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IMPLICITANDEXPLICITSTANCESINLOGIC

AbstractWeidentifyapervasivecontrastinlogicbetweenwhatwecallimplicitandexplicitstancesindesign.Implicitsystemschangethemeaningoflogicalconstantsandsometimesalsothedefinition

ofconsequence,whileexplicitsystemsconservativelyextendclassicalsystemswithnewvocabulary.

Weillustratethecontrast inthetraditionalsettingof intuitionisticandepistemiclogic,thentakeit

further to information dynamics, default reasoning, logics of games and other areas, to show the

wide scope of these complementary styles of logical analysis and system design. Throughout we

showhowawarenessoftheimplicit–explicitcontrast leadstonewlogicalquestions, fromstraight-

forward technical issues to when implicit and explicit systems can be translated into each other,

raisingnewfoundationalissuesaboutidentityoflogicalsystems.Butwealsoshowhowapractical

facilitywiththesecomplementaryworkingstyleshasphilosophicalconsequences,asitthrowsdoubt

onstrongphilosophicalclaimsmadebyjusttakingonestanceandignoringthealternativeone.We

willillustratethelatterbenefitforthecaseoflogicalpluralismandhyper-intensionalsemantics.

KeywordsLogic,Modality,Implicit,Explicit,Translation.

1 Explicitandimplicitstancesinlogicalanalysis

Thehistoryoflogichasthemesrunningaspectrumfromdescriptionofontological

structures in the world to elucidating patterns in inferential or communicative

behavior.Themathematicalfoundationaleraaddedthemethodologyofformalsys-

temswith semantic notions of truth and validity andmatchingproof calculi. This

modusoperandiisstandardfare,enshrinedinthemajorsystemsofthefield.Butlivedisciplinesarenotfinishedfieldsbutadvancingquests.Logichasagrowing

agenda, including the study of information, knowledge, belief, action, agency, and

otherkeytopicsinphilosophicallogicorcomputationallogic.Howaresuchtopics

tobebrought into thescopeof theestablishedmathematicalmethodology?There

arebothmodificationsandextensionsofclassicallogicforthesepurposes,andthe

aimofthispaperistopointattwomainlines,representingasortofwatershed.Onelineofenrichingclassicallogicaddsnewoperatorsfornewnotionstobeanaly-

zed,leavingoldexplanationsofexistinglogicalnotionsuntouched.Atypicalcaseis

modallogic,addingoperatorsformodalities,whilenothingchangesintheproposi-

tional base logic. Let us call this the explicit style of analysis, though the label

‘conservative’makessense,too:wedonottouchnotionsalreadyinplace.

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But there is another line, where we use new concepts to modify or enrich our

understanding ofwhat the old logical constantsmeant, orwhat the old notion of

validconsequencewasmeanttodo.Thisleadstonon-standardsemantics,perhaps

rethinkingtruthas‘support’or‘forcing’,andtoalternativelogicswhoselawsdiffer

fromthoseof classical logicon theoriginalvocabularyof connectivesandquanti-

fiers.Heretherichersettingisreflected,notinnewlawsfornewvocabulary,butin

deviationsonreasoningpatternsintheoldlanguage–andfailuresofoldlawsmay

besignificantandinformative.Aparadigmaticcaseisintuitionisticlogic,butfurther

instanceskeepemerging.Letuscall this the implicit styleofanalysis,withoutany

pejorativeconnotation.Implicitnessisahall-markofcivilizedintercourse.Wewilldiscussasequenceofillustrationsdisplayingthecontrast,andanalyzewhat

makes it tick.Weset thescenebyrecallingsomekey factsabout twowell-known

systems:epistemiclogicandintuitionisticlogic,presentedwithafocusoninforma-

tionandknowledge.Afterthatwediscusslessstandardcasessuchaslogicsofinfor-

mation update, default reasoning, games, quantummechanics, and truth making.

Throughout,we take explicit and implicit approaches seriously as equallynatural

stances,andwediscussnewlogicalquestionssuggestedbytheirco-existence.Our

finalconclusionfromallthiswillbethattheinterplayofthetwostancesneedstobe

graspedandappreciated,asitraisesmanynewpointsandopenproblemsconcer-

ningthearchitectureoflogic,whileitalsohasphilosophicalrepercussions.Thismaynotbeaneasypapertoclassifyquastyleorresults,butwehopethatthe

readerwillbenefitfromlookingatlogicalsystemdesigninourbroadmanner.2 Information,knowledge,andepistemiclogic

Anaturaladditiontotheheartlandoflogicarenotionsofknowledgeandinforma-

tionforagents,thathavebeenpartofthedisciplinefromancienttimesuntiltoday,

[35], [7]. Inwhat followswedonotneed intricatecontemporary logics forepiste-

mology, [32], interesting and innovative though these are. The contrast inmodus

operandiweareaftercanbeseenatmuchsimplerlevel,datingbacktothe1960s.Hereisamajorexplicitwayoftakingknowledgeandinformationseriously.Weadd

modal operators for knowledge to propositional logic, and study the laws of the

resultingepistemic logicson topofclassical logic.Theseconservativeoperatorex-

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tensionsofclassicallogicalsystemshaveinterestingstructureandmodelingpower,

alsofornotionsbeyondknowledge,suchasbelief.Henceepistemiclogicisusedin

manydisciplines:philosophy,linguistics,computerscience,andeconomics.Inmoredetail,theclassic[31]proposesananalysisofknowledgethatinvolvesan

intuitive conception of information as a range of candidates for the real situation

(‘world’,‘state’).Thisrangemaybelarge,andweknowlittle,orsmall(perhapsasa

result of prior informationupdates eliminatingpossibilities) and thenweknowa

lot.Inthissetting,anagentknowsthatϕatacurrentworldsifϕistrueinallworlds

in the current range of s, the epistemically accessibleworlds from s via a binary

relations~t.Toexpressreasoninginamatchingsyntax,wetakestandardproposi-

tionallogicasabase,andaddaclauseforformulasoftheformKϕ–subscriptedto

Kiϕfordifferentindicesiincasewewanttodistinguishbetweendifferentagentsi.

Thentheprecedingintuitionbecomesthefollowingtruthdefinition: M,s|=p iff s∈V(p)

M,s|=¬ϕ iff notM,s|=ϕ

M,s|=ϕ∧ψ iff M,s|=ϕandM,s|=ψ

M,s|=Kϕ iff M,t|=ϕforalltwiths~t.This extends classicalpropositional logic: thebase clauses are standard,withone

operatorclauseadded.Epistemicaccessibility~isoftentakentobeanequivalence

relation–butwecanvaryonthisifneeded.TheresultinglogicisS5foreachsingle

agent, without non-trivial bridge axioms relating knowledge of different agents.

Thus, basic epistemic logic is a conservative extension of classical logic, and the

same holds for variations like S4 or S4.2 that encode other intuitions concerning

knowledge,[46].Moreintricatelawsholdformodalitiesofcommonordistributed

knowledgeingroups,butagainthesewillnotbenotneededhere.Fewpeople today see the epistemicmodality as a conclusive analysis for the full

philosophical notion of knowledge. But even so, this system is a perfect fit for

anotherbasicnotion,the‘semanticinformation’thatanagenthasatherdisposal,cf.

theclassicsource[6].And,thesimpleperspicuousexplicitsyntaxofepistemiclogic

is still in wide use as a lingua franca for framing epistemological debates, for

instance,fororagainstsuchbasicprinciplesofreasoningaboutknowledgeas

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omniscience,orclosure K(ϕ→ψ)→(Kϕ→Kψ)

introspection Kϕ→KKϕSignificantly,thesearedebatesaboutintuitivelyacceptablereasoningprinciplesfor

knowledge,notaboutthelawsoftheunderlyingpropositionallogic.Moresophisticatedphilosophicalaccountsdefineknowledgeasanotion involving

structurebeyondmeresemanticranges,suchasrelevanceorder,plausibilityorder

ofworldsforbelief(whichwediscusslateron),orsimilarityorderforconditionals.

Evenso, logicsfortheseextendedsettingstendtobemulti-modalsystems,thatis,

theystillfallunderwhatwehavecalledtheexplicitapproach.Allthisistypicalfor

manyareasofphilosophicallogic,suchastemporal,deontic,orconditionallogic.RemarkThisbriefexpositionmaybemisleadingabouttheagendaofthefield.Epis-

temiclogichascomeintowideuseingametheoryandcomputersciencebecauseof

its potential for describing multi-agent interactions in communication or games.

SeetherecentHandbook[19]forthestateoftheart.3 Intuitionisticlogic

Next,considerasecondwayoftakingknowledgeandinformationseriously,which

issometimespresentedasarevoltagainstclassicallogic.Wenolongertaketheold

notionsforgranted,butredefinethemeaningsofthelogicalconstants,perhapsalso

thenotionofconsequence,togetatcrucialaspectsofknowledge.A typical instance of this second approach is intuitionistic logic that does not add

knowledgeoperators,butencodesbehaviorofknowledgeinitsdeviationsfromthe

lawsof classical consequence.This approach seemsmore radical, asbreaking the

classical laws has an iconoclastic appeal, andmore subtly, the absence of explicit

expressionsforepistemicnotionsmakesthebehaviorofknowledgenowshow,not

innewlaws,but implicitly, inabsenceofold laws,or inmodificationsthereof.For

instance, thewell-known intuitionistic failure of ExcludedMiddleϕ∨¬ϕ tells us

somethingessentialabouttheincompleteness,ingeneral,ofourknowledge.Buton

thepositiveside,thecontinuedintuitionisticvalidityof¬ϕ↔¬¬¬ϕrevealsamore

delicateformofintrospectionforknowledgethanthesimpleS4lawwehadabove–

wherenegationnowtalksaboutknowledgeinanimplicitmanner.

