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Anoverviewofthequantumcognitionresearchprogramme

EmmanuelM.Pothos1

JenniferS.Trueblood2

ZhengWang3

JamesM.Yearsley1

JeromeR.Busemeyer4

Affiliationsandemail1.DepartmentofPsychology,City,UniversityofLondon,LondonEC1V0HB,UK.Emmanuel.Pothos.1@city.ac.uk(correspondence);James.Yearsley@city.ac.uk2.DepartmentofPsychology,VanderbiltUniversity.jennifer.s.trueblood@vanderbilt.edu3.SchoolofCommunication,OhioStateUniversity.wang.1243@osu.edu4.DepartmentofPsychologicalandBrainSciences,IndianaUniversity.jbusemey@indiana.eduRunningtitle:quantumcognitionRunningtextwordcount:20,639

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AbstractThequantumcognitionresearchprogrammeconcernstheapplicationofquantumprobabilitytheory(therulesforhowtoassignprobabilitiestoeventsfromquantummechanics,withoutanyofthephysics)tocognitivemodelling.Quantumcognitivemodelshavebeenappliedtoseveralareasofpsychologyandweprovidearepresentativeoverview,withcoverageinperception,memory,similarity,conceptualprocesses,causalinference,constructiveinfluencesinjudgment,decisionordereffects,conjunction/disjunctionfallaciesindecisionmaking,andotherjudgmentphenomena.Achallengeinthisreviewisthattheapplicationofquantumtheoryineachareacomeswithitsownuniquechallengesandassumptions.Wepresenttheempiricalfindingsdrivingtheapplicationofquantummodelsforeachareawithouttheory,wherethisispossible,toallowanappreciationoftheempiricaldriversofmodelapplication;wediscussthequantumcognitivemodelsinawaywhichallowsconsiderationoftheirkeyfeatures;wethencastacriticaleyeonthemodelsandexplorecomparisonswithnon-quantummodels.Ourcriticalassessmentofquantummodelingworkisaimedathelpingusanswerquestionssuchasisitworthperseveringwithquantumcognitionmodels,whatarethemainweaknessesofsuchmodels,andwhatistheirpromise.

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1.IntroductionIn2018thequantumcognitionresearchprogrammeturned23yearsold(onecanconsiderAerts&Aerts,1995,asthefirstmajorpublicationinthisprogramme).Itisagoodtimetoconsiderquestionssuchaswhathasthequantumcognitionresearchprogrammeachieved,whatareitsfurtherprospects,andhowdoesitrelatetootherinfluentialresearchtraditionsincognitivescience. Wecallquantumcognitiontheapplicationofquantumprobabilitytheorytocognitivetheory,wherequantumprobabilitytheoryreferstotherulesforprobabilisticinferencefromquantummechanics,withoutanyofthephysics(forintroductionssuitedtopsychologists,indecreasingcomplexity,seeBusemeyer&Bruza,2011,Yearsley,inpress,Pothos&Busemeyer,2013;forintroductorytreatmentsforphysicistsseeHughes,1989,Isham,1989).Inalltheworkwereview,thereisnoassumptionregardingphysicalquantumstructureattheneurophysiologicallevel:weassumeafullyclassicalbrain,suchthatneuronalprocessescangivetoquantum-likestructureatthemacroscopiclevel.Thus,thequantumcognitionresearchprogrammeasdefinedhereisnon-overlappingwiththecontroversialquantumbrainhypothesis(Hameroff,2007;Littetal.,2006).Onemaywonderwhytheissueofneuralimplementationischallengingparticularlyforthequantumcognitionresearchprogramme.Forexample,onewouldnotstartanoverviewofclassicalprobabilitytheoryincognitionwithastatementregardingneuronalapplication.Theansweristhatcharacteristicquantumeffects,suchasonticuncertaintyorspookyactionatadistance,areoftenthoughttorequireaquantumphysicalsystem,whichwouldprecluderelevanceinaclassicalbrain.However,weshallseethattheapplicationofquantumtheoryinpsychologyeschewssuchissuesand,indeed,effectswhichmayappearweirdforphysicalsystemsmayhavefamiliarinterpretations(suchascontextuality). Quantumprobabilitytheoryisasetofrulesforhowtoassignprobabilitiestoevents.Epistemicallyitishardlydifferentfromthemoreinfluentialclassical/Bayesianprobabilitytheory.Thelatterhasledtoafertileresearchtradition,successfulbothinitsdescriptivecoverageandapriorijustification(e.g.,Griffithsetal.,2010;Lakeetal.,2015;Oaksford&Chater,2007;Tenenbaumetal.,2011).However,classicalprobabilitytheoryisjustonewaytoassignprobabilitiestoevents.Evenoutsidepsychologicaleffortstoextendprobabilitytheory(e.g.,Shafer,1976;Tversky&Köhler,1994),thereareseveralsystemsforprobabilisticassignmentavailabletopsychologists(potentiallyinfinite,e.g.,Sorkin,1994).So,evenifclassicalprobabilitytheorywereuniformlysuccessfulinitsapplicationtocognitivetheory,whyrestrictourselvestothefirstreasonablesolutionforprobabilisticinferencewedeveloped?Analternativeprobabilitytheorymightbesometimesmoresuccessfulthanclassicalprobability.

Infact,classicalprobabilitytheoryhasnotgoneunchallengedincognitivemodelling.TverskyandKahnemanhavebeenmostinfluentialinpointingoutdiscrepanciesbetweentheprinciplesofclassicalprobabilitytheoryandbehavior,butmanyothershavefollowed.TheimpactofTverskyandKahneman’sworkispartlyduetoexperimentalteststhatchallengedthemostbasicprinciplesofBayesianinferenceandpartlyduetothepersistenceintheconflictbetweenBayesianprescriptionandintuitionintheirexperiments–evenwhenwearetoldwhatistherelevantBayesianprinciple,itissometimesdifficulttoovercomethe(classicallyerroneous)intuition(cf.Gilboa,2000).InconsistenciesbetweenBayesianprinciplesandbehavioraretypicallycalledfallaciesandthemostcommontheoreticalrouteinvolvesso-calledheuristicsandbiases,thatis,individualprinciplesthatcanguidecognition,that

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arenotpartofaformalmathematicalframework,butrathertypicallyrelatetoothercognitiveprocesses,suchasattention,memory,orsimilarity.Suchheuristicsandbiaseshavethemselveshadanextremelyprominentplaceinpsychologicaltheory(NobelprizesforKahnemanin2002andin2017forThaler,bothforeconomics;e.g.,Kahnemanetal.,1982;Tversky&Kahneman,1973,1974,1983;Sloman,1996). Theongoingdebatebetweenclassicalprobabilityexplanationsandonesbasedonheuristicsandbiaseshasbeeninfluential.Beyondpurelydescriptivearguments(whicharenotalwaysstraightforwardtoconclusivelyevaluatebecauseofmimicries),ageneralepistemicreasonforpreferringtheorybasedonheuristicsandbiasesisthatitoftenprovidesbridgesbetweenpsychologicalprocesses,forexample,decisionmakingandsimilarity.Ageneralepistemicreasonforpreferringmodelsbasedon(classical)probabilitytheoryisthatitprovidesacollectionofcoherent,interrelatedprinciplesandeitherallornoneofthemhavetobeinvolvedinpsychologicaltheory.Note,evenforpsychologicalmodelsbasedonclassicalprobabilitytheory,somenon-probabilisticassumptionsareneeded,e.g.,intermsofhowtobuildrepresentationsfromtheavailableinformation.Thisisreasonable,unlesstherearesomanysuchassumptionsthatthedistinctionbetweenaformalclassicalprobabilisticmodelandaheuristics/biasesoneisblurred(Jones&Love,2011). Regardingquantumtheory,itisworthnotingthatphysicistshadbeeninitiallyextremelyreluctanttoadoptquantumtheory–theywereforcedintoitbyempiricalfindings.Quantumprobabilitytheoryworksinphysicsbecausethestructureofthetheorymatchesthewaytheuniverseworks,evenifwedon’tunderstandwhy.Inphysics,quantumprobabilityhasledtoseveraldiscoveries,whichwerepreviouslyunthinkable,partlybecausequantumprobabilitytheoryembodiesawayofthinkingaboutprobabilitiesdifferentfromtheclassicalone.Themotivationforconsideringquantumprobabilitytheoryincognitionisanalogous:ourmostbasicpointisthattherehavebeenseveralempiricalfindingsinpsychologywhich,superficiallyatleast,indicatequantumstructure.Suchfindingsareoftenexplainedbyheuristics,buttheapplicationofquantumtheorycancomplementsuchexplanationsintermsofquantitativepredictionsandfurthergenerativevalue.Thequantumcognitionresearchprogrammeisaboutexploringthepotentialofquantumprobabilityinpsychology,notingthatitmaybethecasethatcertainbehavioralfindingsmaybeoutsidethescopeofbothclassicalandquantumprobabilitytheoriesandmayrequiremodelsbasedone.g.heuristics/biases. Afinalpreliminaryquestioniswhyisareviewofquantumcognitivemodelsneeded.Therearetwoanswers.First,theapplicationofquantumtheoryincognitionisstillrelativelynew,soitisimportanttopause,evaluate,andconsiderthemeritsofcontinuingwithsuchapplications.Second,thereisanopportunityatthispointtoprovideareasonablycomprehensivereview.2.Whatisquantumprobabilitytheory?Webeginwithaninformalpresentationofquantumprobabilitytheoryfocusedonitskeypropertiesanddifferencescomparedtoclassicalprobabilitytheory.Wedosobyemployingoneofthemostfamousdecisionfallacies,TverskyandKahneman’s(1983)conjunctionfallacy.Inoneoftheirconditions,participantsweretoldofahypotheticalperson,Linda(thearguablymostfamoushypotheticalpersonindecisionresearch),whowasdescribedinawaytomakeherlooklikeafeminist,butnotlikeabankteller.ParticipantswereaskedtorankorderanumberofstatementsaboutLinda,accordingtohowlikelythesewere.ThecriticalstatementswerethatLindaisafeminist,Lindaisabankteller,andLindais

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afeministandbankteller.Theresultsindicatedthat𝑃𝑟𝑜𝑏 𝑓𝑒𝑚𝑖𝑛𝑖𝑠𝑡 > 𝑃𝑟𝑜𝑏 𝑓𝑒𝑚𝑖𝑛𝑖𝑠𝑡 & 𝑏𝑎𝑛𝑘 𝑡𝑒𝑙𝑙𝑒𝑟 > 𝑃𝑟𝑜𝑏 𝑏𝑎𝑛𝑘 𝑡𝑒𝑙𝑙𝑒𝑟 . Theconjunctionfallacyisthefindingthattheconjunctionisjudgedmoreprobablethanoneofthemarginals.Note,theconjunctionfallacyhasbeenextensivelyreplicatedandwithstoodthetestofproposalsforpossibleconfoundsoralternativeexplanations(notablyconversationalimplicatures;e.g.,Dulany&Hilton,1991;Moro,2009). Theconjunctionfallacyisproblematicforclassicalprobabilitytheorybecausetheprobabilisticmodelinclassicaltheoryisessentiallyvolumetric,whichmeansthattheuniverseofpossibilitiescanbethoughtofasageneralizedvolumeandmorespecificquestions/possibilitiesaresubsetsofthisvolume.Thus,amorecomplexpossibility(theconjunctionbetweentwopredicates)canneverbemoreprobableacorrespondingsimplerone(eitherofthetwopredicates;Figure1).Theclassicalimpossibilityoftheconjunctionfallacyimmediatelybecomesapparentifwerecasttheproblemwithcountableinstances,forexample,comparenumberofdaysonwhichitrainedandsnowedinLondonin2017comparedtothenumberofdaysonwhichitjustrained.InTverskyandKahneman’s(1983)seminaldemonstration,wehaveclassicalprobabilitiesassubjectivedegreesofbelief,butclassicalprobabilitiesbasedonfrequenciesandclassicalprobabilitiesbasedonsubjectivebeliefsareequivalent(deFinettietal.,1993).

Figure1.ConsiderasamplespaceofallpossibleLinda’sconsistentwiththestory.Theonesforwhichbothpropertiesofbanktellerandfeministaretruecanneverbemorethanthoseforeitherindividualpossibility.

Figure2.AcaricatureofBusemeyeretal.’s(2011)modelfortheconjunctionfallacy,basedonone-dimensionalsubspaces. 𝜓isthementalstateandF,BTthefeminist,banktellerproperties.

Quantumprobabilitytheoryisbasedonsubspacesofalargespacerepresentingtheuniverseofpossibilities.Inthatuniversewecanhavesubspacesofvaryingdimensionalitiescorrespondingtodifferentpossibilities/questions.Therelationbetweensubspacestypicallyneedstobespecifiedandanyprobabilisticcomputationdependsonthestateofthesystemunderconsideration.Psychologicallythesystemtypicallycorrespondstothemindsetoftheparticipantpriortoaprobabilisticinference.Classically,themindsetofparticipantsistypicallynotexplicitlyspecified,thoughsuchinformationcanbeintroducedthroughconditionalization. Figure2showsacaricatureofthequantummodelfortheconjunctionfallacy(Busemeyeretal.,2011).Severalsimplificationsaremade:first,one-dimensionalsubspacesareemployedtorepresentthepossibilitiesthatLindaisabanktellerorafeminist.ForthequestionofwhetherLindaisabankteller,

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wehaveaso-calledbasisvectorforthepossibilitythatsheisindeedabanktellerandanotheronefortheoppositepossibility.Thebasisvectorscorrespondingtoalloutcomesofaquestionarecalledabasisset.Second,theoverallspaceisatwo-dimensionalrealvectorspace,insteadofann-dimensionalcomplexspace.Third,themindsetofparticipantspriortoadecisionisrepresentedbyastatevector,insteadofadensitymatrixwhichisthemoregeneralapproach.Whenthestateofthesystemisrepresentedasastatevector,thenprobabilisticcomputationinvolvescomputingthesquaredlengthoftheprojectionofthestatevectorontotherelevantsubspace.Forexample,ifweareinterestedinhowprobablepossibilityAis,whenthesystemisrepresentedas|𝜓 1(orjust𝜓),thenwehavetocompute|𝑃!|𝜓 |!,where𝑃!islinearoperatorcomputingtheprojectionofavectortosubspaceA(itisoftencalledaprojector,projectionoperator,ormeasurementoperator),|𝜓 isanormalizedcolumnvector,and|𝑃!|𝜓 |!expressedasmatrixoperationsgives𝜓!𝑃!𝜓,where𝜓!istheconjugatetransposeof𝜓.OthertechnicaldetailsarethatthesevectoroperationstakeplaceinaHilbertspace,whichisacomplexvectorspacewithsomeadditionalproperties,andthatifthesystemwererepresentedasadensitymatrix𝜌,probabilityforAwouldbegivenas𝑇𝑟𝑎𝑐𝑒(𝑃!𝜌),wherethetraceoperationsumsdiagonalelementsofamatrix.

TheFigure2representationshowsthattheinitialmindsetofparticipantsafterreadingtheLindastoryissuchthattheyarelikelytoconsiderLindaafeministandnotabankteller,consistentlywithTverskyandKahneman’s(1983)design.TherelationbetweenthefeministandbanktellerraysisdeterminedbyimaginingafeministLinda(soplacingthestatevectoralongthefeministray)andconsideringwhetherforsuchapersonwewantsmallorlargeprojections(probabilities)tothebanktellerray.TheassumptionmadeinFigure2isthatafeministLindaisneitherparticularlylikelynorparticularunlikelytobeabankteller.

Thisbringsustothefirstuniquepropertyofquantumprobabilitytheory(hereandelsewhere)relativetoclassicalprobabilitytheory.Questions/possibilitiesinquantumtheorycanbeincompatible,sothatcertaintyaboutoneintroducesunavoidableuncertaintyabouttheotherandviceversa(inFigure2astatevectorcontainedwithinthebanktellersubspacehasnon-zeroprojectionstoboththefeministandnon-feministpossibilities).Incompatibleeventsinquantumtheorygiverisetouncertaintyrelations.Inphysicssuchuncertaintyrelationscanimply,forexample,thatitisimpossibletoconcurrentlyknowthepositionandmomentumofaparticleandhasledtoextensivedebate(foranintroductorydiscussionseeIsham,1989).Inpsychology,abasicwaytoapproachuncertaintyrelationsisthatacceptingonepossibility(e.g.,thatLindaisafeminist)createsauniqueperspectiveormindsetforother,incompatibleones(e.g.,thatLindaisabankteller).Forincompatiblepossibilities,ajointprobabilitydistributiondoesnotexist(sincewecannotbecertainforallcombinations).Inclassicalprobabilitytheory,possibilitiescanonlybecompatible,forwhichwemustalwaysspecifyacompletejointprobabilitydistribution.Compatiblepossibilitiesexistinquantumprobabilitytheorytoo,butforsuchpossibilitiesquantumandclassicalpredictionscanconverge.

Becauseuncertaintyrelationsrequirethenon-existenceofjointprobabilitydistributions,conjunctionsinvolvingincompatiblequestionshavetobeassessedthroughasequentialprojectionoperation,forexample,𝑃𝑟𝑜𝑏 𝐹 & 𝑡ℎ𝑒𝑛 𝐵𝑇 = |𝑃!"𝑃!|𝜓 |!,where𝑃!" and𝑃! areprojectorstothe

1Thisistheso-calledDirac’sBracketnotation,because 𝑎|isabra(arowvector),|𝑏 (acolumnvector)andtogethertheymakeabra-ket, 𝑎|𝑏 ,whichisthedotproductbetweenvectorsa,b.

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correspondingpossibilities;theyarenon-commuting,i.e.,𝑃!"𝑃! ≠ 𝑃!𝑃!",astheF,BTquestionsareassumedincompatible.InFigure2,thedirectprojectiontotheBTsubspaceisshorter(lowerprobability)thantheprojectiontotheFsubspacefirstandthentotheBTone,correspondingto𝑃!"𝑃!|𝜓 .Thoughthiscaricatureillustrationisquitesimple,itstillprovidesanexistencedemonstrationforhow|𝑃!"𝑃!|𝜓 |! > |𝑃!!|𝜓 |!accordingtoquantumprobabilitytheory.OnequestioniswhethertheprobabilityofFandthenBTinquantumtheoryareequivalenttoaconditionalprobabilityBTgivenFinclassicaltheory.Thisisnotthecase,sinceinquantumtheory|𝑃!"𝑃!|𝜓 |! = |𝑃!"|𝜓! |!|𝑃!|𝜓 |! ≡𝑃𝑟𝑜𝑏 𝐵𝑇 𝐹 𝑃𝑟𝑜𝑏(𝐹),sothatthenotionofconditionalprobabilityinquantumtheory(embodiedintheLuder’slaw,e.g.,Hughes,1988)isdistinctfromsequentialconjunction.Thereasonwhysequentialconjunctioncannotbereducedto(somenotion)ofclassicalconditionalprobabilityisuncertaintyrelations,thatrequirere-introductionofuncertaintywitheveryprojection.

Uncertaintyrelationsleadtointerferenceeffects.Forexample,|𝑃!"|𝜓 |! = |𝑃!"𝐼|𝜓 |! = |𝑃!"(𝑃! + 𝑃~!)|𝜓 |! = |(𝑃!"𝑃! + 𝑃!"𝑃~!)|𝜓 |!

= |𝑃!"𝑃!|𝜓 |! + |𝑃!"𝑃~!|𝜓 |! + |𝑃!𝑃!"𝑃~!|𝜓 |! + |𝑃~!𝑃!"𝑃!|𝜓 |!

