Advances in proof-theoretic semantics PDF
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Trends in Logic 43 Thomas Piecha Peter Schroeder-Heister E ditors Advances in Proof- Theoretic Semantics Trends in Logic Volume 43 TRENDS IN LOGIC Studia Logica Library VOLUME 43 Editor-in-chief Heinrich Wansing,Ruhr-University Bochum, Bochum, Germany EditorialAssistant Andrea Kruse,Ruhr-University Bochum, Bochum, Germany Editorial Board Aldo Antonelli,University ofCalifornia, Davis, USA ArnonAvron,University ofTelAviv, TelAviv, Israel Katalin Bimbó,University ofAlberta, Edmonton, Canada Giovanna Corsi,University ofBologna,Bologna,Italy JanuszCzelakowski, University ofOpole, Opole, Poland RobertoGiuntini, University of Cagliari,Cagliari,Italy Rajeev Goré,Australian National University, Canberra, Australia Andreas Herzig, University of Toulouse, Toulouse,France AndrzejIndrzejczak, University of Łodz,Łodz, Poland Daniele Mundici,University of Florence,Florence,Italy Sergei Odintsov,Sobolev Institute of Mathematics, Novosibirsk,Russia EwaOrłowska,Institute of Telecommunications, Warsaw, Poland PeterSchroeder-Heister, University ofTübingen,Tübingen,Germany YdeVenema, University ofAmsterdam, Amsterdam, TheNetherlands AndreasWeiermann, University ofGhent, Ghent, Belgium FrankWolter, University of Liverpool,Liverpool, UK Ming Xu,Wuhan University,Wuhan, People’s Republicof China Founding editor RyszardWójcicki, Polish Academyof Sciences, Warsaw, Poland SCOPE OF THE SERIES ThebookseriesTrendsinLogiccoversessentiallythesameareasasthejournalStudiaLogica, thatis,contemporaryformallogicanditsapplicationsandrelationstootherdisciplines.The series aims at publishing monographs and thematically coherent volumes dealing with importantdevelopmentsinlogicandpresentingsignificantcontributionstologicalresearch. Theseriesisopentocontributionsdevotedtotopicsrangingfromalgebraiclogic,model theory,prooftheory,philosophicallogic,non-classicallogic,andlogicincomputerscience to mathematical linguistics and formal epistemology. However, this list is not exhaustive, moreover, the range of applications, comparisons and sources of inspiration is open and evolves overtime. More information about this series at http://www.springer.com/series/6645 Thomas Piecha Peter Schroeder-Heister (cid:129) Editors Advances in Proof-Theoretic Semantics Editors ThomasPiecha PeterSchroeder-Heister Department ofComputer Science Department ofComputer Science University of Tübingen University of Tübingen Tübingen Tübingen Germany Germany ISSN 1572-6126 ISSN 2212-7313 (electronic) Trends inLogic ISBN978-3-319-22685-9 ISBN978-3-319-22686-6 (eBook) DOI 10.1007/978-3-319-22686-6 LibraryofCongressControlNumber:2015949264 SpringerChamHeidelbergNewYorkDordrechtLondon © The Editor(s) (if applicable) and The Author(s) 2016. The book is published with open access at SpringerLink.com. Open Access This book is distributed under the terms of the Creative Commons Attribution NoncommercialLicense,whichpermitsanynoncommercialuse,distribution,andreproductioninany medium,providedtheoriginalauthor(s)andsourcearecredited. All commercial rights are reserved by the Publisher, whether the whole or part of the material is concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformationstorageand retrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknown orhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. 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Printedonacid-freepaper SpringerInternationalPublishingAGSwitzerlandispartofSpringerScience+BusinessMedia (www.springer.com) Contents Advances in Proof-Theoretic Semantics: Introduction. . . . . . . . . . . . . . 1 Thomas Piecha and Peter Schroeder-Heister On the Relation Between Heyting’s and Gentzen’s Approaches to Meaning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Dag Prawitz Kreisel’s Theory of Constructions, the Kreisel-Goodman Paradox, and the Second Clause . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Walter Dean and Hidenori Kurokawa On the Paths of Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Kosta Došen Some Remarks on Proof-Theoretic Semantics. . . . . . . . . . . . . . . . . . . . 79 Roy Dyckhoff Categorical Harmony and Paradoxes in Proof-Theoretic Semantics. . . . 95 Yoshihiro Maruyama The Paradox of Knowability from an Intuitionistic Standpoint . . . . . . . 115 Gabriele Usberti Explicit Composition and Its Application in Proofs of Normalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Jan von Plato Towards a Proof-Theoretic Semantics of Equalities . . . . . . . . . . . . . . . 