TEMPORAL LOGIC
Studies in Linguistics and Philosophy Volume 57 Managing Editors GENNARO CHIERCHIA, University of Milan PAULINE JACOBSON, Brown University FRANCIS J. PELLETIER, University of Alberta Editorial Board JOHAN VAN BENTHEM, University of Amsterdam GREGORY N. CARLSON, University of Rochester DAVID DOWTY, Ohio State University, Columbus GERALD GAZDAR, University of Sussex, Brighton IRENE HEIM, M./.T., Cambridge EWAN KLEIN, University of Edinburgh BILL LADUSAW, University of California at Santa Cruz TERRENCE PARSONS, University of California, lrvine The titles published in this series are listed at the end of this volume.
TEMPORAL LOGIC From Ancient Ideas to Artificial Intelligence by PETER OHRSTROM Department of Communication, Aalborg University, Denmark and PER F. V. HASLE Department of Information Science, University of Aarhus, Denmark KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON
Library of Congress Cataloging-in-Publication Data ~hrstrem, Peter. Temporal logic : From ancient ideas to artlflclal Intelligence / by Peter Ohrstrem and Per F.V. Hasle. p. cm. -- (Studles in linguist~cs and philosophy ; v. 57) Includes bibliographical references and index. ISBN 0-7923-3586-4 (hb. alk. paper) 1. Tense (Logic) 2. Logic, Symbolic and mathematical. 3. Time. I. Haste, Per F. V. II. Title. III. Series. BC199.T4037 1995 160--dc20 95-22191 ISBN 0-7923-3586-4 Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. Kluwer Academic Publishers incorporates the publishing programmes of D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, The Netherlands. Printed on acid-free paper All Rights Reserved 1995 Kluwer Academic Publishers 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. Printed in the Netherlands
TABLE OF CONTENTS PREFACE... ~i INTRODUCTION... 1 PART 1. 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 1.9. 1.10. 1.11. PART 2. 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 2.9. TIMEAND IA~C... 6 The Sea-fight Tomorrow... 10 The Master Argument of Diodorus Cronus... 15 The Study of Tenses in the Middle Ages... 33 Temporal Ampliation... 39 The Duration of the Present... 43 The Logic of Beginning and Ending... 52 Time and Consequentia... 65 Temporalis - the Logic of"while'.... 71 Human Freedom and Divine Foreknowledge... 87 The Downfall of Medieval Tense-logic... 109 Logic as a Timeless Science... 114 TIME AND LOGIC REUNITED... 118 The 19th century and Boolean logic... 122 C.S. Peirce on Time and Modality... 128 Lukasiewicz's Contribution to Temporal logic... 149 A Three-Point Structure of Tenses... 155 A. N. Prior's Tense-logic... 167 The Idea of Branching Time... 180 Tense Logic and Special Relativity... 197 Some Basic Systems of Temporal Logic... 203 Four Grades of Tense-Logical Involvement... 216 V
vi 2.10. Metric Tense Logic... 231 PART 3. 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. MODERN ISSUES IN TEMPORAL LOGIC... 241 Two Paradigms of Temporal Logic... 243 Indeterministic Tense Logic... 257 Leibnizian Tense Logic... 270 Tense Logic and Counterfactual Reasoning... 282 Logic of Durations... 303 Graphs for Time and Modality... 320 Temporal Logic and Computer Science... 344 4. CONCLUSION... 366 APPENDIX... 373 BIBLI~... 386 INDEY~... 405 SOME IMI~RTANT LOGICIANS... 413
PREFACE Time is ubiquitous. Look to such diverse fields as literature and computers, ethics and physics, logic and rhetoric, philosophy and natural science. If you are studying any of these subjects, professionally or con amore, you are very likely to come across temporality as a crucial factor to your studies. For this reason, people are led into the study of time from a variety of highly different disciplines. For the same reason, the study of time is useful and enlightening, both for its own sake and for a large number of specific purposes. The rather ambitious goal of this book is to comprehend time in its diversity, and yet to do this in a focused manner. Our study stretches from Antiquity to the present day, and spans the field from literature to computer science. It thus comprises a historical as well as a systematical dimension. We believe that such a comprehensive approach is necessary in order to achieve a fuller understanding of time. The cost of this approach is that not all aspects can be given a treatment quite as thorough as they deserve. Just for example, there is much more to say about such fields as program verification, trivalent and many-valued logic, and quantified temporal logic than what we have managed to cover here. There are also relevant topics which have been entirely left out: for instance, the current discussions on indexicals, and the logic of truth value gaps, to mention two of the most important omissions. With these disclaimers we wish to make it clear that we are ourselves aware of some limitations of this book. But we also believe that it does contain an unusually comprehensive exposition of the study of time. We must say a few words about the genesis of this book. Peter Ohrstr m was the first of us to do research on the concept of time, leading to his 1988 dr. scient, thesis on this subject. This thesis put a special emphasis on the relation between the logic of time and the general history of natural science. Per Hasle began studies on the logic of time in 1988, adding to our incipient common project perspectives from linguistics and information vii
viii PREFACE science. Differences in our backgrounds notwiths anding, the contribution contained in this volume is the result of essentiality joint work. The book contains entirely new results as well as previously published material, which has been reworked and put into the wider context of our exposition here. A particularly important source for our book has been several interviews with Dr. Mary Prior, who has graciously provided valuable and interesting information on the work of her late husband Arthur Norman Prior - the founder of modern temporal logic. Furthermore, Mary Prior has granted us access to/k N. Prior's papers kept at the Bodleian Library in Oxford, also a crucial source for some of the new findings presented here. A very special thanks must go to her. We are indebted to many persons for advice, criticism and inspiring discussions. We especially thank Mogens Wegener and Marta Ujvari for carefully reading and constructively criticising our manuscript. We also want to thank Harmen van den Berg, Knud Capion, Jack Copeland, M. J. Cresswell, Dick Crouch, Sten Ebbesen, Milea Angela Simoes Froes, Claudine Engel-Tiercelin, Antony Galton, Richard Gaskin, Nils Klarlund, Inger Lytje, Claus Myltoft, Jakob M~ller, Stig Andur Pedersen, Amir Pnueli, Anne Rasmussen, Stephen Read, Jan Schmidt, Peter Simons, and Jan Tapdrup. All these persons have in various ways been helpful and inspiring for our work. Peter OhrstrCm Per Hasle Aalborg, January 1995