The Age of Alternative Logics: Assessing Philosophy of Logic and Mathematics Today

338 Pages·2006·3.47 MB·English

by  Johan van Benthem

THE AGE OF ALTERNATIVE LOGICS
LOGIC, EPISTEMOLOGY, AND THE UNITY OF SCIENCE
VOLUME 3
Editors
Shahid Rahman, University of Lille III, France
John Symons, University of Texas at El Paso, U.S.A.
Editorial Board
Jean Paul van Bendegem, Free University of Brussels, Belgium
Johan van Benthem, University of Amsterdam, the Netherlands
Jacques Dubucs, University of Paris I-Sorbonne, France
Anne Fagot-Largeault, Collège de France, France
Bas van Fraassen, Princeton University, U.S.A.
Dov Gabbay, King’s College London, U.K.
Jaakko Hintikka, Boston University, U.S.A.
Karel Lambert, University of California, Irvine, U.S.A.
Graham Priest, University of Melbourne, Australia
Gabriel Sandu, University of Helsinki, Finland
Heinrich Wansing, Technical University Dresden, Germany
Timothy Williamson, Oxford University, U.K.
Logic, Epistemology, and the Unity of Scienceaims to reconsider the question of the unity
of science in light of recent developments in logic. At present, no single logical, semantical
or methodological framework dominates the philosophy of science. However, the editors
of this series believe that formal techniques like, for example, independence friendly logic,
dialogical logics, multimodal logics, game theoretic semantics and linear logics, have the
potential to cast new light on basic issues in the discussion of the unity of science.
This series provides a venue where philosophers and logicians can apply specific technical
insights to fundamental philosophical problems. While the series is open to a wide variety
of perspectives, including the study and analysis of argumentation and the critical
discussion of the relationship between logic and the philosophy of science, the aim is to
provide an integrated picture of the scientific enterprise in all its diversity.
The Age of Alternative Logics
Assessing Philosophy of Logic
and Mathematics Today
Edited by
Johan van Benthem
ILLC, University of Amsterdam, The Netherlands
Gerhard Heinzmann
Archives Henri Poincaré, Nancy University, France
Manuel Rebuschi
Archives Henri Poincaré, Nancy University, France
and
Henk Visser
ILLC, University of Amsterdam, The Netherlands
AC.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN-10 1-4020-5011-9 (HB)
ISBN-13 978-1-4020-5011-4 (HB)
ISBN-10 1-4020-5012-7 (e-book)
ISBN-13 978-1-4020-5012-7 (e-book)
Published by Springer,
P.O. Box 17, 3300 AADordrecht, The Netherlands.
www.springer.com
Printed on acid-free paper
All Rights Reserved
© 2006 Springer
No part of this work may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, microfilming,
recording or otherwise, without written permission from the Publisher, with the
exception of any material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work.
CONTENTS
Acknowledgments ix
1
Introduction: AlternativeLogicsandClassicalConcerns 1
JohanvanBenthem
PartI Proof,KnowledgeandComputation
2
EpistemicModels,LogicalMonotonyandSubstructuralLogics 11
MikaëlCozic
3
SemanticsasBasedonInference 25
JaroslavPeregrin
4
Effectiveness 37
StewartShapiro
5
DoesGödel’sIncompletenessTheoremProvethatTruthTranscendsProof? 51
JosephVidal-Rosset
6
Transpositions 75
HenkVisser
PartII TruthValuesBeyondBivalence
7
Many-ValuedandKripkeSemantics 89
Jean-YvesBéziau
vi Contents
8
TheLogicofComplementarity 103
NewtonC.A.daCosta,DécioKrause
9
SemanticsforNaiveSetTheoryinMany-ValuedLogics 121
ThierryLibert
PartIII Category-TheoreticStructures
10
ContinuityandLogicalCompleteness: AnApplication 139
ofSheafTheoryandTopoi
SteveAwodey
11
WhatisCategoricalStructuralism? 151
GeoffreyHellman
12
CategoryTheoryasaFrameworkforaninreInterpretation 163
ofMathematicalStructuralism
ElaineLandry
13
Categories,SetsandtheNatureofMathematicalEntities 181