153 Reinhard Kahle On the Proof-Theoretic Foundations of Set Theory. . . . . . . . . . . . . . . . 161 Lars Hallnäs A Strongly Differing Opinion on Proof-Theoretic Semantics? . . . . . . . . 173 Wilfrid Hodges v vi Contents Comments on an Opinion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Kosta Došen On Dummett’s “Proof-Theoretic Justifications of Logical Laws”. . . . . . 195 Warren Goldfarb Self-contradictory Reasoning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Jan Ekman Completeness in Proof-Theoretic Semantics . . . . . . . . . . . . . . . . . . . . . 231 Thomas Piecha Open Problems in Proof-Theoretic Semantics. . . . . . . . . . . . . . . . . . . . 253 Peter Schroeder-Heister Advances in Proof-Theoretic Semantics: Introduction ThomasPiechaandPeterSchroeder-Heister Abstract Asdocumentedbythepapersinthisvolume,whichmostlyresultfromthe secondconferenceonproof-theoreticsemanticsinTübingen2013,proof-theoretic semanticshasadvancedtoawell-establishedsubjectinphilosophicallogic. Keywords Proof-theoreticsemantics In the mid-1980s, the term “proof-theoretic semantics” (Schroeder-Heister 1991 [13],andbeforeinlectures)wasproposed(1)toexplainmeaningintermsofproof ratherthandenotationortruthand(2)togiveasemanticsforproofs.Thoughrelated to the meaning-as-use approach in the philosophy of language, and belonging to whatinamoregeneralsettinghasbeencalled“inferentialism”(Brandom1994[1]), theintention of proof-theoretic semantics was tocapture and continue thespecific line of research that originated from the work of Gentzen (1934/35) [5] (and also Jas´kowski1934[6])andwastakenupanddeveloped,amongstothers,byLorenzen (1955) [8], Prawitz (1965, 1971) [11, 12], von Kutschera (1968) [15], Martin-Löf (1975,1984)[9,10],andDummett(1975,1991)[2,3].Whereasinthe1980sproof- theoreticsemanticswasalmostexclusivelythebusinessofproof-theorists,thefield hassinceexpandedintothewiderareaofphilosophicallogic.Thefirstconference onproof-theoretic semantics washeld in1999 inTübingen withaspecial issueof Synthese originating from it, which was published in 2006 (Kahle and Schroeder- Heister2006[7]).Atthetimeofthisfirstconference,thesubjectstillbelongedtoa relativelysmallcommunityoflogiciansandphilosophers.Thishaschangedinthe meantime. One only needs to look at the multitude of papers published on issues ofproof-theoreticsemanticsinthepastdecadeandatthewidespreadusageofthis term. “Proof-theoretic semantics” is no longer the provocative title it used to be, containinganallegedcontradictioinadjectobetweenprooftheoryasdealingwith syntax,andsemanticsasdealingwithmeaning.Thelinkbetweenproofsandmeaning is well-established now. Given the growing interest in the subject, we organised a B T.Piecha( )·P.Schroeder-Heister DepartmentofComputerScience,UniversityofTübingen,Tübingen,Germany e-mail:[email protected] P.Schroeder-Heister e-mail:[email protected] ©TheAuthor(s)2016 1 T.PiechaandP.Schroeder-Heister(eds.),AdvancesinProof-TheoreticSemantics, TrendsinLogic43,DOI10.1007/978-3-319-22686-6_1 2 T.PiechaandP.Schroeder-Heister secondconferenceonproof-theoreticsemanticsinTübingenin2013todiscussthe advances in the now well-established field (for overviews see Wansing 2000 [16], Schroeder-Heister2012[14]).Somespeakersofthesecondconferencehadalready spoken at the first, namely Došen, Dyckhoff, Hallnäs, Kahle, Prawitz, Sundholm, TaitandUsberti. Thepresentationsgivenattheconferencewerethefollowing. • SergeiN.Artëmov:OnBrouwer-Heyting-Kolmogorovprovabilitysemantics • WalterDeanandHidenoriKurokawa:Kreisel’ssecondclauseandtheTheoryof Constructions • KostaDošen:Twowaysofgeneralprooftheory • RoyDyckhoff:Generalisedeliminationrules • LarsHallnäs:Ontheproof-theoreticfoundationsofsettheory • WilfridHodges:Thechoiceofsemanticsasamethodologicalquestion • ReinhardKahle:Themodeofpresentation • YoshihiroMaruyama:Onparadoxesinproof-theoreticsemantics • JanvonPlato:Explicitcompositionanditsapplicationinnormalizationproofs • DagPrawitz:RemarksonrelationsbetweenGentzenandHeytinginspiredPTS • GiovanniSambin:Unificationoflogicsbyreflection • GöranSundholm:BHKandBrouwer’stheoryofthecreativesubject • WilliamW.Tait:Compositionalsemanticsforpredicatelogic:Eliminatingbound variablesfromformulasanddeductions • GabrieleUsberti:Intuitionism,theParadoxofKnowabilityandempiricalnegation • HeinrichWansing:Atwo-sortedtypedlambda-calculus Thisvolumecollectsthecontributionsofmanyoftheparticipants,notnecessarily