Triangular Norms
The history of triangular norms started with the paper "Statistical metrics" [Menger 1942]. The main idea of Karl Menger was to construct metric spaces where probability distributions rather than numbers are used in order to de scribe the distance betwee
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TRENDS IN LOGIC Studia Logica Library VOLUME8 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 Queens land, 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 6I, 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, camparisans and sources of inspiration is open and evolves over time.
The titles published in this series are listed at the end of this volume.
ERICH PETER KLEMENT Johann es Kepler University, Linz, Austria
RADKO MESIAR Slovak University ofTechnology, Bratislava, Slovakia and Czech Academy of Sciences, Prague, Czech Republic and
ENDRE PAP University of Novi Sad, Yugoslavia
TRIANGULAR NORMS
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-5507-1 DOI 10.1007/978-94-015-9540-7
ISBN 978-94-015-9540-7 (eBook)
Printedon acid-free paper
All Rights Reserved
© 2000 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2000 Softcoverreprint ofthe bardeover 1st edition 2000 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
Bernhard, Sissy, Katharina, Janka, Andrea, and Danijela
Table of Contents
Acknowledgments . . . . . . .
...
.. ......
.. . .. ......
. . . xi. . . . . . . .
Introduction .. ............. .. ......................... . . . ....... xiii Notations used in this book ..................................... xvii Part I 1.
. . . .. . . . .. Basic definitions and properties . . . . . . . . . . . . . . . . . . . .... . . . . .. . . . . . . . . .. ..... 1.1 Triangular norms . . . . . . . . . . ... . . .. . . . . . . . . . . . . . . . . 11 1.2 Triangular conorms . . . . . . . . . . . . . ..... .. . . . . . . . . . . . .14 1.3 Continuity . . . . . . . . . . . . . . . . . . . . . .
2.
. . . . . . . . . . . . . . . . . .21. Algebraic aspects . . . . . . . . . . . . . . . . . . . . . .. . . 22 . . . . . . . . . . . . . . . . . . . . . . properties 2.1 Elementary algebraic 35 .. . . . . ... 2.2 Sem
