Tools and Techniques in Modal Logic PDF
Preview Tools and Techniques in Modal Logic
Tools and Techniques in Modal Logic Marcus Kracht II. Mathematisches Institut Freie Universita¨t Berlin Arnimallee 3 D – 14195 Berlin [email protected] About this Book Thisbookisintendedasacourseinmodallogicforstudentswhohavehadprior contactwithmodallogicandwishtostudyitmoredeeply. Itpresupposestraining inmathematicsorlogic. Verylittlespecificknowledgeispresupposed,mostresults whichareneededareprovedinthisbook. Knowledgeofbasiclogic—propositional logic,predicatelogic—aswellasbasicmathematicswillofcoursebeveryhelpful. Thebooktreatsmodallogicasatheory,withseveralsubtheories,suchascomplete- ness theory, correspondence theory, duality theory and transfer theory. Thus, the emphasis is on the inner structure of the theory and the connections between the subdisciplinesandnotoncoverageofresults. Moreover,wedonotproceedbydis- cussingonelogicaftertheother;rather,weshallbeinterestedingeneralproperties oflogicsandcalculiandhowtheyinteract. Onewillthereforenotfindsectionsde- voted to special logics, such as G, K4 or S4. We have compensated for this by a specialindexoflogics,bywhichitshouldbepossibletocollectallmajorresultson aspecificsystem. Heavyuseismadeofalgebraictechniques;moreover,ratherthan startingwiththeintuitivelysimplerKripke–frameswebeginwithalgebraicmodels. The reason is that in this way the ideas can be developed in a more direct and co- herentway. Furthermore,thisbookisaboutmodallogicswithanynumberofmodal operators. Althoughthismayoccasionallyleadtocumbersomenotation,itwasfelt necessarynottospecializeonmonomodallogics. Forinmanyapplicationsoneop- eratorisnotenough,andsomodallogiccanonlybereallyusefulforothersciences ifitprovidessubstantialresultsaboutpolymodallogics. Nobookcantreatasubjectareaexhaustively,andthereforeacertainselection hadtobemade. Thereaderwillprobablymissadiscussionofcertainsubjectssuch asmodalpredicatelogic,provabilitylogic,prooftheoryofmodallogic,admissibility of rules, polyadic operators, intuitionistic logic, and arrow logic, to name the most importantones. Thechoiceofmaterialincludedisguidedbytwoprinciples: first,I prefertowriteaboutwhatIunderstandbest;andsecond,aboutsomesubjectsthere alreadyexistgoodbooks(see[182],[43],[31],[157],[224]),andthereisnoneedto addanotherone(whichmightevennotbeasgoodastheexistingones). IgotacquaintedwithmodallogicviaMontagueSemantics,butitwasthebook [169]byWRthatreallyhookedmeontothissubject. Itisapity that this book did not get much attention. Until very recently it was the only book whichtreatedmodallogicfromamathematicalpointofview.(Meanwhile,however, v vi Aboutthisbook the book [43] has appeared in print, which is heartily recommended.) However, twentyyearshavepassedfromitspublicationandmanystrongandimportantresults havebeenfound,andthiswasthereasonforwritingthisbook. MyintellectualcreditsgonotonlytoWRbutalsotoS B—whoseearlydeathsaddenedmegreatly—forteachingmealge- bra,HSforhisinspiringintroductiontogeometryandlinearalgebra, andtoWFforhisintroductiontologicandexactmathematics.Further- more,IwishtothankKFformakinganexceptionandtakingmeashisstudent inEdinburgh. Hetootaughtmelogicinhisratherdistinctway. Morethananyone in the last years, F W has been an inspiration and collaborator. Without him,thisbookwouldnothavebeenwritten. ThankstoCGforhishelp both with some of the pictures as well as modal logic, and thanks also to A B¨ and M M. Thanks to M B, S D, K F, C H, C I, T K and T S for careful proofreading and R G´ and M Z for their ad- viceinmanymatters. ThefinaldraftwascarefullyreadbyHMandB N. SpecialthanksgotoAEforhisneverendingmoralsupport. Noendeavourcansucceedifitisnotblessedbyloveandunderstanding. Iam fortunatetohaveexperiencedboththroughmywifeJD,myparents, my brother and my sister. This book is dedicated to all those to whom it gives pleasure. May it bring — in its own modest way — a deeper understanding of the humanspirit. Berlin,March1999 MarcusKracht Added. Anumberoferrorsintheprintedversionhavebeenbroughttomyattention byGB,LH,andTK. Overview The book is structured as follows. There are