This page intentionally left blank ParadoxandParaconsistency ConflictResolutionintheAbstractSciences In a world plagued by disagreement and conflict, one might expect thattheexactsciencesoflogicandmathematicswouldprovideasafe harbor. In fact, these disciplines are rife with internal divisions be- tweendifferent,oftenincompatiblesystems.Dothesedisagreements admit of resolution? Can such resolution be achieved without dis- turbingassumptionsthatthetheoremsoflogicandmathematicsstate objectivetruthsabouttherealworld? In this original and historically rich book, John Woods explores apparentlyintractabledisagreementsinlogicandthefoundationsof mathematicsandsetsoutconflictresolutionstrategiesthatevadeor disarm these stalemates. Among the conflicts to which these strate- giesareappliedare:thedisagreementbetweenclassicalandrelevant logicians;Quine’sattackonquantifiedmodallogic;disagreementas towhetherdialethiclogichasanadequatemotivation;conflictsabout how to understand the paradox of sets and the Liar paradox; the vexedrelationshipbetweenmodernlogicandtheoriesofreasoning; andconditionsunderwhichlogicallawsexhibitnormativeforce. An important subtheme of the book is the extent to which plu- ralisminlogicandthephilosophyofmathematicsunderminesrealist assumptions.Woods’sresponseisanaccountoftruthinwhichrealism isanirresistibleepiphenomenon. Thisbookmakesanimportantcontributiontosuchareasofphilos- ophyaslogic,philosophyoflanguage,andargumentationtheory,but italsowillbeofinteresttomathematiciansandcomputerscientists. JohnWoodsisDirectoroftheAbductiveSystemsGroup,University ofBritishColumbia. Paradox and Paraconsistency Conflict Resolution in the Abstract Sciences JOHN WOODS UniversityofBritishColumbia    Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge  , United Kingdom Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521810944 © John Woods 2003 This book is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2002 -  isbn-13 978-0-511-06435-7 eBook (NetLibrary) -  isbn-10 0-511-06435-7 eBook (NetLibrary) -  isbn-13 978-0-521-81094-4 hardback -  isbn-10 0-521-81094-9 hardback isbn--13 978-0-521-00934-8 paperback -  isbn-10 0-521-00934-0 paperback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. InMemoryof RichardSylvan(ne´ Routley),1936–1996. RobGrootendorst,1944–2000. Contents Preface pageix Prologue:PostmodernLogic xi 1 ConflictintheAbstractSciences 1 2 Modalities 40 3 ManagingInconsistency 80 4 SemanticIntuitions 120 5 SetsandTruth 155 6 Fiction 196 7 CurryingLiars 228 8 Normativity 277 Notes 327 Bibliography 341 Index 359 vii