Constructive Negations and Paraconsistency
This book presents the author’s recent investigations of the two main concepts of negation developed in the constructive logic: the negation as reduction to absurdity (L.E.J. Brouwer) and the strong negation (D. Nelson) are studied in the setting of
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TRENDS IN LOGIC Studia Logica Library VOLUME 26 Managing Editor Ryszard Wójcicki, Institute of Philosophy and Sociology, Polish Academy of Sciences, Warsaw, Poland Editors Vincent F. Hendricks, Department of Philosophy and Science Studies, Roskilde University, Denmark Daniele Mundici, Department of Mathematics “Ulisse Dini”, University of Florence, Italy Ewa Orłowska, National Institute of Telecommunications, Warsaw, Poland Krister Segerberg, Department of Philosophy, Uppsala University, Sweden Heinrich Wansing, Institute of Philosophy, Dresden University of Technology, Germany
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.
Volume Editor Heinrich Wansing
The titles published in this series are listed at the end of this volume.
Sergei P. Odintsov
Constructive Negations and Paraconsistency
123
Sergei P. Odintsov Russian Academy of Sciences Siberian Branch Sobolev Institute of Mathematics Koptyug Ave. 4 Novosibirsk Russia
ISBN 978-1-4020-6866-9
e-ISBN 978-1-4020-6867-6
Library of Congress Control Number: 2007940855 © 2008 Springer Science+Business Media B.V. 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
Contents 1 Introduction
I
1
Reductio ad Absurdum
2 Minimal Logic. Preliminary 2.1 Definition of Basic Logics 2.2 Algebraic Semantics . . . 2.3 Kripke Semantics . . . . .
13 Remarks 15 . . . . . . . . . . . . . . . . . . . . 15 . . . . . . . . . . . . . . . . . . . . 21 . . . . . . . . . . . . . . . . . . . . 28
3 Logic of Classical Refutability 31 3.1 Maximality Property of Le . . . . . . . . . . . . . . . . . . . 32 3.2 Isomorphs of Le . . . . . . . . . . . . . . . . . . . . . . . . . 35 4 The Class of Extensions of Minimal Logic 4.1 Extensions of Le . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Intuitionistic and Negative Counterparts for Extensions of Le . . . . . . . . . . . . . . . . . 4.2 Intuitionistic and Negative Counterparts for Extensions of Minimal Logic . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Negative Counterparts as Logics of Contradictions 4.3 Three Dimensions of Par . . . . . . . . . . . . . . . . . . .
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45
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48 52 53
5 Adequate Algebraic Semantics for Extensions of Minimal Logic 5.1 Glivenko’s Logic . .
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