Fuzzy Logic Mathematical Tools for Approximate Reasoning
Fuzzy logic in narrow sense is a promising new chapter of formal logic whose basic ideas were formulated by Lotfi Zadeh (see Zadeh [1975]a). The aim of this theory is to formalize the "approximate reasoning" we use in everyday life, the object of investig
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TRENDS IN LOGIC Studia Logica Library VOLUME 11 Managing Editor
Ryszard W6jcicki, Institute of Philosophy and Sociology, Polish Academy of Sciences, Warsaw, Poland Editors
Daniele Mundici, Department of Computer Sciences, University of Milan, Italy Graham Priest, Department of Philosophy, University of Queensland, Brisbane, Australia
Krister Segerberg, Department of Philosophy, Uppsala University, Sweden
Alasdair Urquhart, Department of Philosophy, University of Toronto, Canada Heinrich Wansing, Institute of Philosophy, Dresden University of Technology, Germany 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 of this volume.
GIANGIACOMO GERLA Department of Mathematics and Computer Sciences, University of Salerno, Italy
FUZZY LOGIC Mathematical Tools for Approximate Reasoning
Springer-Science+Business Media, B.Y.
A c.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-5694-8 ISBN 978-94-015-9660-2 (eBook) DOI 10.1007/978-94-015-9660-2
Printed on acidjree paper
All Rights Reserved © 2001 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2001. Softcover reprint of the hardcover 1st edition 2001 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.
To my wife Loredana, to my daughters Brunella and Francesca
CONTENTS
CONTENTS ................................................... VII PREFACE .................................................... XI CHAPTER 1. Abstract logic in a lattice 1 Introduction............................................. 2 Lattices, Boolean algebras, triangular norms. . . . . . . . . . . . . . . . . . . . 3 Closure operators and closure systems. . . . . . . . . . . . . . . . . . . . . . . . . 4 A Galois connection between operators and classes .. . . . . . . . . . . . . 5 Abstract logic in a lattice .................................. 6 Continuity for abstract logics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Step-by-step deduction systems ............................. 8 Logical compactness .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9 Product of two abstract deduction systems . . . . . . . . . . . . . . . . . . . .. 10 Duality principle for ordered sets .......
