Table Of ContentPureInductiveLogic
Pure inductive logic is the study of rational probability treated as a branch of
mathematical logic. This monograph, the first devoted to this approach, brings
togetherthekeyresultsfromthepastseventyyears,plusthemaincontributions
of the authors and their collaborators over the last decade, to present a compre-
hensiveaccountofthedisciplinewithinasingleunifiedcontext.Theexposition
is structured around the traditional bases of rationality, such as avoiding Dutch
Books, respecting symmetry and ignoring irrelevant information. The authors
uncoverfurtherrationalityconcepts,bothintheunaryandinthenewlyemerging
polyadiclanguages,suchasconformity,spectrumexchangeability,similarityand
language invariance. For logicians with a mathematical grounding, this book
providesacompleteself-containedcourseonthesubject,takingthereaderfrom
thebasicsuptothemostrecentdevelopments. Itisalsoausefulreferencefora
wideraudiencefromphilosophyandcomputerscience.
Jeffrey Paris is a professor in the School of Mathematics at the University
of Manchester. His research interests lie in mathematical logic, particularly set
theory,modelsofarithmeticandnon-standardlogics.In1983hewasawardedthe
LondonMathematicalSociety’sJuniorWhiteheadPrizeandin1999waselected
a Fellow of the British Academy in the Philosophy Section. He is the author of
TheUncertainReasoner’sCompanion(CambridgeUniversityPress,1995).
Alena Vencovská received her PhD from Charles University, Prague. She has
heldastringofresearchandlecturingpositionsintheSchoolofMathematicsat
theUniversityofManchester.Herresearchinterestsincludeuncertainreasoning,
nonstandardanalysis,alternativesettheoryandthefoundationsofmathematics.
PERSPECTIVESINLOGIC
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centralthemeliesinanyareaoraspectoflogic.Booksthatpresentnewmaterial
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DepartmentofMathematics,UniversityofCaliforniaBerkeley
EditorialBoard:
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DepartmentofComputingScience,UniversityofOxford
StevenA.Cook
ComputerScienceDepartment,UniversityofToronto
MichaelGlanzberg
DepartmentofPhilosophy,UniversityofCaliforniaDavis
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DepartmentofMathematics,UniversityofChicago
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PERSPECTIVES IN LOGIC
Pure Inductive Logic
JEFFREY PARIS
UniversityofManchester
ALENA VENCOVSKÁ
UniversityofManchester
association for symbolic logic
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
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©AssociationforSymbolicLogic2015
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Firstpublished2015
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
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Paris,J.B.(JeffB.),author.
Pureinductivelogic/JeffreyParis,UniversityofManchester;
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pages cm.–(Perspectivesinlogic)
Includesbibliographicalreferencesandindex.
ISBN978-1-107-04230-8(Hardback)
1. Induction(Logic) I. Vencovská,Alena,author. II. Title.
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CONTENTS
Preface.......................................................... ix
Part1. TheBasics
Chapter1. IntroductiontoPureInductiveLogic............ 3
Chapter2. Context............................................ 9
Chapter3. ProbabilityFunctions............................. 11
Chapter4. ConditionalProbability........................... 21
Chapter5. TheDutchBookArgument........................ 25
Chapter6. SomeBasicPrinciples.............................. 33
Chapter7. SpecifyingProbabilityFunctions.................. 39
Part2. UnaryPureInductiveLogic
Chapter8. IntroductiontoUnaryPureInductiveLogic..... 49
Chapter9. deFinetti’sRepresentationTheorem............... 55
Chapter10. RegularityandUniversalCertainty............. 61
Chapter11. Relevance........................................ 69
Chapter12. AsymptoticConditionalProbabilities............ 73
Chapter13. TheConditionalizationTheorem................. 81
Chapter14. AtomExchangeability............................ 87
Chapter15. Reichenbach’sAxiom.............................. 93
Chapter16. Carnap’sContinuumofInductiveMethods...... 99
Chapter17. Irrelevance.......................................103
v
vi Contents
Chapter18. AnotherContinuumofInductiveMethods...... 125
Chapter19. TheNP-Continuum............................... 135
Chapter20. TheWeakIrrelevancePrinciple..................143
Chapter21. EqualitiesandInequalities....................... 155
Chapter22. PrinciplesofAnalogy............................165
Chapter23. UnarySymmetry..................................171
Part3. PolyadicPureInductiveLogic
Chapter24. IntroductiontoPolyadicPureInductiveLogic. 181
Chapter25. PolyadicConstantExchangeability..............183
Chapter26. PolyadicRegularity..............................189
Chapter27. SpectrumExchangeability........................193
Chapter28. Conformity....................................... 199
Chapter29. TheProbabilityFunctionsup¯,L...................205
Chapter30. TheHomogeneous/HeterogeneousDivide........ 213
Chapter31. RepresentationTheoremsforSx..................223
Chapter32. LanguageInvariancewithSx.................... 239
Chapter33. SxwithoutLanguageInvariance.................247
Chapter34. AGeneralRepresentationTheoremforSx....... 257
Chapter35. TheCarnap-Stegmu¨llerPrinciple................267
Chapter36. InstantialRelevanceandSx..................... 269
Chapter37. Equality..........................................275
Chapter38. ThePolyadicJohnson’sSufficientnessPostulate. 285
Chapter39. PolyadicSymmetry................................291
Chapter40. Similarity.........................................303
Chapter41. PIPandAtomExchangeability...................311
Chapter42. TheFunctionsup¯,L ............................... 317
E¯
Contents vii
Chapter43. LessWellTravelledRoads.......................323
Bibliography....................................................327
Index............................................................337
SymbolsandAbbreviations......................................341