= 𝑃𝑟𝑜𝑏 𝐹 & 𝑡ℎ𝑒𝑛 𝐵𝑇 + 𝑃𝑟𝑜𝑏 ~𝐹 & 𝑡ℎ𝑒𝑛 𝐵𝑇 + ΔSo,weendupwithanexpressionwhichappearslikethequantumanalogueofthelawoftotalprobability,plusΔ.Δisaninterferenceterm,whichcanbepositiveornegative,andallowsquantumprobabilitiestoviolatethelawoftotalprobability. Anotherkeycharacteristicofquantumprobabilitytheoryconcernsthekindofuncertaintyreflectedinastatesuchas|𝜓 = 𝑎|𝐵𝑇 + 𝑏|~𝐵𝑇 ,whichiscalledasuperposition(a,barecalledamplitudesandarecomplexnumberswhosesquaredmoduliareprobabilities).Suchuncertaintyisontic,sothatpriortoameasurement(decision)thereisnorealityregardingwhetherLindaisBTor~BT,andameasurementcreatesapossibilityintobeing(cf.Atmanspacher&Primas,2003).Forexample,postmeasurementthestatemightbe|𝜓 = |𝐵𝑇 .Thus,resolvingaquestionchangesthestateofthesystemandthisprocessiscalledthecollapseofthestatevector.Inquantumtheorysomestatescanincorporateepistemicuncertaintytoo,whichreflectslackofknowledge.Bycontrast,classicalprobabilitytheoryonlyinvolvesepistemicuncertainty(foramoresubtleexpressionofthedifferencebetweenclassicalandquantumuncertaintyseeGriffiths,2013,andSpekkens,2007).Inphysics,themeaningofsuperpositionsandthecollapseofthestatevectorhasledtoextensivedebate.Forexample,whatdoesitmeanforanelectrontonothaveapositionpriortomeasurement?Inpsychology,ifanything,thesituationistheconverse,sinceitisoftenreasonabletoassumethatdecisionschangethementalstate. Thestateofaquantumsystemcanchangethroughtimeevolution.Wedistinguishbetweentwosituations,whenasystemisisolatedfromandwhenitinteractswithitsenvironment.Psychologically,thiscanberelatedtowhetherataskissolvedwithoutorwithinfluencefromtheexperiencesorgeneralknowledgeofaperson.UnitarydynamicsandSchrodinger’sequationconcernisolatedsystems;open-systemsdynamicsandLindblad’sequationwhenthereisinteractionwiththeenvironment.Akeypropertyofunitaryevolutionisthatprobabilitieskeeposcillating,thatis,thereisalwayschangewithtime.Bycontrast,withopen-systemsdynamicsprobabilitieseventuallystabilizetosomepattern.For

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bothkindofdynamics,theHamiltonianoperatorcontainsinformationforhowprobabilitieschangewithtime. Finally,superpositionstatescanbecompositionalornon-compositional.ConsideraparticipantdecidingnotjustwhetherLindaisaBT,butwhetherJaneisaBTtoo.SupposethatthejudgmentforLindaisindependentfromthejudgmentforJane.Thenwecanwrite

|𝜓 = 𝑎𝑏|𝐵𝑇!"#$% |𝐵𝑇!"#$ + 𝑎𝑏′|𝐵𝑇!"#$% |~𝐵𝑇!"#$ + 𝑎′𝑏|~𝐵𝑇!"#$% |𝐵𝑇!"#$+ 𝑎′𝑏′|~𝐵𝑇!"#$% |~𝐵𝑇!"#$= (𝑎|𝐵𝑇!"#$% + 𝑎′|~𝐵𝑇!"#$% )⨂(𝑏|𝐵𝑇!"#$ + 𝑏′|~𝐵𝑇!"#$ )

Thatis,thestateforwhetherLinda,Janearebanktellersisgivenasthetensorproductforthestateforeachpersonindividually.Alternatively,supposeweknowJaneisaverygoodfriendofLinda’s–sothatwethinkthatLinda,Janewouldbeverysimilar.Then,anappropriatementalstatewouldbe|𝜓 =𝑥|𝐵𝑇!"#$% |𝐵𝑇!"#$ + 𝑦|~𝐵𝑇!"#$% |~𝐵𝑇!"#$ .SuchastateiscalledaBellstateanditillustratesthekeyquantumpropertyofentanglement.Here,adecisionforeitherLindaorJaneforcesanoutcomefortheother.Quantumentanglementprecludesthespecificationofajointprobabilitydistributionforallpossiblequestioncombinations,thatcanfactorizeintotheprobabilitydistributionsforeachcomponentquestion.Putdifferently,entanglementmeansthat,forcomponentquestionsAandB,wecannotwritethestateforthecombinedquestionas|𝐴 ⨂|𝐵 ,thatisthecombinedstatecannotbeconstructedbyindependentlycombiningpartsoftherepresentationfrom|𝐴 andthen|𝐵 .Inphysics,entanglementhasledtoendlessdebate,becausethetwosystemscanbespatiallyseparated.However,whendealingwithmentalstates,entanglementislessphilosophicallychallenging(itcanbeattributedtosomeconnectednessbetweenthesystems,e.g.,thedifferentcomponentconceptsofacompositeone). Entanglementcanallowsupercorrelations.ConsidertwosystemsAandB,andtwobinaryquestionsforeachsystem,A1,A2andB1,B2.Wecancomputeexpectationvalues(labellingthebinaryoutcomesforeachquestionas±1)forallpairscomposedofonequestionfromsystemAandonequestionforsystemB.AningeniousresultbyBellshowsthatclassicallytheseexpectationvalueshavetobeboundedasfollows 𝐸[𝐴1,𝐵1] + 𝐸[𝐴1,𝐵2] + 𝐸[𝐴2,𝐵1] − 𝐸[𝐴2,𝐵2] ≤ 2(thisistheCHSHformoftheBellinequality,Clauseretal.,1969;Bell,2004).Here,classicalmeansthatthetwosystemscanbeperfectlycorrelatedoranticorrelated,butthatthetwosystemsdonotinteractwhenmeasured.BellshowedthatifA,BaredescribedwithaquantumentangledsystemtherearepairsofquestionsforwhichtheBellboundisexceeded(andanotherboundapplies,theso-calledTsirelsonbound).Inthissense,entangledquantumsystemssupercorrelate. Overall,thekeyfeaturesofquantumtheorythatlendthemselvestopsychologicalapplicationareprimarilyinterferenceeffects,thecollapseofthestatevector,entanglement,andsupercorrelations,thoughofcoursethesefeaturesarecloselyinter-dependent.Physicistshavesoughttoreduceallofquantumtheoryintoasinglefundamentalfeature(e.g.,superposition;Harding2001),butsuchconsiderationsarebeyondthepresentscope.Table1summarizesthemainterms.

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Table1.Asummaryofthemaintermsinquantumtheory.

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3.Howcanquantumstructureemergefromaclassicalbrain?Wecouldalsoask‘howcanclassicalprobabilitiesemergefromthebrain’.Thereisnomoremysteryforhowquantumprobabilitiesemergeatthecognitivelevel,thanfore.g.classicalprobabilities.Alltheapplicationsconsideredinthisworkemployquantumtheoryasasetofcomputationalprinciplestobuildcognitivemodels.Inthesamewayquantumprocessescanbeprogrammedonaclassicalcomputer,soweassumethatquantumprocessesatthecognitivelevelcanemergefromclassicalbrainneurophysiology.Forexample,superpositionsatthecognitivelevel(i.e.,thelevelofconsideringcognition/behavior,withoutreferencetoneuronalprocesses)areconsideredepiphenomenalandtheirquantuminterpretationisvalidonlyatthecognitivelevel(Yearsley&Pothos,2014). Nonetheless,someresearchershaveattemptedtospecifytheneurophysiologicalorotherwiseoriginsofquantumstructureincognition.OneidearelatestoSuppesetal.’s(2012)neuraloscillatorproposalforhowoscillationpatternsinvolvingsynchronizedneuronscancorrespondtostimulus,responseassociations.Note,therehasbeenevidencethatsynchronizationoffiringratesamongstneuronscanberelatedtocognitiveprocessing(Eckhornetal.,1988;Friedrichetal.,2004).Neuraloscillatorshavewave-likepropertiesandsocanproduceinterferencepatterns,arisingfromphasedifferencesbetweenoscillators.DeBarros(2012;deBarros&Suppes,2009)arguedthatsuchinterferencepatternscanproducethedynamicsrequiredforquantummodels(theydidthisspecificallyforquantummodelsofquestionordereffects,e.g.,Wang&Busemeyer,2013;seealsoKhrennikov,2011).Onequestionforsuchproposalsiswhetherthereisevidenceforquantumstructurebeyondtheobservationthatwave-likebehaviorcanproduceinterference. Busemeyeretal.(2015)showedthatquantumcomputationscanbeimplementedwithastandardneuralnetwork.Theneuralnetworkimplementedthreesteps,acomputationofamplitudesfromunitaryevolution,theconversionofamplitudestoprobabilities,andstatereduction.Notwithstandingtheinputandoutputmappingcapabilitiesofneuralnetworks(Churchland,1990),thisworkprovidesanillustrationofhowafamiliar,algorithmicframeworkcanproducequantum-likeoutput. BeimGrabenandAtmanspacher(2006;Atmanspacher&Scheingraber,1987)consideredhowincompatibilitybetweenclassicalquestionscansometimesemerge,ifthedescriptionofthequestionsiscoarse,sothatquestionshavesomeirreduciblefuzzinessregardingpossibleoutcomes.Oneissueiswhetherthekindofcoarsenessrequiredtoproduceincompatibilityarisesinthewayquestionsarementallyrepresented.

Finally,AertsandSassolideBianchi(2015)proposedthatthequantumprobabilityruleincognitionarisesfromaveraging.Consideratypicaldecisionexperiment,inwhichparticipantsareaskedtoassignprobabilitiestoevents.Supposethatdifferentparticipantsemploydifferentprobabilityrules–note,itisunlikelythateachparticipantwillhavehis/heridiosyncraticsystemforassigningprobabilitiestoevents,neverthelessmoreplausiblescenarioscanbeenvisagedasspecialcasesofthisgeneralone.Invariably,theresearcheranalyzingthedataaveragesresultsacrossparticipants.Arethereexpectationsforwhatwouldbetheformofsuchanaverage?AccordingtoAertsandSassolideBianchi(2014)such

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anaveragewillbeequivalenttothequantumruleforprobabilisticassignmentandthisiswhyquantumtheoryappearssuccessfulincognitiveapplications.4.Arequantumcognitivemodelsmorecomplexthanclassicalones?Inquantumprobabilitytheorytherearetwotypesofquestions(incompatible,compatible)andonlyoneinclassicaltheory(compatible).Thisfactmaytempttheinferencethatquantumtheoryisallofclassicaltheory(forcompatiblequestions)andalittlebitmore(forincompatibleones),sothatquantumcognitivemodelsarenecessarilymorecomplexthanclassicalones.Thisisincorrect.Quantumcognitivemodelstypicallyemployincompatiblequestionssothatthecomplexitycomparisoninvolvesaquantummodelwithincompatiblequestionsandabroadlymatchedclassicalone.Theissueofcomplexityconcernsthenhowwellconstructedthequantum/classicalmodelsare.Onthefewoccasionswhenthishasbeenstudiedindetail,theresultsfavoredthequantummodel.Busemeyeretal.(2015)werethefirsttoemployaBayesianmethodtoquantitativelycomparematched,classicalmodelsfordataondynamicconsistency(whichconcernswhetherdecisionmakersfollowthroughwithplansmadeinadvance)andthecomparisonfavoredthequantummodel.Truebloodetal.(2017)comparedahierarchyofmodelsinacausalinferencesituation,includingfullyquantumandfullyclassicalones,withdevianceinformationcriterion(DIC),andanalogouslyforTruebloodandHemmer(2017)forresultsconcerningepisodicmemory–inbothcasesquantummodelswerefavored.Arelatedissueconcernsparameterrecoveryandquantummodelshaveyettobeevaluatedinthisrespect(forapromisingdemonstrationseeMistryetal.,inpress).Onepossibleissueisthatforsomemodelslineartransformationsonparameterscangivethesameoutput,becauseinvariablyprobabilitiesaresinusoidalfunctions(sothatadding2𝜋doesnotalterresults). Beyondcomparisonsbetweenspecificquantum,classicalmodels,AtmanspacherandRomer(2012)attemptedtoprovideageneralcomplexityanalysis(focusedonparameternumbers)concerningquantumandclassicalmodelsforquestionordereffects,wherethenumberofquestionoutcomesandquestionscanvary.Theyobservedthatasthisnumberincreased,agenericquantumapproachwouldbeincreasinglylesscomplexthanamatchedclassicalone,becauseintheformercaseincompatibilitycankeeptheoverallrepresentationaldimensionalitylow,butnotinthelattercase. Overall,thereisnoevidencethatquantumcognitivemodelsaresystematicallymorecomplexthanclassicalonesandsomearguments(likeAtmanspacher&Romer’s,2012)tothecontrary.5.EmpiricalresearchThemainobjectiveofthisworkistocriticallyconsidertherangeofcognitiveapplicationsofquantumtheory.Wehaveidentifiednotablequantummodelsacrossperception,memory,similarity,conceptualprocesses,causalinference,constructiveinfluencesinjudgment,decisionordereffects,conjunction/disjunctionfallaciesindecisionmaking,andotherjudgmentphenomena.Thefinercategorizationrelatingtodecisionmakingsimplyreflectsthefocusofquantummodelsandevensothereislargevarianceinsectionsize.Note,thesedivisionsareforconvenienceofexposition.Foreachempiricaldomain,wewillconsidertherelevantpsychology,describethequantummodelwithafocusontheaspectsofquantumtheorycarryingtheexplanatoryburden,andpresentacriticalevaluationandanycontroversy.Oneissuewillconcernwhetheraquantummodelisaimedatprovidinganostensiblybetter

12 Quantumcognition

explanationforpreviouslyknownresultsorwhethertheyhavebeenemployedinagenerativeway.Thispointdeservesafewremarks.Themainwayinwhichquantummodelswereintroducedinpsychologyconcernsprobabilisticresults,suchastheconjunctionfallacy,whichareincorrectwithbaselineclassicalprobabilityinference,butcouldbeshowncorrectwithquantuminference(Busemeyeretal.,2011;Pothos&Busemeyer,2009).Itisworthappreciatingthesignificanceofthispoint:aresult‘impossible’fromaclassicalperspective,suchastheconjunctionfallacy,wouldbeshown‘correct’whenemployingquantumprobabilities.Animportantcontributionofthequantumcognitionprogrammeisexactlythatitrevealedanalternativeintuitionforprobabilisticinferenceandcorrectness(butnote,correctnessdoesnotimplybeingnormative,whichisaseparateissue,Pothosetal.,2017).Notwithstandingthispoint,resultssuchastheconjunctionfallacyarenownearly35yearsold,andthereisanonusonthequantumcognitionprogrammetorevealgenerativevaluetoo. Thefinalpreliminarypointisthatcertainempiricalfindingsforwhichquantummodelsareimplicatedhaveattractedintensedebateanditisdifficulttomakefulljusticetoeachofthesedebates.Inevitablyourcoveragewillbeselective,focusedonthepresentaim,whichisevaluatingthecontributionofquantummodels.5.1Perception5.1.1RelevantpsychologyWewillconsidertwostudiesinthissection,bothrelatingtobistableperception.Bistableperceptionoccurswithambiguousfigures,suchastheNeckercube,whichcanbeperceivedinoneofeithertwoways,suchthateachwayrevealsadifferentinterpretationofthefigure.Typicallythereisasensationofeffortinswitchingbetweeninterpretations. Conteetal.(2009)employedaparadigminvolvingthesequentialpresentationoftwoambiguousfigures(eachfigurecouldbeperceivedintwodifferentways)orthepresentationofjustoneofthefigures.ItispossiblethatseeingonefigurefirstmayresultinsomebiasinperceivingthesecondfigureandindeedConteetal.(2009)reportedaviolationofthelawoftotalprobability,sothat𝑝 𝐴! ∧ 𝐵! + 𝑝 𝐴! ∧ 𝐵! ≠ 𝑝(𝐴!)(AandBrefertothetwofiguresandthe+and–signstothetwopossiblewaysofperceivingthem). AtmanspacherandFilk(2010)consideredtheconsistencyofchangesinbistableinterpretationacrossdifferenttimepoints.Letusfirstdefinethreequantities,𝑁! 𝑡!, 𝑡! ,𝑁! 𝑡!, 𝑡! ,and𝑁! 𝑡!, 𝑡! ,whichcorrespondtothenumberofcasesinwhichthestimulusinterpretationisdifferentacrossthereferencedtimepoints.Settheoryrequiresthat𝑁! 𝑡!, 𝑡! ≤ 𝑁! 𝑡!, 𝑡! + 𝑁!(𝑡!, 𝑡!),whichisaformofthetemporalversionoftheBellinequality.AtmanspacherandFilk(2010)outlinedthekindofempiricaltestswhichwouldberequiredtodemonstrateaviolationofsuchatemporalBellinequality,basedonmanipulationswhichostensiblyleadtointerpretationswitcheswithtime.5.1.2QuantumcognitivemodelsConteetal.(2009)arguedthattheviolationofthelawoftotalprobabilityintheirbistableperceptionresultsindicatesthatthementalstatescorrespondingtotheinterpretationofeachfigurearesuperpositions,sothatinterferenceeffectscanarise.Theypresentedasimplequantummodelwhichprovidedgoodempiricalcoverage.

13 Quantumcognition

AtmanspacherandFilk’s(2010)quantummodelforbistableperceptionspecifiesthedynamicsofprocessingabistablestimulusfocusingonhowthedecayofthementalstateinteractswiththedynamicsofobservingthestimulus.Themodelinvolvestwocomponents,onecorrespondingtochangesinthementalstatewhenthestimulusisnotobservedandanother,calledacognitiveupdateprocess,changingthestateasaresultofobservation.ThedecayprocessisassociatedwiththeHamiltonianandtheobservationonewiththedeterminationofstimulusinterpretation.AslongastheHamiltonianandtheobservationoperatordonotcommute(sinceiftheydowehavetrivialdynamics),itisinprinciplepossibletoviolatethetemporalBellinequality.AtmanspacherandFilk(2010)presentedanentangledmentalstateforbistableperceptionandsuitableHamiltonian,observationoperators,whichcanleadtoviolationsoftemporalBell(thecorrelationbetween𝑡!, 𝑡!couldexceedthoseacrossintermediatetimepoints).5.1.3CriticalevaluationandcontroversyForConteetal.(2009),thereisageneralpointtomakerelatingtoanyresultostensiblyinconsistentwithclassicalprobabilitytheory,toincludeviolationsofthelawoftotalprobability(conjunction,disjunctionfallacies;thedisjunctioneffect),questionordereffects,andanycontextualityeffects.Alltheseresultscanbereconciledwithclassicalprobabilitytheorythroughappropriateconditionalization.Forexample,regardingtheconjunctionfallacy,onecouldwrite𝑃𝑟𝑜𝑏(𝐴&𝐵 𝑓𝑚1 > 𝑃𝑟𝑜𝑏(𝐴|𝑓𝑚2),wherefm1,fm2cancorrespondtodifferentframeofminds(Dzhafarov&Kon,inpress).However,onehardlyeverencounterssucharguments,becauseanarbitrary,posthocconditionalizationcarrieslowexplanatorypower.Wewillthereforeautomaticallydiscountsuchpossibilitiesinsubsequentdiscussion. Notethatclassicallyconditionalizingone.g.frameofmindindicatesanassumptionthatmentalprocessingiscontextualandcontextualityisessentiallyhowquantumtheoryprovidesanaccountofsuchresultstoo:iftheA,Bquestionsarecontextual,processingtheAquestionfirstcreatesacontextforthesubsequentBquestionwhichisdifferentfromthatwhentheBquestionisconsideredfirst.KujalaandDzhafarov(2014,p.2)notedforcontextualvariablesthat‘‘theserandomvariablescannotbesewntogetherintoasinglesystemofjointlydistributedrandomvariablesifoneassumesthatallorsomeofthempreservetheiridentityacrossdifferentconditions”.Quantumtheoryishardlyuniqueinincorporatingcontextualinfluences.Anadvantageofthequantumapproachisthattherearespecificrulesforrelatingdifferentprobabilityspacesarisingfromcontextually(quantumtheorycanbethoughtofasaclassicalprobabilityframework,butwherethereisaneedtointegratetogethermultipleprobabilityspaces;e.g.,Hughes,1989). AtmanspacherandFilk’s(2010)workmakesaboldprediction:inordertoviolatethetemporalBellinequality,ithastobethecasethatparticipantsdonotswitchinterpretationacross𝑡!, 𝑡!andalsodonotswitchbetween𝑡!, 𝑡!,butthenswitchacross𝑡!, 𝑡!,whichviolatesanobviousintuitionoftransitivity.Theonlyplausibleexplanationforsuchapatternofresultswouldbethattherelevantmentalstateisextendedtime-wise,thatis,itdoesnothaveawell-definedtrajectorythroughtime.Atimewisenon-localmentalstatemayextend,forexample,acrossbothperiods𝑡!, 𝑡!,sothatonecannolongerassumethataneventat𝑡!causedchangesat𝑡!.So,ifthepredictionsfromthisworkweretobeempiricallyconfirmed,thiswouldhaveimplicationsforourunderstandingofcausalityacrosstime(Yearsley&Pothos,2014).