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Intuitionistic logic arose in the analysis of constructivemathematical proof, with

logicalconstantsacquiringtheirmeaningsinproofrulesviatheBrouwer-Heyting-

Kolmogorov interpretation. In the 1950s, Beth andKripke proposedmodels over

treesoffiniteorinfinitesequences,andinlinewiththeideaofproofasestablishing

aconclusion,intuitionisticformulasaretrueatanodeofsuchatreewhen‘verified’

insomeintuitivesense.Ageneraltopologicalframeworkforplacingalltheseideas

ispresentedin[15].Astandardversionthatsufficesforourpurposesusespartially

orderedmodelsM=(W,≤,V)withavaluationV,setting: M,s|=p iff s∈V(p)

M,s|=ϕ∧ψ iff M,s|=ϕandM,s|=ψ

M,s|=ϕ∨ψ iff M,s|=ϕorM,s|=ψ

M,s|=¬ϕ iff fornot≥s,M,t|=ϕ

M,s|=ϕ→ψiff forallt≥s,ifM,t|=ϕ,thenM,t|=ψInsuchpartialorders,wecanthinkoftheobjectssasinformationstagesorinfor-

mationpieces,whilemodelsunraveledtotreesgiveatemporalpictureofarecord

ofpossibleinvestigations.Next,inlinewiththeideaofaccumulatingcertaintyinthe

processofinquiry,thevaluationVinthesemodelsispersistent,i.e., ifM,s|=pands≤t,thenalsoM,t|=p.Thetruthdefinitionasstatedhereliftsthispersistencepropertytoallformulasϕ.Inthismodusoperandi,incontrastwithepistemiclogic,thereisnoseparatesyntax

forknowledgeorinformation–butoldlogicalconstantsarere-interpreted,making

negationandimplicationsensitivetotheinformationstructureofnewmodelswith

an inclusion order that is absent in models for classical logic. In particular, an

intuitionisticnegation¬ϕsaysthattheformulaϕisnotjust‘nottrue’,butthatitwill

neverbecometrueatanyfurtherstagealongtheinclusionordering.Also,failureof

classical definability equivalences leads to fine-structure for classical notions like

implication,whichcannowbeviewedinseveralnon-equivalentways.This ‘meaning loading’of the classical operatorsmakes the intuitionistic laws for

negation and implication deviate from classical logic. Now earlier points become

precise.Famously,thissemanticsinvalidatesthelawofExcludedMiddleϕ∨¬ϕ,as

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thisdisjunctionfailsatstateswhereϕisnotyetverifiedthoughitwilllaterbecome

so.Thesedeviationsfromclassicallogicareinformativeintellingusimplicitlyabout

propertiesofknowledge.FailureofExcludedMiddlesaysthatagentscannotdecide

everythingapriori.Thusmeaningloadingmakestheremainingvaliditiesinforma-

tive(theynowsaysomethingnew),andmoremysteriously,itpacksinformationin

theabsenceofclassicallaws–likedogsthatdonotbarkinthenight-time.Atthesametime,eventhoughtheclassicallanguageisnotextended,thereisanin-

creaseinexpressivepower,sinceclassicallawsfailnow.Forinstance,ϕ→ψisnot

equivalenttoeither¬ϕ∨ψor¬(ϕ∧¬ψ):intuitionisticlogichasmoreimplications

thanclassicallogic.Thisisanimplicitcounterparttoexplicitlanguageextension.RemarkThisbrief expositionof intuitionistic logicdoesnotdo justice to itsdeep

connections with proof theory, universal algebra, and category theory, or to the

many surprising effects of working in mathematical theories on top of a weaker

baselogic.Seetheencyclopedicsource[47]forawide-rangingexposition.4 Theexplicit/implicitcontrast:epistemiclogicversusintuitionisticlogic

So, nowwehave encountered twomajor researchparadigms in the fieldof logic,

both meant to take information and knowledge seriously – but doing so in very

differentways.Letushighlightthemajordifferencesthatshowedintheabove:epistemiclogic explicit,conservativelanguageextensionofclassicallogic

intuitionisticlogicimplicit,meaningchangeoldlanguage,non-classicallogicHighlighting thedistinction, consider the fact thatwedonot know the answer to

everyquestion,andmaybeneverwill.Thisshowedasfollowsinintuitionisticlogic.

ExcludedMiddleϕ∨int¬intϕwasnotvalid–whereindiceshighlightthefactthatthe

failure occurs on the intuitionistic understanding of negation and disjunction –

though special cases of this principlemay, and do, remain valid. In contrastwith

this, the law of Excluded Middle is unrestrictedly valid in epistemic logic, but it

shouldnotbeconfusedwiththeinvalidepistemicformulaKϕ∨classK¬classϕ.Muchmorecanbesaidaboutthesetwoapproachestoknowledgeandinformation.

But for the purposes of this paper, wewill just stipulate that both are based on

stableinterestingsetsofintuitions,bothhavegeneratedarichmathematicaltheory,

andbothseembonafideinstancesofalogicalmodusoperandiinsystemdesign.

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With this single illustration, we hope the reader has grasped themethodological

pointweareafter–andinlatersections,wewillnowexplorethe‘implicit’versus

‘explicit’contrastinothercases,addingmoredepthtowhatitinvolves.5 Choiceorco-existence:translationsandmerges

Butfirstitmayseemtimeforachoice.Isintuitionisticlogicorepistemiclogicbetter

ordeeperasananalysisofinformationandknowledge?Shouldwepreferoneover

the other? Many philosophers think this way, but we feel that this adversarial

attitude is not very productive, and it also runs counter to knownmathematical

factsaboutsystemconnections(forasimilar,butmoregeneralcriticism,cf.[29]).Already in Gödel’s seminal [26], there is a faithful translation from intuitionistic

logic into the modal logic S4 whose underlying intuition follows the present

knowledgeperspective.Wenowlookatthisconnectiontoseewhatitachieves.Translating IL into EL TheGödel translation t turns the intuitionistic truth defi-

nitionintoasyntacticrecipe.Atomspgotomodalformulas☐p,upwardpersistent

on partial orders, conjunctions remain conjunctions, disjunctions remain disjunc-

tions,intuitionisticnegations¬ϕgotomodalizedclassicalnegations☐¬ϕ,andintui-

tionisticimplicationsϕ→ψgotomodalizedclassicalimplications☐(ϕ→ψ).Using

ILforthestandardproofsystemofintuitionisticpropositionallogic,wethenhave:Fact IL|–ϕiffS4|–t(ϕ),forallpropositionalformulasϕ.Thisexplainskeyfeaturesofintuitionisticlogicinmodalterms.E.g.,varietiesofim-

plicationplacedifferentdemandsonknowledge:intuitionisticϕ→ψis☐(ϕ→ψ),

¬ϕ∨ψthestronger☐¬ϕ∨ψ,and¬(ϕ∧¬ψ)theweaker☐(ϕ→♢ψ).Also,intuitio-

nisticlawslike¬ϕ↔¬¬¬ϕarespecialcasesofthefactthatS4has14non-equiva-

lent iterations ofmodalities. But intuitively, themodal setting is richer, as it also

supports reasoning about non-persistent formulas that can become false at later

stages.Thus,itstheoryofinquiryallowsforrevision,notjustcumulativeupdate.Usesof translationsSomepeopleviewtranslations like thisasmere tricks,espe-

cially thosewhoseedifferent logicsasseparatereligions.But the translation faci-

litatesaresoundingtransfer:everythinganintuitionistsaysorinferscanbeunder-

stoodbyaclassicalmodallogician.Thisfacilitatestrafficofideasbetweenintuitio-

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nisticandepistemic logic,andmeaningfulcontactsbetweentheiragendas.For in-

stance,keypropertiesofS4 suchasdecidabilitycarryoverautomatically to intui-

tionistic logic,andapplicationskeepemerging,suchasusesofmodalbisimulation

inintuitionisticlogic,[39].Butalsoconceptually,ideasfromepistemiclogiccannow

enterintuitionisticlogic,suchasthestudyofmulti-agentscenarios.TranslatingELintoILOurdiscussionsofarmayhavegiventheedgetoepistemic

logic,asitembedsintuitionisticlogic.Whatabouttheotherwayaround?Intuitively,

aswenoted,thesemanticsofS4seemsricher,allowingnon-persistentnotions,but

the two logics have the same computational complexity (their SAT problems are

Pspace-complete), so there is no a priori obstacle to mutual translation. In fact,

surprisingly, [22]gaveaconverse translation(withacorrection in [27]),which is

muchlessknown.ItworksquitedifferentlyfromGödel’st,bymimickingevaluation

ofmodalformulasinfinitemodelsinsidetheintuitionisticlanguage.Thus, translations between stances occur, and they are significant asmanuals for

communicationandinteraction.So,areintuitionisticlogicandepistemiclogicreally

just the same system in different guises because of their faithful mutual embed-

dings?Thisfurtherquestionraisesdelicateissuesofsystemidentity.Translation and system identity Despite the clear benefits of translations, they

neednotreduceonelogictoanotherineveryrelevantaspect.TheGödeltranslation

encodesoneparticularmodaltakeonthelogicalconstants,whichmaynotbewhat

anintuitionistconsiderstheiressence.Andthereismore.TolettheGödeltransla-

tionbefaithful,deductivepowermustberestrictedtoS4orlogicsclosetoit.Thisis

relevant, since so far,weusedS5 as anepistemic logic, and theGödel embedding

doesnotworkthere:ILisPspace-complete,andhencemorecomplexthanS5,which

ismerelyNP-complete.Andalsoconversely,studyingthesyntacticaldetailsof the

encodingfromELintoIL,onedoesnotgetafeelingofstrongresemblancebetween

thetwosystems:itseemsmorelikeacaseofintuitionisticlogichatchingS4eggs.RemarkOnewayofmakingfinerdifferencesconcreteisintermsofcomputational

complexity. Theories that are equivalent under translation, perhaps an inefficient

translation,mayhavedifferent computationalproperties.Wewilnotpursue such

anglesinthispaper,butcomplexityisanaturalwayofdrivingafinerwedge.

