Hybrid logic and its proof theory PDF
Preview Hybrid logic and its proof theory
Hybrid Logic and its Proof-Theory APPLIED LOGIC SERIES VOLUME37 ManagingEditor DovM.Gabbay,DepartmentofComputerScience,King’sCollege,London,U.K. Co-Editor JonBarwise† EditorialAssistant JaneSpurr,DepartmentofComputerScience,King’sCollege,London,U.K. SCOPEOFTHESERIES Logic is applied in an increasingly wide variety of disciplines, from the traditional subjects of philosophy and mathematics to the more recent disciplines of cognitive science,computerscience,artificialintelligence,andlinguistics,leadingtonewvigor inthisancientsubject. Springer,throughitsAppliedLogicSeries,seekstoprovidea homeforoutstandingbooksandresearchmonographsinappliedlogic,andindoingso demonstratestheunderlyingunityandapplicabilityoflogic. Forfurthervolumes:http://www.springer.com/series/5632 Hybrid Logic and its Proof-Theory by Torben Braüner RoskildeUniversity,Denmark 123 Dr.TorbenBraüner RoskildeUniversity Programming,LogicandIntelligentSystems ResearchGroup(PLIS) 4000Roskilde Denmark [email protected] ISSN1386-2790 ISBN978-94-007-0001-7 e-ISBN978-94-007-0002-4 DOI10.1007/978-94-007-0002-4 SpringerDordrechtHeidelbergLondonNewYork LibraryofCongressControlNumber:2010938944 (cid:2)c SpringerScience+BusinessMediaB.V.2011 Nopartofthisworkmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformorby anymeans,electronic,mechanical,photocopying,microfilming,recordingorotherwise,withoutwritten permissionfromthePublisher,withtheexceptionofanymaterialsuppliedspecificallyforthepurpose ofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthework. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface Thisbookisacollationoftheresearchonhybridlogicanditsproof-theoryIhave doneoveranumberofyears.Tobemoreprecise,thebookpresentsacollectionof research results originally published in the journal papers listed below. After each paperisanindicationofwhereinthebooktheresultsofthepaperarepresented. T.Brau¨ner. Naturaldeductionforhybridlogic.JournalofLogicandComputation, 14:329–353,2004a.Chapter2. T.Brau¨ner. Axioms for classical, intuitionistic, and paraconsistent hybrid logic. Journal of Logic, Language and Information, 15:179–194, 2006. Chapters 2 and8. T.BolanderandT.Brau¨ner Tableau-based decision procedures for hybrid logic. JournalofLogicandComputation,16:737–763,2006.Chapter3. T.Brau¨ner. Twonaturaldeductionsystemsforhybridlogic:Acomparison.Journal ofLogic,LanguageandInformation,13:1–23,2004b.Chapter4. T.Brau¨ner. Proof-theoreticfunctionalcompletenessforthehybridlogicsofevery- whereandelsewhere.StudiaLogica,81:191–226,2005c.Chapter5. T.Brau¨ner. Natural deduction for first-order hybrid logic. Journal of Logic, Lan- guageandInformation,14:173–198,2005b.Chapter6. T.Brau¨ner. Adding intensional machinery to hybrid logic. Journal of Logic and Computation,18:631–648,2008.Chapter7. T.Brau¨nerandV.dePaiva Intuitionistic hybrid logic. Journal of Applied Logic, 4:231–255,2006.Chapter8. T.Brau¨ner. Why does the proof-theory of hybrid logic work so well? Journal of AppliedNon-ClassicalLogics,17:521–543,2007.Chapters9and10. Thenotationandterminologyofthepapershasbeenrevisedwiththeaimofgiving auniformpresentation,andmoreover,interdependencieshavebeenpointedout.In some cases more substantial revisions as well as omissions have also taken place. Furthermore, new material has been added. In particular, some material from my article Brau¨ner (2005a) in The Stanford Encyclopedia of Philosophy has been in- corporatedinChapter1andsomematerialfrommypartofthechapterBrau¨nerand v vi Preface Ghilardi (2007) in Handbook of Modal Logic has been incorporated in Chapter 6. Material from the Chapters 1, 2, 3, 9, and 10 of this book is incorporated in my forthcomingchapterBrau¨ner(2011)inHandbookofPhilosophicalLogic. ThepresentbookisbasedonthethesisBrau¨ner(2009)acceptedinfulfillmentof therequirementsfortheDanishhigherdoctoratedr.scient.