METAMATHEMATICS OF FUZZY LOGIC

METAMATHEMATICS OF FUZZY LOGIC

TRENDS IN LOGIC Studia Logica Library VOLUME 4 Managing Editor Ryszard Wojcicki, Institute ofphilosophy and Sociology, Polish Academy ofsciences, Warsaw, Poland Editors Petr Hajek, Institute ofcomputer Sciences, Academy ofsciences ofthe Czech Republic, Prague, Czech Republic David Makinson, Ville d'avray, France Daniele Mundici, Department ofcomputer Sciences, University ofmilan, Italy Krister Segerberg, Department ofphilosophy, Uppsala University, Uppsala, Sweden Alasdair Urquhart, Department ofphilosophy, University oftoronto. Canada Assistant Editor Jacek Malinowski, Box 61, UPT 00-953, Warszawa 37, Poland SCOPE OF THE SERIES Trends in Logic is a bookseries covering essentially the same area as the journal Studia Logica - that is, contemporary formal logic and its applications and relations to other disciplines. These include artificial intelligence, informatics, cognitive science, philosophy of science, and the philosophy of language. However, this list is not exhaustive, moreover, the range of applications, comparisons and sources of inspiration is open and evolves over time. The titles published in this series are listed at the end ofthis volume.

PETRHĂJEK Institute o/computer Science, Academy of Sciences of the Czech Republic, Prague, Czech Republic METAMATHEMATICS OF FUZZY LOGIC SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-1-4020-0370-7 ISBN 978-94-011-5300-3 (ebook) DOI 10.1007/978-94-011-5300-3 Printed on acid-free paper AII Rights Reserved 1998 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1998 Softcover reprint of the hardcover 1 st edition 1998 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, incjuding photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner

To my wife Marie, to my daughter Marie, to my son Jonas, to my grandson Jonas with love.

CONTENTS CONTENTS PREFACE v vii CHAPTER ONE I PRELIMINARIES 1 1.1 Introduction 1 1.2 A survey of Boolean propositional logic 6 1.3 Boolean predicate calculus 10 1.4 Function symbols; varieties ofalgebras 15 1.5 Lattices and Boolean algebras 20 1.6 Ordered Abelian groups 22 CHAPTER TWO I MANY-VALUED PROPOSITIONAL CALCULI 27 2.1 Continuous t-norms and their residua 27 2.2 The basic many-valued logic 35 2.3 Residuated lattices; a completeness theorem 46 2.4 Some additional topics 56 CHAPTER THREE I LUKASIEWICZ PROPOSITIONAL LOGIC 63 3.1 Getting Lukasiewicz logic 63 3.2 MV-algebras; a completeness theorem 70 3.3 Rational Pavelka logic 79 CHAPTER FOUR I PRODUCT LOGIC, GODEL LOGIC 89 4.1 Product logic 89 4.2 GOdellogic 97 4.3 Appendix: Boolean logic 103 CHAPTER FIVE I MANY-VALUED PREDICATE LOGICS 109 5.1 The basic many-valued predicate logic 109 5.2 Completeness 119 5.3 Axiomatizing GOdellogic 124 5.4 Lukasiewicz and product predicate logic 127 5.5 Many-sorted fuzzy predicate calculi 139 5.6 Similarity and equality 141 v

Vi CONTENTS CHAPTER SIX I COMPLEXITY AND UNDECIDABILITY 149 6.1 Preliminaries 149 6.2 Complexity of fuzzy propositional calculi 154 6.3 Undecidability of fuzzy logics 161 CHAPTER SEVEN I ON APPROXIMATE INFERENCE 167 7.1 The compositional rule of inference 168 7.2 Fuzzy functions and fuzzy controllers 177 7.3 An alternative approach to fuzzy rules 189 CHAPTER EIGHT I GENERALIZED QUANTIFIERS AND MODALI- TIES 195 8.1 Generalized quantifiers in Boolean logic 195 8.2 Two-valued modal logics 205 8.3 Fuzzy quantifiers and modalities 215 8.4 On "probably" and "many" 228 8.5 More on "probably" and "many" 238 CHAPTER NINE I MISCELLANEA 249 9.1 Takeuti-Titani fuzzy logic 249 9.2 An abstract fuzzy logic 261 9.3 On the liar paradox 265 9.4 Concluding remarks 271 CHAPTER TEN I HISTORICAL REMARKS 277 10.1 Until the forties 277 10.2 The fifties 278 10.3 The sixties 279 10.4 The seventies 279 10.5 The eighties 280 10.6 The nineties 281 REFERENCES 283 INDEX 295

PREFACE This book presents a systematic treatment of deductive aspects and structures of fuzzy logic understood as many valued logic sui generis. Some important systems of real-valued propositional and predicate calculus are defined and investigated. The aim is to show that fuzzy logic as a logic of imprecise (vague) propositions does have well developed formal foundations and that most things usually named "fuzzy inference" can be naturally understood as logical deduction. There are two main groups of intended readers. First, logicians: they can see that fuzzy logic is indeed a branch of logic and may find several very interesting open problems. Second, equally important, researchers involved in fuzzy logic applications and soft computing. As a matter offact, most ofthese are not professional logicians so that it can easily happen that an application, clever and succcessful as it may be, is presented in a way which is logically not entirely correct or may appear simple-minded. (Standard presentations ofthe logical aspects of fuzzy controllers are the most typical example.) This fact would not be very important ifonly the bon ton oflogicians were harmed; but it is the opinion ofthe author (who is a mathematical logician) that a better understanding of the strictly logical basis of fuzzy logic (in the usual broad sense) is very useful for fuzzy logic appliers since if they know better what they are doing, they may hope to do it better. Still more than that: a better mutual understanding between (classical) logicians and researchers in fuzzy logic promises to lead to deeper cooperation and new results. The book has been developed from a series of lectures which I held first in the Institute of Computer Science of the Academy of Sciences of the Czech Republic (for postgraduate students) and then also at the Technical University of Vienna, Austria. My paper [75] was the first draft of the contents of the book. Later I had the opportunity to give tutorials on fuzzy logic at the Helena Rasiowa minisemester at Warsaw in winter 1996, for Italian PhD students at Ravello in spring 1997 and at the IFSA congress in Prague, as well as the FUZZ-IEEE congress at Barcelona in summer 1997. There is written material that appeared as ref. [81]. I am indebted to my collaborators Ivan Kramosil, Ms. Dagmar Harmancova, Milan Daniel, David Coufal and David Svejda for stimulating comments and cooperation. I have extremely enjoyed long-term cooperation with my Spanish (Catalan) colleagues F. Esteva and L. Gado. I have learned from the work of and contact with S. Gottwald, D. Mundici, U. Hohle, M. Baaz, VB

viii PREFACE J. Paris, J. Shepherdson, Ms. Isabel Ferreirim and several other colleagues. Finally, the criticism of the anonymous referee was helpful in the final stage of writing this book. The text was typed in LaTex by Ms. Iva Sindelkova, 1. Oliverius and Ms. I. Baranovska. My Tex experts were A. Stedry and R. Neruda. My thanks go to all of them. I also recognize partial support by the COST Action 15 and the grant No A103060111966 of the Grant Agency of the Academy of Sciences of the Czech Republic; this support has been crucial for attending conferences, meeting people and buying relevant books. Finally let me mention two related monographs in preparation: Cignoli, D'Ottaviano and Mundici [30] and Gottwald [66] (English version of his [67].) The overlap with the present book is small and the books complement each other. Prague, September 1997.