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