ten chapters, which are grouped intothreeparts. ThefirstpartcontainstheChapters1–3,thesecondparttheChap- ters4–7andthethirdparttheChapters8–10. Thefirstpartcontainsroughlythe equivalent of a four hour one semester course in modal logic. Chapter 1 presents thebasicsofalgebraandgeneralpropositionallogicinasmuchastheyareessential for understanding modal logic. This chapter introduces the theory of consequence relations and matrix semantics. From it we deduce the basic completeness results inmodallogic. Thegeneralityoftheapproachisjustifiedbytwofacts. Thefirstis thatinmodallogicthereareseveralconsequencerelationsthatareassociatedwitha givenlogic,sothatacquaintancewiththegeneraltheoryofconsequencerelationsis essential. Second, manyresultscanbeunderstoodmorereadilyintheabstractset- ting.AfterthefirstchapterfollowtheChapters2and3,inwhichweoutlinethebasic terminology and techniques of modal logic, such as completeness, Kripke–frames, generalframes,correspondence,canonicalmodels,filtration,decidability,tableaux, normal forms and modal consequence relations. One of the main novelties is the methodofconstructivereduction. Itservesadualpurpose. Firstofall,itisatotally constructivemethod, whencethename. Itallowstogiveproofsofthefinitemodel property for a large variety of logics without using infinite models. It is a little bit more complicated than the filtration method, but in order to understand proofs by constructivereductiononedoesnothavetounderstandcanonicalmodels,whichare rather abstract structures. Another advantage is that interpolation for the standard systems can be deduced immediately. New is also the systematic use of the dis- tinctionbetweenlocalandglobalconsequencerelationsandtheintroductionofthe compoundmodalities, whichallowsforratherconcisestatementsofthefacts. The latterhaslargelybeennecessitatedbythefactthatweallowtheuseofanynumber ofmodaloperators. Also,thefixedpointtheoremforGofDJandG- Sisproved. Here,wededuceitfromtheso–calledBeth–property,which inturnfollowsfrominterpolation. ThisproofisoriginallyduetoCS´ [200]. Thesecondpartconsistsofchaptersondualitytheory, correspondencetheory, transfer theory and lattice theory, which are an absolute necessity for understand- ing higher modal logic. In Chapter 4 we develop duality theory rather extensively, starting with universal algebra and Stone–representation. Birkhoff’s theorems are vii viii Overview proved in full generality. This will establish two important facts. One is that the lattice of normal modal logics is dually isomorphic to the lattice of subvarieties of the variety of modal algebras. Secondly, the characterization of modally definable classesofgeneralizedframesintermsofclosurepropertiesisreadilyderived. After thatwegiveanoverviewofthetopologicalandcategorialmethodsofGS- andVV,developedin[186]and[187]. Furthermore,westudythe connectionbetweenpropertiesoftheunderlyingKripke–frameandpropertiesofthe underlyingalgebrainadescriptiveframe. Wewillshow,forexample,thatsubdirect irreducibilityofanalgebraandrootednessofdualdescriptiveframeareindependent properties. (This has first been shown in [185].) We conclude this chapter with a discussionofthestructureofcanonicalframesandsomealgebraiccharacterizations of interpolation, summarizing the work of L M. An algebraic char- acterizationofHallde´n–completenessusingcoproductsisderived,whichisslightly strongerthanthatof[153]. Chapter5developsthethemeoffirst–ordercorrespon- dence using the theory of internal descriptions, which was introduced in M K[121]. WewillprovenotonlythetheorembyHS[183]but alsogiveacharacterizationoftheelementaryformulaewhicharedefinablebymeans ofSahlqvistformulae. Thisisdoneusingatwo–sidedcalculusbymeansofwhich correspondencestatementscanbesystematicallyderived. Althoughthiscalculusis atthebeginningsomewhatcumbersome,itallowstocomputeelementaryequivalents ofSahlqvistformulaewithease. Moreover,wewillshowmanynewcorollaries; in particular,weshowthatthereisasmallerclassofmodalformulaeaxiomatizingthe Sahlqvistformulae. Ontheotherhand,wealsoshowthattheclassofformulaede- scribedbyBin[10]whichislargerthantheclassdescribedbySahlqvist