Table Of Content

a companion to MODAL LOGIC By the same authors An Introduction to Modal Logic companion to a MODAL LOGIC G.E. HUGHES M.J. CRESSWELL METHUEN LONDON and NEW YORK First published in 1984 by Methuen & Co. Ltd 1 1 New Fetter Lane, London EC4P 4EE Published in the USA by Methuen & Co. in association with Methuen, Inc. 733 Third Avenue, New York, NY 10017 © 1984 G.E. Hughes and M.J. Cresswell Printed in Great Britain by Richard Clay (The Chaucer Press), Ltd Bungay, Suffolk All rights reserved. No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. British Library Cataloguing in Publication Data Hughes, G.E. A companion to modal logic. 1. Modality (Logic) I. Title II. Cresswell, Mi. 160 BC199.M6 ISBN 0-416-37500-6 ISBN 0-416-37510-3 Pbk Library of Congress Cataloging in Publication Data Hughes, G.E. (George Edward), 1918- A companion to modal logic. Bibliography: p. Includes index. I. Modality (Logic) I. Cresswell, M.J. II. Title. BX199.M6H795 1984 160 84-8939 ISBN 0-416-37500-6 ISBN 0-416-37510-3 (pbk.) To BERYL and MARY Contents Preface xi Note on references xvii Normal propositional modal systems I The propositional calculus (1) Modal propositional — logic (3) Normal modal systems (4) Models (7) — — — Validity (9) Some extensions of K (10) Validity- — — preservingness in a model (12) Notes (14) — 2 Canonical models and completeness proofs 16 Completeness and consistency (16) Maximal consistent — sets of wff (18) — Canonical models (22) The complete- — ness of K, T, S4, B and S5 (27) — Three further systems (29) Dead ends (33) Exercises (38) Notes (39) — — — 3 More results about characterization 41 General characterization theorems (41) Conditions not — corresponding to any axiom (47) Exercises (51) Notes — — (51) vi' A COMPANION TO MODAL LOGIC viii 4 Completeness and incompleteness in modal logic 52 Frames and completeness (53) — An incomplete normal modal system (57) General frames (62) What might — — we understand by incompleteness? (65) — Exercises (66) — Notes (66) 5 Frames and models 68 Equivalent models and equivalent frames (68) Pseudo- — epimorphisms (70) Distinguishable models (75) Gene- — — rated frames (77) S4.3 reconsidered (8 1) Exercises — — (86) Notes (87) — 6 Frames and systems 89 Frames for T, S4, B and S5 (90) The frames of canonical — models (94) Establishing the rule of disjunction (98) A — — complete but non-canonical system ( 00) Compactness 1 — (103)—Exercises (110)—Notes (110) 7 Subordination frames 1 12 The canonical subordination frame Proving (1 1 3)— completeness by the subordination method (1 1 5) — Tree frames ( 1 1 8) — S4.3 and linearity (1 23) — Systems not containing D ( 27) Exercises 33) Notes (134) 1 — ( 1 — 8 Finite models 135 The finite model property 35) Filtrations 36) ( 1 — (1 — Proving that a system has the finite model property (141) The completeness of KW (145) Characterization — — by classes of finite models (148) Thefinite frame property — (149) Decidability (152) Systems without the finite — — model property (154) Exercises (161) Notes (162) — — 9 Modal predicate logic 164 Notation and formation rules for modal LPC (164) — Modal predicate systems (165) Models (167) Validity — — and soundness (168) The V property (172) Canonical — — CONTENTS ix models for S + BF systems (176) General questions — about completeness in modal LPC (179) Exercises — (184)—Notes (184) Bibliography 186 Glossary 189 List of axioms for propositional systems 195 Index 197