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Thus,mutualtranslation,thoughastrongbond,neednotimplysystemequivalence

inallrelevantaspects.Itisgoodtosearchforsuchconnections,butinwhatfollows,

wewillkeepanopeneyeforintensionaldifferencesbetweentranslatedsystems.Digression:proofstrength inmodal logicsTranslationsonlyworkproof-theore-

ticallywiththerightamountofdeductivepoweronbothsides.Thisraisesfurther

issues.Thevarietyofaxioms insystems likeS5or S4 fine-tunesdeductivepower,

anditcanbeanalyzedbysemanticcorrespondences,butitdoesmore.Movingfrom

onelogictoanothermaybeaswitchbetweenconceptualframeworks.Inparticular,

thereflexivetransitiveaccessibilityrelationofS4doesnot justencodeanS5-style

epistemicrange:itslackofsymmetryalsosuggestsmovingforwardthroughtimein

inquiry. Like with intuitionistic logic, an S4-model suggests a process where an

agent learns progressively about the actualworld. Since atomic facts can become

falsealongtheway,S4-modelsalsomodelnon-intuitionisticinformationretraction

orrealworldchange.Butevenwiththisdifference,intranslatingintuitionisticlogic

into the modal world of S4, we have gone a certain way toward adopting the

intuitionisticmindsetofinquiry.Wewillreturntothisthemepresently.FromtranslatingtomergingFinally,movingaway fromreduction,aweakerbut

stillsignificantcontactbetweenexplicitandimplicitlogicsiscompatibility.Cansuch

systems be merged in meaningful ways? Intuitionistic modal logics have long

existed, and hybrids of explicit and implicit logics keep emerging, as wewill see

lateron.Oftenthisjuxtapositionseemsroutine,buthybridscanalsobenatural.6 Dynamiclogicofinformationchange

Havingintroducedourexplicit/implicitcontrastfortwowell-knownlogics,wenow

movetomorerecentdevelopmentsandseewhereitleads.Westartbynotingthat

inquiryliesattheheartofbothepistemicandintuitionisticlogic.Clearly,knowled-

geandinformationdonotfunctioninisolation,butinanongoingdynamicprocess

ofinformationalaction,orinasocialsetting,interactionbetweenagents.Statics and dynamics Key informational actions that guide agents come in three

kindsthatworktogetherinmanynaturalscenarios.Actsofinferencematter–but

equally importantareactsofobservation,andof communication.Suchactions,or

othereventsthatembodythem,arestudiedincurrentdynamic-epistemiclogics,by

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adding an explicit vocabulary for the core actions to existing logical systems, and

thenanalyzingthemajorlawsofknowledgechange,cf.thebroadstudy[11].ModelupdateHereisasystemmakingthedynamicactionsbehindbasicepistemic

logic explicit by representing informational action asmodel change. The simplest

caseofsuchachangeoccurswithapublicannouncementorasimilarpublicevent

!ϕthatproduceshardinformation,whereone learnswithtotalreliabilitythatϕ is

thecase.Thiseliminatesallworldsinthecurrentmodelwhereϕisfalse:

fromMs toM|ϕs

ϕ ¬ϕAswesaidwhenmotivatingepistemicmodels, getting informationbyshrinkinga

rangeofoptionsisacommonideainmanydisciplines,thatworksforinformation

flowbybeingtoldorthroughobservation.Wecancallthishardinformationbecau-

seofitsirrevocablecharacter:theupdatestepeliminatesallcounter-examples.Public announcement logic Public announcements are studied in PAL, a system

thatextendsepistemiclogicwithadynamicmodalityfortruthfulannouncements:

M,s|=[!ϕ]ψiffifM,s|=ϕ,thenM|ϕ,s|=ψ Thisdynamicmodalityhasacomplete logic thatcananalyzedelicatephenomena,

suchascomplexepistemicassertions,sayofcurrentignorance,changingtruthvalue

underupdate.Thistypicallyshowsinorderdependence:asequence!¬Kp;!pmakes

sense, but !p ; !¬Kp is contradictory. Herewe only display the ‘recursion law’ for

knowledgeafterupdate,whichisthebasicdynamicequationofhardinformation:[!ϕ]Kψ↔(ϕ→K(ϕ→[!ϕ]ψ))

TogetherwiththeS5-lawsforepistemiclogicplussimpleaxiomsforBooleancom-

poundsafterupdatethisgivesacompleteaxiomatizationforPAL.Anotherinteres-

tinglawdemonstratingthedynamicsofPALgovernsiteratedupdates:

[!ϕ][!ψ]χ↔[!(ϕ∧[!ϕ]ψ)]χ

Recursionaxiomsreduceformulaswithdynamicoperatorstostaticbaseformulas,

sotheextensionofourclassicalbaselogicisconservativeintheusualexplicitstyle.

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General dynamics There is a method here. One ‘dynamifies’ a given static logic,

makingitsunderlyingactionsexplicitanddefiningthemasmodeltransformations.

Theheartof thedynamic logic is thenacompositionalanalysisofpost-conditions

forthekeyactionsviarecursionlaws.Thisleadstoconservativeextensionsofthe

baselogic,thoughsomedynamic-epistemicsystemsforceredesignofthebaselan-

guage,whilesomerecentsemanticsnolongersupportall-outreduction.Manyfur-

thernotionscanbetreatedinthisstyle,suchaschangesinagents’beliefs,inferen-

ces,agendaissues,orpreferences–whereoneoftenchangestheorderingofacur-

rentmodelratherthaneliminatingworlds.Moreover,extendeddynamiclogicsalso

dealwithpublicandprivateeventsinrichmulti-agentscenariossuchasgames.Digression: explicitizing the explicit Evenwhere a dynamic logic conservatively

extendsabaselogicitmayaffectourviewofthestatics.Recallthevarietyofmodal

logicswithaxiomsmatchingconditionsonaccessibilityrelations.Onecanaskwhy

suchconditionsholdindynamicterms.Say,transitiverelationsarisefromanacttc

of transitiveclosureofarbitrarymodels, say: reflectionorexploration.But thena

K4-modalityKϕ isanordinarymodality£overmodelsresulting fromthisaction,

makingitacompound[tc]£ϕ.ThisfaithfullyembedsK4overtransitivemodelsinto

propositionaldynamic logicoverarbitrarymodels. In thisdynamics-inspiredway,

varietyofmodallogicsdissolvesinfavorofonebaselogicplusmodalitiesformodel

change,explaininginsteadofpostulatingspecialrelationalproperties.7 Implicitdynamicsinintuitionisticlogic

We have now extended epistemic logic, an explicit approach to knowledge, to a

dynamic logicwithexplicit informationalactions. Is therean implicitcounterpart?

Givenourearlierdiscussion, itmakessensetosearchintherealmof intuitionism.

WecouldjustaddtheactionsofPALtointuitionisticlogic,[3].Butcanwebemore

implicitaboutactions,withoutputtingthemexplicitlyintothesyntax?Teasing out the hidden actions Intuitionistic models represent a process of

inquiry,withendpointsasfinalstageswhereweknowthetruthaboutallproposi-

tionletters.Whataretheimplicitstepsinthebackgroundofsuchaprocesstaking

usfromnodetonode?Movesfromonestatetoasuccessorcomeintwokinds.Example Thehiddendynamicsofintuitionisticmodels.

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Consider twomodelsM1,M2,where the first refutes the classicaldoublenegation

law¬¬p→p,andthesecondthelawof‘weakexcludedmiddle’¬p∨¬¬p:

M1 #p M2!¬p!p

p pTheannotationssaythatthetwobranchesofM2maybeviewedaspublicannounce-

ments of which endpoints, viewed as classical valuations, the process can get to.

This is likePAL-style learningbyeliminationofworlds.But inother intuitionistic

steps, liketheoneinM1, thereisnosucheliminationofendpoints,andwemerely

getmorepropositionletterstrueatthenextstage.Onemightexplainthismoveasa

newtypeofinformationalaction,namely,‘awarenessraising’#ϕthatsomefactϕis

thecase,whereawarenessinvolvessyntacticinadditiontosemanticinformation.Factualandprocedural informationBut there ismore thanmere transposingof

concerns fromdynamic-epistemic logic.Thetreestructureof intuitionisticmodels

registerstwonotionsofsemantic informationonapar,adistinctionalso found in

epistemicinquirywithlong-termscenariosinlearningtheory,[33]:(a) factualinformationabouthowtheworldis,

(b) proceduralinformationaboutourcurrentinvestigativeprocess.

Howwecangetknowledgematters,not justwhatisthecase.Whileendpointsre-

cord eventual factual information states, the branching tree structure of intuitio-

nisticmodels,bothitsavailableanditsmissingintermediatestages,encodesfurther

non-trivialinformation:viz.agents’knowledgeabouttheprocessofinquiry.Thischallengessimpleviewsofhowintuitionisticandepistemiclogicconnect.The

epistemic logic forsemantic information isS5, and the fact that theGödel transla-

tiontakesusintoS4reflectsaviewofintuitionisticmodelsastemporalprocessesof

inquiry.Thus,anexplicitcounterparttointuitionisticlogicneedsatemporalversion

of dynamic epistemic logic. Indeed, temporal `protocol models’ with designated

admissiblehistoriessatisfyingconstraintsoninquiry,[11],modelproceduralinfor-

mationinlong-termprocessesofinquiryorlearningbeyondlocaldynamicsteps.

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Thus,bothepistemic logicand intuitionistic logichavedynamicextensionshaving

todowithinquiry,andthesecanbedevelopedinbothexplicitandimplicitstyles.

Moreover,thisprocessisnotroutineandinterestingnewissuescometothefore.8 Dynamicsemantics,meaningasinformationchangepotential

Intuitionistic logic is not the only vehicle for ameaningful comparisonwithPAL.