does not axiomatize a larger class of logics. Next we turn to the classic result by K F [65] that a logic which is complete and elementary is canonical, but also the result that a modally definable first–order condition is equivalent to a positive restricted formula. This has been the result of a chain of theorems developed by SF,RGandmainlyJB,see[10]. In Chapter6wediscusstransfertheory,arelativelynewtopic,whichhasbroughtalot of insights into modal logic. Its aim is to study how complex logics with several operatorscanbereducedtologicswithlessoperators. Thefirstmethodisthatofa fusion. Given two modal logics, their fusion is the least logic in the common lan- guagewhichcontainsbothlogicsasfragments. Thisconstructionhasbeenstudied by F W in [233], by K F and G S [67], and by F WandMKin[132]. FormanypropertiesPitisshownthatafu- sionhasPiffbothfragmentshaveP. Inthelastsectionaratherdifferenttheoremis proved. Itstatesthatthereisanisomorphismfromthelatticeofbimodallogicsonto anintervalofthelatticeofmonomodallogicssuchthatmanypropertiesareleftin- variant. ThisisomorphismisbasedonthesimulationsdefinedbyS.K.Tin [208,210]. Someuseofsimulationshasbeenmadein[127],butthistheoremisnew inthisstrongform. OnlythesimulationsofThavethesestrongproperties. Overview ix Extensive use of these results is made in subsequent chapters. Many problems in modallogiccanbesolvedbyconstructingpolymodalexamplesandthenappealing to this simulation theorem. Chapter 7 discusses the global structure of the lattices ofmodallogics. ThisinvestigationhasbeeninitiatedbyWBandW R,whosesplittingtheorem[170]hasbeenagreatimpulseintheresearch. WestateithereinthegeneralformofFW[234],whobuilton[120]. The lattergeneralizedthesplittingtheoremof[170]tonon–weaklytransitivelogicsand finitelypresentablealgebras. [234]hasshownthisusetobeinessential;weshowin Section 7.5 that there exist splitting algebras which are not finitely presentable. In the remaining part of this chapter we apply the duality theory of upper continuous lattices,whicharealsocalledframesorlocales(see[110])tomodallogic. Onere- sult is a characterization of those lattices of logics which admit an axiomatization base. ThisquestionhasbeenputandansweredforK4byACand MZ[42]. Theargumentusedhereisrathersimpleandstraight- forward.WeproveanumberofbeautifultheoremsbyWBaboutthedegreeof incompletenessoflogics. Thewaytheseresultsareproveddeservesattention. We donotmakeuseofultraproducts,onlyofthesplittingtheorem. Thisisratheradvan- tageous, since the structure of ultraproducts of Kripke–frames is generally difficult to come to terms with. Finally, the basic structure of the lattice of tense logics is outlined. Thisistakenfrom[123]. Thelastpartisaselectionofissuesfrommodallogic. Sometopicsaredevel- oped in great depth. Chapter 8 explores the lattice of transitive logics. It begins withtheresultsofKFconcerningthestructureoffinitely generatedtransitive framesandtheselectionprocedureofMZ, leadingtothecofi- nalsubframelogicsandthecanonicalformulae. Thecharacterizationofelementary subframelogicsbyKFisdeveloped. Afterthatweturntothestudyoflogics offinitewidth. Theselogicsarecompletewithrespecttonoetherianframessothat thestructuretheoryofKF[66]canbeextendedtothewholeframe. Thisisthe starting point for a rich theory of transitive logics of finite width. We will present somenovelresultssuchasthedecidabilityofallfinitelyaxiomatizabletransitivelog- icsoffinitewidthandfinitetightnessandtheresultthatthereexist13logicsoffinite widthwhichboundfinitemodelpropertyinthelatticeofextensionsofS4. Thefirst resultisasubstantialgeneralizationof[247],inwhichthesameisshownforexten- sionsofK4.3. InChapter9weproveaseriesofundecidabilityresultsaboutmodal logicsusingtwomainingredients. ThefirstisthesimulationtheoremofChapter6. AndtheotheristheuseofthelogicsK.alt . ThelatterhavebeenstudiedbyK n S[197]andFB[5]andtheirpolymodalfusionsbyC G [91]. The latter has shown among other that while the lattice of K.alt is 1 countable, the lattice of the fusion of this logic with itself has 2ℵ0 many coatoms. Moreover, the polymodal fusions of K.alt can be used