Piotr Labenz - Common knowledge.pdf

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Oncommonknowledgeinconversation
PiotrLabenz
20thMay2004
Abstract
Thisworkingpaperaimstowardsacoherentaccountofthesort
ofcommonknowledgethatisnecessaryformakingfelicitousutter-
ancesinaconversation.Thehypothesistothate ect,originating
fromLewis[29]andSchi er[37]ispresentedinSection1alongwith
thestandardde nitionofcommonknowledge.However,asClarkand
Marshall[10]observed,thatde nitionleadstoaparadox;inSection
2Idiscussthepossibilitiesofavoidingit.Iargue(contraClark)in
favourofa xed-pointapproachalaBarwise[7],which,reliesonthe
notionofcoordinationdevices.Sincetheargumentaboutthecommon
knowledgeparadoxresortstothecognitiveplausibilityofthenotion,
inSection3Itrytoclarifyitsempiricalstatusbyconsideringsome
relevantdata.
Thisbeingdone,theoutlookofthispaperissketchedinSection
4,beingageneticaccountofcoordinationdevicesintermsofsome
simplegames.Itspurposeistosupplementthe xed-pointapproachto
commonknowledgeinparallelwithClark’s[9]psychologicalaccount.
Thisconcludeswithanattemptatacognitivelyplausiblede nitionof
commonknowledge.
Ithastobestressedthatasitstands,thispaperisaninformal,
workingsketchmeantasabasisforfurtherwork.
1Introduction.Thehypothesis
Havingaconversation,wequiteordinarilyassumeourinterlocutortoshare
someinformationwithus.Hadn’twesuchanassumption,itwouldberea-
sonabletobeentirelyandthoroughlyexplicitinourutterances.Butnot
onlywouldthisbebizarre,butimpossibleaswell.Forinstance,insaying:
(1)TheonlyrecipeIhavetoavoidfatigueisnottodotoomuchwork.
1
Iassumethatmyinterlocutorknowswhatwork,fatigueandarecipeare,
whatconstitutesdoingwork,havingarecipeandsoforth.Itwouldhardly
bepossibleformetobemoreexplicit;indeed,anyattempttowouldbe
ridiculous(andsoinfelicitous).Theinterlocutor,inherturn,mustknow
thatIknowwhatarecipeisetc.,inordertobesurethatImeantwhat(1)
meansratherthansomethingelse.Afterall,hadInotknownthecorrect
meaningsofthewordsin(1),Imighthavemistakenlyusedthemtoexpress,
forinstance
(2)Ilikekids,butIdon’tthinkIcouldeatawholeone.
orsomethingevenmorepreposterous.Thirdly,Imustknowthatsheknows
thatIknowwhatapaperisetc.,becauseotherwiseIwouldnotbesure
thatinhavinguttered(1)IhadconveyedtheinformationIhadintendedto.
Indeed,hasshenotknownthemeaningsofwordsconstituting(1),utteringit
wouldnotbefelicitous.But,bythesametoken,shemustknowthatIknew
andsoonadin nitum.Socommonknowledgeofalanguage’smeaning
postulatesisaprerequisiteofusingthatlanguage 1 .
Indeed,thisphenomenonisevenmoreconspicuousinthecaseofde nite
reference,beitbydeixis,byanaphoraorbypropername.Takinganinstance
ofthe rst:
(3)I’vemetthatblondebefore.
presupposessomepropositionidentifyingtheindividualreferredtolike,tak-
ingtheexpressionontheleft-handsidetostandforatokenexpression(ut-
teredasapartof(3)byacertainagentatcertaintime),andthatonthe
right-handsidetobeanameuniquelyreferringtothatblonde:
(4)thatblonde=Lydia.
Haditnotbeenforthesepresuppositions,thereferenceof(3)wouldnotbe
(uniquely)determined.InutteringthesesentencesIassumethatmyinter-
locutoridenti eswhatisreferredto namely,Lydia thesamewayasIdo.
Furthermore,asClarkandMarshall[10]noticed,thesepresuppositionsmust
becommonknowledge.Toutter(3)felicitously,Imustnotonlypresuppose
(4),butknowthatmyinterlocutorpresupposesitaswell thusidentifying
thatblondethesamewayIdo.Tounderstand(3)thewayImeant,she
mustknowthatIknowthatshepresupposes(4).AndImustknowthatshe
AlreadyAjdukiewicz[1]observedthatthesepostulatesmustbeknown.Thatcommon
knowledgeisrequiredhasbeen rstnotedbyLewis[29].Aboutcommonknowledgeand
presuppositionscf.Stalnaker[40].
