Proof Theory for Fuzzy Logics
Fuzzy logics are many-valued logics that are well suited to reasoning in the context of vagueness. They provide the basis for the wider field of Fuzzy Logic, encompassing diverse areas such as fuzzy control, fuzzy databases, and fuzzy mathematics. This bo
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APPLIED LOGIC SERIES VOLUME 36
Managing Editor Dov M. Gabbay, Department of Computer Science, King’s College, London, U.K. Co-Editor Jon Barwise† Editorial Assistant Jane Spurr, Department of Computer Science, King’s College, London, U.K.
SCOPE OF THE SERIES Logic is applied in an increasingly wide variety of disciplines, from the traditional subjects of philosophy and mathematics to the more recent disciplines of cognitive science, computer science, artificial intelligence, and linguistics, leading to new vigor in this ancient subject. Springer, through its Applied Logic Series, seeks to provide a home for outstanding books and research monographs in applied logic, and in doing so demonstrates the underlying unity and applicability of logic.
For further volumes: http://www.springer.com/series/5632
Proof Theory for Fuzzy Logics
by George Metcalfe Vanderbilt University, Nasville, TN, USA
Nicola Olivetti Paul C´ezanne University, Marseille, France
Dov Gabbay King’s College, London, UK
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Dr. George Metcalfe Vanderbilt University Department of Mathematics 1326 Stevenson Center Nashville TN 37240 USA [email protected]
Dr. Nicola Olivetti Universit´e Aix-Marseille III LSIS-UMR CNRS 6168 Campus de Saint Jerˆome Avenue Escadrille Normandie-Niemen 13397 Marseille CX 20 France [email protected]
Prof. Dr. Dov M. Gabbay King’s College London Department of Computer Science Strand London United Kingdom WC2R 2LS [email protected]
ISBN: 978-1-4020-9408-8
e-ISBN: 978-1-4020-9409-5
Library of Congress Control Number: 2008938185 c Springer Science+Business Media B.V. 2009 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com
Acknowledgements
A book like this is not written without a great deal of help and encouragement. In particular, we are grateful to our friends and colleagues, Agata Ciabattoni, Petr Cintula, Christian Ferm¨uller, Rostislav Horˇc´ık, and Francesco Paoli, for taking the time to read and comment on earlier drafts of the manuscript. Their attention to detail and insightful suggestions are very much appreciated. For thought-provoking discussions and support at various stages in the process, we would also like to thank distinguished researchers in the Mathematical Fuzzy Logic community: Arnon Avron, Francesc Esteva, Llu´ıs Godo, Petr H´ajek, Franco Montagna, Daniele Mundici, Hiroakira Ono, and Constantine Tsinakis. The first author would also like to offer some personal thanks. First, for two years of stimulating conversations and camaraderie, to members of the logic group at the Vienna University of Technology: Matthias Baaz, Rosalie Iemhoff, Norbert Preining
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