(doctorscientiarum). ALittleBackgroundonHybridLogicandProof-Theory Hybrid logics are obtained by adding further expressive power to ordinary modal logic.ThehistoryofhybridlogicsgoesbacktothephilosopherArthurPrior’swork in the 1960s. The most basic hybrid logic is obtained by adding nominals, which arepropositionalsymbolsofanewsortinterpretedinarestrictedwaythatenables referencetoindividualpointsinaKripkemodel(wherethepointsrepresentpossible worlds,times,locations,epistemicstates,statesinacomputer,orsomethingelse). Another addition is the satisfaction operator @, which enables the evaluation of formulasatparticularpoints.Itisnotablethatnominalsandthesatisfactionoperator do not disturb the local character of the Kripke semantics. The extra expressive powerisusefulformanyapplications,forexample,whenreasoningabouttimeone oftenwantstoformulateaseriesofstatementsaboutwhathappensatspecifictimes, andordinarymodallogicsimplydoesnotallowthis. The addition of hybrid-logical machinery increases the expressive power, but often decidability is retained. Hybrid logics are closely related to description log- ics, which are a family of decidable logics used for knowledge representation in ArtificialIntelligence.IndescriptionlogicsthepointsinaKripkemodelrepresent individualsinthespecificationofanontology.Atpresent,significantresearcheffort isputintoexploringtheborderlinebetweendecidableandundecidablelogics,one majorreasonbeingthatdecidabilityisimportantforcomputationalapplications. The subject of proof-theory is the notion of proof and formal systems for rep- resentingproofs.Thereareanumberofdifferenttypesofproofsystems.Someof themostimportanttypesarenaturaldeductionsystems,Gentzensequentsystems, tableau systems, and axiom systems. They are motivated in different ways: Proof systems of the first three types are suitable for actual reasoning. (Here the word ”actual” has a broad meaning, not restricted to actual human reasoning. The logic does not care whether it is a human that carries out the reasoning, or the reason- ingtakesplaceinacomputer,orinsomeothermedium.)Axiomsystemsareusu- ally not meant for actual reasoning, but are of a more foundational interest. When a decidable logic is considered, Gentzen and tableau systems have the desirable feature of often giving rise to decision procedures in a very direct way, therefore Gentzenandtableausystemslendthemselvestowardcomputerimplementations.In fact, during the last couple of decades, tableau systems have become a highly ac- tiveresearcharea,involvingbasicresearchaswellaspracticallyappliedwork,for exampletableausystemsforhigh-speedtheoremprovingindescriptionlogics. Thereislittleconsensusaboutproof-theoryforordinarymodallogic,especially inconnectionwithnaturaldeductionsystems,Gentzensequentsystems,andtableau Preface vii systems.Manymodal-logicalproofsystemslackimportantproof-theoreticproper- tiesandtherelationshipsbetweenproofsystemsfordifferentmodallogicsareoften unclear.InthequotationbelowHeinrichWansinggivesasuccinctsummaryofthe statusofmodal-logicalproof-theory. Comparedwiththemultitudeofnotonlyexistingbutalsointerestingaxiomaticallypre- sentablenormalmodalpropositionallogics,thenumberofsystemsforwhichsequentcal- culus presentations (of some sort) are known is disappointingly small.In contrast to the axiomaticapproach,thestandardsequent-styleprooftheoryfornormalmodallogicsfails tobe‘modular’,andtheverymechanismbehindthesmallrangeofknownpossiblevaria- tionsisnotveryclear.(Wansing 1994,p.128) Inthepresentbookweshalldemonstratethathybrid-logicalproof-theoryremedies thislackofuniformityinmodal-logicalproofsystems. TheContentofThisBook Themainissueofthisbookistheproof-theoryofhybridlogic.Tobemoreprecise: Natural deduction, Gentzen, tableau, and axiom systems for hybrid logic. We first deal with the propositional case, that is, we describe sound and complete natural deduction,Gentzen,andaxiomsystemsforpropositionalhybridlogic.Thenatural deduction and Gentzen systems satisfy the requirements that such systems are ex- pectedtosatisfy:Thenaturaldeductionsystemsatisfiesnormalization,andnormal derivationssatisfyaversionofthesubformulaproperty.TheGentzensystemiscut- freeandalsotheGentzenderivationssatisfyaversionofthesubformulaproperty. Moreover,wegivetableau-baseddecisionproceduresfortwodecidablefragments ofhybridlogic,oneofthesebeingadecisionprocedureincludingtheveryexpres- siveuniversalmodality.Afterhavingdealtwiththepropositionalcase,wedescribe proof-theoryforfirst-orderhybridlogic,includingintensionalmachinery.Further- more,wedescribeproof-theoryforintuitionistichybridlogic. Thus, we consider a spectrum of