2
1
willunderstanditthus,henceImustknow...,etc.IfIwasunsureabout
anylevelofthisregress,thenIcouldnotbesurethatmyutterancewillbe
felicitous. 2
Thereforeithasbeenclaimed, rstbyLewis[29]andSchi er[37]that
commonknowledgeisaprerequisiteoflinguisticcommunication,becauseit
isindispensableformakingfelicitousutterances.Commonknowledgecanbe
de nedthus:
De nition1(Iteratedcommonknowledge). 3 Apropositionpiscom-
monknowledgeinasetCofagentsi
8x2C
2
y pand
8x2C8y2C8z2C
2
x
2
2
x
2
y
2
z pand
etc.adin nitum.
Thenifapropositionqpresupposespropositionsp 1 ,p 2 ...p n ,thesemustbe
commonknowledgeinasetofagentscontainingthespeakerandthehearer
inorderforptobeutteredfelicitously.Thisview,whichmaybecalled
commonknowledgehypothesis (CKH),isthereceivedview 4 .
Ofcourse,itissomewhatunsettlingtopostulateanin niteseriesof
knowledgeattributionsinordertoexplainhowisitpossibletouttersen-
tencesfelicitously;afterallitisnotthatdi cult.Thussomehavesuggested
thatthemaximumrequiredknowledge-operatordepthbereducedfromin-
nitetothreeorfour 5 .Whiledoubtlessmoreplausiblepsychologically,this
would,however,belogicallyinadmissible.Theconsecutivestepsinthein-
niteregressyieldedbyde nition1form,asLewis[29,p.53]remarks,a
chainofimplications.Cuttingitatanypointwould,byasortofdomino
e ect,invalidatetheinitialelementsaswell:forinstance,shoulddepthnfail
toobtainof(4),thensowoulddepthn−1,etc.Moreover,itispossibleto
contriveexamplesshowingthatcommonknowledgeofanyarbitrarydepth
canbeexplicitlyrequiredforthefelicityofsomeutterances;cf.[37],[10],
[41].
2
Cf.also[19],[9].
Thisde nitionwas rstsuggestedbySchi er[37]andisstandardinepistemiclogic
andAI;cf.[14],[31].Onde nitionsofcommonknowledgegenerally,see[7],[14],[42].
4
Eventhoughthereisaconsiderableamountofconceptualconfusion: common , mu-
tual and shared ; knowledge , belief and ground .See[28],[26,p.29],[9,p.99].
5
Forreferences,see[9,p.100],[10].Amajorattempttodispensewiththenotion
ofcommonknowledgealtogetherwasSperberandWilson’srelevancetheory[38],which,
however,seemstohavesomedrawbacks inparticulartoberatherunderdevelopedon
theformalside.
3
x pand
8x2C8y2C
3
2Theparadoxandthede nitions
y ...,sothatthede niensisan
in niteconjunction.Hencetocheckwhetherpiscommonknowledge,onehas
tocheckwhethereachofthein nitenumberofconjunctsistrue.Now,onthe
computationalconceptionofthemind,checkingwhetherasentenceistrue
takessometime perhapssmall,yetnonzero.Thereforetolearnwhetherp
iscommonknowledgetakesanin nitetime.But,byCKH,inordertomake
felicitousutterances,weneedtoknowthatsomepresuppositionsarecommon
knowledge.Yetwemakefelicitousutterancesin nitetime(notwithstanding
thatoneadmittedlymighttakequitesometimeponderingsentenceslike
(3)).Acontradiction.
Tobeginwith,notethatDe nition1canbeequivalentlyrephrasedin
asemanticmanner:
x
2
y
2
x
2
y
2
x
2
De nition2(Iteratedcommonknowledge).LetR betheancestral
closureoftheaccessibilityrelationsofallagentsinC.Thenpiscommon
knowledgeinCattheworldw(assumewistheactualworld)i forall
possibleworldsv,R (w,v)!v|=p.
