modal logic, on the cusp of philosophy and...

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ModalLogic,onthecuspofphilosophyandmathematics

roadsnottaken,shiftingperspectives,andinterdisciplinaryinfluences

JohanvanBenthem,Amsterdam&Stanford,http://staff.fnwi.uva.nl/j.vanbenthem

Philosophicaloriginsofmodallogic

1Aristotle’smodalsyllogistic,varietiesoftruth,ontologicalstructureuniverse.2Frege’sBegriffsschriftdropsmodalities,withanimplicitrejectionofKant.3Lewis&Langford:strictimplication,connectedtomodallogics.Primary:proofintuitions.Striking:diversityoflogicalsystems–cf.Bolzano’sProgram.4Meta-proofintuitionsfromontology:H.B.Smithontheontologicalladderofmodalities.

Mathematicsenters

5Poland.Algebraicsemantics.Topologicalsemantics,☐ =∃∀.Highlight:S4thelogicofIR.Twomainexamples:metricspaces,trees.Motivationsnowincludeintuitionism.6Jónsson&Tarski,BAOsandrepresentationinrelationalmodels.Interplayalgebraandmodeltheoryhaspersistedinmodallogicuntiltoday.7Furthermathematicalsourcesofmodalities:provabilitylogic.Now☐ϕisexistential.

Confluenceofstreams

8KangerHintikkaKripke:graphmodels,☐ =∀.Analyzemodallogicsbyframeconstraints.Specialcaseoftopologicalsemantics:orrather,ofneighborhoodsemantics.9Manymorenotionsconsideredmodal,andtreatedinthissemanticparadigm:time(Prior),knowledge,belief(Hintikka),deontic(Hansson),conditionals(Lewis),etc.10Influencesfromlinguistics.Modalityinnaturallanguage:highzoomlevelforreasoning.Frege’s‘eyeversusmicroscope’.SeeHandbookofPhilosophicalLogic.11Generalapproach:ordinarytruth,modality~multiplereference,atpoints/tuples.

M,s|=ϕ12Discussion:criticismofR,andof‘possibleworlds’.Abstraction,take‘holisticview’.

Theorylinesinthe1970s

13Translation:notchoose,butconnect.Importclassicalmodeltheory,Gestaltswitch.☐ϕ:∀y(Rxy→[y/x]ϕ)

Fragmentsofclassicallanguages,modallogicasfine-structure,or‘zoomingout’.14Expressivepower,invarianceandsimulation.ErlangerProgram.Bisimulationinvariancecharacterizationofthemodallanguage.15Completenesstheory:stillamajorenterprise,startfromlogics–orfromstructures.Beautifulexamples:semanticsprovabilitylogic,completenessMinkowskispace-time.16Frametruth,monadicsecond-orderlogic.Correspondencetheory,SahlqvistTheorem.17Enteruniversalalgebra.Goldblatt&Thomasontheoremmodallydefinableframeclasses.‘TheThreePillars’:CompletenessTheory,CorrespondenceTheory,AlgebraicDuality.

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Theeraofapplicationsinthe1980s

18Computerscience,economics,moremathematics,etc.Bringtheirownfurtherstructure.19Aswellasgeneralthemesthatentertheself-understandingofmodallogic.From‘applied’to‘pure’.Fromdecidabilitytocomputationalcomplexity.Bisimulationandprocessinvariance.‘TheBalance’ofexpressivepowerandcomplexityinlogicdesign.

Theorylinesinthe1990s–2010s

20Propositionallanguageextensions:Prior,Bulgaria,HybridLogic,=fine-structureFOL.GuardedFragment,admitallquantifiedforms∃y(G(x,y)&ϕ(x,y)),andmore.Open-ended.Mostrecentdiscovery:decidable‘UnaryNegationFragment’.21Ongoing:modalpredicatelogic.Howtocombinemodality,predication,quantifiers?Findingthebestsemanticsisbothaphilosophicalandamathematicalproblem.22Propositionaldynamiclogic,modalmu-calculus,fixed-pointlogicsforinductionandrecursion.ConnectionswithAutomataTheory,ProcessAlgebra,gamesemantics.23Multi-agentsystems.Knowledge,belief,action,games,preference.Modallogiccombination,surprisinglycomplicated.Fromtreestogrids.24Moregeneralframeworks.Neighborhoodsemanticsrevival.25Returntothehistoricalmethodology.Prooftheoryformodallogics.26Alltheseresearchlinesinterconnectinthecurrentliterature.Richerlanguagesforneighborhoods,Fix-pointviewsofoldmodallogics.

Challenges

27Algebraiclogictodayoftenaheadofmodalmodeltheory.28Co-algebra:theriseofcategorytheory.29Metaphysics,hyperintensionality.Backtothelostphilosophicalparadise?Richeruniverses,notionsoftruth,hypergraphs.

Whatismodallogic?

30Majorturninthinking:modalityasexoticadditiontoclassicallogicorasfine-structure.31Modallogicas‘classicallogicplusoperators’versustrendtoward‘alternativelogics’.Statusofsystemtranslation.Manyopenproblemsevenintheareaofmodallogics.32Nothinginlogiciswhatitseems.MLafragmentofFOL,ormoregeneralsemantics?Modalfoundationsofpredicatelogic:Tarskisemanticsitselfhasamodalflavor.

M,s|=∃xϕiffforsomed∈D,M,s[x:=d]|=ϕLeadstounusualdecidablesystems,andfirst-orderextensionsoffirst-orderlogic.