Table Of Content

Fuzzy 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.