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1 Modal Logic, on the cusp of philosophy and mathematics roads not taken, shifting perspectives, and interdisciplinary influences Johan van Benthem, Amsterdam & Stanford, http://staff.fnwi.uva.nl/j.vanbenthem Philosophical origins of modal logic 1 Aristotle’s modal syllogistic, varieties of truth, ontological structure universe. 2 Frege’s Begriffsschrift drops modalities, with an implicit rejection of Kant. 3 Lewis & Langford: strict implication, connected to modal logics. Primary: proof intuitions. Striking: diversity of logical systems – cf. Bolzano’s Program. 4 Meta-proof intuitions from ontology: H. B. Smith on the ontological ladder of modalities. Mathematics enters 5 Poland. Algebraic semantics. Topological semantics, ☐ = ∃∀. Highlight: S4 the logic of IR. Two main examples: metric spaces, trees. Motivations now include intuitionism. 6 Jónsson & Tarski, BAOs and representation in relational models. Interplay algebra and model theory has persisted in modal logic until today. 7 Further mathematical sources of modalities: provability logic. Now ☐ϕ is existential. Confluence of streams 8 Kanger Hintikka Kripke: graph models, ☐ = ∀. Analyze modal logics by frame constraints. Special case of topological semantics: or rather, of neighborhood semantics. 9 Many more notions considered modal, and treated in this semantic paradigm: time (Prior), knowledge, belief (Hintikka), deontic (Hansson), conditionals (Lewis), etc. 10 Influences from linguistics. Modality in natural language: high zoom level for reasoning. Frege’s ‘eye versus microscope’. See Handbook of Philosophical Logic. 11 General approach: ordinary truth, modality ~ multiple reference, at points/tuples. M, s |= ϕ 12 Discussion: criticism of R, and of ‘possible worlds’. Abstraction, take ‘holistic view’. Theory lines in the 1970s 13 Translation: not choose, but connect. Import classical model theory, Gestalt switch. ☐ϕ : ∀y (Rxy → [y/x]ϕ) Fragments of classical languages, modal logic as fine-structure, or ‘zooming out’. 14 Expressive power, invariance and simulation. Erlanger Program. Bisimulation invariance characterization of the modal language. 15 Completeness theory: still a major enterprise, start from logics – or from structures. Beautiful examples: semantics provability logic, completeness Minkowski space-time. 16 Frame truth, monadic second-order logic. Correspondence theory, Sahlqvist Theorem. 17 Enter universal algebra. Goldblatt & Thomason theorem modally definable frame classes. ‘The Three Pillars’: Completeness Theory, Correspondence Theory, Algebraic Duality.

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Page 1: Modal Logic, on the cusp of philosophy and mathematicslogic.berkeley.edu/logic@UCB/van_Benthem_logic@UCB.pdf · 27 Algebraic logic today often ahead of modal model theory. 28 Co-algebra:

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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.

Page 2: Modal Logic, on the cusp of philosophy and mathematicslogic.berkeley.edu/logic@UCB/van_Benthem_logic@UCB.pdf · 27 Algebraic logic today often ahead of modal model theory. 28 Co-algebra:

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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.


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