14 Quantumcognition

PrecisepredictionsfromAtmanspacherandFilk’s(2010)frameworkforwhenthetemporalBellinequalityislikelytobeviolateddependonthedetailsofhowtheHamiltonianandobservationoperatorsarespecified.Furtherworkisneededtomotivate(ideallypre-fitwithpilotdata)thevariouscomponentsofthequantummodel.AnotherissueishowtightlycanwemakeanassociationbetweenaputativeviolationoftemporalBellandquantumprocesses.WefullydiscussthisissueinSection5.4.2but,briefly,evenaclassicalsystemwithdisturbing,‘clumsy’measurementscanleadtoviolationsofBell.Therefore,beforeaviolationoftemporalBellcanbeuniquelyassociatedwithquantumstructureatthecognitivelevel,anumberofpreconditionsneedtobetested(Wilde&Mizel,2012).5.2Memory5.2.1RelevantpsychologyWefirstconsiderBruzaetal.(2009)whooutlinedanempiricalset-upsuitablefortestingviolationsoftheBellinequalityincuedrecallmemory.Consideracuedrecallexperimentinvolvingcuesthatcanhavemultiplesenses,e.g.,thecue“bat”canhaveananimalandasportssense.Studyisdividedintoparts,sothatineachpartacuewordispresentedwithwordswhichactivatedifferentsenses(e.g.,inonestudypartthecuebatmightappearwiththewordsballandglove,soastoactivatethesportscontext).Poststudy,twocuewordsarepresentedindividuallywitharequesttorecallotherwordsfromthelistsjuststudied,withaviewtoexaminethewordsensesactivatedbythecues,byinterpretingtherecalledwords.Theauthorsoutlinea2x2designofcues(twocues,eachofwhichhastwosenses),whichcanpotentiallyleadtoviolationsoftheBellbound. Anumberofresearchershaveexploredaso-calledmemoryoverdistributioneffectandvariantsinmemoryrecognition.Inthetypicalparadigm,participantsencodeasetofmemorytargets,forexample,awordlist.Intestparticipantsarepresentedwiththetargets,relateddistractorsthataresemanticallyrelatedtothetargets,andunrelateddistractors.Recognitioninstructionscanbevariedfactoriallywiththetestitems,forexample,acceptrelateddistractorsbutrejecttargetsandunrelateddistractors.IfTsymbolizestargetsandRrelateddistractorsandprobesareincludedforthedisjunctionandthemarginals,thenakeyempiricalfindingisthatrecognitionprobabilitiesindicate𝑃𝑟𝑜𝑏 𝑇 +𝑃𝑟𝑜𝑏 𝑅 > 𝑃𝑟𝑜𝑏(𝑇 ∪ 𝑅),whichisadisjunctionfallacy(alsocalledepisodicoverdistribution).Classically,sincethecategoriesofTandRaremutuallyexclusive,thisisimpossible,andweinsteadrequire𝑃𝑟𝑜𝑏 𝑇 + 𝑃𝑟𝑜𝑏 𝑅 = 𝑃𝑟𝑜𝑏(𝑇 ∪ 𝑅).Thisempiricalfindingillustratesthatsomeitemsarebeingrememberedasbothpresentedandnotpresented(Brainerd&Reyna,2008;Brainerdetal.,2010).Togetherwiththedisjunctionfallacy,wealsohaveasubadditivityeffect,whichiswhentheprobabilityofacceptinganiteminmutuallyexclusiveandexhaustivecategoriesisgreaterthanone(so,subadditivityalsoindicates𝑃𝑟𝑜𝑏 𝑇 + 𝑃𝑟𝑜𝑏 𝑅 > 𝑃𝑟𝑜𝑏(𝑇 ∪ 𝑅),butinthiscasethereisnoovert𝑇 ∪ 𝑅cue).5.2.2QuantumcognitivemodelsBruzaetal.(2009)aimedtoillustratethataquantumrepresentationbasedonanentangledstate,matchedtotheassumptionsoftheirparadigm,allowsforviolationsoftheBellbound.Allwordswererepresentedasrays.

15 Quantumcognition

Regardingthememoryoverdistributioneffect,Brainerdetal.(2013)developedaquantumrepresentationmodel,forsubadditivityinaparticularthreelistvariantoftherecognitionparadigm(Brainerdetal.,2012).EachitemintheempiricaltestwasrepresentedinafivedimensionalHilbertspace,suchthatthreedimensionscorrespondedtoverbatimfeatures(mostlysuperficialcharacteristics)foreachofthethreelists,onedimensiontogistfeatures(mostlysemantic/abstractcharacteristics,butalsorelational,contextualinformation),andonedimensiontodistractorfeatures.ThemodelintheBrainerdetal.(2013)workwastermedtheQEMmodel(quantumepisodicmemorymodel).Brainerdetal.(2015)providedanelaborationoftheBrainerdetal.(2013)modelandpresenteditasaformalizationoffuzzytracetheory(Reyna,2008;Reyna&Brainerd,1995),whichwecallQEM+,abusingnotationforsimplicity.IntheQEM+,eachitemisrepresentedinathreedimensionalHilbertspace,withbasisvectorscorrespondingtoverbatim,gist,andotherinformation(informationnotmatchingthecue’seithersurfaceorsemanticcontent).AccordingtoQEM+,subadditivityisdirectlypredictedbecausetheretrievalprobabilitiesarecomputedinawaythatthecontributionfromthegistinformationappearstwiceinverbatimquestions.NotethattheQEM+couldapplyeitherforincompatibleorcompatiblememorymeasures(e.g.,regardingthevariousprobes),creatinganeedfordeterminingcompatibility(Section7).

DenolfandLambert-Mogiliansky(2016)notedthatintheQEM/QEM+modelsverbatiminformation,gistinformationetc.arerepresentedwithorthogonalvectors,whichmakesthemmutuallyexclusive(i.e.,completelyacceptinggistinformationforatestcuemeanscompletelyrejectingverbatiminformation).Instead,DenolfandLambert-Mogiliansky(2016)proposedaquantummemorymodelcalledComplementaryMemoryTypes(CMT)inwhichtheverbatiminformationandgistinformationaremodeledasincompatible,sothattheynotcommeasurableandperfectknowledgeofoneintroducessomeuncertaintyfortheother.DenolfandLambert-Mogilianskymodel(2016)madeasymmetryassumptionfortherepresentationofthegistinformation.Inafourdimensionalspace,threebasisvectorscorrespondedtoverbatiminformationforeachoneofthethreememorizedlists(intheexperimentcoveredbythemodel)andonetounrelatedinformation.Analternativebasissetcorrespondedtogistinformation.Gistinformationwasmodeledwithasinglevector,withequalamplitudesalongthebasisvectorsforeachmemorizedlistandzeroforunrelatedinformation.

AnapproachbasedonincompatibilitywasfurtherdevelopedbyTruebloodandHemmer(2017;seealsoBroekaert&Busemeyer,2017).IntheirGeneralizedQuantumEpisodicMemoryModel(GQEM),verbatim,gist,andnewinformationareassumedincompatible.Eachofthethreetypesofinformationcorrespondstoatwo-dimensionalbasis,obviatingtheproblemofdifferingdimensionalitiesinDenolfandLambert-Mogiliansky(2016).Acceptanceprobabilitiesunderverbatimandgistinstructionsarecomputedbyfirstevaluatingacceptancebasedongistinformationand,ifthisproducesanegativeoutcome,evaluatingacceptanceonverbatiminformation(gistinformationhasbeenarguedtobeprocessedmorerapidly;Brainerdetal.,1999).Ananalogousassumptionwasemployedwithparadigmsinwhichtrainingitemsarepresentedindifferentcontexts.5.2.3CriticalevaluationandcontroversyBruzaetal.’s(2009)workawaitsempiricalexamination,whichifprovidedwouldindicatethatitisnotpossibletohaveafour-wayclassicalprobabilitydistributionforthedifferentsensesoftherecallwords.

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Thedisjunctionfallacy/subadditivityinmemoryrecognitionhavechallengedstandardrecognitionmodels,suchastheone-processsignaldetectionmodel(Glanzer&Adams,1990),andstandardsourcerecognitionmodels,suchasthesourcemonitoringmodel(Batchelder&Riefer1990),whichunelaboratedcannotpredicttheseeffects.Bycontrast,Brainerdetal.(2013)reportedacceptablefitsfortheQEMfortheBrainerdetal.(2012)data,whencomparedtotheirownpredominantmodelfortheepisodicsubadditivityeffect(theOverdistributionmodel).Brainerdetal.(2015)reportedresultssupportingpredictionsfromtheirQEM+,notablythatitisspecificallysubadditivityandnotsuperadditivitythatispredictedandalsothatempiricalmanipulationswhichenhancerelianceongistinformationand/ordecreaseinfluencefromverbatiminformationshouldbothincreasesubadditivity.BothQEMandQEM+areelegantintheirparsimony,butmakelimiteduseofquantumfeaturesquestioningthenecessityforquantumtheory.Forexample,Brainerdetal.(2015)notedthatakeyassumptioninfuzzytracetheoryisthatgistinformationallowsthesameitemtobeperceivedasbelongingtodifferentcategories,whichiswhyintheQEM+gistinformationappearstwiceinretrievalprobabilities,producingsubadditivity.

DenolfandLambert-Mogiliansky’s(2016)andTruebloodandHemmer’s(2017)proposalsthatfuzzytracetheorycanbeelaboratedwithanassumptionofincompatibilitybetweenverbatim,gistinformationisarguablyamoresignificantcontributionofquantummodelsinthisarea.However,DenolfandLambert-Mogiliansky’s(2016)symmetryassumptionishardtojustify.

DenolfandLambert-Mogiliansky(2016)reportedcoverageofbothsubadditivityandfavorablecomparisonswithQEM+.TruebloodandHemmer(2017)notedsomeproblematicpredictionsfromtheQEM+.Forexample,retrievalprobabilitiesarepredictedtobeequalunderverbatimandverbatimplusgistinstructions,butempiricalresultsshowthelattertobehigher.Moreover,theQEM+predictsahigherprobabilityofacceptingatestcuewithverbatiminstructions,comparedtogistinstructions,againcontrarytoempiricaldata.RegardingtheOverdistributionmodel,theynotedthatthemodelcancovertheepisodicoverdistributioneffect,butnotthesubadditivityeffect(themodelisunderspecifiedregardingacceptanceprobabilitiesunderunrelatednewinstructions).TheyconductedahierarchicalBayesianmodelcomparisonbetweentheirGQEMmodelandtheOverdistributionmodel,basedonanewexperimentusingtheitemmemoryvariantofthememorytask(testcuescanbeoldornew)andresultsfromKellenetal.(2014)usingthesourcememoryvariant(trainingcuespresentedindifferentcontexts).TheformercomparisonfavoredtheGQEM.ThelattercomparisonfavoredtheOverdistributionmodel,butGQEMwasshowntoproduceequivalentfitstothedata.5.3Similarity5.3.1RelevantpsychologyGeometricmodelsofsimilaritywherebyobjectsarerepresentedaspointsandsimilarityissomefunctionofthedistancebetweenthemhavebeenhugelyinfluential(e.g.,Nosofsky,1984;Shepard,1987).Tversky(1977)reportedsomekeychallengestosuchmodels.Hearguedthatsimilarityjudgmentscanviolateminimality,symmetry,andthetriangleinequalityaswellasbeingsubjecttocontextualinfluencesfromtherangeofstimuliconcurrentlyconsidered(adiagnosticityeffect).Heproposedthatviolationsofsymmetryarisefromdifferencesinprominence/degreeofknowledge,sothate.g.𝑆𝑖𝑚 𝐶ℎ𝑖𝑛𝑎,𝐾𝑜𝑟𝑒𝑎 < 𝑆𝑖𝑚 𝐾𝑜𝑟𝑒𝑎,𝐶ℎ𝑖𝑛𝑎 ,whereparticipantsareassumedtohavemore

17 Quantumcognition

knowledgeofChinathanKorea.ForviolationsofthetriangleinequalityheusedWilliamJames’sexample,wherebySimilarity(Russia,Jamaica)islow,butSimilarity(Russia,Cuba)ishigh(becauseofpoliticalaffiliation)andSimilarity(Cuba,Jamaica)isalsohigh(becauseofgeographicalproximity),sothatSimilarity(Russia,Jamaica)<Similarity(Russia,Cuba)+Similarity(Cuba,Jamaica).Finally,anexampleofthediagnosticityeffectisthatoutofHungary,Sweden,Poland,thecountrychosenasmostsimilartoAustriaisSweden,butoutofHungary,Sweden,Norway,itwasHungary.

Violationsofsymmetryhavebeendemonstratedonmultipleoccasions(e.g.,Aguilar&Medin,1999).However,despitethestrongintuitionforthetriangleinequality,Tversky’s(1977)exampleconcernsviolationsofthetriangleinequalityondissimilarities,notsimilarities.Thatis,thetriangleinequalityconstraint(fromanassumedmetricrepresentation)isthatDissimilarity(Russia,Jamaica)<Dissimilarity(Russia,Cuba)+Dissimilarity(Cuba,Jamaica),wheredissimilaritycanbestraightforwardlyassociatedwithdistance.Itisnotstraightforwardtotranslateaninequalityondissimilaritiestooneonsimilarities.Tverskyseemedtobeawareofthisissue(cf.Tversky&Gati,1982),butithasbeenignoredinmuchofsubsequentliterature.Consideringoptionsforhowtoconvertthetriangleinequalityondissimilaritiestosimilarities,Yearsleyetal.(2017)derivedaso-calledmultiplicativetriangleinequality(basedonShepard’s,1987,functionforrelatingdistancestosimilarities).Additionally,replicationsofthediagnosticityeffecthavebeenrare(Evers&Lakens,2014).

Finally,eventhoughmuchofthesimilarityliteraturehasdevelopedintermsofpoint-wisecomparisons,researchershaverecognizedthatmatchingpartsbetweenobjectrepresentationscanhaveagreaterinfluenceonsimilarityjudgments,thanmismatchingparts(e.g.,Gentner,1983;Goldstone,1994).5.3.2QuantumcognitivemodelsApriormotivationforapplyingquantumtheoryinsimilarityisthatitinvolvesgeometricrepresentationsandgeometricrepresentationshaveconsistentlyfeaturedprominentlyinsimilaritymodels(Nosofsky,1984;Shepard,1987).But,whereasintraditionalgeometricmodelsrepresentationispoint-wise(eachobjectcorrespondstoasinglepoint),inquantummodelsrepresentationscanbesubspaces.So,quantummodelshaveanaturalwaytocapturedifferencesinknowledgebetweenconcepts.Additionally,quantumprobabilitiesembodyordereffectsandarecontextual,whicharebaselinerequirementsrespectivelyforviolationsofsymmetryandthediagnosticityeffect.

Pothosetal.(2013)modeledsimilarityjudgmentsasquantumconjunctiveprobabilities,involvingaprojectionordermatchingtheorderinwhichthecomparedstimuliarereferencedandaninitialstatewhichisneutral,thatissetsothatwhencomparingstimuliA,B,wehave 𝑃! ∙ |𝜓 ! =𝑃! ∙ |𝜓 !.Thenthemodelimmediatelyproducese.g.𝑆𝑖𝑚 𝐶ℎ𝑖𝑛𝑎,𝐾𝑜𝑟𝑒𝑎 < 𝑆𝑖𝑚 𝐾𝑜𝑟𝑒𝑎,𝐶ℎ𝑖𝑛𝑎 ,aslongasthedimensionalityofthesubspacerepresentingChinaisgreaterthantheoneforKorea.ThiswasprovedinPothosetal.(2013)fortwo-dimensionalvs.one-dimensionalsubspacesandargumentswereofferedastowhythisresultgeneralizes.Regardingviolationsofthetriangleinequality,differentregionsinaquantumspaceareassociatedwithdifferentcontexts(e.g.,features/concepts).So,usingTversky’s(1977)example,ifaregionofaquantumspaceisconsistentwiththepropertyofCommunism,thenrepresentationsinorclosetothatregionwouldreflectvaryingdegreesofconsistencywithCommunism.Therefore,ifbasissetsarearrangedinatwo-dimensionalspacesothatoneregionreflectsCuba,sandwichedbetweenaregionforRussiaandoneforJamaica,itisstraightforwardtoseehow

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triangleinequalityviolationscanoccur(atleastondissimilarities).Coverageofthediagnosticityeffectreliesonordereffectsinnon-commutingprojectors.Pothosetal.(2013)assumedthateachpossiblepairwisesimilaritywascomputedwiththequantumsimilarityrule,butmodifiedtoincludepriorprojectionstoanycontextstimuli.So,inasequenceofprojectionsinvolvingboththecomparedstimuliandthecontextones,differencesineventualselectionwereobservedconsistentwiththediagnosticityeffect. Regardingsimilaritycomparisonssensitivetostructure,PothosandTrueblood(2015)adaptedthequantumsimilaritymodelusingtheideaofSmolensky(1990)forstructureinlinguisticrepresentationsbasedontensorproducts.Forexample,forstimulihavingthreeparts(top,middle,bottom),suchthateachpartcandifferincolorandshape,wecanwritearepresentationvectoras|𝑡𝑜𝑝 𝑝𝑎𝑟𝑡 ⨂|𝑠ℎ𝑎𝑝𝑒 ⨂|𝑐𝑜𝑙𝑜𝑟 + |𝑚𝑖𝑑𝑑𝑙𝑒 𝑝𝑎𝑟𝑡 ⨂|𝑠ℎ𝑎𝑝𝑒 ⨂|𝑐𝑜𝑙𝑜𝑟 +|𝑏𝑜𝑡𝑡𝑜𝑚 𝑝𝑎𝑟𝑡 ⨂|𝑠ℎ𝑎𝑝𝑒 ⨂|𝑐𝑜𝑙𝑜𝑟 . The‘part’vectorskeeptrackofmatchingparts(andcanbeadjustedtocaptureinfluencefrommatchesinplaceormatchesoutofplace,Goldstone,1994).For

exampleif|𝑡𝑜𝑝 𝑝𝑎𝑟𝑡 =100

,|𝑚𝑖𝑑𝑑𝑙𝑒 𝑝𝑎𝑟𝑡 =010

etc.thenincomputingsimilaritythereare

contributionsonlyacrossmatchingparts;brieflythisisbecause𝑥⨂𝑦⨂𝑧|𝑥′⨂𝑦′⨂𝑧′ = 𝑥|𝑥′ 𝑦|𝑦′ 𝑧|𝑧′ ,thatistensorproductstructurerespectsorthogonalityineachspaceindividually.5.3.3CriticalevaluationandcontroversyPothosetal.’s(2013)assumptionofamentalstatethatisneutralinrelationtothecomparedstimuliwasjustifiedasoneofuniformedpriors.Forthediagnosticityeffect,incorporatingcontextualinfluenceaspriorprojectionstakesadvantageofquantumtheory’scontextualfeatures,inthatthesamequestionhastobeconsidereddifferentlydependingonwhetheritisconsideredinisolationorinthecontextofpriorincompatiblequestions;notethefinalprojectiondependsontheentireprojectionsequence.However,thereisanissueofscalabilitywhendealingwithmultiplecontextualitems. Onecanquestionarepresentationschemewhichassumesthatconceptsarerepresentedasincompatible.Therationaleisthatasequenceofprojectionscorrespondstoatrainofthought,sothatsuccessiveprojectionscorrespondtothementalstategoingthroughdifferentconcepts.Dependingonwhichconceptiscurrentlyconsidered,incompatibilitymeansthattherearedifferingperspectivesonthelikelihoodofthinkingaboutotherconcepts.Forexample,ifoneisthinkingaboutChinathereisuncertaintyforwhetherhe/shewillnextthinkaboutKorea,dependingontherelationbetweenthesubspaces. Themaincontributionofthequantumsimilaritymodelwastoshowhowasymmetriesintheextentofknowledgebetweenconceptscanbecapturedwithdifferencesinsubspacedimensionalityandthatthisproducessimilarityasymmetriesintheobserveddirection.Eventhoughthemodelhasnothadgenerativevalue,thecompletenessofcoverageofTversky’s(1977)seminalresultscomparesfavorablytothatfromotherpredominantapproaches.Forexample,themainissuewithTversky’s(1977)contrastmodelisthatitinvolvestwoindependentparameters.Forviolationsofsymmetry,thesehadtobesetinacircumventedway(basedonassumptionsabouttherelativeweightingofthefeaturesofthesubjectandthereferentinasimilaritycomparison).Krumhansl’s(1978,1988)distance-density

19 Quantumcognition

modelencountersaproblemregardingviolationsofsymmetry,sinceadequatecoveragerequirestheassumptionthatChinaissimilartomoreothercountriesthanKorea.Additionally,regardingthetriangleinequality,themodelincorporatesamechanismofcomputingsimilaritywithinreducedsubspaces,correspondingtodifferentcontexts,butthisisunderspecified(e.g.,howtodeterminesuchsubspaces).Forsymmetry,AshbyandPerrin’s(1988)GeneralRecognitionTheoryrequiredanassumptionofhowmanyothercountiesaresimilartoKoreavs.ChinaoppositetothatofKrumhansl(1978)2.Additionally,forviolationsofthetriangleinequality,inequivalentperceptualdistributionswererequiredforRussia,Jamaica,andCuba,andtherearenoapriorireasonstoanticipatesuchanassumption.