Explicit logicsneednothaveuniqueimplicitcompanions,theremaybemoremat-

chings. Indeed, themorestriking implicit counterpart todynamicepistemic logics

maywellbeanotherlogicalparadigm,thatwewilldiscussnow,raisingnewissues.Hereisafundamentalideafromtheareaofdynamicsemanticsofnaturallanguage,

goingbacktoclassicalsourceslike[28],[48].Theguidingintuitionofthisapproach

tolanguageinvolvescommunication-oriented‘informationchangepotential’:

Themeaningofanexpressionisitspotentialforchanginginformation

statesofsomeonewhoacceptstheinformationconveyedbytheexpression.Thissounds likeapleafortaking informationalactionsseriously,aswedid inthe

precedingsection.Butthistime,theyaretreated,notbyaddingnewoperators,but

implicitly,byloadingthemeaningsofclassicalvocabularywithdynamicfeatures.Dynamic semantics comes in many forms. We will use Veltman’s propositional

updatelogicUSforacomparisonwiththeexplicitPALapproach.Here,onauniverse

of information states (in the simplest case, sets of valuations), eachpropositional

formulaϕinducesastatetransformation[[ϕ]]bythefollowingrecursion: [[p]](S) = S∩[[p]]

[[ϕ∨ψ]](S) = [[ϕ]](S)∪[[ψ]](S)

[[ϕ∧ψ]](S) = [[ψ]][[ϕ]](S),

[[♢ϕ]](S) = S,if[[ϕ]](S)≠∅,and∅,otherwise.Conjunction now stands for a dynamic notion: sequential composition of actions,

whileanexistentialmodalitybecomesa‘test’onthecurrentinformationstate.Aswith intuitionistic logic, thesenewmeanings for old operators result in devia-

tionsfromclassicallogic.Inparticular,conjunctionisnolongercommutative,reflec-

ting the typical order dependence of dynamic acts. Facts about the informational

process arenowencoded in the logicof the logical operators in this system.This

14

encodingbecomesevenmorepronouncedwithanewdynamicconsequencesaying

that,afterprocessingthesuccessivepremises,theconclusionhasnofurthereffect:

ϕ1,…,ϕn|=ψiffforeveryinformationstateXinanymodel,ϕn(…(ϕ1(X)))

isafixedpointfor[[ψ]]:i.e.,thissetstaysthesameunderanupdate[[ψ]].Dynamicconsequencediffersfromclassicalconsequence,anditsdeviationsencode

typicalfeaturesoftheupdateprocess,likesensitivitytoorderormultiplicityofpre-

mises.Buttypicallyfortheimplicitstyle,whatchangesherearetheclassicallawsof

logic,notitsmethodology.Completenesstheoremsexistfordynamicconsequence.Remark There are much more sophisticated systems of dynamic semantics for

other classes of expressions, with different notions ofmeaning, state change and

dynamicconsequence–buttheaboveisafairdescriptionofthebasicmechanics.9 Anewcontrast:dynamicsemanticsversusdynamiclogicofinformation

PALanddynamicsemanticsDynamicepistemiclogicslikePALandupdateseman-

ticsforpropositional logicbothtakeinformationchangeseriously,withanalogous

scenariosandintuitions.Andbothsystemshaveapreciseaccountforthedynamics

ofinformationalactions.Butonedoessoexplicitly,andtheotherimplicitly:

Dynamicsemanticskeepstheactionsimplicit,whilegivingtheoldlanguageof

propositionallogicricherdynamicmeaningssupportinganewnotionofconse-

quence,withatechnicaltheorythatdiffersfromstandardpropositionallogic.

Dynamicepistemiclogicmakestheactionsexplicit,providesthemwithexplicit

recursionlaws,extendstheoldbaselanguagewhileretainingtheoldmeanings

forit,andinallthis,itstillworkswithstandardconsequence.As before, this is not just amatter of attaching two labels `implicit’ and `explicit’.

Seeingthings in termsofourcontrast leads tonewquestionsandopenproblems.

Onestraightforwardconsequenceconcernssystemdesign.Inquiry and questions A current innovation in dynamic semantics is inquisitive

semantics for natural language, [18], where formulas get richer ‘inquisitivemea-

nings’ reflecting their role in,not just conveying information,butalso indirecting

discourse.Theresultinglogicisanon-classicalintermediatelogicrelatedtoMedve-

dev’slogicofproblemsfromtheintuitionistictradition.Ouranalysisthensuggests

15

thedesignofanexplicitcounterpart.Suchdynamic-epistemiclogicsofinquiry–not

tied tonatural language,but closer toepistemologyand learning theory– involve

explicit acts of ‘issue management’, where questions and related actions modify

currentissuestructuresontopofepistemicmodels,[14],[30].Intheremainderofthissection,wegointomoredepthonthefoundationalissueof

howthetwoviewsofdynamicsarerelated,andshownewissuesthatemerge.TranslationsbetweenUSandS5Aswithepistemicand intuitionistic logic, there

aretranslationsbetweendynamicsemanticsanddynamic-epistemiclogic,butthey

involvenewissues.Ourfirstobservationcomesfrom[9]:Fact Thereisafaithfultranslationfromupdate-validityintothemodallogicS5.Thefollowingisarecursivemaptrfrompropositionalformulasϕtomodalformu-

lastr(ϕ)(q),whereqisafreshpropositionletter(notetheclauseforconjunction): tr(p) = q∧p

tr(¬ϕ) = q∧¬tr(ϕ)

tr(ϕ∨ψ) = tr(ϕ)∨tr(ψ)

tr(♢ϕ) = q∧♢tr(ϕ)

tr(ϕ∧ψ) = [tr(ψ)/q]tr(ϕ)Thenthefollowingholds,formodelsMwhosedomainisasetSdenotedbyq, [[ϕ]](S)={s∈S|M,s|=tr(ϕ)}Asacorollary,forupdatevalidity,wehavethat ϕ1,…,ϕn|=USψiff|=S5tr(ϕ1∧…∧ϕn∧ψ)↔tr(ϕ1∧…∧ϕn)Infact,connectionsrunbothways.Thereisalsoaconverseembedding:Fact ThereisafaithfultranslationfromS5-validityintoupdatevalidity.Toseethis,transformS5-formulasϕintotheirnormalformnf(ϕ)ofmodaldepth1.

Then,forS5-validities,theupdatefunction[[nf(ϕ)]]istheidentityonallsets,while

fornon-validities,onanycounter-model,[[nf(ϕ)]]returnstheemptyset.SystemidentityNowanearlierissuereturns.DotheprecedingresultssaythatUSis

thesamesystemasS5?Ourtranslationsreducevalidconsequencebothways,which

isenoughforthestandardnotionofsystemequivalence.Buttheintuitivenoveltyof

16

US is that it does somethingmore: it can express the dynamics ofmodel change.

However,thedetailsofourfirsttranslationgiveinformationaboutthisaspect,too:

S5candefinemodelchangesinambientsetsqusingtheformulastr(ϕ)asindicated,

andthisprocessevensimulatestheworkingofUSinarecursivemanner.Andyetthetwosystemsfeelintensionallydifferent,andUSseemsanewdiscovery.

Imustleavethismatteroffinerintensionaldifferencesopenhere,butwillreturnto

theissueofcomparingdynamiccomponentsbydrawinginthelogicPAL.PALandS5Similarpointscanbemadeconcerningpublicannouncementlogic.Fact TherearefaithfultranslationsbetweenPAL-validityandmodalS5.This time, the reason is that the recursion lawsprovideaneffective truth-preser-

vingtranslationfromallPAL-formulaswithdynamicmodalitiesintotheS5baselan-

guage,whileforthatstaticfragment,PALisaconservativeextensionofS5.ComparingUSandPALdirectlyComposingtheirmutualtranslationswithS5gives

faithful embeddings betweenUS and PAL, our paradigms of implicit and explicit

dynamics.Butdespitewhatwassaidbefore,goingvia thestatic logicS5doesnot

relatethedynamiccharacterofbothapproachesdirectly.Canwedobetter?There is anobstacle.Update semantics recurses on the structureof propositional

formulasviewedasupdates,whereasPALdoesnotrecurseoninnerstructureofan-

nouncements !ϕ, but on post-conditionsψ formodalities [!ϕ]ψ. Hence,we enrich

PALwith‘conversationalprograms’builtfromactions!ϕbystandardoperationsof

unionandsequentialcomposition.Thefollowingtranslationcanthenbedefined: Tr(p) = !p

Tr(¬ϕ) = !¬<Tr(ϕ)>T

Tr(ϕ∨ψ) = Tr(ϕ)∪Tr(ψ)

Tr(♢ϕ) = !♢<Tr(ϕ)>T

Tr(ϕ∧ψ) = Tr(ϕ);Tr(ψ)Nowitiseasytoshowthat,formodelsMwhosedomainisthesetS, [[ϕ]](S)={s∈S|M,s|=<Tr(ϕ)>T}Toseehowthisworks,comparethePALprogram!♢¬<!p>T;!pfortheconsistentUS

formula♢¬p∧pwiththeprogram!p;!♢¬<!p>Tfortheinconsistentp∧♢¬p.

17

TrdoesnottranslateUSupdatesintosinglePALactions,butitcomesclose.Earlier

on,we saw how public announcements are closed under sequential composition,

andhenceTr(ϕ∧ψ)amountstoannouncingjustonesuitableS5-formula.OpenproblemIstherealsoadirecttranslationfromPALactionsintoUSupdates?DiscussionThesetranslationsagainhavevarioususes.Decidabilityofdynamiccon-

sequencefollowsfromthatforS5.AndwecanuseresultsaboutPALasroadsigns

forUS.E.g.,thelogicofPALextendedwithconversationalprogramsthatallowfinite

iterations isnon-axiomatizableandnotarithmeticallydefinable, [38]. So,dynamic

semanticsfordiscourseratherthansentencesmightrunthesamecomplexityrisk.But earlier reservations apply: despite the translations,US andPAL seem intuiti-

velydifferent.Forinstance,recallournotionof‘proceduralinformation’inintuitio-

nisticmodels.ViewingPALasalogicofinquiry,ageneralizedsemanticsof‘protocol

models’withrestrictedtemporalhistoriesofupdatesmakessense,[11].Thisnatu-

ralchange inmodelschangesthe lawsofPAL,and itblocksthetranslationofPAL

intoS5.However,itisunclearifprotocolmodelsmakesenseindynamicsemantics.

