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Outstanding Contributions to Logic 14 Melvin Fitting Brian Rayman Editors Raymond Smullyan on Self Reference Outstanding Contributions to Logic Volume 14 Editor-in-chief Sven Ove Hansson, Royal Institute of Technology, Stockholm, Sweden Editorial Board Marcus Kracht, Universität Bielefeld Lawrence Moss, Indiana University Sonja Smets, Universiteit van Amsterdam Heinrich Wansing, Ruhr-Universität Bochum More information about this series at http://www.springer.com/series/10033 Melvin Fitting Brian Rayman (cid:129) Editors Raymond Smullyan on Self Reference 123 Editors Melvin Fitting BrianRayman City University of NewYork (CUNY) NYCDepartment of Education System Brooklyn, NY City University of NewYork USA NewYork City,NY USA ISSN 2211-2758 ISSN 2211-2766 (electronic) Outstanding Contributions to Logic ISBN978-3-319-68731-5 ISBN978-3-319-68732-2 (eBook) https://doi.org/10.1007/978-3-319-68732-2 LibraryofCongressControlNumber:2017956336 ©SpringerInternationalPublishingAG2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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Coverillustration:MelvinFitting Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface ThisvolumebeganasatributetoRaymondSmullyan,withtheadditionalhopethat it would please him. It ended as a tribute to his memory. Along the way, it did please him, in a pre-publication version. Raymond Smullyan had a lifelong interest in self-reference. Actually, this is a considerable understatement. He returned to the subject over and over. Self-reference and fixpoints are closely related, after all. His first technical publi- cations, preceding his Ph.D., had to do with self-reference, as did many of his technical books. Outside of mathematics, he is best-known for his popular puzzle books, and self-reference is a fundamental theme of many of these. The present volume is not a detailed formal study of Smullyan’s work. Rather it consists of commentary on and extensions of this work, by colleagues, friends, and admirers. We hope it provides some enlightenment and some entertainment, which is what Raymond would have hoped for too. New York City, USA Melvin Fitting Brooklyn, USA Brian Rayman v Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Melvin Fitting 2 Formal Systems, Logics, and Programs . . . . . . . . . . . . . . . . . . . . . 23 Robert L. Constable 3 Adaptive Fault Diagnosis using Self-Referential Reasoning. . . . . . . 39 Robert Cowen 4 Russell’s Paradox, Gödel’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . 47 Melvin Fitting 5 Dance of the Starlings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Henk Barendregt, Jörg Endrullis, Jan Willem Klop and Johannes Waldmann 6 Some Tweets About Mockingbirds . . . . . . . . . . . . . . . . . . . . . . . . . 113 Rick Statman 7 Knights, Knaves, Truth, Truthfulness, Grounding, Tethering, Aboutness, and Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Stephen Yablo 8 What I Tell You Three Times Is True . . . . . . . . . . . . . . . . . . . . . . 141 Martin Davis 9 Gödel, Lucas, and the Soul-Searching Selfie . . . . . . . . . . . . . . . . . . 147 Vann McGee 10 An Island Tale for Young Anthropologists . . . . . . . . . . . . . . . . . . . 165 Andrew G. Buchanan and John H. Conway vii viii Contents 11 Making The ‘Hardest Logic Puzzle Ever’ a Bit Harder . . . . . . . . . 181 Walter Carnielli 12 Bibliography of Raymond Smullyan . . . . . . . . . . . . . . . . . . . . . . . . 191 Raymond M. Smullyan Chapter 1 Introduction MelvinFitting Raymond Smullyan was born in 1919, after the Great War and before the Great Depression.IhaveknownhimsinceIwasagraduatestudentandhewasmyadvisor. IgotmyPhDin1968,soitisclearthatithasbeenalongtimethatIhaveknown him.InthisintroductionIwillcallhim“Raymond.”Thismaysoundformaltothose whoknowhimas“Ray,”butitiswhatIhavealwayscalledhim,andIdon’tfeellike changingnow. Raymondhashadtwolong,different,butoverlappingcareers:oneasanacademic mathematicianandphilosopher,andanotherasamorepopularfigure.1Mathematics andphilosophyareintertwined,eachinfluencestheother,anditisessentiallyimpos- sibletodiscussonewithouttheother.Further,largepartsofbothrelatedirectlyto our chosen topic of self-reference. We begin with a broad overview, followed by sectionsonhistechnicalandonhismorepopularwork. 