RegardingPothosandTrueblood’s(2015)proposalforstructuralsimilarity,currentlythisisdescriptiveanditstheoreticalmeritrestsinrevealingacommonframeworkforstructurebetweenlanguage(asinSmolensky,1990)andsimilarity.5.4Conceptualreasoning5.4.1RelevantpsychologyItissometimesthecasethataparticularinstancecanbeagoodexampleofacompositeconcept,butapoorexampleofeitherindividualconcept.Forexample,agoldfishisagoodexampleofthecombinedconceptpet-fish,butapoorexampleofeitherpetorfish(e.g.,Hampton1988a,b;Osherson&Smith,1981;Stormsetal.,1999).Specifically,anoverextensioneffectiswhenthestrengthofcategorymembershipforacombinedconceptisgreaterthanforotherindividualconcepts;andanunderextensioneffectoccurswhentheconverseistrue.Boththeseeffectsindicateviolationsofthelawoftotalprobability.Analogouseffectshavebeenobservedfordisjunctiveconceptualcombinations. Bruzaetal.(2015)addressedthequestionofwhetherthereiscompositionalstructureinconceptualcombination.Theyemployednovel,ambiguousconceptualcombinationscomposedoftwowords,suchthateachwordhadtwo(fairly)distinctmeanings,e.g.,in‘boxerbat’,boxercanrefertoapersonoradogandbattoasportingequipmentorananimal.Eachparticipantreceivedoneoffourprimes(asingleword),correspondingtothe2x2meaningpossibilities,andsubsequentlywasaskedtointerprettheambiguousconceptandratethesenseofeachcomponentwordthatwasemployed.Theauthorsthencompiledconditionalprobabilitiesfortheinterpretationofeachconceptcombination,givenparticularprimes.Theseprobabilitiescanbeemployedtotestforquantumentanglement,withavariantoftheCHSHinequality.ViolationsofCHSHwereobserved,indicatingnon-compositionality,thatis,thesemanticsfortheconceptualcombinationcannotbereconstructedbylookingatthesemanticsofeachcomponentconceptindependently.

Non-compositionalityinconceptassociations(ratherthanconceptualcombinationassuch)wasfurtherexploredbyCervantesetal.(inpress).ParticipantswerepresentedwithoneAquestionandoneBquestion,suchthatA1:GerdaorTroll,A2:SnowQueenorOldFinnwoman,B1:BeautifulorUnattractive,B2:KindorEvil.Participantresponseswereusedtocomputeexpectationse.g.𝐸[𝐴1,𝐵1],2AshbyandPerrin(1988,p.133)notedthat“…formanypeopleNorthKoreaisverysimilartoseveralothercountries.”But,recall,Krumhansl(1978,p.454)madetheexactoppositeassumption,“Ifprominentcountries…arethosestimulihavingrelativelymanyfeatures,thentheseobjectshavefeaturesincommonwithalargernumberofdifferentobjects….”.Inotherwords,Krumhansl(1978)assumedthatitisChina,notKorea,whichissimilartoagreaternumberofothercountries.

20 Quantumcognition

bycodingquestionanswerswith±1.Cervantesetal.(inpress)reportedviolationsoftheCHSH,hencedemonstratingnon-compositionalityinconceptassociationsofthiskind.Note,theconclusionofsupercorrelationfollowsbyconsideringtherelationbetweendifferentpairsofA,Bquestions.

Achallengingissueinconceptualreasoningconcernsborderlinevagueness,theideathatformanyconceptstherearenoclearlydefinedboundariesandthereareborderlinecasesforwhichitisunclearwhetherthepredicateapplies.TheSoritesparadoxexemplifiessuchproblems:StartbystatingthatifXisaheapofsand,thenremovingonegrainwillsurelyresultinaheap.However,therepeatedapplicationofthisruleresultsintheparadoxthatthelastgrainleftmuststillcountasaheap.Thepresentfocusisontheacceptanceofborderlinecontradictionsalongthelinessuchas‘Xistallandnottall’,whereXreferstoaborderlinecase(AlxatibandPelletier,2011).5.4.2QuantumcognitivemodelsWefocusonAerts’s(2009;seealsoAerts&Gabora,2005a,2005b;Aerts,Sozzo,&Veloz,2015)modelforconceptualapplication,whichisararecognitiveapplicationofquantumfieldtheoryandFockspaces.AFockspaceisasuperpositionofdifferentHilbertspaces.ForaconceptualcombinationbetweenconceptsA,B,Aerts(2009)proposedapartbasedonthetensorproductbetweenthe(quantum)representationsforA,Bandapartbasedonsuperposition,sothattheresultingstatewould

begivenby𝜓 𝐴,𝐵 = 𝑚𝑒!"|𝐴 ⨂|𝐵 + !!!"

!(|𝐴 + |𝐵 ),with𝑚! + 𝑛! = 1.TheFockspacestructureis

essentially𝑋⨂𝑌⊕ (𝑋 + 𝑌),whereX,Yarestates.Note,thiscombinationruleisintendedforconjunctionsandtheonefordisjunctionsfollowsfromthisone.Themodelpurportstoencompassapartcorrespondingtoclassicalconceptualcombination(thetensorproductspace)andanon-classicalpart(thesuperposition).Interferenceeffectsarisefromthesuperpositionpart,whichallowthemodellingofoverextensionsandunderextensions.Thismodelpurportstoformalizetheideaoftwoseparateroutestoconceptcombination,aclassicaloneandaquantumone.

Regardingnon-compositionalityinconceptualcombinations,Bruzaetal.’s(2015)applicationofquantumtheorymorebroadlyrelatestotheformalizationofcorrespondingtestsfromthequantumliterature(Bell,2004).Bruzaetal.(2015)providedexamplesofnon-compositionalquantumrepresentationsforconceptualcombinations. Cervantesetal.(inpress)generalizedtheCHSHinequalityinawaythatpossibleinfluencesfromsignalingcanbediscounted.TheideaisthatexceedingtheCHSHboundcanarisefromsignaling,whichistheextenttowhichthemeasurementofe.g.A1isdisturbedbythemeasurementofB1vs.B2,quantifiedasthedifferenceinthemarginaldistributionofA1ineachofthetwomeasurementssituations(B1vs.B2).Whenthesemarginaldistributionsarethesame,thenwecanassumethatthemeasurementcontextisnotdisturbing,whichCervantesetal.(inpress)calltheno-signalingormarginalselectivitysituation.Undersuchcircumstances,accordingtoCervantesetal.(inpress)wehaveapurertestofcontextuality,tomeanalteringthemeaningoftherelevantconcepts(e.g.,kindvs.evil),dependingoncontext(e.g.,whichcharactersarebeingevaluated;Dzhafarovetal.,2015,2016;Kujala&Dzhafarov,2013,2014;cf.Wilde&Mizel,2012).TheexpressionforthegeneralizedCHSHinequalityis

𝑚𝑎𝑥!,!∈ !,! 𝐸 𝐴!!𝐵!! − 2𝐸 𝐴!! 𝐵!!

!,!∈ !,!

− 𝐸 𝐴!! − 𝐸 𝐴!!

!∈ !,!

− 𝐸 𝐵!! − 𝐸 𝐵!!

!∈ !,!

≤ 2

21 Quantumcognition

whereEindicatesexpectation. Blutneretal.(2013)developedaquantummodelwhereprobabilitiescorrespondedtoactivationpatternsinaHopfieldnetworkwithasingleinputunitrepresentingthestatementofinterest(e.g.,“Johnistall”)andtwooutputunitsrepresentingtruth,falsity.Thestatementcouldbecharacterizedinoneofthreeways,definitetruth,definitefalsity,andtruthandfalsityatthesametime–thislastpatternexpressesa‘glut’andisafeatureofborderlinevagueness.ProbabilitieswereassignedtoeachofthesethreepossibilitiesusingtheBoltzmanndistribution.Themainassumptionwasthattheprojectorsfortruthandfalsitywerenotorthogonal,allowingemergenceofinterferenceterms,whichaffectedthenormalizationfactorsfortheBoltzmanndistributionprobabilities.5.4.3CriticalevaluationandcontroversyAerts(2009)reportedclosefitstooverextension,underextensioneffectsinconceptualcombination,coveringbothconjunctiveanddisjunctivecategories(mostlyusingHampton’s,1988a,1988b,data),thoughthenumberofmodelparametersdoesnotcomparefavorablytodatadegreesoffreedom.Standardconceptualcombinationapproacheshavedifficultywithsucheffects.Forexample,Hampton(2011)discussedfuzzylogicaveragingapproachestooverextensions,underextensions,andnegationstoconcludethattheyfailtoappropriatelycaptureinteractionsbetweenconceptsand,instead,anapproachbasedonprototypesismorepromising.Hisprototypesapproachinvolvesvariousheuristicassumptionsandsoamoreformalizedapproach(suchasAerts’s,2009)mayprovideatighterexplanation.

Acriticalpointisthattheformsemployedforconjunctionanddisjunctionandthepostulatedrelationbetweencategorymembershipintheconjunctiveconceptandinthedisjunctiveconcept(theprobabilityofmembershipinadisjunctiveconceptisoneminusthatforaconjunctiveconcept)donotcleanlymatchexpectationsforsuchoperations.ThisissuealsoarisesinAerts’s(2009)applicationofhisconceptualcombinationmodeltothedisjunctioneffect,whichisaviolationofthesurethingprincipleindecisionmaking(Tversky&Shafir,1992;Section5.8).Thetwoconceptswereequatedwiththevariousconditionsrequiredfortestingthesurethingsprinciple.Thiscreatesaconfusingpicture,sinceinaconceptualcombinationwehave𝑀𝑒𝑚𝑏𝑒𝑟𝑠ℎ𝑖𝑝 𝐴&𝐵 = 𝑓 𝑀𝑒𝑚𝑏𝑒𝑟𝑠ℎ𝑖𝑝 𝐴 ,𝑀𝑒𝑚𝑏𝑒𝑟𝑠ℎ𝑖𝑝 𝐵 .However,forthedisjunctioneffectwerequire𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑎𝑐𝑡𝑖𝑜𝑛 = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑎𝑐𝑡𝑖𝑜𝑛&𝑘𝑛𝑜𝑤𝑛 𝑋 + 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦(𝑎𝑐𝑡𝑖𝑜𝑛&𝑘𝑛𝑜𝑤𝑛 𝑌).Despitethesequalifications,Aerts’s(2009)modelformalizesadualrouteforconceptualcombinationandithasfurthervalueasoneofthepioneeringquantumcognitionmodels(inAerts&Gabora,2005a,2005b).

Bruzaetal.(2015)identifiedsomeconceptualcombinationsforwhichtherewasevidenceofnon-compositionality,regardlessoftheoriginsofthisnon-compositionality,i.e.,contextualityvs.disturbingmeasurements.Theseresultsimpactonapproachestoconceptualcombinationrequiringcompositionality(e.g.,Fodor’s,1994),lesssoononesforwhichacompositionalconstraintislesscentral,e.g.,proposalsbasedonemergentproperties(Hampton,1997).Non-compositionalityandweakerformsofcompositionalityarenowwellestablishedintheconceptualcombinationliterature(Swinneyetal.,2007).So,thecontributionofBruzaetal.(2015)isprovidingaformalframeworkforassessingcompositionality/contextuality.Ademonstrationofcontextuality,however,doesnotnecessitateaquantummodel.Also,onecanenvisageamodelwithanentangledstate(perhaps

22 Quantumcognition

analogoustothatinBruzaetal.,2015)andconsidertheintriguingquestionofwhatwouldbethegenerativevalueofsuchamodel.

Cervantesetal.(inpress;Dzhafarovetal.,2015)arguedthatthereisnoevidenceforpurecontextualityinanypsychologicaldemonstrationoftheCHSHinequality,apartfromCervantesetal.(inpress).Theywerethefirsttodemonstratethatcontextualityispossibleinbehavior,withoutdisturbingmeasurements/judgments,withthehelpoftheirtechnicallysophisticatedelaborationoftheCHSHinequality. Blutneratal.(2013)showedthatAlxatibandPelletier’s(2011)experimentaldatacouldbefitbytheirquantummodel,butnotacloselymatchedclassicalone(forwhichprobabilityassignmentwasidenticaltothequantumone,withtheexceptionofthenormalizationfactorforwhichinterferencetermsaroseinthequantumcase).So,thisisanotherexampleofhowinterferencetermsinquantumprobabilitiesareemployedtoaccommodateeffectivelyaviolationofthelawoftotalprobability.5.5Causalinference5.5.1EmpiricalresearchThissectionconcernsthewayknowledgeofthecausalstructurelinkingcausesandeffectsforaparticularsituation,andsomerelatedinformation,affectsinference.Truebloodatal.(2017)presentedthreeexperiments,broadlybasedonthecausalstructuresandproceduresfromRehder(2014),seekingtochallengeclassicalintuitionincausalinference.Theyreportedthreenovelempiricalfindings.First,theyobservedevidenceforreciprocity,thatis,𝑃𝑟𝑜𝑏 𝐴 𝐵 = 𝑃𝑟𝑜𝑏(𝐵|𝐴),whichissurprisingbecauseoftheasymmetrybetweencausesandeffects(reciprocityshowsinsensitivityincausaldirection).Theinversefallacy(Kahneman&Tversky,1972)isrelatedtoreciprocity,buttherehavenotbeendemonstrationsincausalinference.Second,therewasevidenceformemorylessness,whichoccurswhenconditionalizationinprobabilityestimationdependsonlyonthelatestinformationtobeconsidered.Forexample,insomecasesconditionalizationonA&thenBwouldappearequivalenttojustB.Finally,theyobservedevidencethatclassicalinconsistenciesweremorelikelytobeadoptedinunfamiliarsituationsandforparticipantswhoapproachedthetaskinamoreintuitiveway(asopposedtodeliberative,measuredusingtheCognitiveReflectionTest;Frederick,2005).Togetherwiththesenovelempiricalfindings,Truebloodetal.(2017)alsoreportedviolationsoftheMarkovcondition,anti-discountingbehavior,andordereffects,partlyreplicatingRehder(2014).5.5.2QuantumcognitivemodelsTheevidenceforinconsistencieswithclassicalprobabilityprinciplesincausalinferencemotivatestheapplicationofquantumtheory.Equally,thereisevidenceforsomeclassicalbehaviorincausalinference(Rehder,2014).Accordingly,Truebloodetal.(2017)specifiedahierarchyofrepresentations,fromfullyquantumtofullyclassical,whichincludedintermediatelevelswithsomeeffects/causesrepresentedincompatiblewaysandsomeinincompatibleways.Thefullyclassicalrepresentationrequiredaneightdimensionalspace,sinceallcausalreasoningscenariosinvolvedthreebinaryvariables,whilethefullyquantumrepresentationrequiredonlyatwodimensionalspace(cf.Trueblood&Busemeyer,2012).Theadvantageofthehierarchicalapproachwasthatquantumandclassicalputativeinfluencescouldbeintegratedinthesameframeworkinacoherentmanner.Regardingthequantumpartofthemodel,

23 Quantumcognition

differentcauses,effectswererepresentedasraysandrotationmatriceswerefittedtothedatatoachieveempiricalfits.Probabilitieswerecomputedbyprojectorsinnearlyallcases,exceptforthefullyquantumone,forwhichaversionwithpositive-valuedoperatormeasures(POVMs)outperformedaversionwithprojectors.APOVMprojectstoasubspace,butwhereasusingaprojectorisnoiseless,POVMprojectionisassociatedwithasmallerrorthatprojectionwillgotoanothersubspace.Thatis,POVMsincorporatetheideathattheremaybeotherpossibilitieseithercausing,orbecausedby,aparticularevent.InTruebloodetal.’s(2017)modelling,POVMsallowedthecircumventionofreciprocity(whichwasnotobservedconsistently).5.5.3CriticalevaluationandcontroversyTruebloodetal.(2017)employedmodelcomparisontechniquespenalizingfornumberofparametersandsodeterminedthemostappropriatelevelofquantumnessfordescribingthedata,foreachexperiment.Inallcases,adegreeofquantumnesswasrequiredforbestfit.PriortoTruebloodeta.’s(2017)work,ithadalreadybeenrecognizedthatcausalinferencecanviolateclassicalprinciples,whichunderminedthethenpredominantdescriptiveandnormativeframeworkforcausalreasoningbasedBayesiannetworks(Pearl,1988).Rehder’s(2014)resultsprovidedstrongevidencethat,atleastinsomecases,thereisadiscrepancybetweenpredictionsfromBayesnetsandhumanbehavior(seealsoFernbach&Sloman,2009;Park&Sloman,2013;Rottman&Hastie,2016).Rehder’s(2014)approachfortherangeofbehaviorsincausalinference,includingbothclassicalandnon-classicalbiases,wastospecifythreeheuristicmodels,whichtogetherwithaBayesianonecouldadditivelycombinetopredictparticipantbehavior.Truebloodetal.(2017)arguedthattheirmaincontributionwastoshowhowindividualheuristicprincipleswerenotneeded,butrathernon-classicalbehaviorincausalinferencecouldbeexplainedbyemployingincompatibility/interferenceeffectsfromquantumtheory(seealsoMistryetal.inpress).Note,evenifthereisnotechnicalneedforseparateheuristics,heuristicspotentiallyprovidedescriptiveinterpretationsforsubsetsoftherelevantbehaviorunderquantumtheory(Mistryetal.,inpress).Also,Truebloodetal.(2017)showedthatmorecompatiblerepresentationsweremorelikelytobeobservedforparticipantsadoptingamoreanalyticmodeofreasoning(Frederick,2005)andaftermoreexperiencewiththetask,sosupportingtheirviewforthesourceofincompatibilityincognition.5.6Constructiveinfluencesinjudgment5.6.1EmpiricalresearchSometimesanopinionorjudgmentappearstoaltertheunderlyingmentalstate(Brehm,1956;Lichtenstein&Slovic,2006;Schwarz,2007;Sharotetal.,2010).Forexample,anearlierjudgmentcanactivatethoughtsorperspectivesthatalterperceptionofsubsequentones.Oritispossiblethatachoicebiasesare-interpretationofpreferencestoavoidcognitivedissonance(Festinger,1957).Forexample,Glöckneretal.(2010)demonstratedcoherenceshifts,changesinsubjectivecuevaliditiesrelatedtoadecision,inadirectionindicatinggreaterconsistencywiththedecision.Likewise,inahypotheticallegalcase,HolyoakandSimon(1999;Simonetal.,2001)showedthattheevaluationofargumentschangedtobecomemoreconsistentwiththeproducedverdict.Therehavebeensomestudiesattemptingtoharnesssuchintuitionsintoparadigmssuitableforapplicationsofquantumtheory.