Theproblemonwhichtheparadoxhingesisofsyntacticnature;itcould
beeasilyalleviatediftherewasa nitemethodofcheckingwhetherpis
commonknowledge(CK-checkingforbrevity)semantically.Butthereis
not.Namely,byDe nition2,pmusteitherbetrueatallworldsinthe
model,orfalseonlyatsuch‘solitary’worldsthatnoagentseesthemfrom
anyotherworld.Then,trivially,foreveryagentx,
2
x pistrueateachworld
y petc.,soeventuallyp
iscommonknowledgeinC.ThusforCK-checkingonemustcheckthatpis
trueatallnon-solitaryworlds.Butunlessthemodelis nite,thisrequiresan
in nitenumberofsteps(note,however,thattocheckthatsomethingisnot
commonknowledge,alwaysa nitenumberofstepssu ces).Ifthemodelis
niteandptrueatallnon-solitaryworlds,andthemodeldoesn’tsplitinto
submodelssuchthatnoworldbelongingtoonesubmodelcanbeseenfrom
theotherandconversely,thenpiscommonknowledge.Theseconditionsare
nitelyveri able;thereforeonewaytoavoidtheparadoxistoassumethe
iteratedde nitionlimitedto nitemodels.
Whethertherestrictionto nitemodelsisaplausibleoneisphilosoph-
icallydebatable.Ontheonehand,thephysicalworldseemstobe nite;
ontheother,therearein nitelymanynumbers,counterfactualpossibilities
2
x
2
4
ClarkandMarshall[10]arguethatDe nition1leadstoaparadox.Namely,
evenifCis nite,operatorsforitseverymembercanbeiteratedalongthe
formula,forinstancethus:
2
exceptsolitaryworldsand,bythesametoken,
andsoon.However,regardlessofthatthereareotherreasonstorejectthe
iterateapproach.Firstly,evenrestrictedto nitemodelsitiscognitively
untenable,becauseCK-checkingontheabovelinesrequiresprocessingan
immensesearchspace.Itseemsmostunlikelythatprocessingeveryutter-
ancewecheckwhethereachofits(numerous)presuppositionsistrueateach
of(verynumerous)non-solitarypossibleworlds.Secondly,asMcCarthyet
al.[30]observed,theiterateaccountdoesnotentailthatcommonknowledge
ofpshouldalwaysbecommonknowledgeitself.Thatisadrawbackinso-
faritseemsreasonabletoexpectasortofgroupintrospection:itshouldbe
commonknowledgeamongthemembersofCwhatisthecommonknowledge
theyshare.
Thereforeanalternativede nitionofcommonknowledgewouldbemost
desirable.Onealternativeistoexplicitlymentionsharedbasis,thesetof
beliefsthatgiverisetothecommonknowledgeinagivengroup:
De nition3(Shared-basiscommonknowledge). 6 Thepropositionp
iscommonknowledgeamongtheagentsinCi thereisabasisBsuchthat
8x2C8p2B
2
x p,where
B|=(p^8x2C8p2B
2
x p).
Thisde nitionnicelycorrespondswithClark’spsychologicalprincipleofjus-
ti cation: inpractice,peopletakeapropositiontobecommongroundin
communityonlywhentheybelievetheyhaveapropersharedbasisforthe
propositioninthatcommunity [9,p.96].Clarkalsoclaimsthatitavoids
theparadoxicalregress.
Ordoesit?Let
2
x p2B;thenbyDe nition3foranyy2C,
y p2B.OnecaniteratethisstepobtaininginBboxstringsofany
length:
x
2
y ...,justlikeontheiterateapproach.Belongingto
B,propositionsstartingwiththesestringsmustbeknowntoallmembers
ofC;thereforesyntacticallythereisthesamesortofregressaspreviously.
However,semanticallyDe nition3doesmuchbetter:itsu cestosupply
anexampleofsharedbasisB,whichisdoableinasinglestep,andcommon
knowledgefollows.However,asBarwise[7]remarks,therecanbemanydif-
ferentbasesBthatyieldthesamecommonknowledge(are informationally
equivalent ).Indeed,theremaybe unintended onesthatwewouldnot
thinkof,thatdonotrepresentanypsychologicallyplausiblesharedbasis.
Andtheremaybein nitelymanysuchunintendedbases;looselyspeaking,
theshared-basisapproachisnotcategorical.Thereforeonemaycheckin
nitetimethatpiscommonknowledge,butnotthatitisnotcommon
2
x
2
y
2
x
2
y
2
x
2
6
FirstproposedbyLewis[29]andAumann[4].
5
2

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