Algebraic Foundations of Many-Valued Reasoning
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TRENDS IN LOGIC Studia Logica Library VOLUME7 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 ofTechnology, 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 SOUfces of inspiration is open and evolves over time.
The titles published in this series are listed at the end afthis valurne.
ROBERTO L.O. CIGNOLI Department of Mathematics. University of Buenos Aires. Argentina
ITALA M.L. D'OTfAVIANO Department of Philosophy and The Centre for Logic. Epistemology and the History of Science. State University of Campinas. Brazil
and
DANIELE MUNDICI Department of Computer Science. University of Milan. Italy
ALGEBRAIC FOUNDATIONS OF MANY-VALUED REASONING
Springer-Science+Business Media, B.V.
A c.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-5336-7 ISBN 978-94-015-9480-6 (eBook) DOI 10.1007/978-94-015-9480-6
Printed on acid-free paper
All Rights Reserved © 2000 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2000. Softcover reprint of the hardcover 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, incIuding photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.
To the memory
0/
ROLANDO CHUAQUI ENNIO DE GlORGI ANTONIO MONTEIRO
great Scientists and Teachers
Contents Introduction
1
1 Basic not ions 1.1 MV-algebras . . . . . . . . . . . . 1.2 Homomorphisms and ideals '" 1.3 Subdirect representation theorem 1.4 MV-equations . . 1.5 Boolean algebras . . . . 1.6 MV-chains . . . . . . . . 1.7 Bibliographical remarks
7 7 12 19
20 24 27
29
2 Chang completeness theorem 2.1 The functor r . . . . . . . . 2.2 Good sequences . . . . . . . 2.3 The partially ordered monoid M A 2.4 Chang's f-group GA' . . . . . 2.5 Chang completeness theorem. 2.6 Bibliographical remarks
31
3 Free MV-algebras 3.1 McN aughton functions . . . . . . . . 3.2 The one-dimensional case . . . . . . . 3.3 Decomposing McNaughton functions 3.4 Ideals in free MV-algebras 3.5 Sim
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