24 Quantumcognition

Whiteetal.(2014,2015)providedthemostdirectdemonstrationofconstructiveinfluencesinjudgment,aspredictedbyaquantumframework.Theyemployedaparadigmofpresentingtwostimuliofoppositepositive,negativevalence(e.g.,correspondingtoimaginedsmartphoneads).Thepairsofstimuliwerealwayspresentedinthesameorder.Inallcasesparticipantsprovidedanaffectiveratingforthesecondstimulus.Insomecases,participantsprovidedanaffectiveratingforthefirstoneaswell.Whiteetal.(2014,2015)observedwhattheycalledanevaluationbias,accordingtowhichthesecondstimuluswasratedinamoreextremewaywhenthefirststimulushadbeenratedtoo,thanwhenithadnot.Putdifferently,theintermediatejudgmentcreatedanimpressionofgreatercontrastbetweenthefirstandsecondstimuli.Whiteetal.(2014,2015)presentedseveralmodificationstotheirprocedureandcontrols,andgeneralizedthefindingwithotherkindsofjudgments(e.g.,trustworthiness).

Kvametal.(2014)examinedconstructiveinfluencesinaprisoner’sdilemmatask(Shafir&Tversky,1992).Insuchatask,participantshavetodecidewhethertodefectorcooperatewitha(usuallyhypothetical)associate,andthenapayoffisassignedtodifferentcombinationsofactions(Section5.8.1).Thepayoffsusuallyencouragedefection,onanassumptionofassociatecooperation.Kvametal.(2014)employedasequentialprisoner’sdilemmataskwiththeaddedmanipulationthatin-betweenthetwotasksparticipantswereaskedtostatewhethertheywereintendingtocooperate.Theyfoundthatastatementofcooperationimpactedbehavioronthesecondprisoner’sdilemmatask.

TheideathatdecisionscanforcetheidentificationofthementalstatewiththedecisionoutcomewasexploredbyYearsleyandPothos(2016).Theyemployedahypotheticalmurdermystery,suchthatthesuspectwouldbeoriginallyconsideredinnocent.Allparticipantswouldthenreceive12piecesofevidenceindicatingthesuspecttobeguilty,suchthateachpieceofevidencewasindividuallyweakbutcollectivelytheywouldmakeastrongcaseofguilt.Withabetweenparticipantsmanipulation,YearsleyandPothos(2016)variedthenumberofintermediatejudgmentsofguiltforthesuspect.Theyfoundthatincreasingthenumberofintermediatejudgmentssloweddownopinionchange.Note,thisisthequantumZenopredictioninphysicalsystems,astranslatedtoopinionchange.

InKvametal.(2015)participantswereaskedtojudgethedirectionofmotioninadynamicdotdisplay(forthesubsetofdotswhichweremovingcoherently)andprovideconfidenceratingstoo.Themainempiricalfindingwasthatconfidenceratingswerelessextremewhenparticipantsmadeanintermediatechoicepriortoratingconfidence,thanwhentheydidnot,aslongastherewassometimedelaybetweenthejudgmentandconfidenceratings.5.6.2QuantumcognitivemodelsWhiteetal.’s(2014,2015)quantummodelfortheevaluationbiasassumedabasisforpositive,negativeaffectandaninitialstateclosetooneoftheserays,dependingonthevalenceofthefirststimulus(e.g.,ifthefirststimuluswaspositive,thementalstatewasclosetothepositiveaffectray).Thesubsequentoppositelyvalencedstimuluswasmodeledwithanappropriaterotation(e.g.,ifthesubsequentstimuluswasnegative,therotationwastowardsthenegativeray).Theimpactoftheintermediateratingwasaprojectionofthementalstatetothecorrespondingsubspace,whichmeantthatthesubsequentrotationwouldbringthementalstateclosertotheoppositelyvalencedray.Therefore,thecollapseassumptioninquantumtheorydrivesprediction.

Analogously,Kvametal.(2014)employedaquantummodelforbehaviorintheprisoner’sdilemmatask,forhowinternalstatesofintentiontocooperateornotrelatetoaction,andsoassociated

25 Quantumcognition

anintermediatestatementofcooperativenesstoperformanceinthesecondPDtask.Theintermediatestatementofcooperationbetweenthetwotaskscollapsesthementalstate. YearsleyandPothos(2016)employedatwo-dimensionalspace,withabasissetcorrespondingtotheguilt,innocenceofthesuspect.Theeffectofeachpieceofevidenceonthementalstatewasmodeledbyatime-dependentunitaryoperator,sothattheimpactofparticularpiecesofevidenceonopinionchangedependedontheirserialposition(Hogarth&Einhorn,1992).IntermediatejudgmentsonthementalstateweremodeledthroughPOVMs,sincewithseveraljudgments(insomeconditions)theremightbelargepotentialerrorinanysinglejudgment.YearsleyandPothos(2016)computedananalyticexpressionforthesurvivalprobability(probabilityofno-change)asafunctionofintermediatejudgments.Themainfeatureoftheirquantummodelisthecollapsepostulate. Kvametal.(2015)employedaquantumrandomwalkmodel,involvingatridiagonalHamiltonian,fromtheFeynmancrystalmodel(Feynman&Hibbs,1965).Offdiagonalentriesdiffuseamplitudetoadjacentstates,whichcorrespondtoallpossibleconfidencelevels.TheparameterizationoftheHamiltonianwasconsistentwiththatoftheintensitymatrixforamatchedclassicalMarkovrandomwalkmodel(Pike,1966).Withoutanintermediatechoice,amplitudewaspushedtowardsextremeconfidencelevels.Withanintermediatechoiceandsubsequentprocessingofthestimuli(secondstageprocessing),theflowofamplitudetowardsextremeconfidencelevelswaslesspronounced,asempiricallyobserved.Interferencearisesinthemodelbecausethesecondstageprocessingmakestheinitiallycommutingprojectorsforjudgmentandconfidenceratingsnoncommuting.5.6.3CriticalevaluationandcontroversyWhiteetal.(2014,2015)aimedtoshowhowtheprojectionassociatedwiththeintermediatejudgmentcouldaccountforachangeinsecondstimulusevaluation,aspredictedbyquantumtheory.Nospecificfitswerecarriedout,butthefindingoftheevaluationbiaswasconsideredtoconfirmthequantumprediction,basedoncertainassumptions(e.g.,therelativeplacementofrays,whichWhiteetal.,2014,justifiedwithconsistencyarguments).Whiteetal.(2014)reportedthattheevaluationbiaswasanaprioripredictionofquantumtheory.TheyexaminedHogarthandEinhorn’s(1992)anchoringandadjustmentmodel,asaneventualjudgmentfrommultiplepiecesofevidencecanbeinfluencedbyintermediatejudgments.However,Whiteetal.(2014)showedthatreasonableparameterizationsofHogarthandEinhorn’s(1992)modeldonotallowtheevaluationbias.Memoryorattentionprocessescouldalsopotentiallyaccountfortheevaluationbias.Forexample,theintermediatejudgmentpotentiallycreatesastrongermemorytraceforthefirststimulus,whichthencreatesastrongercontrastwiththesecondstimulus.Whiteetal.(2014,2015)arguedthatsuchpotentialexplanationsfortheevaluationbiasarecomplementarytotheformaldescriptionfromquantumtheory. Kvametal.’s(2014)quantummodelfittedtheirprisoner’sdilemmadatabetterthanamatchedclassicalmodel,whichlackedaconstructiveinfluencefromtheintermediatejudgment.Note,whetheraquantummodelcanpredictacontrasteffect(asinWhiteetal.,2014,2015)ornot(cf.Kvametal.,2014)dependsonthespecificmodelassumptions(subspaces,initialstateetc.);quantumtheorycanaccommodatebothpossibilities. YearsleyandPothos(2016)comparedthequantummodelwithamatchedclassicalprobabilitymodel,whichalsoincorporatederror-pronedecisions;thetwomodelsprimarilydifferedinthatthe

26 Quantumcognition

quantummodelallowsforconstructivejudgments.Becausetheempiricalfindingsshowedanimpactfromthenumberofintermediatejudgments,theproposedclassicalmodelperformedpoorly. Kvametal.(2015)comparedaclassicalrandomwalkdriftdiffusionmodelandacorrespondingquantumversion.Driftdiffusionmodelsprovidethepredominantapproachtoevidenceaccumulation(Ratcliff&Smith,2004)and,asclassicalmodels,assumeknowledgeoftherelativeevidencefortheavailablehypothesesateachtimepointexists(i.e.,anyuncertaintyisepistemic).Insuchmodels,thereisnodefaultmechanismforhowanintermediatejudgmentaffectsperformance.Kvametal.(2015)notedthatboththeclassicalandthequantummodelscanbeinformedbyintermediatechoices.However,intheformercase,thechoicedoesnotchangetheevidence/state,onlytheinformationthatmaybeavailabletothedecisionmaker.ABayesianmodelcomparisonfavoredthequantummodel.Atechnicalissueisthattheinitialstatesforthequantumandclassicalmodelswerenotmatched.IfbothmodelsinvolvedaninitialGaussianstate,thenbothclassicalandquantumevolutionwouldresultinastateconformingtoaGaussiandistribution.Then,anyfinalstatedifferencecouldbeattributedtothemeasurement.Also,onecouldquestionhowapplicabletheMarkovrandomwalkmodelwasinthiscase,sinceaccumulatortypemodelsaremoretypicalinperceptualdecisionmakingtasks,unlikeKvametal.’s(2015)approach.Forexample,Markovrandomwalkmodelshavenomechanismforarrivingatadecisionwithoutanexternalprompt,butstimuluspresentationinKvametal.’s(2015)paradigmwassometimesverylong,suggestingthatstoppingbehaviorwouldbeinternallydriven.5.7Decisionordereffects5.7.1EmpiricalresearchThereisabundantevidenceforordereffectsindecisionmaking.Forexample,inawell-knowninvestigationbasedonGalluppolls,theprobabilitytoansweryestoaquestionaboutwhetherClintonistrustworthydependedonwhetherananalogousquestionaboutGoreprecededorfollowedtheClintonone(Moore,2002).Therearesimilarresultsinmedicaldiagnosis,wherebytheassessmentfortheprobabilityofadiseasedependedontheorderofconditionalizingevidence,evenformedicaltrainees(Bergusetal.,1998),andinajurydecisiontask(McKenzie,Lee,&Chen,2002;Trueblood&Busemeyer,2011).Arelatedfindingconcernsdifferencesintheevaluationofpublicserviceannouncements,whetherfromtheperspectiveoftheselfvs.theperspectiveofanotherobserver(Wang&Busemeyer,2015).Questionordereffectscanbeidentifiedasprimacyvs.recencyeffectsorcontrastvs.assimilationeffects,referringtothevariouswaysquestionsorpiecesofevidenceinteract/combine(Hogarth&Einhorn,1992;Payneetal.,1993;Wang&Busemeyer,2013). Isitpossibletoestablisharelationbetweenthepatternforyesandnoresponses,fortwobinaryquestions,presentedinallpossibleorders?Abbreviating𝐴 = 𝑦𝑒𝑠 to𝐴!"#etc.,considerthefollowingquantity:𝑃𝑟𝑜𝑏 𝐴!"#,𝐵!"# + 𝑃𝑟𝑜𝑏 𝐴!" ,𝐵!" − 𝑃𝑟𝑜𝑏 𝐵!"#,𝐴!"# + 𝑃𝑟𝑜𝑏 𝐵!" ,𝐴!" = 𝑃𝑟𝑜𝑏 𝐴!"#,𝐵!" +𝑃𝑟𝑜𝑏 𝐴!" ,𝐵!"# − 𝑃𝑟𝑜𝑏 𝐵!"#,𝐴!" + 𝑃𝑟𝑜𝑏 𝐵!" ,𝐴!"# .Wangetal.(2014;Wang&Busemeyer,2013;Yearsley&Busemeyer,2016)proposedthatthisquantityiszeroandcalledtheresultingequalitythequantumquestion(QQ)equality.Wangetal.(2014;Wang&Busemeyer,2013)examinedtheQQequalityacross70surveys,withparticipantsvaryingbetween651and3,006,andreportedconsistency

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withtheQQequalityinnearlyallcases.YearsleyandTrueblood(2018)consideredavariantbasedonconditionalprobabilities,𝑃𝑟𝑜𝑏 𝑁 𝐵,𝐴 − 𝑃𝑟𝑜𝑏 𝑁 𝐴,𝐵 = 𝑃𝑟𝑜𝑏 𝑁 𝐴,𝐵 − 𝑃𝑟𝑜𝑏 𝑁 𝐵,𝐴 = 0,andprovidedsupportingempiricalevidence. YearsleyandTrueblood(2018)providedanoveltestoftheco-occurrencebetweenordereffectsandconjunctionfallacies.TheauthorscollectedjudgmentsregardingfivemaincandidatesfortheRepublicanandDemocratic2016presidentialnominations,concerningtheprobabilityofeachcandidatewinningvariousprimariesandtheeventualnomination.Ordereffectsintheconditionalizinginformationwereobserved(𝑃𝑟𝑜𝑏 𝑁 𝐵,𝐴 ≠ 𝑃𝑟𝑜𝑏 𝑁 𝐴,𝐵 )andconjunctionfallacies(𝑃𝑟𝑜𝑏 𝐴 ∧ 𝐵 >min {𝑃𝑟𝑜𝑏 𝐴 ,𝑃𝑟𝑜𝑏 𝐵 }),butnodoubleconjunctionfallacies.Importantly,botheffectswereobservedforthemajorityofparticipants.Note,asforTruebloodetal.(2017),amoreanalyticstyleofthinkingwasmorelikelytobeassociatedwithcompatiblerepresentations. Finally,theideathatpreviousquestionscanprovideauniquecontextorperspectiveforsubsequentones(e.g.,Schwarz,2007)wasconsideredusingtheso-calledABAparadigm(Khrennikov,Basieva,Dzhafarov,&Busemeyer,2014).SupposethatordereffectsareidentifiedforquestionsA,Binaninitialexperiment.Theninanotherexperiment,questionAispresented,followedbyB,followedbyAagain.Khrennikovetal.(2014)suggestedthatnaïveobserverswouldaimtobeconsistentacrossthecopiesofA,regardlessofthepresenceofBinbetweenortherelationbetweentheA,Bquestions.BusemeyerandWang(2017)conductedanABAexperimentwith325participantsonsetsofquestionsforwhichordereffectshadbeenpreviouslyidentifiedandreportedamoderateamountofopinionchangebetweenthefirstandseconditerationoftheAquestion.5.7.2QuantumcognitivemodelsRegardingquestionordereffects,TruebloodandBusemeyer’s(2011)approachencompassedBergusetal.’s(1998)medicaldiagnosistask,McKenzieetal.’s(2002)jurydecisiontaskfrom,andnewexperiments.Theyspecifiedamentalstateforthepossiblecombinationsbetweenthepresence,absenceofe.g.adiseaseandpositiveornegativeevidence,inacomposite(directsum)space,suchthatonesubspacecorrespondedtothediseasebeingpresentandtheotherabsent.Theinitialstoryandsubsequentpiecesofevidenceweremodeledwithdifferentrotationsofthismentalstate.ThespecificationoftheHamiltonianswasbroadlyanalogoustothatofPothosandBusemeyer(2009)forthedisjunctioneffect(Shafir&Tversky,1992),thatis,therewasapartthatjustoperatedwithineachsubspaceandapartthatmixedamplitudesacrossthetwosubspacesandcouldleadtoviolationsofthelawofthetotalprobability.Failuresofcommutativityinconditionalizingconjunctions(e.g.,𝑃𝑟𝑜𝑏(𝑑𝑖𝑠𝑒𝑎𝑠𝑒|𝑒𝑣𝑖𝑑𝑒𝑛𝑐𝑒1 & 𝑒𝑣𝑖𝑑𝑒𝑛𝑐𝑒2))intheirmodelarosebecauseHamiltoniansdidnotcommutewiththeprojectorsforintermediatejudgmentsaboutthepresenceofthediseasegiventhepresentedevidenceateachstep. AnalternativeapproachtoquestionordereffectswasthatofWangandBusemeyer(2015;seealsoWang&Busemeyer,2016a),whoemployedaquantumrandomwalkmodelformodellingtheeffectivenessofpublichealthserviceannouncementsfromtheperspectiveoftheselfandthenanotherpersoninoneordervs.thereverseorder.ThemodelwasanalogoustoKvametal.’s(2015). TheQQequalityisanaprioripredictionofquantumtheoryindecisionmaking.Itcanbederivedassumingthatnonewinformationisincludedbetweenquestions.Intwodimensions,theQQequalityrelatestothewaythepositiveandnegativeanswersfortwoquestions(twodifferentbasissetsin

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Hilbertspace)arelinkedup–theunitarytransformthattakesusfromthebasissetforonequestiontothatofanotherisfixedbytherelationbetweenanyoneanswertoonequestionandanothertotheotherquestion.YearsleyandTrueblood’s(2018)QQequalityvariantrequiresone-dimensionalsubspacesandreciprocity(reciprocityisautomaticallytruewhenA,Barerepresentedbyone-dimensionalsubspaces). YearsleyandTrueblood’s(2018)maincontributionwastoidentifyincompatibilityasthedrivingfactorforbothordereffectsandconjunctionfallaciesandsopredictaco-occurrencefortheseeffects.Theypresentedasimplequantummodelwithone-dimensionalsubspacesfortheeffectsofinterestandextractedconstraintsonthesizeoftheeffects. TheABAparadigmwasproposedasatestofhowincompatibilitycanmoderateourexpectationofquestionordereffects.ThecorrelationbetweenthetworesponsesofAshouldrevealidentitywhenA,Barecompatible,butitwouldbeotherwiseboundeddependingontheuncertaintyrelationbetweenA,B(Khrennikovetal.,2014).Thatis,ifaresponsetoquestionAisfollowedbyanincompatiblequestionB,thentheanswertoquestionAhastobecomeuncertainagainandthecorrelationbetweenthetwoAcopiesshouldbelessthan1.TheseexpectationswererefinedwithWangandBusemeyer’s(2016)quantummodelforratings,whichshowedthatthesquaredcorrelationbetweenthefirstandsecondAmeasurementsshouldbepositivelyrelatedtothesquaredcorrelationbetweentheA,Bmeasurements.5.7.3CriticalevaluationandcontroversyTruebloodandBusemeyer(2011)reportedexcellentfitsofthequantummodelwithexperimentalresults.Itisappealingthatthesamequantumformalismwasemployedforbothordereffectsandthedisjunctioneffect(cf.Pothos&Busemeyer,2009).

