Displaying Modal Logic
The present monograph is a slightly revised version of my Habilitations schrift Proof-theoretic Aspects of Intensional and Non-Classical Logics, successfully defended at Leipzig University, November 1997. It collects work on proof systems for modal and c
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TRENDS IN LOGIC Studia Logica Library VOLUME3 Managing Editor Ryszard W6jcicki, Institute of Philosophy and Sociology, Polish Academy of Sciences, Warsaw, Poland Editors Petr Hajek, Institute of Computer Sciences, Academy of Sciences of the Czech Republic, Prague, Czech Republic David Makinson, Ville d'Avray, France Daniele Mundici, Department of Computer Sciences, University of Milan, Italy Krister Segerberg, Department of Philosophy, Uppsala University, Uppsala, Sweden Alasdair Urquhart, Department of Philosophy, University of Toronto, Canada 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 sources of inspiration is open and evolves over time.
Volume Editor Ryszard W6jcicki
HEINRICH WANSING Institute ofLogic and Philosophy of Science, University of Leipzig, Leipzig, Germany
DISPLAYING MODAL LOGIC
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-017-1280-4 (eBook) ISBN 978-90-481-5079-3 DOI 10.1007/978-94-017-1280-4
Printed on acid-free paper
All Rights Reserved Springer Science+Business Media Dordrecht Originally published by K1uwer Academic Publishers in 1998 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
© 1998
CONTENTS
v
CONTENTS PREFACE
IX
CHAPTER ONE I INTRODUCTION 1.1 The problem of Gentzenizing modal logic 1.2 Standard sequent systems for normal modal logics 1.3 Rules as meaning assignments 1.4 Uniqueness 1.5 Modularity and the Dosen Principle 1.6 Subformula property, cut, and analytic cut 1. 7 Generality
1 2 4 7 10 10 11 13
CHAPTER TWO I SEQUENTS GENERALIZED 2.1 Modal signs 2.2 Higher-level sequent systems 2.3 Higher-dimensional sequent systems 2.4 Higher-arity sequent systems 2.5 Hypersequents 2.6 Natural deduction systems
15 15 16 17 18 22 24
CHAPTER THREE I DISPLAY LOGIC 3.1 Gentzen terms 3.2 Residuation 3.3 The Display Theorem 3.4 Introduction rules 3.5 Completeness
27 27 30 34 37 43
CHAPTER FOUR I PROPERLY DISPLAYABLE LOGICS, DISPLAYABLE LOGICS AND STRONG CUT-ELIMINATION 4.1 Properly displayable logics 4.2 A case distinction and primitive reductions 4.3 Strong normalization 4.4 Displayable logics
47 47 48 51 55
v
CONTENTS
Vl
4.5 4.6
Characterization of the properly displayable logics Scope of the method
57 62
CHAPTER FIVE I A PROOF-THEORETIC PROOF OF FUNCTIONAL COMPLETENESS FOR MANY MODAL AND TENSE LOGICS 5.1
