IMPLICIT AND EXPLICIT STANCES IN LOGIC · from those of classical logic on the original vocabulary...
Transcript of IMPLICIT AND EXPLICIT STANCES IN LOGIC · from those of classical logic on the original vocabulary...
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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.
18
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
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