1.1 TheBroadBackground Asanacademic,Raymondhaswrittenseveralbooksandnumerousresearchpapers thathavebeeninfluential,ofteninunexpectedways.Hisfirstbook,TheoryofFor- malSystems(Smullyan1961)wasanovelpresentationofrecursiontheoryandthe Gödelincompletenesstheorems,topicsherevisitedoftenthroughouthiscareer.The bookintroducedideasthatwereinfluentialinautomatatheory,aswellasadecidedly abstractapproachtoincompletenessandundecidabilityresultsconcerningthefoun- dationsofmathematics.Remarkablyandunexpectedly,italsoanticipatedsomeof 1Iomitdiscussionofhislivesaspianistandasmagician. B M.Fitting( ) CityUniversityofNewYork(emeritus),Montrose,NY10548,USA e-mail:melvin.fi[email protected] ©SpringerInternationalPublishingAG2017 1 M.FittingandB.Rayman(eds.),RaymondSmullyanonSelfReference, OutstandingContributionstoLogic14,https://doi.org/10.1007/978-3-319-68732-2_1 2 M.Fitting theideasunderlyingthecomputerlanguageProlog,developedmuchlaterforusein artificialintelligence.NeitheratthattimenortodayhasRaymondclaimedtoknow the first thing about computer science, but of course that is not a prerequisite for havinganinfluence. Raymond’ssecondbook,First-OrderLogic(Smullyan1968)wasintendedtobe an elegant and beautiful presentation of formal logic.2 It was indeed elegant, and influencedseveralgenerationsoflogicians.Afewyearsagoitwasreprintedinthe Doverbookseries,andcontinuesitsinfluence.Butstartinginthe1980sithasalso had an unanticipated life in computer science, in the field of automated theorem proving. There is a long-running annual conference, Tableaux, with methods and programs covering many areas of logic and mathematics, and many a paper is, in away,adirectorindirectdescendentofRaymond’s1968book.First-OrderLogic, alongwithajournalpaperofRaymond,AUnifyingprincipleinquantificationthe- ory,(Smullyan1965),introducedwhatbecamecalledtheModelExistenceTheorem, whichlaterbecameafundamentaltoolininfinitarylogic,againanunexpectedout- comeofRaymond’ssearchforelegance.ButtheareascoveredinFirst-OrderLogic do not involve self-reference, and we will not discuss it further here. There have been several other of Raymond’s books devoted to his research in mathematical logic: Gödel’s Incompleteness Theorems (Smullyan 1992a); Recursion Theory for Metamathematics(Smullyan1993)DiagonalizationandSelf-Reference(Smullyan 1994); and jointly with Melvin Fitting, Set Theory and the Continuum Problem (Smullyan and Fitting 1996). All but the last of these do have self-reference as a centraltopic.Inadditiontothebooksmentionedabove,therehavebeennumerous researchpapers,afewofwhichwillbediscussed. In the 1970s, even while continuing his formal mathematical logic researches, Raymond’scareertookaratherunexpectedturn.Hedevelopedaninterestinpuzzles, especiallythosethatwerebasedonmathematicallogic.MartinGardnerdevoteda column inScientific American to some of these puzzles, and they turned out to be quitepopular.ThisledtoWhatistheNameofThisBook?(Smullyan1978).Itturned outtobeunexpectedlysuccessful,andwasfollowedbyaverylargenumberofothers: ThisBookNeedsNoTitle(Smullyan1980);AliceinPuzzle-Land(Smullyan1982a), and many others. I have lost track of foreign translations—there have been a very large number. As a result of the popularity of these books, Raymond appeared on theTonightShowwithJohnnyCarson,andonWilliamF.Buckley’sfiringline,and othertelevisionshowsaswell. At some point Raymond decided that puzzles could be used to teach some of thefundamentaldiscoveriesofmodernlogic.Thisideawasfirstimplementedinhis bookTheLadyortheTiger?(Smullyan1982c),whichactuallyledreadersthrough the basic ideas of Gödel’s incompleteness theorem, via a series of puzzles. That was followed by To Mock a Mockingbird (Smullyan 1985), which explored the Lambda-calculusviapuzzlesaboutbirds.3TheLambdacalculusisasystemofformal 2ElegantisthewordmostoftenusedaboutRaymond’sworkinreviews. 3RaymondalsoexploredthetopicmoretechnicallyinhisbookDiagonalizationandSelf-Reference, (Smullyan1994).