Proof Theory of Modal Logic

Proof Theory of Modal Logic is devoted to a thorough study of proof systems for modal logics, that is, logics of necessity, possibility, knowledge, belief, time, computations etc. It contains many new technical results and presentations of novel proof pro

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APPLIED LOGIC SERIES VOLUME 2

Managing Editor

Dov M. Gabbay, Department of Computing, Imperial College, London, U.K. Co-Editor Jon Barwise, Department of Philosophy, Indiana University, Bloomington, IN, U.S.A.

Editorial Assistant Jane Spurr, Department of Computing, Imperial 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. Kluwer, 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.

Proof Theory of Modal Logic edited by

HEINRICH WANSING University ofLeipvg, Institute ofLogic and Philosophy ofScience

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

Library of Congress Cataloging-in-Publication Data Proof theory of modal logic 1 ed1ted by Heinr1ch Wans1ng. p. cm. -- Proceed1ngs of a workshop held at the Univers1ty of Hamburg. Nov. 19-20, 1993.

Includes index. ISBN 978-90-481-4720-5 ISBN 978-94-017-2798-3 (eBook) DOI 10.1007/978-94-017-2798-3

1. Modal log1c--congresses. 2. Proof theory--Congresses. I. Wansing, H. II. Series. OA9.46.P76 1996 511.3--dc20

96-9019

ISBN 978-90-481-4720-5

Logo design by L. Rivlin

Printed on acid-free paper

All Rights Reserved

1996 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1996 No part of the material protected by this copyright notice may be reproduced or utilized in any fonn or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written pennission from the copyright owner.

©

EDITORIAL PREFACE The editors of the Applied Logic Series are pleased to present the second volume in the series, this one on Proof Theory and Modal Logic. In recent years the topic of multi-modal logics has had significant applications in a variety of disciplines, including Artificial Intelligence, theoretical computer science, philosophy, linguistics, Peano arithmetic, and generalized quantifiers. In this volume, leading researchers examine the proof theory of this rich and active field. The Editors

CONTENTS

Preface

ix

I Standard Proof Systems JORG HUDELMAIER A Contraction-free Sequent Calculus for S4

3

GRIGORI MINTS, VLADIMIR OREVKOV AND TANEL TAMMET Transfer of Sequent Calculus Strategies to Resolution for S4

17

HAROLD SCHELLINX A Linear Approach to Modal Proof Theory

33

TOMASZ SKURA Refutations and Proofs in S4

45

II Extended Formalisms EWA ORLOWSKA Relational Proof Systems for Modal Logics

55

NUELBELNAP The Display Problem

79

MARCUS KRACHT Power and Weakness of the Modal Display Calculus

93

HEINRICH WANSING A Proof-theoretic Proof of Functional Completeness for Many Modal and Tense Logics 123

viii RAJEEVGORE On the Completeness of C