Also,PAL update has a natural extension to dynamic-epistemic logics withmuch

moredrasticmodel changesmodeling thedynamicsofpartlyprivate information,

anditisunclearifthisricherdynamicshasanyroleinadynamicsemantics.Whatthesetwoexamplessuggestisamoredemandingcriterionofsystemidentity:

equalityordifferencein‘naturalgeneralizations’.Butisthereaformalbasistothis,

orwouldthecriterionmerelyconcernourcurrentpowersofimagination?RemarkTherearealsoother translationsbetweensystemsofdynamic semantics

and explicit logics, such as the translation of dynamic predicate logic, [28], into

propositionaldynamiclogicofprograms,givenin[21],andyetotherkindsexist.Wefoundnaturaltranslationsbetweendynamicsemanticsanddynamic-epistemic

logics.Still,implicitandexplicitapproachesdonotcollapse,andagainwemightbe

contentwithcreatingmerges.Eitherway,therealmofinformationdynamicsseems

arichsourceforourexplicit/implicitcontrast,raisinginterestingissuesofitsown.

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10 Dynamiclogicsofsoftinformation

Ourdiscussionsofarcenteredonthestaticsanddynamicsofknowledge.However,

theimplicit/explicitcontrastappliesjustaswelltologicsofbelief,perhapsthemore

importantattitudeinagency.Thecaseofbeliefshowsinterestingnewfeaturesand

suggests new comparisons between implicit and explicit logic systems. We start

withbeliefdynamicsinexplicitstyle,movingtoimplicitcounterpartslater.BeliefandconditionalbeliefEpistemic-doxasticmodelsforbeliefordertheearlier

bareepistemicrangesbyarelationof‘relativeplausibility’≤xybetweenworldsx,y.

Thesemodelsinterpretoperatorsofabsoluteandconditionalbelief: M,s|=BϕiffM,t|=ϕforallt~smaximalintheorder≤on{u|u~s}

M,s|=BψϕiffM,t|=ϕforall≤-maximaltin{u|s~uandM,u|=ψ}Butthereisaricherrepertoireofepistemicnotionsavailableonthismodels.Forin-

stance, on binary world-independent orderings, a good addition is ‘safe belief’, a

standardmodalityintermediateinstrengthbetweenknowledgeandbelief: M,s|=[≤]ϕiffM,t|=ϕforalltwiths≤tLogicsforconditionalbeliefarelikethoseof[36],[17].Foramoregeneralpicture

ofnaturalmodalitiesthatcanbedefinedonthesemodels,see[4].Belief change under hard information Beliefs guide our decisions and actions,

goingbeyondwhatweknow.But beliefs canbewrong, andnew information can

leadtocorrectionandlearning.Onetriggerforbeliefrevisionaretheearlierpublic

announcements.Hereistherecursionlawgoverningthematchingmodelchanges: [!ϕ]Bψ↔(ϕ→Bϕ[!ϕ]ψ)Asimilarprincipleforupdatingconditionalbeliefsaxiomatizesthesystemcomple-

tely.Thereisalsoarecursionlawforsafebeliefunderpublicannouncement,which

isevensimpler.Thefollowingequivalenceholdsonplausibilitymodels: [!ϕ][≤]ψ↔(ϕ→[≤][!ϕ]ψ)Belief changeunder soft informationBut richerbeliefmodels also supportnew

transformations.Inadditiontohardinformation,thereissoftinformation,whenwe

takeasignalasserious,butnotinfallible.Itsmechanismisnoteliminatingworlds,

butchangingplausibilityorder.Awidelyusedsoftupdateis‘radicalupgrade’:

19

⇑ϕchangesacurrentmodelMtoM⇑ϕ,whereallϕ-worldsbecome

betterthanall¬ϕ-worlds;withinthesezones,theoldorderremains.Thedynamicmodalityforradicalupgradeisinterpretedasfollows:

M,s|=[⇑ϕ]ψiffM⇑ϕ,s|=ψanditsdynamiclogiccanagainbeaxiomatizedcompletelyusingrecursionlaws.LogicsofbeliefchangeRecursionlawsexistforbeliefchangesunderawidevariety

ofsofteventsrepresentingdifferentlevelsoftrustoracceptancefornewinforma-

tion, [4], [25].Anareawhere thisvarietymakesspecialsense isLearningTheory,

[24]:differentupdaterules inducedifferentpolicies forreachingtruebelief inthe

limit. The Handbook [19] has details on the landscape of modal logics for belief

change,plusconnectionswithAGM-stylepostulationalapproaches.Thesystemspresentedhereareexplicit inadoublesense.Notonlydoeventsand

actsthatusuallystayinthebackgroundoflogicalsystemsbecomefirst-classobjects

ofstudy,butalso,dynamiclogicsforknowledgeandbeliefhaveexplicitsyntaxand

lawsfortheseactions.Thenewstructureisnotputintothemeaningsoftheoriginal

language, and so we get conservative extensions of earlier static logics, although

sometimesthereisaneedforsomeredesignoftheoriginalstaticvocabulary.11 Non-monotonicconsequencerelationsasimplicitdevices

Next,howcanwedobeliefrevisioninanalternativeimplicitstyle?Onelinerunsvia

dynamicsemantics,withnewmeaningsforlinguisticexpressionssuchasepistemic

modals, [42], [48], [49].All our earlierpoints apply, butwewill notpursue them

here.Instead,weshowhowourcontrastcanalsotakeus,perhapssurprisingly,to

anareaofimplicitlogicthatseemsquitedifferentfromthosediscussedsofar.VarietiesofconsequenceInthe1980s,thestudyofdifferentconsequencerelations

modeling varieties of common sense-based problem solving started in Artificial

Intelligence, and it has since entered other fields. In particular, the consequence

notionof circumscription [37], [45] says that, inproblem solvingor related tasks,

thefollowinginferencesareallowed:

20

Aconclusionneednotbetrueinallmodelsofthepremises,

butonlyinthemostpreferred,ormostplausiblemodels.Thus,problemsolvinginvolvesonlyinspectionofcurrentlymostrelevantcases.Thisstyleof reasoningdeviates fromclassical logic. Inparticular, it is ‘non-mono-

tonic’:aconclusionCmay followfromapremiseP in thissense,but itmayfail to

followfromtheextendedsetofpremisesP,Q.For,themaximalmodelswithinthe

setofmodelsfortheconjunctionP∧QneednotbemaximalamongthemodelsofP.Manyformsofdefeasibleinferencehavebeenstudied,giventhelargerepertoireof

humanreasoningstyles–andcompletestructuralrulesorproofsystemshavebeen

found, following what Bolzano and Peirce already advocated in the 19th century.

These systems, that usually drop some classical rules, while retaining modified

variants,aretypicallytakentoencodebasicfeaturesofsuchstylesofreasoning.Non-standardconsequenceasimplicitlogic Thislookslikeanimplicitapproach

inoursense.What isnewaboutareasoningpractice isnotexplicitlyonthetable,

butitshowsindifferencesandanalogieswithclassicalconsequenceforaclassical

logical language. But non-standard consequence relations have concrete motiva-

tions,theydonotjustarisebytinkeringwithclassicalstructuralrules.MakingitexplicitCanweprovidealternativeexplicitaccountsleavingthenotionof

consequencestandard,whileaddingvocabularytobringouttheoriginsofthenew

consequencenotions?Ofcourse,weneedaguidingsemanticperspectivefordoing

so,andthiswilldependontheprecisemotivationforthenewconsequencerelation.

Inthefollowingcasestudy,weconcentrateontheroleofbeliefincircumscription–

thoughexplicitizingconsequencerelationsmaywellinvolveothernotions,too.FrominferencetobeliefchangeRevisitingtheoriginalAIscenarios,onecanalso

construeproblemsolvingdifferently.Wehavebeliefsaboutaproblemandwherea

solutionmightgo,basedonscenariosthatseemmostplausibletoconsider–either

deep beliefs based on experience in problem solving, or light beliefs as lacking

considerationstothecontrary.Then,aswetakeinnewinformationrelevanttothe

problem, this set of scenarios changes, and beliefs are modified. Now this fits

preciselywithourmodelsofbeliefs.Forinstance,acircumscriptiveconsequence ϕ1,…,ϕn⇒ψ

21

translatesintoourearlierdynamiclogicforbeliefchange,usingtheformula

[!ϕ1]…[!ϕn]BψThistranslationexplainsthedeviationsofnon-monotonic logicfromclassicalcon-

sequence, as the structural rulesof circumscriptionnow follow from thedynamic

logic of belief revision. For instance, it is easy to see how [!ϕ]Bψdoes not imply

[!ϕ][!α]Bψ forall formulasα–exceptforspecialcases.Thisexplanationofthede-

viantinferentialbehaviorhastwosources:thekeyattitudeinpracticalreasoningis

falliblebeliefratherthanknowledge,andalso,wehaveexplicitdynamicevents.Incidentally, this simple analysis is not the only explicit view of belief and non-

monotoniclogic.In[34],defaultreasoningonlysubmitscandidatesforbeliefwhile

furtherinferencesiftsavailableevidencethatsupportstheeventualbeliefs.RemarkTherearewell-knownanalogiesbetweennon-monotonicconsequenceand

conditionalsinthestyleofLewis[36].Insteadof[!ϕ]Bψ,thismightfavorconditio-

nalbeliefBψϕpre-encodingeffectsoflearningthatψ.Thetwoversionsarenotquite

thesame,asupdate!ψrestrictsamodeltoitsψ-worlds,whileaconditionalcanstill

lookat¬ψ-worldswhenevaluatingϕ.Butthesedetailsneednotconcernushere.Eitherway,ourgeneralpointsapply.Thejuxtapositionofperspectivesraisesinte-

restingissues.Againweseeatrade-offbetweenimplicitandexplicitapproaches: nonstandardconsequence oldclassicallanguage,deviantrulesofreasoning explicitdynamicreanalysis newlanguagewithbeliefandactionmodalities,

consequenceisjustclassicalconsequence.Onthesecondapproach,non-standardreasoningisamixtureofclassicalreasoning

and further features of informational actions, not a family of radical alternatives.