intheformpresented,andalsosomeadditionalpapersbyauthorswhodidnotspeakat theconference.Thereforeitexemplifiesfrommanyperspectiveswhatproof-theoretic semantics is about. The papers by Prawitz and Dean and Kurakowa confront the proof-theoreticapproachwiththeconstructivesemanticsofHeytingandofKreisel andGoodman,respectively.Došenpleadsforacategorialapproachtoproof-theoretic semantics,arguingthatitbestexhibitsthestructuresofdeductions.Dyckhoff’spaper isacriticalexaminationofapproachesinproof-theoreticsemanticsbasedongeneral elimination rules of a certain form. Maruyama gives a taxonomy of various forms of paradoxes based on a categorial approach to proof-theoretic harmony. Usberti proposesasolutiontotheepistemicknowabilityparadoxfromthestandpointoflogic. Von Plato investigates the significance of a rule of explicit composition in natural deduction which makes the substitution of a derivation for an open assumption an inferencestepofitsown.Kahleappliesproof-theoreticsemanticstothetreatmentof equalitybyelucidatingthedifferencebetweenextensionalandintensionalequality in a non-denotational way. Hallnäs sketches some ideas towards a proof-theoretic foundationforsettheoryusinggeneralisationsofdefinitionalreflection.Thepaperby Hodges,whichistheonlyonenotwrittenfromtheperspectiveofaproof-theoretic semanticist or constructivist, defends model theory against false claims from the proof-theoreticsemantics‘camp’.ItisfollowedbyareplybyDošen. AdvancesinProof-TheoreticSemantics:Introduction 3 Goldfarb’spaperhasbeencirculatingformorethan15yearsandhasbeenreferred to repeatedly. It was the subject of Dummett’s presentation at the first conference in 1999. It represents significant results on the proof-theoretic semantics of intu- itionisticlogic,inparticularonthequestionofcompleteness.Anothercontribution notpresentedattheconferenceisachapterofEkman’sthesis(Ekman1994[4]).It uses an interesting way of associating labels with formulas that are proved, which is different from standard Curry–Howard term-annotation and particularly suited toanalysenon-well-foundedandparadoxicalreasoning,atopicwhichhasrecently gained much attention in the proof-theoretic semantics community. There are also papersbyus,theeditors:Anoverviewonresultsconcerningcompletenessinproof- theoretic semantics and a presentation of three open problems that are considered significantforthefurtherdevelopmentofthesubject. We would like to thank all participants of the conference, and in particular all contributors to this volume, who made this work possible, as well as those who helped with reviewing and editing the papers. We are also very grateful toMarine Gaudefroy-Bergmann for invaluable organisational assistance. The second editor is particularly grateful to Reinhard Kahle and Thomas Piecha, who organised the conference as a present for his 60th birthday including a special colloquium with friendsandcolleagues. Acknowledgments ThisworkwassupportedbytheFrench-GermanANR-DFGproject“Hypo- theticalReasoning—ItsProof-TheoreticAnalysis”(HYPOTHESES),DFGgrantSchr275/16-2.Its completionwassupportedbytheFrench-GermanANR-DFGproject“BeyondLogic:Hypothetical ReasoninginPhilosophyofScience,Informatics,andLaw”,DFGgrantSchr275/17-1. OpenAccess ThischapterisdistributedunderthetermsoftheCreativeCommonsAttribution NoncommercialLicense,whichpermitsanynoncommercialuse,distribution,andreproductionin anymedium,providedtheoriginalauthor(s)andsourcearecredited. References 1. Brandom,R.B.:MakingItExplicit:Reasoning,Representing,andDiscursiveCommitment. HarvardUniversityPress,Cambridge(1994) 2. Dummett,M.:Thejustificationofdeduction.In:ProceedingsoftheBritishAcademy,pp.201– 232(1975).SeparatelypublishedbytheBritishAcademy1973.ReprintedinDummett,M.: TruthandOtherEnigmas,Duckworth,London(1978) 3. Dummett,M.:TheLogicalBasisofMetaphysics.Duckworth,London(1991) 4. Ekman,J.:Normalproofsinsettheory.Ph.D.thesis,UniversityofGöteborg(1994) 5. Gentzen,G.:UntersuchungenüberdaslogischeSchließen.MathematischeZeitschrift39,176– 210,405–431(1934/35).Englishtranslationin:Szabo,M.E.(ed.)TheCollectedPapersof GerhardGentzen,pp.68–131.NorthHolland,Amsterdam(1969) 6. Jas´kowski,S.:Ontherulesofsuppositionsinformallogic.Stud.Log.1,5–32(1934).Reprinted in:McCall,S.(ed.)PolishLogic1920–1939,pp.232–258.Oxford(1967) 7. Kahle,R.,Schroeder-Heister,P.(eds.):Proof-TheoreticSemantics.Springer.Specialissueof Synthese148(3),503–743(2006) 8. Lorenzen,P.:EinführungindieoperativeLogikundMathematik.Springer,Berlin(1955).2nd edn.1969
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