logic Programs with Context-dependent Preferences 381
Gerhard Brewka
An Overview of Inconsistency-tolerant Inferences in 395
Prioritized Knowledge Bases
Salem Benferhat, Didier Dubois and Henri Prade
Index 419
EDITORIAL PREFACE
We welcome Volume 15, Fuzzy Sets, Logics and Reasoning about Knowledge
on fuzzy and many-valued 10gics. The volume editors and contributors are from
among the most active front-line researchers in the area and the contents shows
how wide and vigorous this area iso There are strong scientific connections with
earlier volumes in the series.
I am confident that the appearance of this book in our series will help kindIe
the interest of more and more researchers from formallogic in the foundations of
fuzzy logic.
D. M. Gabbay
DIDIER DUBOIS, HENRI PRADE AND ERICH PETER
KLEMENT
INTRODUCTION: BRIDGING THE GAP
BETWEEN MULTIPLE-VALUED LOGICS,
FUZZY LOGIC, UNCERTAIN REASONING AND
REASONING ABOUT KNOWLEDGE
The term 'Fuzzy Logic', although very well-known, is ambiguous as it
refers to several only loosely related concerns ranging from rule-based sys
tem control to various multiple-valued logics. The most popular acception
of the term 'fuzzy logic' is generally not related to logic proper, since it is
primarily employed by control engineers that use fuzzy rule-based systems
like a sort of neural network capable of approximating non-linear functions.
However in the recent past, it has been stressed that fuzzy logic in the nar
row sense can be envisaged from the point of view of logic provided that
fuzzy sets are considered as stemming from the multiple-valued logic tra
dition. The relationship between multiple-valued logic and fuzzy sets had
been noticed by Moisil [1972] in the late sixties. At the same period, in
Eastern Germany, Klaua independently built up a multiple-valued set the
ory (see Gottwald [1984] for an extensive bibliography of Klaua's papers).
The current trend relating fuzzy sets and multiple-valued logic actually
dates back to a seminal paper by Goguen [1969]. In this paper Goguen
insists on an algebraic structure he calls a 'closg' (for commutative lattice
ordered semi-group) and shows that the lattice-theoretical concept of resid
uation can be generalized to operations other than the minimum. Following
Goguen's program, Pavelka [1979] has definitely anchored fuzzy logic in the
multiple-valued tradition, emphasizing a link with Lukasiewicz logic already
pointed out by Giles [19761. Since then, a large amount of work has been
carried out whose aim is to equip fuzzy logic with a syntactic component,
and several algebraic structures have been laid bare as potential candidates
for supporting fuzzy logics.
The aim of this book is to report on recent results pertaining to gen
uinely logical aspects of fuzzy sets in relation to algebraic considerations,
to knowledge representation and common-sense reasoning. It proposes a
state-of-the-art glance at the current status of multiple-valued and fuzzy
set-based logics.
An intriguing state of facts is the almost non-existing role of multiple
valued logic in Artificial Intelligence (AI). As of today there is very little
activity that lies at the intersection of both fields, if we except some very
limited use of three-valued logics in logic programming, natural language
processing, and non-monotonic reasoning. Perhaps one reason for this lack
D. Dubois er al. (eds.). Fuu;y Sets. Logics and Reasoning about Knowledge. 1-6.
© 1999 Kluwer Academic Publishers.
2 DIDIER DUBOIS, HENRI PRADE AND ERICH PETER KLEMENT
of interest in multiple-valued logic from AI is that in AI, the central prob
lem is to represent incomplete knowledge and draw plausible condusions.
This problem is naturally addressed via logics of uncertainty, and modal
logics. Multiple-valued logics are not typically concerned with the issue of
incomplete information nor belief revision. They are tailored to the logi
cal representations of statements about non-Boolean entities, propositions
involving variables that can take more than two values. On the contrary,
logics of uncertainty are supposed to express statements about an agent's in
complete ability to know the truth status of standard Boolean propositions.
Thrner's [1984] survey book mentions three-valued logics in connection with
partial knowledge, and multiple-valued logics in connection with fuzzy logic.
However since then, non-monotonie logics, modallogics and temporallogics
have blossomed in AI, rather than multiple-valued logic as such.
Another reason for the lack of a multiple-valued logic tradition in AI is the
suspicion that a multiple-valued logic is a form of number-crunching device
that offends the taste of AI researchers for symbolic reasoning. Noticeably,
Zadeh's proposals for approximate reasoning patterns have encountered in
difference or even hostility from mainstream AI in the past, and it is only
recently that fuzzy logic has become an official keyword for Artificial Intel
ligence conferences. It is conjectured that Artificial Intelligence will become
more interested in multiple-valued logics once the specificity of the latter,
that is, a logicallanguage to speak about non-Boolean variables, is acknowl
edged. However it seems to be not so easy to bring many fuzzy set scholars
back to AI, especially now that the 'Soft Computing' school sometimes
presents itself in opposition to symbolic AI.
This volume is an attempt to show that rigorous research in the frame
work of fuzzy logic is taking place on topics that may be of interest to
researchers in Artificial Intelligence. Some rigorous formal systems for ap
proximate reasoning about non-Boolean sentences are currently under study.
Some new forms of common-sense reasoning not so popular in Artificial In
telligence, such as similarity-based reasoning and interpolative reasoning
seem to be naturally cast in the setting of multiple-valued logics. It is dear
that multiple-valuedness play an important role in such forms of reasoning.
One original feature of this volume is precisely to bring together an overview
of multiple-valued logics and their mathematical underpinnings, and a col
lection of contributions on reasoning about vagueness, similarity, interpola
tion, uncertainty, priority, and partial inconsistency. Moreover the volume
sheds light on the links between fuzzy sets and multiple-valued logics, in
duding the question of devising a proper syntax for fuzzy logic. Lastly, it
offers improved communication between Logics of AI and Fuzzy Sets, whose
respective programs have many common concerns, while researchers in each
area usually do not publish in the same journals.
The book is divided in 4 sections that go from a review of multiple-valued
systems to issues in approximate and practical reasoning.