different versions of hybrid logic (proposi- tional, first-order, intensional first-order, and intuitionistic) and a spectrum of dif- ferenttypesofproof-systemsforhybridlogic(naturaldeduction,Gentzen,tableau, andaxiomsystems).Allthesesystemscanbemotivatedindependently,butthefact that the systems can be given corroborates the point of view that hybrid logic and hybrid-logicalproof-theoryisanaturalenterprise.Thislineofthinkingisexpressed brieflyandtothepointinthefollowingquotationbyNuelD.Belnap. Itseemstobegenerallyconcededthatformalsystemsarenaturalorsubstantialiftheycan belookedatfromseveralpointsofview.Wetendtothinkofsystemsasartificialoradhoc ifmostoftheirformalpropertiesarisefromsomeonenotationalsystemintermsofwhich theyaredescribed.(AndersonandBelnap 1975,p.50) Besidessatisfyingtheabovegeneralrequirements,hybrid-logicalproof-theoryfur- thermoresatisfiesthemoreconcreterequirementthatproofsystemsforwideclasses ofhybridlogicscanbegiveninauniformway,forexample,naturaldeductionsys- temsforawideclassofhybridlogicscanbeobtainedinauniformwaybyadding viii Preface derivationrulesasappropriate.Thisissimplynotpossibleinconnectionwithstan- dardproof-theoryforordinarymodallogic. This leads us to the following question: Why does the proof-theory of hybrid logic behave so well compared to the proof-theory of ordinary modal logic? Be- fore we give an answer to this question, we shall make two remarks: Firstly, we remarkthatthemetalinguisticsemantic,thatis,model-theoretic,machineryofhy- bridlogicisinternalizedinthehybrid-logicalobjectlanguage(viasatisfactionop- erators).Secondly,weremarkthatmodal-logicalrulesforreasoningdirectlyabout models(calledlabelledrules)areproof-theoreticallywell-behaved,thatis,theysat- isfy the proof-theoretic requirements such rules are expected to satisfy (but at the expenseofmakinguseofmetalinguisticmachinery).Inthepresentbookwedemon- stratethatthegoodbehaviouroflabelledrulesispreservedbyinternalization.Tobe moreprecise,ournaturaldeduction,Gentzen,andtableaurulesforhybridlogiccan be seen as internalized rules for reasoning directly about models, and what we do inthepresentbookisthatweprovideaproof-theoreticanalysisoftheinternalized rules,bywhichitisdemonstratedthattheinternalizedrulesareproof-theoretically well-behaved. The answer to the question above is accordingly that internalization of model- theoretic machinery in the object language enables us to give well-behaved proof- theory for hybrid logic. So model-theory is a prerequisite for our proof-theoretic analysis—in this sense the present book has proof-theory as well as model-theory asitsstartingpoint. Acknowledgements I have benefited greatly from discussions and collaboration with a number of re- searchers. First I would like to thank Peter Øhrstrøm for introducing me to Arthur Prior’s workandforincludingmeinhisPriorprojectatAalborgUniversitybackin1997. AlsothankstoPerHasleforconversationsonPrior’sworkandmanyothersubjects. Prior’sworkwasmyfirstcontactwithwhatnowisknownashybridlogic. My work on the proof-theory of hybrid logic has benefited in particular from discussions with Carlos Areces, Patrick Blackburn, Thomas Bolander, Balder ten Cate,MelvinFitting,ValentinGoranko,JensUlrikHansen,MaartenMarx,Valeria de Paiva, and Jørgen Villadsen. Special thanks to Thomas for fruitful collabora- tioninconnectionwiththeprojectsHyLoMOL(HybridLogicMeetsOtherLogics, 2005–2008)andHYLOCORE(HybridLogic,Computation,andReasoningMeth- ods,2009–2012).IamgratefultoPatrickformanyconversationswheneverIvisited France or he visited Denmark. Thanks to Thomas Mu¨ller for discussions on some ofthemorephilosophicalmaterialinthisbookandthankstoSaraL.Uckelmanfor helpfulcommentsonapreliminaryversionofthebook. Preface ix I would like to thank the evaluation committee—Stig Andur Pedersen, Patrick Blackburn,andMelvinFitting—formanyusefulcommentsandsuggestionsonmy dr.scient.thesisBrau¨ner(2009)onwhichthepresentbookisbased. I wish to acknowledge the financial support received from The Danish Natural ScienceResearchCouncilasfundingfortheabovementionedprojectsHyLoMOL andHYLOCORE. Thanks to my wife Anne and our children, Sara, Lea, Thomas, and Maria, for supportandforbeingthere. Roskilde,Denmark TorbenBrau¨ner
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