Questionordereffectsareclassicallypuzzling,becauseofcommutativityinconjunction.SupposeA,Brepresenttheansweryestocorrespondingquestions.Then,theeventansweringyestothefirstquestionandthenyestothesecondoneisgivenby𝑃𝑟𝑜𝑏 𝐴 𝑃𝑟𝑜𝑏 𝐵 𝐴 = 𝑃𝑟𝑜𝑏 𝐴&𝐵 =𝑃𝑟𝑜𝑏 𝐵 𝑃𝑟𝑜𝑏(𝐴|𝐵)andsoaccommodatingordereffectscanonlybedonebyconditionalizingone.g.order.Aswasthecaseforconstructiveinfluencesinjudgment,thereareseveralbaselineintuitionsforhowearlierjudgmentscouldaffectlaterones,e.g.,earlierquestionsmaycreateauniquecontextforlaterones(Schwarz,2007).HogarthandEinhorn’s(1992)anchoringandadjustmentmodelcandescribeordereffects,butMcKenzieetal.(2002)arguedthatacrossreasonableparameterizationsthemodelcouldnotreproducetheirresults.McKenzieetal.(2002)proposedavariantcalledtheminimumacceptablestrengthmodel.Inthestandardanchoringandadjustmentmodelevidenceforahypothesisdependsonthemostrecentevidenceandtheaccumulatedweightforthehypothesis.Theimpactofthemostrecentpieceofevidencedependsonafixedreferencepoint.McKenzieetal.’s(2002)extensioninvolvedavariablereferencepoint,achievingbetterdescription,butattheexpenseofmoreparameters.TruebloodandBusemeyer(2011)comparedthisminimumacceptablestrengthmodelwiththequantummodelandstillconcludedinfavorofthelatter. WangandBusemeyer(2015)comparedtheirquantummodelwithacloselymatchedMarkovrandomwalkmodelandreportedsuperiorfitsforthequantummodelfortheirempiricalresults.Thiswaslargelybecauseinthequantummodelthetwoquestionswererepresentedasincompatible,sothat

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inconsecutivejudgmentsorder/interferenceeffectscanemerge;thiswasnotpossibleintheMarkovrandomwalkmodel. Overall,weseetwoquantumapproaches(Trueblood&Busemeyer,2011;Wang&Busemeyer,2015)forquestionordereffects,bothbasedondynamicalevolutionofamentalstate,butinvolvingdifferentspecification.Arguably,thiscanbejustifiedthroughafocusoncomparisonswithparticularnon-quantummodels.Note,violationsofcommutativityinquantumconjunctionscanalsobecapturedwithnon-dynamicalapproaches. TheQQequalityisconsideredoneofthemostimpressivepredictionsfromthequantumcognitionresearchprogramme,becauseitisaparameterfree,stringentaprioriconstraint.Kellenetal.(inpress)presentedaclassofmodelsallowingvaryingformatsanddegreeforthedependenceoflaterquestionsontoearlierones.TheyidentifiedversionswhicheitherproducedtheQQequalityexactlyorshowedtheQQequalityasalikelyprediction,thuscontestingthenecessityforthequantumformalismtoaccountfortheQQequality.AnadvantageofKellenetal.’s(inpress)approachisthatitenablesadeeperprocessunderstandingoftherelevantpsychologicalmechanism,inthattheirvariousmodelsarespecifiedmoredirectlyintermsofpsychologicalprocesses.Bycontrast,formalprobabilisticmodels(includingquantumones)requirefurtherinterpretationalstepsbeforeprocesspartscanberecastintraditionalpsychologicalterms.AnadvantageofthequantumapproachisthattheQQequalitycanbepredictedapriori,aswellastheconditionsunderwhichnon-conformityisexpected.Overall,itisunsurprisingthatheuristicprinciplescanbeemployedtoreproducetheQQequality.But,wewouldarguethatanexplanationbasedonheuristicsandoneonformalprobabilityprinciplesservedifferentpurposes.Note,itisunclearwhetherKellenetal.’s(inpress)analysiscanreproducetheQQequalityvariantthatYearsleyandTrueblood(2018)reported. YearsleyandTrueblood(2018)reportedthepredictedco-occurrenceofordereffectswithconjunctionfallaciesandconsistencybetweenempiricalresultsandthesizeoftheseeffects.Theyalsoobservedadependenceofstyleofthinking(analyticvs.reflexive)andpoliticalidentityonthelikelihoodofincompatiblerepresentations.YearsleyandTrueblood(2018)examinedwhetherHogarthandEinhorn’s(1992)anchoringandadjustmentmodelcanreproducetheirQQequalityvariant,butthiswasnotpossiblewithoutimplausibleassumptions.TheyalsoconsideredCostelloandWatts’s(2014)probabilitytheoryplusnoisemodelfortheconjunctionfallacy,buttheyderivedaconstraintfromthistheorynotsatisfiedbytheirdata.Theynotedthattherearenoexistingnon-quantummodelswhichrequireaco-occurrenceofordereffectsandconjunctionfallaciesorthatpredictadependencyofsucheffectsoncognitivestyleofthinking. TheresultsoftheABAparadigmshowedthesquaredcorrelationbetweenthefirstandsecondAmeasurementstopositivelyrelatetothesquaredcorrelationbetweentheABmeasurements,aspredictedbythequantummodel(BusemeyerandWang,2017).Eventhoughthereisnon-quantumpsychologicaltheoryforhowpreviousquestionscanalterthecontextforsubsequentones(e.g.,Schwarz,2007),thedependenceofthiseffectonthecompatibilityofA,B,aswellastheprecisequantitativepredictions,arebeyondanysuchbaselineintuitions.Classicaltesttheory(e.g.,Lord&Novick,2013)alsopostulatesvariabilityinmeasurementsacrossoccasions.Accordingtothisview,themeasurementresultfrome.g.aquestionratingrepresentsbothatrueopinionthatisfixedacrossrepeatedmeasurementsplussomemeasurementerrorthatrandomlyvariesacrossrepeatedmeasurements.However,themeasurementerrormodeldiffersfromthequantumapproachina

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fundamentalmanner.WhenconsideringanAAsequence(i.e.,respondingtoAtwicewithoutBinbetween),theprojectionprincipleofquantumprobabilitypredictsperfectcorrelationbetweenthefirstandsecondmeasurementsofA,whereasclassicaltesttheoryallowsmeasurementerrortooccuracrossrepetitions;empiricalresultssupportthequantumview(Busemeyer&Wang,2017)3.ClassicaltesttheoryalsocannotpredicttheroleofcompatibilityinwhetherthereisidentitybetweenthetwoiterationsofAinanABAsequence.5.8Conjunction/disjunctionfallaciesindecisionmaking5.8.1EmpiricalresearchTherearearangeofrelatedresults,includingconjunctionfallacies(Tversky&Kahneman,1983;Moro,2009),disjunctionfallacies(Bar-Hillel&Neter,1993;Carlson&Yates,1989;Fisk,2002),unpackingeffects(Rottenstreich&Tversky,1997;Sloman,Rottenstreich,Wisniewski,Hadjichristidis,&Fox,2004),andmorecomplexconjunctions(e.g.,Winman,Nilsson,Juslin,&Hansson,2010).UsingtheterminologyfromTverskyandKahneman’s(1983)paradigmaticLindaexample,whereBT=bankteller,F=feminist,theconjunctionfallacyis𝑃𝑟𝑜𝑏 𝐹 ∧ 𝐵𝑇 > 𝑃𝑟𝑜𝑏(𝐵𝑇).Thedisjunctionfallacyis𝑃𝑟𝑜𝑏 𝐹 ∨ 𝐵𝑇 < 𝑃𝑟𝑜𝑏(𝐹)(note,thisisnottobeconfusedwiththedisjunctioneffect,seeshortly;Shafir&Tversky,1992).Conjunctioneffectsoftenarisewhencombininganunlikelyconjunctwithalikelyoneand/orwhentheconjunctshaveastrongcausalassociation.Unpackingeffectsareviolationsofthelawoftotalprobabilitye.g.instancesofsubadditivity,whereby𝑃𝑟𝑜𝑏 𝐴 < 𝑃𝑟𝑜𝑏 𝐴 ∧ 𝐵 + 𝑃𝑟𝑜𝑏(𝐴 ∧ ~𝐵). GronchiandStrambini(2017)presentedavariantoftheLindaproblemtotesttheWigner–d’Espagnatinequality,𝑃𝑟𝑜𝑏 𝐴 ∩ 𝐶 ≤ 𝑃𝑟𝑜𝑏 𝐴 ∩ 𝐵 ∪ 𝐵 ∩ 𝐶 (d’Espagnat,1979).TheWigner–d’EspagnatinequalityisaversionofBell’sinequality(Bell,2004)bettersuitedtoLinda-styleparadigmsandaVenndiagramillustrateshowtrivialitisfromaclassicalperspective(i.e.,ifallpossibilitiesarerepresentedinthesamesamplespace).Bellnotablystated“trivialasitis,theinequalityisnotrespectedbyquantummechanicalprobabilities’’(Bell,1981,p.52).GronchiandStrambini(2017)reportedtwoexperimentsconcerningtheprobabilityofpickingdifferentobjectsfromanurnorofmalevs.femalestudentsmorelikelytobeplayingsoccer.InthefirstexperimenttheresultsindicatedviolationsoftheWigner-d’Espagnatinequality.Inasecondexperiment,whenparticipantsdirectlyestimatedunpackedprobabilities(𝑃𝑟𝑜𝑏(𝐴 ∩ 𝐵)and𝑃𝑟𝑜𝑏(𝐵 ∩ 𝐶)separately)Wigner–d’Espagnatviolationsdisappeared,thusindicatingthatasubadditivitypattern(𝑃𝑟𝑜𝑏 𝐴 ∩ 𝐵 ∪ 𝐵 ∩ 𝐶 ≤ 𝑃𝑟𝑜𝑏 𝐴 ∩ 𝐵 + 𝑃𝑟𝑜𝑏 𝐵 ∩ 𝐶 )possiblyaccountsforWigner-d’Espagnatviolations. Thedisjunctioneffectisaninfluentialdemonstrationofaviolationofthelawoftotalprobability(Shafir&Tversky,1992;Tversky&Shafir,1992).Inonevariant,participantswerepresentedwithaprisoner’sdilemmagame,suchthatthepayoffsrecommendeddefectionontheassumptionthatthehypotheticalotherplayercooperates(Section5.6.1).Participantsreceiveddifferentinformationabouttheintentionoftheotherplayer:thathe/shewouldcooperate,defect,ornoinformationwouldbegiven.Ineachofthe‘known’conditions,mostparticipantspreferredtodefect,butinthe‘unknown’conditionmanyreversedtheirjudgmenttocooperate,revealingapatternof

3NoteusingPOVMsinsteadofprojectorswouldpredictnon-identityforconsecutivejudgments.CurrentlythereisachallengeinunderstandingthecircumstanceswhenPOVMsvs.projectorsaremoresuitable.

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𝑃𝑟𝑜𝑏 𝑑𝑒𝑓𝑒𝑐𝑡; 𝑢𝑛𝑘𝑛𝑜𝑤𝑛 < 𝑃𝑟𝑜𝑏 𝑑𝑒𝑓𝑒𝑐𝑡 𝑐𝑜𝑜𝑝𝑒𝑟𝑎𝑡𝑒 ∙ 𝑃𝑟𝑜𝑏 𝑐𝑜𝑜𝑝𝑒𝑟𝑎𝑡𝑒 + 𝑃𝑟𝑜𝑏 𝑑𝑒𝑓𝑒𝑐𝑡 𝑑𝑒𝑓𝑒𝑐𝑡 ∙𝑃𝑟𝑜𝑏 𝑑𝑒𝑓𝑒𝑐𝑡 ,wheretheconditionalizationsrefertotheprovidedinformation(𝑃𝑟𝑜𝑏 𝑑𝑒𝑓𝑒𝑐𝑡; 𝑢𝑛𝑘𝑛𝑜𝑤𝑛 justreferstothemarginalwhenthereisnoinformationabouttheotherplayer’saction).Themarginalintheunknownconditionhastobeboundedbytheconditionalsineachoftheknownconditions,sothattheresultsindicatedaviolationofthelawoftotalprobability.Othervariantswereproposed(e.g.,atwostagegamblingtask)andthefindinghasbeenwidelyreplicated.Also,extensionstotheprisoner’sdilemmataskhavebeendeveloped,e.g.,asequentialversion(Blancoetal.,2014)oronetostudypriorcommitmentstodefectorcooperate(Kvametal.,2014). AnotherviolationofthelawoftotalprobabilitywasreportedinTownsendetal.(2000;Busemeyeretal.,2009).Participantswerefirstpresentedwithaface.Someparticipantswereaskedtocategorizethefaceasfriendlyorhostile,whileotherparticipantssimplyobservedit.Inasecondstep,allparticipantswereaskedtoindicateanaction,attackorwithdraw.Acrossseveralsuchstimuli,weexpect𝑃𝑟𝑜𝑏 𝑎𝑡𝑡𝑎𝑐𝑘 = 𝑃𝑟𝑜𝑏 𝑎𝑡𝑡𝑎𝑐𝑘 ∧ 𝑓𝑟𝑖𝑒𝑛𝑑𝑙𝑦 + 𝑃𝑟𝑜𝑏 𝑎𝑡𝑡𝑎𝑐𝑘 ∧ ~𝑓𝑟𝑖𝑒𝑛𝑑𝑙𝑦 ,butthiswasnotobserved.WangandBusemeyer(2016b)replicatedthiscategorization-decisioneffect.5.8.2QuantumcognitivemodelsBusemeyeretal.(2011)employedastaticmodelinwhichaconjunctionfallacyemerges,mostlyonthebasisofsuitablerepresentationassumptions.Specifically,|𝑃!"|𝜓 |! = |𝑃!"𝑃!|𝜓 |! + |𝑃!"𝑃~!|𝜓 |! +𝜓!",~! 𝜓!,! + 𝜓!",! 𝜓!",~! ,wheretheinterferencetermisΔ = 𝜓!",~! 𝜓!",! + 𝜓!",! 𝜓!!,~! (Section2).Note,thisdecompositionimpliesthattheconjunctivestatementF&BTisevaluatedintermsofaprojectionfirsttotheFsubspaceandthentotheBTone,anassumptionjustifiedthroughappealtoheuristicsprioritizingtheprocessingofmorelikelyinformation(Gigerenzer&Todd,1996).Iftheinterferencetermisnegative,thentheconjunctionfallacyfollows.Anegativeinterferencetermobtainswithtwoconditions.First,|𝑃!"𝑃~!|𝜓 |!hastobesmall,whichmeansthatthe~FpropertymakestheBToneunlikely.IntheLindaproblem,wecanask:doesknowledgethatLindaisnotafeministmaketheBTpropertylikelyorunlikely?Thisisareasonableintuition,sincetheknowledgethatapersonisnotafeministdoesnotindicateaparticularprofession.Second,theinterferencetermhastobenegative,whichisthecaseifthestateproducedfromthesequence𝑃!"𝑃~!𝜓isoppositecomparedto𝑃!"𝑃!𝜓.Thisisreasonable,sincethecharacteristicsofafeministbanktellerareplausiblyoppositetothoseofanon-feministbankteller. GronchiandStrambini(2017)producedtwoquantummodelstoaccountforviolationsofWigner–d’Espagnat.OnemodelisbasedonincompatiblequestionsinthesamespaceandsothisapproachisanalogoustothatofBusemeyeretal.(2011).TheothermodelassumesthatthetwoquestionsforeachlogicaloperationintheWigner–d’Espagnatequationareevaluatedinseparatespacesandsotheyarecompatible.AnyviolationsofWigner–d’Espagnatthenemergefromentanglement.Thetwomodelswereshowntobeequivalent. PothosandBusemeyer(2009)proposedadynamicalmodelforthedisjunctioneffect.Eachpossibilityfortheopponent’saction(defect,cooperate)wasmodeledwithaseparateHamiltonian/unitarycorrespondingtohowpayoffaffectsprobabilitytodefect.Followingaclassicalintuition,thedynamicalprocessintheunknowncasecanbespecifiedfromthatoftheknowncasesas𝑈!(𝑡)⊕𝑈!(𝑡),sothat

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𝑃𝑟𝑜𝑏 𝐷; 𝑢𝑛𝑘𝑛𝑜𝑤𝑛 𝑐𝑎𝑠𝑒 = |𝑀!,! ⊕𝑀!,!𝑈!(𝑡)⊕ 𝑈!(𝑡)𝜓! ⊕ 𝜓!|! =|𝑀!,!𝑈!(𝑡)𝜓! ⊕𝑀!,!𝑈!(𝑡)𝜓!|! = 𝑃𝑟𝑜𝑏 𝐷&𝐷 + 𝑃𝑟𝑜𝑏(𝐷&𝐶),where𝑀!,!isameasurementoperator(aprojector)fortheprobabilitytodefect,whentheparticipantknowstheopponentwilldefect.Thus,ifthementalstatechangesbyadynamicalprocessoftheform𝑈!(𝑡)⊕ 𝑈!(𝑡)thelawoftotalprobabilityisobeyed.QuantumtheoryallowsmorecomplexdynamicalprocessesandPothosandBusemeyer(2009)employedaHamiltonianwithaform𝐻! ⊕ 𝐻! + 𝐻!"#$% sothatthecorrespondingunitarycannolongerbewrittenaslike𝑈!(𝑡)⊕ 𝑈!(𝑡).Interferenceeffectscanthenemerge,allowingforviolationsofthelawoftotalprobability.Psychologically,𝐻!"#$% alignsbeliefsandactions,thusreducingcognitivedissonance(Festinger,1957).Denolfetal.(2017)proposedananalogousquantummodelforasequentialversionofthePrisoner’sDilemmagame(fromBlancoetal.,2014).Themainfeaturesintheirmodelwereanelaborationofassumptionsregardingwhetheractionsorbeliefscouldbeconsideredcompatibleorincompatible. Forthecategorization,decisionparadigm,WangandBusemeyer(2016b)alsoemployedaHamiltonianwithstructure𝐻!"#$%&'(⨁𝐻!"#$%&',correspondingtohowknowledgeofeachpossiblecategorizationcanaffectdecision,whichwasextendedto𝐻!"#$%&'(⨁𝐻!"#$%&' + 𝐻!"#$%,toallowinterferenceeffectsandviolationsofthelawoftotalprobability;themixeralignscategorizationwithdecision.MoreiraandWichert(2016)adoptedanalternativeapproach,employingTucci’s(1995)quantum-likenetworktheory,whichpropagatesamplitudesinsteadofprobabilities.Becauseamplitudespropagateacrossmultiplepaths,quantumnetscangiverisetointerferenceeffects,notpossibleinBayesiannetworks.MoreiraandWichert(2016)computedtheinterferencetermsintheirquantumnetworkfromthefaceimagesinthecategorization,decisionparadigm(fromBusemeyeretal.,2009).Theyconvertedtheimagesintovectors,stochasticallydeterminedassignmentintogoodandbadfaces,andcomputedcosinesimilaritiesbetweenimagesinthedifferentgood/badcategories.Thesesimilaritieswereemployedasinterferencetermsinthequantumnetworkmodel. YukalovandSornette(2011)providedanalternativequantumframeworkprimarilyaimedatthedisjunctioneffectandconjunctionfallacy.Theydefinedasetofprospects,treatedasquantumtheoryoperators,whichleadtoaquantityanalogoustoclassicalexpectedutility.Thisclassicalexpectedutilityquantityiscomplementedwithanon-classicalone,calledanattractionfactor,whicharisesfrominterferenceeffectsincompositeprospects.Largeattractionfactorsareassumedtoresultinmoreattractiveprospects.Forthedisjunctioneffect,thereisanattractiontermforeachaction,analogoustoquantuminterferenceterms.Usingauniformityassumptionforsomeofthevariables,uniformpriors,andanuncertaintyaversionprinciple(reluctancetoactunderlargeruncertainty),theyreconstructedprobabilitiesconsistentwiththedisjunctioneffect.Regardingtheconjunctionfallacy,theyusedthequantumlawoftotalprobability(Busemeyeretal.,2011)andinvolvedtheuncertaintyprincipletojustifyanegativeinferenceterm.5.8.3CriticalevaluationandcontroversyBusemeyeretal.’s(2011)quantummodelarguablyrevolutionarizesournotionofcorrectnessinprobabilisticinference,whichhasbeendominatedbyclassicalprobabilitytheorytothepointthatitishardtoenvisagealternatives.Withprobabilitiesassubsetsofsomesamplespace,itisimpossibletoimagineaconjunctionasmorelikelythanamarginal,yetthegeometricpictureofprobabilitiesin

33 Quantumcognition

quantumtheoryallowsanintuitionofhowthiscanbeso.Busemeyeretal.’s(2011)workencompassedseveralrelatedfallacies(thedisjunctionfallacy,unpackingeffects,morecomplexconjunctionsetc.).Iftheconjunctionfallacycanbe‘correct’,withquantumprobabilities,canitalsoberational?Pothosetal.(2017)showedthatquantumtheoryisconsistentwiththeDutchBookcriterionforrationalbehavior(deFinettietal.,1993)andthataconjunctionfallacycanberationalwhenthequestionsarecontextual(thepresenceofonequestionaltersthemeaningoftheother)ordisturbing. InBusemeyeretal.’s(2011)modelaconjunctionfallacyarisesfromincompatibility.Therefore,manipulationsthatencourageprocessingoftheconjunctsinaconcurrent(compatible)wayshouldreduceconjunctionfallacies.Thereissuchevidencefromtrainingonthealgebraofsets(Agnoli&Krantz,1989;seealsoYamagishi,2003,andWolfe&Reyna,2010)tofeedbacktrainingontheclassicalprobabilityrule(Nilsson,2008).FamiliarityisalsothoughttoinducetransitionsfromincompatibletocompatiblerepresentationsandNilssonetal.(2013)reportedsomeevidencefortheconjunctionfallacy(Nilssonetal.,2013).RecallalsoYearsleyandTrueblood(inpress)whodemonstratedtheco-occurrenceofconjunctionfallaciesandordereffects.Busemeyeretal.(2011)assumedthatregardlessofthepresentationorderofthetwoconjuncts,themorelikelyoneisprocessedfirst.Thereissomeevidencethatmanipulatingprocessingorder(e.g.,throughpriming)supportsthequantumaccount(Stolarz-Fantinoetal.,2003,Experiment2;Gavanski&Roskos-Ewoldsen,1991).Acriticalobservationconcernsthedefinitionofdisjunctioninthequantumaccount,whichdoesnotbenefitfromasclearajustificationasthesequentialconjunction.