Jean-PierreMarquis
PartIV Independence,EvaluationGamesandImperfectInformation
14
Truth,NegationandOtherBasicNotionsofLogic 195
JaakkoHintikka
15
SignallinginIFGames: ATrickyBusiness 221
TheoM.V.Janssen,FrancienDechesne
16
Independence-FriendlyLogicandGamesofIncompleteInformation 243
Ahti-VeikkoPietarinen
Contents vii
17
IFandEpistemicActionLogic 261
ManuelRebuschi
PartV DialogueandPragmatics
18
NaturalizingDialogicPragmatics 285
GerhardHeinzmann
19
LogicasaToolofScienceVersusLogicasaScientificSubject 299
KunoLorenz
20
Non-NormalDialogicsforaWonderfulWorldandMore 311
ShahidRahman
PartVI Appendix
LouisJolyasaPlatonistPainter? 337
RogerPouivet
INDEX 343
ACKNOWLEDGMENTS
TheconferenceonPhilosophicalInsightsintoLogicandMathematics(PILM
forshort),ofwhichthisbookistheoutcome,wouldnothavebeenpossiblewith-
out the generous financial support of the following institutions: Evert Willem
Beth Foundation (Amsterdam), Centre National de la Recherche Scientifique
(CNRS),ConseilRégionaldeLorraine,ConseilGénéraldeMeurtheetMoselle,
CommunautéUrbaineduGrandNancy, UniversitéNancy2, UniversitéHenri
Poincaré,InstitutNationalPolytechniquedeLorraine,LaboratoiredePhiloso-
phieetd’HistoiredesSciences–ArchivesHenri-Poincaré(UMR7117),Insti-
tuteforLogic,LanguageandComputation(ILLCAmsterdam),Loria,Goethe
Institut (Nancy). To all these institutions we express our warm gratitude. We
are also grateful to the members of the PILM scientific committee for their
invaluablehelpinpreparingtheprogramandreportingonsomanylectures,as
well as to the staff of the Poincaré Archives for their help in the preparation
oftheconferenceandtheirexpertassistanceinvariousways. Weparticularly
expressourgratitudetoDr. ProsperDoh(PoincaréArchives)whohastakenon
thejoboftechnicaleditorforthisbookandwhohasrealizedtheindexandthe
camera-ready copy. Finally, the editors are indebted to the editorial board of
“Logic,Epistemology,andtheUnityofScience”foracceptingthisvolumein
theirseries. TheywouldalsoliketothankSpringerPublishersand,inparticular,
FloorOosting.
Chapter 1
INTRODUCTION: ALTERNATIVE LOGICS
AND CLASSICAL CONCERNS
JohanvanBenthem
UniversityofAmsterdam/ILLC
[email protected],[email protected]
Modern logic shows a wide variety of perspectives, application areas, and
formalsystems,whichoftengoundertheheadingof‘alternativelogics’. The
lively PILM conference held in Nancy during September 2002 on which the
presentvolumeisbasedwasintendedasanencounterbetweenmodernworkon
alternativelogicsandclassicalissuesinthefoundationsofmathematicsandthe
philosophyoflogic. Thisbookcontainsasubstantialsampleofwhathappened
intheprocess,butitcanalsobereadindependentlyasareportonthestateof
theart.
Actually, terms like ‘alternative’ or ‘non-classical’ logic can easily be mis-
understood. Our aim with the conference and this book is not a simplistic
endorsementofmass-productionand‘anythinggoes’inlogic. Inparticular,we
do not aim for ‘alternative’ pop-art versions of the grand issues that initiated
20th century logic, whose praises used to be sung in the measured classical
strainsofBachandBeethoven-orperhapsWagneriandoomintimesoffoun-
dationalcrisis. Thereisnoneedtoreplacethesebyversionsinmodernlogics
thatsoundofjazz,rock,or(assomecriticswouldprobablyhaveit)punk,disco,
orrap. Tous,thediversityoflogicalsystemstodayrathersignalsanaturaland
respectable process of growth of the discipline, not of replacement or compe-
tition. In terms of our musical metaphor, this development transforms rigid
classicalpartitionsintoamoreopenplaygroundforimprovisation.
Some of the forces driving this process of growth are external. Over time,
and throughout the last century, logic has been confronted with a growing set
1
J.vanBenthem,G.Heinzmann,M.RebuschiandH.Visser(eds.),TheAgeofAlternativeLogics:
AssessingPhilosophyofLogicandMathematicsToday,1–7.
(cid:2)c 2006Springer.