Dynamic logics of belief change enrich the original language with informational

events and attitude changes, but thenwork conservativelywith a classical conse-

quencerelation,explainingdeviantfeaturesofnon-standardconsequencebyattitu-

deandinformationchangethroughtherecursionlawsforthenewdynamicopera-

tors.Inthefollowingsection,weevaluatethisdifferenceinapproaches.

22

12 ComparisonsandswitchesWe have seen that non-monotonic consequence relations can be translated faith-

fullyintoaclassicallogicwithoperatorsforattitudesandinformationalevents.But

asbefore,thisdoesnotidentifythetwoperspectives:onecanstillhaveadvantages

overtheother.Forinstance,implicitapproachesfocusonstructuralrules,whichare

anaturalhigh-levelvantagepointallowingforeleganttheory.Ontheotherhand,an

explicit dynamic approach provides an emancipation of informational events in

problemsolvingthatisofinterestperse,asitaddsneweventsbeyondinference.NewdynamiclogicsAneutraltwo-wayviewhereseesswitchingperspectives,[11].

In one direction, given an implicit non-standard notion of consequence, one can

teaseoutinformationalorothereventsmotivatingit,andwritetheirexplicitdyna-

mic logic. This style of analysis, backed up bymathematical representation theo-

rems, replacesnon-standarddeviation fromclassical logic intodynamicextension

of classical logic. Explicit events behindnon-standardnotions of consequence are

sometimeseasytofind,asintheaboveanalysisofcircumscription,butthereisno

automatic method for this art – and there are unresolved challenges concerning

major substructural logics, [41]. Inparticular,noexplicit-styledynamic reanalysis

seems to be known so far for linear logic, whose non-classical notion of conse-

quenceisprimarilybasedonproof-theoreticresourceintuitions.NewnotionsofconsequenceViceversa,givenanexplicitdynamiclogicofinforma-

tionalevents,onecanpackagesomebasicstructureintheformofnewconsequence

relations,andstudythoseperse.Thelattermovecanevenaddtoourfundofstyles

ofreasoning.Hereisanillustration.Logicsofbeliefchangepredicttheexistenceof

newstylesof inferencebasedontheirrepertoireofdifferent informationalevents

andattitudes.Inparticular,problemsolvingmayinvolvedifferentattitudes,suchas

bothknowledgeandbelief,andalso,itmaytakesomenewinformationashard,and

someintheearliersoftsense,leadingtovariantsofcircumscriptionsuchas

soft-weakcircumscription [⇑ϕ1]…[⇑ϕn]Bψ

soft-strongcircumscription [⇑ϕ1]…[⇑ϕn]Kψwherethepremisesare just takenassoftupgrades,notaspublicannouncements.

Differentstructuralruleswillthenencodedifferencesintheunderlyingprocessof

23

drawing a conclusion. Thus, we generate new notions of consequence, andmore

wouldarisebyusingothermixturesofknowledge,beliefandupdateactions.Thus,inthestudyofconsequencerelations,implicitandexplicitapproacheslivein

harmony, andwe can often performaGestalt switch fromone to the other. Such

switchesalsosuggestprecisemathematicalsystemtranslationsinourearliersense.PhilosophicalrepercussionsWhiletheprecedinganalysismayseemjusttechnical,

well-knownpositionsbasedonnon-standardlogicsmaycomeunderpressurebyan

explicit-style reanalysis. Inparticular, existenceofdifferent consequencerelations

onaparhasledtothethesisofLogicalPluralism,aviewthatlogicshouldacknow-

ledgecompetingviewsonthenatureoflogicalconsequence,andperhapsalsoother

core notions of the field, [8]. But in our view, this grand conclusion depends on

taking the implicitmethodology for granted. On a dynamic explicit re-analysis as

presentedhere,thecompetitionbetweenconsequencerelationsdisappears,andwe

getcompatibleextensionsofclassicallogicwithoutanycommitmenttocompetition.

Thesecondviewneednotbesuperiortothefirst,butitsveryexistenceundermines

strongconclusionsarisingfromlookingatconsequenceinonlyonestance.13 Furtherexamples

Wehaveseenhowtheimplicit/explicitcontrastrunsthroughbothstaticanddyna-

miclogicsforknowledgeandbelief,aswellasforlogicsforconsequencerelations.

Furtherexamples in thisepistemic linecanbe foundbymoving from information

flowtoagencyandgames:inthemonograph[12],implicitlogicgamesandexplicit

game logics are naturally entangled strands throughout. But once one sees the

contrast,itappliestoanypartoflogicwhatsoever,notjustinformationandagency.

Weshow thiswith twoexamples, from thephilosophyofphysicsand frommeta-

physics.Again,theseraisenewissuesoftheirownthatwewillonlytouchupon.QuantumlogicOurfirstexampleconcernsastrongholdofnon-classicallogicsince

the1930s.Considerthefieldofquantumlogic,wheretheclassicaldistributivelaw (p∧(q∨r))↔((p∧q)∨(p∧r))failsinthedomainofphysicalquantumphenomena.Thereareofcoursereasonsfor

thisfailure:measurementsdisturbthecurrentstateofaphysicalsystem–butthis

24

is left implicit inquantum logic.There is a long traditionof research in this area,

whichhasresultedinanextensivealgebraicandmodaltheoryofquantumlogics.The first explicit companion to all this seems the dynamicmeasurement logic of

BaltagandSmets,cf.[5].Theirsystemof‘quantumPDL’hasdynamicmodalitiesfor

measurement actions that satisfy perspicuous laws mirroring physical quantum

facts,butitremainssquarelybasedonclassicallogic.Indoingso,itexplainsallthe

deviantfeaturesofquantumlogicinauniformmanneraspropertiesofasmallfrag-

mentoftheexplicitlanguage.Forinstance,failureofdistributivitybecomesfailure

ofactionstodistributeoverchoice,awell-knownphenomenoninlogicsofcomputa-

tion,whichhasnothingtodowithpropositionallogic.Butanexplicitdynamiclogic

ofmeasurementcanalsoexpressfurthersignificantpropertiesofphysicalsystems,

andanalyzemoretypesofmeasurementactiononthese,makingtraditionalquan-

tumlogicapoorprojectionofwhatgoesonfromaphysicalpointofview.Thisisnotjustreformulation,thereareseriousphilosophicalconsequences.Quan-

tumlogicwasfamouslytoutedbyQuineasacasewherenoteventhelawsoflogic

areimmunetorevisioninscientifictheoryconstruction.Whatwastakenforgran-

tedherewasthatquantumlogicinimplicitstyleistheonlygameintown.Butthis

claimdissolveswhenwehaveamathematicallyelegantandconceptuallyperspicu-

ouslogicthatexplicitlyputsmeasurementwhereitbelongs:atcenterstage.Thisbriefexpositionmaynotdojusticetoexplicitquantumdynamiclogic,butsuf-

fice it to say that this new approach placing measurement actions and quantum

information flow at center stage ismore than just logic-internal system redesign.

Itfitswellwithasubstantivetopic,viz.recentinvestigationsintoanalogiesbetween

thefoundationsofquantummechanicsandtheoriesofcomputation.TruthmakersemanticsOursecondexampleshowsourcontrastatworkinavery

recent development. `Truth maker semantics’ has been touted as a hyper-inten-

sionalparadigmspringingtheboundsofstandardmodallogic,cf.[23]andrelated

papers. Inour terms, truthmaker semantics isan implicit approach todescribing

metaphysical(or,insomeintendedapplications,epistemic)structure,changingthe

meaningsofthe logicalconstants,anddefiningnewnotionsofconsequencebased

onthese.So,itmakessensetolookforatranslationfromtruthmakerlogicintoan

explicitcompanion,namely,astandardmodallogicoverthesameclassofmodels.

25

Wegiveabriefexplanationofhowthiscanbedoneforonesimplesystem.ModelsfortruthmakingMaretuples(S,≤,V)withobjectssinSviewedaspartsof

theworldorabstractstates.Thebinaryrelation≤isapartialorderbetweenstates.

Therelationofsupremums=sup(t,u)(lowestupperbound)saysthatobjectsisa

sumormergeofthetandu. It isoftenassumedinthe literaturethatallsuprema

exist,oftenas‘impossibleworlds’incasethemergedstatesareincompatible.Thesimplestrelevantlanguagehereisapropositionallogicwithconnectives¬,∧,∨.

Foratomicp,avaluationVrecordswhichstatesinSmakeptrue,thesetV+(p),or

false,V-(p). This can bemade subject to further constraints: for instance, that no

statemakesapropositionbothtrueandfalse.Thetruthdefinitionisthis: M,s|=p iff s∈V+(p)

M,s=|p iff s∈V-(p)

M,s|=¬ϕ iff M,s=|ϕ

M,s=|¬ϕ iff M,s|=ϕ

M,s|=ϕ∧ψ iff thereexistt,uwiths=sup(t,u),M,t|=ϕandM,u|=ψ

M,s=|ϕ∧ψ iff M,s=|ϕorM,s=|ψ

M,s|=ϕ∨ψ iff M,s|=ϕorM,s|=ψ

M,s=|ϕ∨ψ iff thereexistt,uwiths=sup(t,u),M,t=|ϕandM,u=|ψOnecanalsodefinefurthernotionsoftruthandfalsemaking:‘exact’or‘partial’.Nextonecandefinevariousnotionsofconsequence,sucheachtruthmakerforall

premisesbeingatruthmakerfortheconclusion,oreachtruthmakerofthepremi-

sesbeingextendabletoonefortheconclusion,aswellasversionsthataddcondi-

tionsonfalsemaking.Allsupportdifferentlawsforthepropositionalbaselanguage.