Therehavebeenseveralmodelsfortheconjunctionfallacy.TverskyandKahneman’s(1983;Shafiretal.,1990)proposedarepresentativenessheuristic,sothatprobabilityjudgmentsareessentiallysimilarityprocesses.Thus,probabilisticinferenceisrelegatedtoamoregenericcognitiveprocess.However,thesimilaritymechanisminrepresentativenessisunderspecified.Becausethequantummodelcomputesprobabilityasrepresentationaloverlapinamultidimensionalspace,ithasbeenadvocatedasaparticularexpressionofprobabilisticinferenceassimilarity(cf.Sloman,1993).Averagingmodelscomputeconjunctionsasaveragesoftheprobabilitiesoftheconjuncts(Abelson,Leddo,&Gross,1987;Fantino,Kulik,&Stolarz-Fantino,1997;Nilsson,2008;Nilssonetal.,2009).Theseaccountspredictconjunctionfallaciesregardlessofcausallinksbetweentheconjuncts(highrateofconjunctionfallacy)ornot(lowerrate).However,weexpectaconjunctionfallacyincasessuchas“Mr.FhashadoneormoreheartattacksandMr.Fisolderthan55”butnotin“Mr.FhashadoneormoreheartattacksandMr.Gisolderthan55”(TverskyandKahneman,1983). Accordingtotheinductiveconfirmationaccount,probabilityestimatesaredrivenbythedifference𝑃𝑟𝑜𝑏 ℎ 𝑒 − 𝑃𝑟𝑜𝑏(ℎ),wherehisahypothesisandeisevidence(Tentorietal.,2013).Thisaccountdissociatesthedegreeofconjunctionfallacyfromtheprobabilityofthemorelikelyconjunct.Tentorietal.’s(2013)taskinvolvedahypotheticalwomanwhoisRussian(R),livesinNewYork(NY),andinadditionisaninterpreter(I).Theirmainfindingwas𝑃𝑟𝑜𝑏 𝐼 ∧ 𝑁𝑌 𝑅 > 𝑃𝑟𝑜𝑏(𝑁𝑌|𝑅),eventhoughtheIpossibilityislesslikelythanthe~Ione(Tentorietal.,2013,expressthisas𝑃𝑟𝑜𝑏 ~𝐼 𝑁𝑌 ∧ 𝑅 >𝑃𝑟𝑜𝑏 𝐼 𝑁𝑌 ∧ 𝑅 ).Thus,inthisscenarioandvariantstheconjunctionfallacyarisesfromalesslikelyconjunct(beinganI).ThequantumconjunctionfallacymodelcancoverTentorietal.’s(2013)results(Busemeyeretal.,2015).Additionally,Busemeyeretal.(2015)arguedthatordereffectsintheconsiderationoftheconjuncts(Stolarz-Fantinoetal.,2003,Experiment2;Gavanski&Roskos-Ewoldsen,1991)supportthequantumaccount,butnottheinductiveconfirmationone.Finally,thequantum

34 Quantumcognition

accountcanaccommodatemanipulationsthatarguablyencouragecompatiblerepresentations(Agnoli&Krantz,1989;Nilsson,2008;Wolfe&Reyna,2009;Yamagishi,2003),includingincreasingfamiliarity(Nilssonetal.,2013),butconsistencywiththeinductiveconfirmationproposalisunclear.

Intheprobabilitytheoryplusnoiseaccountdecisionmakingisbasedonclassicalprobabilitytheory,butprobabilityestimatesaresubjecttoerror(Costello&Watts,2014).Thiserrorarisesbecauseprobabilityestimatesarebasedeitheronenumeratingrelevantmemoryinstancesoramentalsimulationofsuchinstances.CostelloandWatts(inpress)derivedseveralprobabilisticexpressionswhichdivergedependingonwhetheroneadoptsquantumrulesorprobabilityplusnoiserules.Theirresultsappeartouniformlysupporttheirmodel,notquantumtheory.Note,quantumtheoristshavebeenincorporatingnoiseintheirworktoo(e.g.,Truebloodetal.,2017;YearsleyandPothos,2016).However,noisehasnotbeenexploredfortheconjunctionfallacy,becauseinquantumtheorynoiseemergesasasmallprobabilitythattheinternalcomputationmayproduceA,butonemaymistakenlypronounce~A.Therefore,inquantummodelstheroleofnoiseisrelevantinsituationsofmultipledecisions.

Thereareconcernsforthenoisemodel.First,themechanismassumedtoproducethenoiseisbasedonmemoryenumerationormentalsimulation,butespeciallythelatterseemsimplausible,forunfamiliarquestions.Second,samplingindependenceforestimatingrelatedprobabilities(e.g.,marginalsandaconjunction)isimplausible,butwithoutsamplingindependencethenoisemodelcannotpredicttheconjunctionfallacy.Third,CostelloandWatts(2016)producedaconditionalprobabilitynoiseformula,butinamorerecentpaper(Costello&Watts,2018),theyeschewedtheirownconditionalprobabilityformulatomodelordereffectsusingamemoryprimingparameter.Fourth,errorisimplausibleforcompletelyimpossibleorcertainevents.Fifth,CostelloandWatts’s(2014)predictionforaconjunctionfallacyreliesonP(A∧B)beingclosetoP(A),butTentorietal.(2013)reportedconjunctionfallacyresultsatoddswiththisrequirement.Finally,thenoisemodelcannotaccommodateconjunctionfallacies,when𝑃 𝐴 ∧ 𝐵 ~𝑃 𝐴 > 0.5;YearsleyandTrueblood(inpress)reportedsuchresults.

DoCostelloetal.’s(inpress)resultschallengethequantummodel?Costelloetal.(inpress)considerfamiliaritytoequatewithcompatibilityandthereisevidencethatthisisthecase(e.g.,Truebloodetal.,2017;Yearsley&Trueblood,inpress).However,incompatibilitymayexistforfamiliarquestionstoo,wherequestionsmayinducedifferentperspectivestoeachother,asseemsapparentforordereffectsregardingtheClinton,Gorehonestyquestions(Wang&Busemeyer,2013).However,detailedcoverageoftheCostelloetal.(inpress)resultsfromthequantummodelisstillneeded.

Boyer-Kassemetal.(2016)arguedthataquantummodelfortheconjunctionfallacyentailsthreeempiricalpredictions:(1)consistencywiththeso-calledgrandreciprocityequalities(whichtheyderived),(2)theexistenceofordereffects,and(3)consistencywiththeQQequality.Theyreportedresultswhichtheyclaimedgoagainstthequantummodel.Regardingthegrandreciprocityequalities,thesewerederivedassumingreciprocityandsoone-dimensionalrepresentations,butBusemeyeretal.’s(2011)modelisnotrestrictedinthiswayanditseemsunlikelythatcomplexpsychologicalconcepts(likefeminism)wouldhaveone-dimensionalrepresentations(Pothosetal.,2013).

Regardingtheco-occurrenceofordereffectsandconjunctionfallacies,Boyeretal.(2016)consideredquestionsfromsevendifferentscenarios,fromwell-knownconjunctionfallacydemonstrations,butreportedordereffectsinonlytwocases,noordereffectsintwoothercases,and

35 Quantumcognition

ambiguousresultsintheremainingthreecases.Note,theycorrectedforfamily-wiseerrorusingtheBonferronicorrection,whichhasbeencriticizedasconservative(Nakagawa,2004;Perneger,1998).Importantly,theydidnotdemonstrateconcurrentconjunctionfallacies,asYearsleyandTrueblood(inpress)did.Itispossiblethattheirsamplewaslesspronetoordereffectsandconjunctionfallacies(theirparticipantswerestudentsineconomics,management,ormedicine).TheirreportoffailurestosatisfytheQQequalityawaitsfurtherinvestigation.

GronchiandStrambini’s(2017)workbringsintofocusakeyissue,thatviolationsoftheWigner-d’Espagnatinequalitycanariseeitherbecauseofincompatibilityinthesamespaceorcompatiblerepresentationsinseparatespaceswithanentangledstate.Bothincompatibilityandentanglementprecludeajointprobabilitydistribution,butintheformercasebecauseofuncertaintyrelationsandinthelatterbecauseoftheimpossibilitytodescribethecorrespondingentitiesindependentlyofeachother.However,researchersmayrequireacommitmenttoincompatibilityorcompatibilityinanysituation.InGronchiandStrambini’s(2017)experiments,questionsregardingthecolororshapeofobjectsarelikelytobetreatedascompatible(becauseperceptualphysicalpropertieslikethisareunlikelytohaveacontextualinfluenceoneachother),butgenderandlikelihoodofplayingdifferentsportsincompatible(ifdifferentgenderschangeourperspectiveforlikelysports).

PothosandBusemeyer(2009)showedthatthequantummodelcoulddescribethedisjunctioneffectbetterthanamatchedclassicalprobabilitymodel(employingtheKolmogorovforwardequation,whichistheclassicalequivalentofSchrodinger’sequation).InPothosandBusemeyer’s(2009)approachthetransitionfromclassical(nointerference)toquantum(interferencepossible)probabilitiesoccurswithinthesameframework.Arguably,thismodel(firstdescribedinBusemeyeretal.,2006)wasthefirstquantumcognitivemodelbasedonasimpleapplicationofquantumprobabilities,presentedinajournalfollowedbypsychologists.Thisframeworkhasbeenemployedforotherbroadlyrelatedfindings(Trueblood&Busemeyer,2011;Wang&Busemeyer,2016).TverskyandShafir’s(1992)explanationforthedisjunctioneffectwasbasedontheideaoffailureofconsequentialreasoning.Inprisoner’sdilemmaparticipantsmayhaveagoodreasontodefectknowingtheopponentdefectsandagood(different)reasonknowingtheopponentcooperates,butlackclarityofwhytheyshoulddefectintheunknowncase.Clearly,suchanexplanationlooselyconstrainsspecifictechnicalapproaches.OnecriticismofPothosandBusemeyer’s(2009)modelisthat,becauseofitsuseofunitarydynamics,probabilitiesareperpetuallyoscillatingandPothosandBusemeyer(2009)assumedacutofffordecisions.Theissueofstabilizationinquantumdynamicsisformallyaddressedwithopensystemdynamics(e.g.,Asanoetal.,2011a,2011b).

SimilarconsiderationsapplytoWangandBusemeyer’s(2016b)quantummodelforcategorization,decisionresults.Additionally,theseauthorsdiscussedasignaldetectiontheorymodel(Ashby&Townsend,1986),forwhich,followingcategorization,theeventualdecisioncanstilldependonthestimulus;fortheMarkovandquantummodels,onceacategorizationismade,thedecisiondependsonlyonthecategorization.Theauthorsarguedthatsignaldetectiontheorywouldhavetroubleaccountingforinterference,unlessoneintroducesposthocassumptionsregardinghowtheresponseregions(forthecategorizationsand/orthedecisions)changeboundariesdependingonstimuluscategorization.

Denolfetal.(2017)allowedoutcomesofaratingscaletobenon-orthogonalvectors,whichisunorthodoxinquantumtheory.However,theirapproachisbroadlyequivalenttoanassumptionof

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POVM(asopposedtoprojective)measurement.Itsadvantageisthatitenableslowerdimensionalityspacesandnon-repeatabilityofmeasurements. MoreiraandWichert’s(2017)quantummodelforthecategorization,decisionviolationofthelawoftotalprobabilityraisessomequestions.Itisunclearthatthevectorsinthequantumnetworkareequivalenttotheonesemployedtocomputethesimilarities.Thatis,thereissomedoubtinwhethertheequivalencebetweensimilaritiesandquantuminterferencetermsisvalid.Additionally,MoreiraandWichert(2017)assumedarenormalizationoftheimagevectorstooccupythenegativepartofthevectorspace,becausethisleadstoarestrictiontojustdestructiveinterferenceeffects(asempiricallyobserved),butthisassumptioncoulddowithfurtherjustification.

YukalovandSornette’s(2011)frameworkwaspresentedasageneralframeworkforprobabilisticdecisionmaking,withapplicationsrelatingtothedisjunctioneffectandtheconjunctionfallacy.However,therearesomeweaknesses.Regardingthedisjunctioneffect,theuncertaintyaversionprincipleistranslatedtoaconditionthattheuncertaintyfactorforactingispositiveandthefornotactingnegative,anassumptionofquestionableusefulness,becauseinsomeparadigmsitisunclearwhichoptioncorrespondstoactingornotacting.Forexample,inTverskyandShafir’s(1992)prisoner’sdilemma,thereisasuppressionoftheprobabilitytodefectintheunknownconditionandanincreaseintheprobabilitytocooperate.Buthowcanuncertaintyaversioninformahigherpropensitytocooperate?Regardingtheconjunctionfallacy,itisalsounclearhowuncertaintyaversionoperates,eventhoughthisostensiblydeterminestherightsignfortheinterferenceterm.Thatis,inwhatwaycouldoneassumethattheprobabilityofamarginalismoreuncertainthantheprobabilityofconjunctions,sincenormativelythelawoftotalprobabilityrequiresequality?5.9Otherjudgmentphenomena5.9.1EmpiricalresearchBasievaetal.(2017)consideredprobabilisticupdating.Theauthorsdesignedataskbasedonahypotheticalcrimemystery,withseveralpotentialsuspects.Participantswereinitiallypresentedwithsomeinformationforguiltforthevarioussuspectsandthenwithmoreinformation,whichwasmeanttooverturnexpectationsaboutinitiallyunlikelysuspects. Aertsetal.(2018)soughttoprovideademonstrationofentanglementindecisionmaking.Participantswereaskedtopickapairofwinddirections(e.g.,NorthandSouthwestvs.SouthandNortheast),thattheyconsidered‘goodexamplesoftwodifferentwinddirections’.Acrossseveraltrials,participantswerepresentedwithchoicesetsincludingdifferentcombinationsofpairsofwinddirections.Byanalogywithentanglementexperimentsinphysics,afirstsystemwouldinvolvetwodirectionsA,A’andasecondsystemtwodifferentwinddirectionsB,B’,sothatoverallwewouldhavefourcombinationsofwinddirections(AB,AB’etc.),witheachcombinationspecifyingatrial(sincee.g.eachofA,Barebinaryquestions,therearefourpossibilitiesforthecombinedquestionA,Bandparticipantswouldchooseone).Thedesignwasfullywithinparticipants.ThemainempiricalfindingwasthataveragedparticipantresponsesviolatedtheCHSHinequality,sothatacompletejointprobabilitydistributionfortherelevanteventsisimpossible. TheEllsbergparadox(Ellsberg,1961)isadecisiontaskinvolvingdrawingcoloredballsfromanurn.Thetaskistodecidewhichcolorwillbemostlikely.Aparticularcolorisassociatedwithafixed

37 Quantumcognition

probability(sayred),butanotheronewithanunknownprobability(saygreen).Whenparticipantsaretoldtochoosebetweenabetondrawingaredballvs.agreenone,thentheychoosetheformer.Whenparticipantsaretoldtochoosebetweenabetonnotdrawingaredballvs.notagreenone,thentheyalsochoosetheformer,displayingambiguityaversion(aversiontoambiguousprobabilities)andviolatingexpectedutilitytheory.Dimmocketal.(2015)showedthatthereisambiguityaversionformediumandhighprobabilitiesbutambiguityseekingforlowprobabilities.TheAllaisparadox(Allais,1953)involvesachoicebetweentwogamblesA,BandasecondchoicebetweentwofurtheronesC,D.EventhoughAandCareequivalent,inthefirstchoiceAispreferredbutinthesecondCisnotpreferred,violatingtheindependenceaxiomofexpectedutilitytheory.5.9.2QuantumcognitivemodelsBasievaetal.(2017)comparedclassicalprobabilityupdating(basedonBayeslaw)withquantum

probabilityupdating(basedonLuder’slaw).Luder’slawcanbeexpressedas𝑃𝑟𝑜𝑏 𝐴 𝐵 = |!!!!|! |!

|!!|! |!=

!"#$(! ∧ !!!" !)!"#$(!)

andisequivalenttoBayesianprobabilisticupdating,thedifferencebeingthatclassical

conjunctionisreplacedbythequantumequivalentforincompatiblequestions,whichissequentialconjunction. Aertsetal.(2018)representedthequestionsaboutwinddirectionswithtensorproductsoftheform(𝜎 ∙ 𝑎)⊗ (𝜎 ∙ 𝑏),wherea,baredirectionvectorsand𝜎 = 𝜎! + 𝜎! + 𝜎!,thePaulispinmatrices.Inordertodescribecorrelationsbetweenbinarymeasurementsintwosubsystems(i.e.,theresponsesforeachoftwowinddirections,foreachsubsystem),oneneedstoidentifyfourdirectionvectorsa,b,a’,b’andanentangledstate.Theauthorsclaimthatthereisnoguaranteeofasuitablequantumrepresentationofthiskind.But,theyidentifiedanentangledmentalstateandwinddirectionswhichdescribedtheobservedcorrelations. LaMura(2009)providedaquantumframeworkforexpectedutility,calledprojectiveexpectedutilitytheory.Whereasexpectedutilitytheoryisbasedonobjectiveoutcomes,projectiveexpectedutilityisbasedonsubjectiveconsequences,whichareobtainedasabasischangefromobjectiveoutcomes.Theuseofaquantumframeworkforobtainingprobabilitiesallowsforinterferenceeffectsandcontextualdependence,whichcanaccommodatetheEllsbergandAllaisparadoxes,amongstotherresults.AlNowaitiandDhami(2017)focusedontheEllsbergparadox,aimingtocapturebothambiguityaversionandambiguityseeking(Dimmocketal.,2015).TheyemployedFeynman’srulesforstatetransitions,whentheintermediatestepsareknownvs.notknown.Briefly,inatransition𝜑 → 𝜓through𝜒!,𝜒!,whentheintermediatestateisobserved,thentheoveralltransitionprobabilitiesareobtainedbyaddingprobabilities.However,whenitisnotobserved,thentheoveralltransitionprobabilitiesareobtainedbyaddingamplitudes,whichcanleadtointerferenceeffects.ThisapproachcouldaccommodateDimmocketal.’s(2015)findings,dependingonassumptionsofwhethercognitiveprocessingisovert(probabilitiescombine)orcovert(amplitudeswouldcombine),withoutparametermanipulation.EllsbergandAllaishavebeenthefocusofotherproposals(Aertsetal.,2014;Busemeyer&Bruza,2012;Khrennikov&Haven,2009),whichallsharesomekeyelements(theuseofinterferenceterms/contextualitytoaccommodatetheparadoxes).