to code word problems as 1 decidability problems of logics. Using this method, a great variety of theorems on the undecidability of properties is obtained. This method is different from the one x Overview usedbyLC[44],andACandMZ [41]. TheirproofsestablishundecidabilityforextensionsofK4,butourproofsare essentiallysimpler.Theproofsthatglobalfinitemodelproperty(globaldecidability) areundecidableevenwhenthelogicisknowntohavelocalfinitemodelproperty(is locallydecidable),arenew. We conclude the third part with Chapter 10 on propositional dynamic logic (PDL).Thiswillbeagoodillustrationofwhyitisusefultohaveatheoryofarbitrar- ilymanymodaloperators. Namely,weshalldevelopdynamiclogicasaspecialkind ofpolymodallogic,onethathasanadditionalcomponenttospecifymodaloperators. This viewpoint allows us to throw in the whole machinery of polymodal logic and deduce many interesting new and old results. In particular, we will show the finite model property of PDL, in the version of R P and D K [118], of PDL with converse, by D V [217], and of deterministic PDL by M B–A, J I. H and A P, [7]. Again, constructive reduction is used, and this gives an additional benefit with respect to interpolation. WehavenotbeenabletodeterminewhetherPDLhasinterpolation, butsomepre- liminaryresultshavebeenobtained. Moreover,forthelogicoffinitecomputations weshowthatitfailstohaveinterpolationandthatitdoesnothaveafixedpointtheo- rem. Largely,wefeelthatananswertothequestionwhetherPDLhasinterpolation canbeobtainedbycloselyanalysingthecombinatoricsofregularlanguages. Contents AboutthisBook v Overview vii Part1. TheFundamentals 1 Chapter1. Algebra,LogicandDeduction 3 1.1. BasicFactsandStructures 3 1.2. PropositionalLanguages 7 1.3. AlgebraicConstructions 13 1.4. GeneralLogic 17 1.5. CompletenessofMatrixSemantics 22 1.6. PropertiesofLogics 24 1.7. BooleanLogic 29 1.8. SomeNotesonComputationandComplexity 35 Chapter2. FundamentalsofModalLogicI 45 2.1. SyntaxofModalLogics 45 2.2. ModalAlgebras 53 2.3. Kripke–FramesandFrames 57 2.4. FrameConstructionsI 62 2.5. SomeImportantModalLogics 68 2.6. DecidabilityandFiniteModelProperty 72 2.7. NormalForms 78 2.8. TheLindenbaum–TarskiConstruction 86 2.9. TheLatticesofNormalandQuasi–NormalLogics 92 Chapter3. FundamentalsofModalLogicII 99 3.1. LocalandGlobalConsequenceRelations 99 3.2. Completeness,CorrespondenceandPersistence 105 3.3. FrameConstructionsII 112 3.4. WeaklyTransitiveLogicsI 117 3.5. SubframeLogics 119 3.6. ConstructiveReduction 125 xi xii Contents 3.7. InterpolationandBethTheorems 132 3.8. TableauCalculiandInterpolation 139 3.9. ModalConsequenceRelations 149 Part2. TheGeneralTheoryofModalLogic 157 Chapter4. UniversalAlgebraandDualityTheory 159 4.1. MoreonProducts 159 4.2. Varieties,LogicsandEquationallyDefinableClasses 166 4.3. WeaklyTransitiveLogicsII 172 4.4. StoneRepresentationandDuality 180 4.5. AdjointFunctorsandNaturalTransformations 188 4.6. GeneralizedFramesandModalDualityTheory 194 4.7. FrameConstructionsIII 202 4.8. FreeAlgebras,CanonicalFramesandDescriptiveFrames 208 4.9. AlgebraicCharacterizationsofInterpolation 213 Chapter5. DefinabilityandCorrespondence 219 5.1. Motivation 219 5.2. TheLanguagesofDescription 220 5.3. FrameCorrespondence—AnExample 224 5.4. TheBasicCalculusofInternalDescriptions 227 5.5. Sahlqvist’sTheorem 233 5.6. ElementarySahlqvistConditions 239 5.7. PreservationClasses 245 5.8. SomeResultsfromModelTheory 252 Chapter6. ReducingPolymodalLogictoMonomodalLogic 259 6.1. InterpretationsandSimulations 259 6.2. SomePreliminaryResults 261 6.3. TheFundamentalConstruction 265 6.4. AGeneralTheoremforConsistencyReduction 274 6.5. MorePreservationResults 279 6.6. ThomasonSimulations 283 6.7. PropertiesoftheSimulation 292 6.8. SimulationandTransfer—SomeGeneralizations 304 Chapter7. LatticesofModalLogics 313 7.1. TheRelevanceofStudyingLatticesofLogics 313 7.2. SplittingsandotherLatticeConcepts 315 7.3. IrreducibleandPrimeLogics 321 7.4. DualityTheoryforUpperContinuousLattices 328 7.5. SomeConsequencesoftheDualityTheory 334 7.6. PropertiesofLogicalCalculiandRelatedLatticeProperties 342
The list of books you might like
