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PureInductiveLogic 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 ThePerspectivesinLogicseriespublishessubstantial,high-qualitybookswhose centralthemeliesinanyareaoraspectoflogic.Booksthatpresentnewmaterial not now available in book form are particularly welcome. The series ranges from introductory texts suitable for beginning graduate courses to specialized monographs at the frontiers of research. Each book offers an illuminating perspectiveforitsintendedaudience. The series has its origins in the old Perspectives in Mathematical Logic series editedbythe(cid:2)-Groupfor“MathematischeLogik”oftheHeidelbergerAkademie der Wissenschaften, whose beginnings date back to the 1960s. The Association for Symbolic Logic has assumed editorial responsibility for the series and changed its name to reflect its interest in books that span the full range of disciplinesinwhichlogicplaysanimportantrole. ThomasScanlon,ManagingEditor DepartmentofMathematics,UniversityofCaliforniaBerkeley EditorialBoard: MichaelBenedikt DepartmentofComputingScience,UniversityofOxford StevenA.Cook ComputerScienceDepartment,UniversityofToronto MichaelGlanzberg DepartmentofPhilosophy,UniversityofCaliforniaDavis AntonioMontalban DepartmentofMathematics,UniversityofChicago MichaelRathjen SchoolofMathematics,UniversityofLeeds SimonThomas DepartmentofMathematics,RutgersUniversity ASLPublisher RichardA.Shore DepartmentofMathematics,CornellUniversity Formoreinformation,seewww.aslonline.org/books_perspectives.html PERSPECTIVES IN LOGIC Pure Inductive Logic JEFFREY PARIS UniversityofManchester ALENA VENCOVSKÁ UniversityofManchester association for symbolic logic UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learningandresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781107042308 AssociationforSymbolicLogic RichardShore,Publisher DepartmentofMathematics,CornellUniversity,Ithaca,NY14853 http://www.aslonline.org ©AssociationforSymbolicLogic2015 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2015 AcataloguerecordforthispublicationisavailablefromtheBritishLibrary LibraryofCongressCataloginginPublicationdata Paris,J.B.(JeffB.),author. Pureinductivelogic/JeffreyParis,UniversityofManchester; AlenaVencovská,UniversityofManchester. pages cm.–(Perspectivesinlogic) Includesbibliographicalreferencesandindex. ISBN978-1-107-04230-8(Hardback) 1. Induction(Logic) I. Vencovská,Alena,author. II. Title. BC91.P372015 161–dc23 2015004107 ISBN978-1-107-04230-8Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication,and doesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,accurateor appropriate. 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