Thuspropositionallogicisthelocuswherethenewconceptualanalysisshows.ModalinformationlogicNowessentiallythesesamestructureshavebeenaround

inmodal logic since the 1980s asmodels of abstract information states. Theuni-

versalmodality[↑]ϕdescribesupwardstructurefromapoint,anddownward[↓]ϕ

describesweakerinformation.ThelogicisthentemporalS4.Wheresupremaexist

intheorderthelogicdescribesthemusingbinarymodalities:

26

M,s|=<sup>ϕψiffthereexistt,uwiths=sup(t,u),M,t|=ϕandM,u|=ψ

M,s|=<inf>ϕψiffthereexistt,uwiths=inf(t,u),M,t|=ϕandM,u|=ψIt is easy to show that<sup>pq is not definable in the temporalmodal language,

makingthisanaturalextensionoftheordinarymodallogicS4.As for lawsof reasoning, themodal logicof informationhas interestingvalidities,

butdetailsarenotrelevanthere.Oneprinciplethatfailsthoughisassociativity:

<sup><sup>ϕψα→<sup>ϕ<sup>ψαThereasonisthat,unlikeintruthmakersemantics,wedonotdemandexistenceof

all suprema in our partial orders. The modal logic of information structures is

axiomatizable,butamajoropenproblemiswhetheritisdecidable.Itisknownthat

logics with associative modalities can often encode undecidable word problems,

whichmightbeawarningsignforimpossibleworldsassupremaintruthmaking.Translating truth maker logic into modal information logic The models just

describedandtheirmodallogicareanexplicitcompaniontotruthmakerlogic.And

the connection isvery close.Here is a two-component recipe for translating from

implicit truthmaker logic intoexplicitmodal logic,where the simultaneoususeof

variants+and–isastandardtrickinreducingthree-valuedlogictoclassicallogics.Takenewpropositionlettersp+andp-foreachatomicpropositionletterp.Now,for

eachpropositional formulaϕ,werecursivelyextendthisdoubleset-upas follows,

closelyfollowingtheabovetruthdefinition: (¬ϕ)+ =(ϕ)- (¬ϕ)-=(ϕ)+

(ϕ∧ψ)+=<sup>(ϕ)+(ψ)+ (ϕ∧ψ)-=(ϕ)-∨(ψ)-

(ϕ∨ψ)+=(ϕ)+∨(ψ)+ (ϕ∧ψ)-=<sup>(ϕ)-(ψ)-Theoremϕ1,...,ϕn|=ψisvalidintruthmakersemantics

iff(ϕ1)+,...,(ϕn)+|=(ψ)+inmodalinformationlogic.Wedonotprovideaformalproof,butthethetranslationisalmostself-explanatory.The translation can accommodate partial truth making as <↑>ϕ and loose truth

making as <↓>ϕ. Adding strict versions [↑s], [↓s] of the order modalities defines

strict truthmakingas [↓s]¬ϕ∧ϕ∧ [↑s]¬ϕ.Also, theearlier-mentionedvarietiesof

27

consequence aremodally definable, and so are special conditions that have been

consideredfortruthmakerdenotationssuchasclosureundermerge,orconvexity.DiscussionWhatdoesourtranslationachieve?First,itenlistsmethodsfrommodal

logic inthestudyof truthmaking–thoughnotallquestionsaresettledautomati-

cally,suchasdecidabilityorexplicitaxiomatization.Butinaddition,thetranslation

hasaclearphilosophicalpoint: truthmakersemantics isentirelycompatiblewith

classicalmodallogic,refutingclaimsaboutirreducibilityofitshyper-intensionality.

Finally, inexploringmetaphysical intuitions,anexplicitmodal logicmightevenbe

moreappropriatethanpropositionallogicre-interpretedviatruthmaking,asitsets

nolinguisticconstraintsondescribingthestructureofouruniverse.14 Implicitversusexplicitstancesatwork

Afterthisbroadarrayofdifferentexamples,itmightseemtimeforapreciseformal

definitionoftheimplicit/explicitcontrast.Butwedonothaveonetooffer,andwe

doubtthatadefinitionexistscoveringallcasesinausefulmanner.Evenso,wedid

identifyrecognizablegeneralfeatures.Implicitapproachesenricholdmeanings,and

locate important information indeviantnotionsofconsequence–explicitapproa-

chesintroducenewvocabulary,butconservativelyextendclassical logic.Andwith

this difference comes plurality of alternative logics for implicit approaches, and

compatible extensions of classical logic on explicit approaches. These features

shouldbeenoughtorecognizethetwostyleswhenoneseesthem:theyarenatural

approaches toward any subject in logic. Moreover, our terminology is not just a

matter of assigning labels to what already exists. As we have shown by many

examples, seeing the contrast raises interesting new issues, both practical and

theoretical.Wesummarizeafewstrandsthatoccurredintheprecedingsections.Findingcomplementaryanalyses Ifweseeonestanceona topic,wecanusually

findadualone.Thusourcontrastbecomesaforceforlogicalsystemdesign.Wesaw

thiswithdynamic semanticsofquestions,which suggestedanexplicit companion

logic of issue modifying events. And conversely, explicit logics of belief change

suggestednewimplicitnotionsofconsequenceintheareaofnon-monotoniclogic.TransferofideasDifferentstancesonthesamethingfacilitatecreativeborrowing,

sincetheiragendasmaydiffer.For instance,epistemic logichasarichtraditionof

28

multi-agentandgroupknowledge.Intuitionisticlogiccanthenprofitfromthesame

ideas,creatingaccountsofmathematicsclosertoresearchasasocialactivity,cf.[2].

But one can also borrow ideas inside one stance. For instance, intuitionistic logic

startedfromtheproof-theoreticBHK interpretationofthelogicalconstants,which

met up with semantics only afterwards. Could a similar proof-theoretic analysis

apply todynamicsemantics,amajor implicitparadigmfor informationdynamics?

Or, foranotherexample insidethe implicitrealm:canBHK-styleproofanalysisbe

takentonon-monotoniclogics,andthustobeliefratherthanknowledge?TranslationandidentitycriteriaforlogicsTheexplicit/implicitcontrastalsosug-

gestsnewmathematicalissuesoftranslationorreductionbetweenlogicalsystems.

Wehavegivensomenewexamples,andnodoubtmuchmorecanbeprovedabout

translatingbetweenimplicitandexplicitlogics.Evenso,thereisnoautomaticalgo-

rithm for turningone sort of system into theother. Finding illuminating counter-

partsaswehavedoneisanartratherthanascience,anditmaywellremainso.Wehavealsosuggestedthat,evenwhentwoimplicitandexplicitlogicscanbemu-

tuallyembeddedundertranslation,intensionaldifferencesmayremain.Hereween-

counteredageneralissueinthefoundationsoflogic.Thereisnogenerallyaccepted

criterionforwhentwo(presentationsof)logicalsystemsshouldcountasthesame.

Mutual interpretability is a significantnotionof equivalence that allows formuch

transferofinformation,soweshouldalwaysseeifitoccurs,butitneednotbethe

lastword. In fact, one vexing problem thatmakes it hard to judge howgood this

notionishastodowithascarcityofnegativeresults.Therearenogeneralmethods

showingnon-translatability between logical systems. Perhaps, in the end, there is

toomuchtranslatabilityintherealmoflogics,andafinersieveisneeded.MergingWherewecannottranslatedifferentstancesintoeachother,aweakercon-

nectioniscompatibilityinmeaningfulmerges.Manysystemsintheliteraturecom-

bine implicit and explicit features: intuitionistic modal logics, [20], merged logic

gamesandgamelogics,[1],dynamic-epistemicinquisitivelogics,[43], joint linear-

temporal logics, and so on. Often, thesemerges feel natural. A recent case is the

intuitionisticallyflavoredpossibilitysemanticsforclassicallogicin[13],[44].Weconcludewithtwootheraspectsoftheinterplayofimplicitandexplicitstances.

29

Understandingtheco-existenceWehavenotedtheexistenceofourcontrast,but

wehavenotofferedanexplanationofwhyitisthere.Ithasbeensuggestedbyrea-

dersofthispaperthatthebackgroundmaylie insomewell-knowndistinctionsin

logic and philosophy. One is that between reasoningwith, from an internal first-

personperspective,andreasoningabout,fromanexternalthird-personperspective.

Implicitlogicsmightgivethereasoningwith,andexplicitonesreasoningabout.But

while thisseemsappealing, itdoesnotquite fit.For instance,epistemic logicwith

operators can also be used in first-person mode, and on the other hand, say,

dynamicsemanticshasalsobeenappliedtothird-persondiscourse.Anotherdistinctionthatseemsrelevantisthatbetweenobjectlanguageandmeta-

language.Wecanformalizethemeta-languageofthesemanticsforalogicalsystem

insomeotherlogic–the‘standardtranslation’formodallogicisatypicalexample,

[16]. Is themeta-logic then the explicit version of an implicit object logic? Some

studiesattheinterfaceoflogicandgamespointthisway.Forinstance,themodal-

dynamicgamelogicof[40]formalizespartofthemeta-theoryoffirst-orderevalua-

tiongames.Viceversa,game logics induce logicgames, implicitpractices for their

semantics–andthisdesigncyclecanbeiterated,cf.theprogramof[12].Evenso,a

complete identificationdoes not fit all caseswediscussed. It is not at all clear in

whichsense,say,dynamic-epistemiclogicisthemeta-logicofdynamicsemantics.Choosing locally Co-existencemeans that both implicit and explicit stances have

intrinsicvalue,butevenso,particularareasmaybringfurtherreasonsforusingone

ratherthantheother.For instance,arethere favoredstances inhumancognition?