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5.8.3CriticalevaluationandcontroversyBasievaetal.(2017)arguedthatBayesruleprovidesincompletecoverageofeverydayinference,becauseitprecludeslargejumpsfrompriorstoposteriors.Classicallyupdatedprobabilitiesarelinearlydependentonpriorsandtheratiooflikelihoodsisconstrainedbytheratioofpriors,sothatlowpriorswillrestrictposteriors.Additionally,zeropriorsrequirezeroposteriors,asituationcalledbytheauthorsthezeropriorstrap.Thezeropriorstraphasalonghistory.OliverCromwellallegedlysaidtothemembersofthesynodoftheChurchofScotland“tothinkitpossiblethatyoumaybemistaken”(Carlyle,1885).Theideahereisthatasmallprobabilityshouldbeassignedtoevenhighlyimprobablesituations.Shafer’s(1975)modelcancircumventthezeropriorstrapinanotherwiseBayesianframework,byorganizinghypothesesintogroupsandassigningapriorprobabilitytoeachgroup.Crucially,thereisflexibilityinhowgroupprobabilityisdividedamongstindividualhypotheses,andnewinformationcouldleadtoareassignmentthatmeansthatwenolongerneedtoupdatefromazeroprior.However,Basievaetal.(2017)showedthatShafer’s(1975)hypothesiscannotexplaintheirresults,butLuder’slawcan. Aertsetal.(2018)partlymotivatedtheirworkasatestofCHSHviolationconsistentwiththeconditionforno-signalingfromDzhafarovandKujala(2014;Section5.4.2).Theirbaselinedatastillviolatedthiscondition,butatransformationproducedCHSHviolationtogetherwithconsistencytotheno-signalingcondition.Dzhafarovetal.(2016)questionedthistransformationandarguedthatsignalingaccountsforCHSHviolationsinAertsetal.’s(2018)experiments. RegardingcoverageofEllsbergandAllaisparadoxes,quantumapplications(Aertsetal.,2014;AlNowaitiandDhami,2017;Busemeyer&Bruza,2012;Khrennikov&Haven,2009;LaMura,2009)sharetheuseofinterferenceandcontextuality.AlNowaitiandDhami(2017)extractedprobabilitiesconsistentwiththeEllsbergparadox,withoutparametermanipulation(contrastwithBusemeyer&Bruza,2012;LaMura,2009).Aquestionforsuchapproachesistheircapacityforgenerativevalue.Notealsothatdetailedcomparisonswithnon-quantumexpectedutilityalternativeshavenotbeencarriedout.Forexample,theEllsbergparadoxhasalsobeenapproachedthroughprobabilitytransformations(e.g.,Klibanoffetal.,2005).6.Isitworthperseveringwithquantumcognitionmodels?Havingreviewedseveralcognitiveapplicationsofquantumtheory,wenextaddressquestionscriticalforthisresearchprogramme.Isitworthperseveringwithquantummodels?Inevitablyweareledtofundamentalquestionslikewhatconstitutesastrongtheoreticalframeworkandagoodmodel.

Ahealthytheoreticalframeworkarguablyfulfillsthreeconditions.First,itsprinciplesareinterconnected,asthislimitsarbitrariness.Second,thereareclearnarrativesaboutpsychologicalprocess.Third,thereareplausibleassumptionsregardingcomputationaldemandsforthebrain.Wethinktheprospectsregardingquantumtheoryaregood.Thefirstcharacteristicisself-evidentinthatquantumtheory,asaformalprobabilisticsystem,isaxiomaticallydefined.Inanyspecificquantummodelsomeadditionalassumptionsmightbeneeded,asawaytobridgeanabstractmathematicalframeworkandcognition.However,inmostcognitivequantummodelssuchassumptionshavebeenlimited(cf.Jones&Love,2011).Secondly,theelementsofquantummodelsaretypicallyassignedpsychologicalmeaning.Forexample,instaticmodels,representationsaretypicallysetbyreferencetotheempiricalsituationandprojectionsequencesjustifiedasembodyingrelevantcontextualinfluences

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(Busemeyeretal.,2011;Pothosetal.,2013;Wangetal.,2014).Entangledstatesareinterpretedasinstancesofstrongconnectednessbetweentherelevantentities(e.g.,Bruzaetal.,2015).Indynamicalmodels,Hamiltonianparameterstypicallyindicatedriftstothevariousoptionsandofteninterpretedasrelevantutilities(e.g.,Pothos&Busemeyer,2009;Trueblood&Busemeyer,2011).

Finally,theproblemofcomputationaltractabilityissharedbymanymodellingframeworks.Focusingonquantumvs.classicalprobabilities,theformerareoftensimplerthanthelatter,becauseincompatibilitycanreducethecomplexityoftherequiredjointprobabilitydistributions.TouseTverskyandKahneman’s(1983)Lindaexample,aclassicalprobabilityapproachrequiresconjunctionsnotjustinvolvingwhetherLindaisabanktellerorafeminist,butalsowhereshelives,herrelationships,howshelooksetc.Evenifmanyoftheseprobabilitiescanbeautomaticallysetthroughsuitablepriors,thecomplexityoftheprobabilitydistributionsmaystillbeprohibitive.Wethinkthesituationregardingbraintractabilityofthepostulatedoperationsinquantumtheoryisatleastequivalenttoorbetterthanthatforclassicaltheory(note,heuristicssimplifyingclassicalcomputationscouldbeadaptedforuseinquantumtheory,e.g.,Sanbornetal.,2010).Note,theideathatquantumprobabilitiesmaybesimplerthanclassicalonesexactlyunderwritesthepointthattheymaybemorelikelytobeadoptedwhendecisionmakersarelessengagedorfamiliarwithatask(Truebloodetal.,2017;Yearsley&Trueblood,inpress).

Regardingspecificmodelswithinaframework,arguablysuccessdependsonbalancingparsimonywithempiricalcoverage,comparisonswithotherformalisms,andgenerativevalue.Forthefirstcriterion,wedistinguishbetweenmodelsproposedbyresearchersfromthepsychologycommunity(e.g.,Busemeyer,Dzhafarov,Pothos,Trueblood,Wang,Yearsley)andonesfromotherresearchcommunities(physicsandeconomics;e.g.,Aerts,Atmanspacher,Haven,Khrennikov,Sozzo).Amongstpsychologists,wethinkthereisgeneralagreementregardingthecriteriaforgoodmodels,asabove,butinotherdisciplinessuchcriteriadiffer.So,wefocusthisdiscussiononmodelsproposedbypsychologyresearchersandcontestthatmostsuchmodelssatisfytheabovecriteria.First,thereisnoevidencethatquantumcognitivemodelsaremorecomplexthannon-quantumones.Wheredetailedcomplexitycomparisonswerecarriedout,theyfavoredthequantummodels(Busemeyeretal.,2015;Truebloodetal.,2017).

Second,quantumcognitivemodelshavebeenexaminedagainstrelevantnon-quantummodels.Regardingprobabilisticfallacies,suchasthedisjunctioneffect(Shafir&Tversky,1992)orthecategorizationdecisionparadigm(Townsendetal.,2000),quantummodelshavebeencomparedwithcloselymatchedclassicalprobabilitymodels(dynamicalmodelsbasedontheKolmogorovforwardequation,e.g.,Pothos&Busemeyer,2009;Trueblood&Busemeyer,2011;Wang&Busemeyer,2015).Insuchcases,wethinkthemotivationforemployingquantumprobabilityiscompelling:therearequantumandclassicalmatchedprobabilisticapproaches,humanbehaviorindicatesinconsistencywiththelatterbecauseofapparentinterferenceeffects,soweadopttheformer.Inothercases,forexample,similarityjudgments,therehasbeenaperceptionofadequateexistingmodels(e.g.,Ashby&Perrin,1988;Krumhansl,1988)butcloserscrutinyrevealedinconsistenciesandotherproblems(Pothosetal.,2013;Yearsleyetal.,2017).Note,onecanaskwhetherthereareresultsindicatinganecessityforquantumprobabilities.Inconsistencieswithclassicalprinciplescantypicallybeclassicallyaccommodatedthroughconditionalization(Section5.1.3;Dzhafarov&Kon,inpress).However,

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quantumtheoryhasbeendesignedforsuchresultsandcanoftenoffermoreelegantandparsimoniousmodels.

Theconjunctionfallacy(Tversky&Kahneman,1983)hasattractedanenormousamountofattentionanditisherethatthequantummodelhasbeenscrutinizedthemost.Itsmainweaknessesconcerntheassumptionthatthepredicatesintheconjunctionareorderedinacertainwayandtheoriginoftheassumptionofincompatibility.Webelievethatitsmainstrengthsarewidecoverageofrelatedfallaciesandthefactthatitprovidesanalternativeperspectiveforcorrectnessandrationalityinprobabilisticinference(Pothosetal.,2017).Wearguedthatalternativeformalismscansufferfromnarrowscope(Busemeyeretal.,2015)andincoherentassumptions(asforCostello&Watts,2016).

Onerelatedquestionisthis:arethereresultsdisprovingquantumcognitivemodels?Boyer-Kassemetal.(2016)andCostelloetal.(inpress)arguedthatthisthecase,intermsoffailuresofconsistencywiththeQQequality,constraintsthatoughttobeadheredtoifpeoplewereemployingquantumprobabilities,andlackofassociationbetweendifferentresultspredictedfromincompatibility(e.g.,ordereffectsandconjunctionfallacies).Therearetworesponses.First,wearguedthatsomeofthesecriticismsareincomplete(Section5.8.3).Second,wethinkitwouldbeextremelysurprisingifallbehavioralphenomenacanbemodeledwithinthequantumframework.InthesamewaythatincreasinglyresearchersareadvocatingaviewofcognitionasinvolvingaBayesianpartandanon-Bayesianpart(e.g.,Kahneman,2001;Sloman,1996),wethinktherewillbeboundsofapplicabilityinquantumcognitionmodels.Forexample,regardingsuper-correlations,whilequantumrepresentationscanviolatetheBellbound,theyarerestrictedtoanalternativebound–theso-calledTsirelsonbound–lendingthemselvestosimpletests.Inevitably,certaincognitiveprocesseswouldbeoutsidetheremitofprobabilisticframeworks,classicalorquantumandwouldrequire(plausibly)heuristicexplanations.Thequestionisthenwhetherthereareenoughsuccessfulcognitiveapplicationstowarrantfurtherinterestinquantummodels,andouranswerisyes.

Regardinggenerativevalue,manyofthemajorapplicationsofquantumtheorytocognitionhaveinvolvedestablishedresults,suchastheconjunctionfallacy(Tversky&Kahneman,1983),andsoaconcerniswhethertheapplicabilityofquantummodelsmightberestrictedtore-descriptionsofpreviousresults.However,thisreviewhaspresentedexamplesofquantummodelswhichhavehadgenerativevalueinpsychology.First,therehavebeenexaminationsofcompositionalityinmemoryorconceptualcombination,usingtheBell/CHSHinequality,providingnewempiricaltestsandevidence(e.g.,Aerts,2009;Aertsetal.,2015;Bruzaetal.,2015;Cervantesetal.,inpress).Second,eventhoughconstructiveinfluencesinpsychologyhavebeenexploredbefore(e.g.,Sharotetal.,2010),quantumcognitivemodelsmakespecificandconstrainedpredictionsregardingsuchinfluences,becauseoftherequirementofthecollapseofthestatevector.Suchpredictionshavebeenutilizedinempiricaldemonstrations(Kvametal.,2014;Yearsley&Pothos,2016)andhavebeenthebasisforaproposalofanewdecisionbias(Whiteetal.,2014,2015).Third,theQQequalityhasbeenasurprising,aprioripredictionfromquantumtheory,supportedinmanydatasets(Wangetal.,2014).Fourth,thelinksbetweenfamiliarity,cognitivestyleofthinking(Frederick,2005),andtheemergenceoffallacies,suchastheconjunctionfallacyandordereffects(Truebloodetal.,2017;Yearsley&Trueblood,inpress),haveprovidedanovelindividualdifferencesperspectivetodualrouteideasincognition.Thereareotherpredictionswhichhaveyettomeritdetailedempiricalexamination,forexampleregardingsequencingin

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temporalstructure(e.g.,Atmanspacher&Filk,2010).Overall,wethinkthattheapplicationofquantumtheorytocognitionhashadconsiderableempiricalgenerativeimpact.

Regardinggenerativepromise,wefinallynotethatquantumsystemscanallowspeediercomputationsthanclassicalsystems,incertaincases,duetoentanglementandsuperposition(e.g.,Grover,1997;Shor,1994).Thus,thepossibilityarisesthatquantum-likecomputationatthecognitivelevelmaybethefactorpartlyenablingtheapparentspeedwithwhichweareabletomakeinferences.Suchideasarepotentiallypromising,butcurrentlyunderdeveloped.Onedifficultyisthatquantumvs.classicalspeeduphasbeendocumentedonlyforspecificproblemsandthereisachallengetotranslatethemtopsychologicalones.Anotheristhatquantumspeeduptypicallyrequiresquantumphysicalsystems,forexampleitdependsonquantumgates,whichoperateon(real)superpositionorentangledstates(e.g.,Behrmanetal.,2000;Dongetal.,2008).Aswearecommittedtoaclassicalbrain,itisunclearhowtoapplysuchideastoquantumcognition.Analternativewaytoapproachthisissueiswhetherprobabilisticlearningmightbeaidedviaquantuminsteadofclassicalrepresentationsandthereareindicationsthatthismightbethecase(Bond,2018).7.Whatarethemainweaknessesofquantumcognitivemodels?Wehavecreatedthisreviewwithacriticalmindsetandwesummarizesomekeypointshere.First,mostquantumcognitivemodelsdependonincompatibility,asincompatibilityallowstheinterferenceeffectsnecessarytoviolatethelawoftotalprobability,produceordereffectsetc.Howcanincompatibilitybedeterminedapriori?Someinvestigatorsassessincompatibilityempirically,e.g.,withordereffects(Boyer-Kassemetal.,2016;Busemeyer&Wang,2014;Yearsley&Trueblood,inpress).Othershaveemployedproxymeasuresforincompatibility,notablystyleofthinking(amorereflexivevs.reflectivestyleofthinkingismorelikelytoleadtoincompatiblerepresentations;Frederick,2005;Truebloodetal.,2017;Yearsley&Trueblood,inpress).Lowfamiliaritywithquestioncombinationsisalsothoughttomakeincompatiblerepresentationsmorelikely(Busemeyeretal.,2011;Costelloetal.,inpress;Truebloodetal.,2017).Overall,however,amoregeneralapproachisneeded,whichwouldincludeclarityonthenatureofincompatibilityincognitionandhowitarises. Asecondrelatedissuehasbeenwhethertoadoptadynamicalvs.non-dynamicalapproach.Thisquestionarisesinmanyareasofpsychology(e.g.,adecisionresearchercanadoptanon-dynamicalheuristicvs.adriftdiffusionmodel).However,inquantumcognitionthischoiceisparticularlysignificant,sinceitdetermineswhetherinterferencecanarisefromincompatibility(e.g.,Busemeyeretal.,2011)orcompatibilityand‘mixing’amplitudesacrosstwoinitiallywell-separatespaces(e.g.,Pothos&Busemeyer,2009;Trueblood&Busemeyer,2011).Thisinturnsaffectswhetherthereisanexpectationofcooccurrenceinvolvingviolationsofthelawoftotalprobabilityandotherfallacies,suchasquestionordereffects(cf.Boyer-Kassemetal.,2016;Yearsley&Trueblood,inpress).Conceptually,thereseemstobeacleardistinctionbetweenthetwosituations,sinceinterferenceduetoincompatibilityisaboutwhetheronequestioncanchancetheperspectiveorcontextforanotherone(Pothosetal.,2017),whileinterferenceduetomixingfromdynamicalevolutionrelatestowhethertwoperspectiveswhichindividuallyrecommendthesameactionareinconsistentwitheachotherandsoclashwhenbroughttogether(asinfailuresofconsequentialreasoninginShafir&Tversky,1992).Operationally,distinguishingbetweenthesetwosituationsawaitsfurtherwork(cf.Gronchi&Strambini,2017).Note,theadoptionofdynamicalmodelsincognitivepsychologytypicallygoeshandinhandwith

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interestinmodellingresponsetimes,butasyettherehasbeenlimitedsuchapplicationwithquantummodels(e.g.,seeFuss&Navarro,2013,whoprovidedanopen-systemsquantummodelforreactiontimedistributioninatwo-alternativeforcedchoicetask). Third,violationsofBellinequalitieshaveledtocontroversyinpsychology(asinphysics).Thereareseveraldifficulties.Inphysics,CHSHinequalitiesarederivedunderpreciseassumptionssothatviolationsoftheseinequalitiescanbetracedtotherejectionofparticularassumptions.ThisisessentialsoastocharacterizethecauseofBellviolations,particularly‘spookyactionatadistance’,withitsfar-reachingimplicationsforspace-timeseparability.Inpsychology,asweassumeaclassicalbrain,‘spookyactionatadistance’isprecluded.Dzhafarovetal.’s(2015,2016;Dzhafarov&Kujala,2013)generalizationoftheCHSHinequalityenablesatestofwhetheraCHSH/Bellviolationisduetosignaling/disturbingmeasurementsvs.genuinecontextuality.However,someinvestigatorswouldbeemployingBellasatestofcompositionalityanditmaymatterlesswhetherthecauseissignalingvs.contextuality(e.g.,Bruzaetal.,2015).AnotherdifficultyisthataviolationofBelldoesnotmotivateaquantummodelinthestraightforwardwaythatempiricalevidenceforaninterferenceeffectdoes.EventhoughquantummodelscanviolatetheBellbound,thisoccursonlyforcertainstatesandforcertainquestionpairsandsothereisanadditionalonusforresearcherswishingtomakealinkbetweenBellviolationandaquantummodel.Note,signalingisconnectednessanditiseasytoshowviolationsofBellinconnectedclassicalsystems(Aerts,1982). Someotherdifficultiesincreatingquantumcognitivemodelsarenotuniquetosuchmodels(Busemeyeretal.,2014;Yearsley&Busemeyer,2016).Aspaceofstatesneedstobespecified;aninitialstateneedstobemotivated,typicallybyassuminguniformuncertaintyregardingtheavailableoptions;achoiceofstaticvs.dynamicalapproachneedsbemade;therelevantquestionsneedtoberepresented.Thelattercanrequiremoreworkforaquantummodelthanotherwise,sincethepreciserelationbetweenincompatiblerepresentationsneedstobespecified.Sometimestheapproachisgeneralandconditionsfortherequiredeffectareexaminedacrossrepresentationaloptions(e.g.,Busemeyeretal.,2011).Sometimesaspecificrepresentationisassumed,motivatedfromtheformofthestimuli(e.g.,Whiteetal.,2014)orparameterizedandfittedfromempiricalresults(e.g.,Truebloodetal.,2017).ArecentadvancehasbeenthemethodofBusemeyerandWang(inpress)forextractingquantumrepresentationsfromfrequencytablesviolatingclassicalconstraints.Inprinciple,thisprocedurecanbeadoptedmuchlikehowe.g.categorizationresearchershaveemployedmultidimensionalscalingtoextractvectorrepresentations(e.g.,Nosofsky,1984,1992).Afuturechallengeforthistechniqueishowtoaccommodaterepresentationsbeyondrays(Pothosetal.,2013). Inconclusion,therearemanyreasonstobeoptimisticaboutquantumcognitivemodels.Quantumtheoryprovidesasophisticatedtechnicalframeworkforprobabilisticinference,basedonnovelexplanatoryconcepts,whichcomplementsmodelingworkusingclassicalprobabilitytheory.Totheextentthatpsychologyresearchersbelievethatpartofcognitionshouldbeapproachedasprobabilisticinference,itmakesgoodsensetoexplorealternativeprobabilisticframeworks,andquantumtheoryisapossibilitywithconsiderablepotentialandpromise.AcknowledgmentsEMPwassupportedbyLeverhulmeTrustgrantRPG-2015-311andH2020-MSCA-IF-2015grant696331.

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