Indeed, it has been claimed that natural language conveys much information

implicitly,perhapsforeaseofcoding.Implicit logicswouldthenmodelthisreality

directly, whereas explicit logics of information and agency are outside theorists’

views of language. But this does not fit the facts. Natural language is a medium

where both stances occur, in the guise of one might call participating versus

observingmodes.Akeyfeaturetokeepinmindhereistheuniversalityoflanguage.

We switch between these twomodes all the time, while staying inside the same

mediumofcommunication.Theremaybelocalcognitivepreferencesbetweengoing

explicitorimplicit,butwedoubttheycanbejustifiedinasweepingmanner.

30

15 Conclusion

We have identified a significant methodological contrast running through logic,

between implicit and explicit stances. We use the word ‘stance’ here, and not

‘system’, becausewe do not identify logicwith a family of formal systems. Some

logicalsystemscanindeedbecalledimplicitorexplicit,butthecontrastaswehave

discusseditalsoappliestobroaderworkingshabitsinlogicalanalysis.Eitherway,seeingthecontrastrevealspatternsrunningthroughthefieldof logic,

anditsuggestsnewquestionsofaconceptualandtechnicalnature.Wehaveshown

thisinanumberofconcreteinstancesofsystemdesignandintranslationsbetween

systems.So,seeingthecontrastmeansworktobedone,andinfact,weseeitasa

force towardabetterunderstandingof thecoherenceof logic,both in its systems

andworkinghabits.Moreover,wehavepointedoutinvariousillustrationsthatthe

contrast has philosophical consequences, since it undercuts sweeping ideological

viewsthataretacitlybasedontakingonestancewhileignoringtheother.16 References

1.T.Ågotnes&H.vanDitmarsch,2011,‘WhatWillTheySay?PublicAnnouncement

Games’,Synthese(KRA)179,57–85.

2.S.Artemov,2007,‘JustificationLogic’,TechnicalReportTR-2007019,

CUNYGraduateCenter,NewYork.

3.P.Balbiani&D.Galmiche,2016,‘AboutIntuitionisticPublicAnnouncementLogic’,

inL.Beklemishev,S.Demri&A.Maté,eds.,ProceedingsAdvancesinModal

Logic,Budapest2016,CollegePublications,London,58–77.

4.A.Baltag&S.Smets,2008,‘AQualitativeTheoryofDynamicInteractiveBelief

Revision’,inG.Bonanno,W.vanderHoek&M.Wooldridge,eds.,Texts

inLogicandGamesVol.3,AmsterdamUniversityPress,9–58.

5.A.Baltag&S.Smets,2011,‘QuantumLogicasaDynamicLogic’,Synthese179:2,

285–306.

6.Y.Bar-Hillel&R.Carnap,1953,‘SemanticInformation’,TheBritishJournalfor

thePhilosophyofScience4:14,147–157.

7.J.Barwise&J.Perry,1983,SituationsandAttitudes,TheMITPress,Cambridge

MA.

31

8.J.Beall&G.Restall,2006,LogicalPluralism,OxfordUniversityPress,Oxford.

9.J.vanBenthem,1989,‘SemanticParallelsinNaturalLanguageandCompu-

tation’,inH–DEbbinghausetal.,eds.,LogicColloquium.Granada1987,

North–Holland,Amsterdam,331–375

10.J.vanBenthem,1996,‘ModalLogicasaTheoryofInformation’,inJ.Copeland,

ed.,LogicandReality.EssaysontheLegacyofArthurPrior,Clarendon

Press,Oxford,1996,135–168.

11.J.vanBenthem,2014,LogicinGames,TheMITPress,Cambridge(Mass.).

12.J.vanBenthem,2017,‘TruthMakerSemanticsandModalInformationLogic”,

DepartmentofPhilosophy,StanfordUniversity.

13.J.vanBenthem,N.Bezhanishvili&W.Holliday,2015,‘ABimodalPerspectiveon

PossibilitySemantics’,toappearinLogicandComputation.

14.J.vanBenthem&S.Minica,2012,‘TowardaDynamicLogicofQuestions’,

JournalofPhilosophicalLogic41:4,633–669.

15.G.Bezhanishvili&W.Holliday,2017,‘ASemanticHierarchyforIntuitionistic

Logic’,DepartmentofPhilosophy,UCBerkeleyandDepartmentof

Mathematics,NewMexicoStateUniversity.

16.P.Blackburn,M.deRijke&Y.Venema,2000,ModalLogic,CambridgeUniversity

Press,CambridgeUK.

17.J.Burgess,1981,‘QuickCompletenessProofsforsomeLogicsofConditionals’,

NotreDameJournalofFormalLogic22:1,76–84.

18.I.Ciardelli,J.Groenendijk&F.Roelofsen,2012,‘InquisitiveSemantics:aNew

NotionofMeaning’,ILLCUvA,andLanguageandLinguisticsCompass.

19.H.vanDitmarsch,J.Halpern,W.vanderHoek&B.Kooi,eds.,2015,Handbookof

LogicsforKnowledgeandBelief,CollegePublications,London.

20.K.Dosen,1985,‘ModelsforStrongerNormalIntuitionisticModalLogics’,

StudiaLogica44,39–70.

21.J.vanEijck&F-JdeVries,1992,‘DynamicInterpretationandHoareDeduction’,

JournalofLogic,LanguageandInformation1,1–44.

22.D.Fernandez,2006,‘APolynomialTranslationofS4intoIntuitionisticLogic’,

JournalofSymbolicLogic71:3,989–1001.

23.K.Fine,2014,‘TruthmakerSemanticsforIntuitionisticLogic’,Journalof

PhilosophicalLogic43:2,549–77.ReprintedinthePhilosophers’Annual2014.

32

24.N.Gierasimczuk,2010,KnowingOne’sLimits,LogicalAnalysisofInductive

Inference,Dissertation,ILLC,UniversityofAmsterdam.

25.P.Girard,F.Liu&J.Seligman,2012,‘GeneralDynamicDynamicLogic’,

ProceedingsAIMLCopenhagen2012.

26.K.Gödel,1933,‘EineInterpretationdesintuitionistischenAussagenkalküls’,

ErgebnisseeinesMathematischenKolloquiums,4,39–40.

27.R.Goré&J.Thomson,2017,‘ACorrectPolynomialTranslationofS4into

IntuitionisticLogic’,LogicandComputationGroup,AustralianNational

University,Canberra.

28.J.Groenendijk&M.Stokhof,1991,‘DynamicPredicateLogic’,

LinguisticsandPhilosophy14:1,39–100.

29.H.Halvorson,2017,‘TheInvariantContentofEquivalentTheories’,Department

ofPhilosophy,PrincetonUniversity,Talkat6thCSLIWorkshoponLogic,

RationalityandIntelligentInteraction,Stanford.

30.Y.Hamami&F.Roelofsen,2013,LogicsofQuestions,Synthese,specialissue.

31.J.Hintikka,1962,KnowledgeandBelief,CornellUniversityPress,IthacaNY.

32.W.Holliday,2012,KnowingWhatFollows:EpistemicClosureandEpistemic

Logic,Ph.D.thesis,Departmentofphilosophy,StanfordUniversity.

33.T.Hoshi,2009,EpistemicDynamicsandProtocolInformation,Ph.D.thesis,

DepartmentofPhilosophy,StanfordUniversity(ILLC-DS-2009-08).

34.D.Klein,O.Roy&Ch.Shi,2017,‘DefaultReasoningandBeliefRevision’,

DepartmentofPhilosophy,UniversityofBayreuth,andILLC,Universityof

Amsterdam.

35.W.&M.Kneale,1962,TheDevelopmentofLogic,OxfordUniversityPress,

Oxford.

36.D.Lewis,1973,Counterfactuals,Blackwell,Oxford.

37.J.McCarthy,1980,‘Circumscription–AFormofNon-monotonicReasoning’,

ArtificialIntelligence13,27–39.

38.J.Miller&L.Moss,2005,‘TheUndecidabilityofIteratedModalRelativization’,

StudiaLogica97,373–407.

39.G.Olkhovikov,2013,‘Model-theoreticCharacterizationofIntuitionistic

PropositionalFormulas,ReviewofSymbolicLogic6:2,348–365.

33

40.R.Parikh,1985,‘TheLogicofGames’,AnnalsofDiscreteMathematics24,

111–140.

41.G.Restall,2000,SubstructuralLogics,Routledge,London.

42.B.Rodenhaüser,2014,AMatterofTrust:DynamicAttitudesinEpistemicLogic,

Dissertation,ILLC,UniversityofAmsterdam.

43.F.Roelofsen,2011,‘InquisitiveDynamicEpistemicLogic,manuscript,ILLC,

UniversityofAmsterdam.

44.I.Rumfitt,2015,TheBoundaryStonesofThought,OxfordUniversityPress,

Oxford.

45.Y.Shoham,1988,ReasoningAboutChange:TimeandChangefromtheStandpoint

ofArtificialIntelligence,MITPress,Cambridge(Mass.).

46.R.Stalnaker,2006,‘OnLogicsofKnowledgeandBelief’,PhilosophicalStudies

128:1,169–199.

47.A.Troelstra&D.vanDalen,1988,ConstructivisminMathematics,twovolumes,

North-Holland,Amsterdam.

48.F.Veltman,1996,'DefaultsinUpdateSemantics',JournalofPhilosophical

Logic25,221–261.AlsoappearedinThePhilosopher'sAnnual,1997.

49.S.Yalcin,2007,‘EpistemicModals’,Mind116,983–1026.