HowWeDetectLogicalP.N.Johnson-Laird,PaoloLegrenzi,andVittorioGirottoDe - PDF document

HowWeDetectLogicalP.N.Johnson-Laird,PaoloLegrenzi,andVittorioGirottoDepartmentofPsychology,PrincetonUniversity;UniversityofArchitecture,Venice,Italy;andLaboratoiredePsychologieCognitive,UniversityofProvence,Aix-en-Provence,FranceABSTRACT—Howdoindividualsdetectinconsistencies?Accord-ingtothetheorydescribedinthisarticle,theysearchfora AddresscorrespondencetoP.N.Johnson-Laird,DepartmentofPsychology,PrincetonUniversity,Princeton,NJ08544. CURRENTDIRECTIONSINPSYCHOLOGICALSCIENCEVolume13—Number22004AmericanPsychologicalSociety themsegregatedintoseparatesets.Yourbeliefsabout,say,WaltWhitmanhavenothingtodowithyourbeliefsaboutgolf.Inthisway,youhavesomechanceofmaintainingconsistencywithinsets,thoughinconsistenciesmayarisefromonesettoanother.Evenwithseparatesmallsetsofbeliefs,thepsychologicalproblemremains:Howdoyoudeterminewhetherasetisconsistent?AMODELTHEORYOFTHEEVALUATIONOFCONSISTENCYOnewaytoevaluateconsistencyistorelyonformalrulesofinferencefromlogic.Hereisatypicalformalrule:IfAthenB.Therefore,B.canrefertoanypropositionswhatsoever.Somepsychologicaltheoriespostulatesuchrulestoexplaindeductiveinferencesfrompremisestoconclusions(e.g.,Rips,1994).Buttheevaluationofconsistencyisnotthesametask.Toevaluateconsistency,youneedtodeterminewhetherasetofpropositionscanallbetrue.Nevertheless,formalrulescouldbeadaptedtocopewithconsistency.Inthisap-proach,youselectapropositionfromthesetandtrytoproveitsne-gationfromtheremainingpropositions.Ifyousucceed,thenthesetisinconsistent;otherwise,itisconsistent.Theprocedureseemsim-plausiblepsychologically,however.Wehavethereforeproposedadif-ferenttheorybasedonmentalmodels.Werefertothisaccountasthe‘‘model’’theory.Inbrief,themodeltheoryisasfollows(see,e.g.,Johnson-Laird&Byrne,1991):Inreasoning,individualstrytoenvisagewhatispossiblegiventhepremisesandtheirownknowledge.Eachpossibilitythattheyenvisageisrepresentedinaseparatementalmodel.Amentalmodelhasthesamestructureasthepossibilityitrep-resents.Forexample,amodelrepresentingthepossibilitythatPatownsthreecarswillhaveatokenrepresentingPat,threetokensrepresentingcars,andarelationrepresentingownershipholdingbetweenthetokenforPatandeachofthetokensforcars.Mentalmodelsfollowaprincipleof:Amodelrepresentspropositionsinthepremisesonlyiftheyaretrueinthepossibilitythatthemodelrepresents.Consider,forinstance,thefollowingdisjunctionaboutsomeshapesonablackboard:Thereiseitheracircleontheboardoratriangle,orboth.Adisjunctionisasentencemadeupofseparateclausesconnectedbytheword‘‘or.’’Thementalmodelsofthisdisjunctionareasfollows(eachrowdenotesamodelofaseparatepossibility):Thus,therstmodeldoesnotrepresentthatinthispossibility,itisfalsethatthereisatriangle.Theneglectofwhatisfalseisusuallyharmless,but,asweshowlater,itdoesyieldpredictableTodrawaconclusionfromthepremises,reasonersndapropo-sitionintheirmodelsthatisnotassertedexplicitlyintheprem-ises.Theycheckwhetheritholdsinall,some,oraproportionofthemodels,andtheyformulateacorrespondingconclusionaboutitsnecessity,possibility,orprobability.Tocontinuetheexample,supposeindividualsaregiventhefollowingconclusionandaskedwhetheritfollowsfromthepremiseabouttheshapesontheItispossiblethatthereisbothacircleandtriangle.Reasonerscandeterminethatthisconclusiondoesfollowfromthepremise,becausethethirdmodelcorroboratesit.Theyalsotendtoestimatetheprobabilityofbothacircleandatrianglebeingontheboardasonethird;thatis,theyassumethateachpossibilityisequiprobable.Thisaccountthereforeprovidesanintegratedthe-oryofbothlogicalandprobabilisticreasoningbasedonpossibil-ities(Johnson-Laird,Legrenzi,Girotto,Legrenzi,&Caverni,Reasonerscanrefuteaninvalidconclusionusingacounter-example,whichisamodelofthepremisesinwhichtheconclusionisfalse.Oneassumptionextendsthetheorytodealwithconsistency(Johnson-Laird,Legrenzi,Girotto,&Legrenzi,2000):Individualsevaluatetheconsistencyofasetofpropositionsbysearchingforamodelofapossibilityinwhichtheyarealltrue.Iftheyndsuchamodel,theyevaluatethepropositionsascon-sistent;otherwise,theyevaluatethepropositionsasinconsistent.Inanexperiment,weaskedtheparticipantstoconsiderwhetherthesepropositionsaboutwhatisonatablecouldallbetrueatthesametime:1.Ifthereisn’tanapplethenthereisabanana.Ifthereisabananathenthereisacherry.Thereisn’tanappleandthereisacherry.Individualsshouldbeginevaluatingtheconsistencyofthesepropo-sitionsbyconstructingapossibilityfromtherstproposition,whichisofaformknownasa.Accordingtothetheory(Johnson-Laird&Byrne,2002),itsmostsalientpossibilityisapplebananawhere‘‘’’denotesnegation,‘‘apple’’denotesthepresenceofanapple,and‘‘banana’’denotesthepresenceofabanana.Thenextstepistousetheinformationinthesecondconditionalto‘‘update’’theapplebananacherryThethirdpropositionistrueinthispossibility,andsothesetshouldbejudgedasconsistent.Now,considerthisdescription:Volume13—Number2 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2.Thereisanappleorthereisabanana.Thereisn’tabananaorthereisacherry.Thereisn’tanappleandthereisacherry.Individualsinterprettherstproposition,whichisadisjunction,ascompatiblewiththreedifferentpossibilities,shownhereonseparateapplebananaAccordingtothetheory,reasonersshouldbeginbyenvisagingtherstofthesepossibilities:Thenextstepistoupdatethispossibilitywiththerstmodeloftheseconddisjunction:Thispossibility,however,isnotconsistentwiththethirdproposition,andsoreasonersshouldretracetheirstepsandconsiderthesecondmodeloftherstdisjunction:Theseconddisjunctionupdatesthepossibility.Thepresenceofabananaeliminatesthepossibilitycorrespondingtothedisjunction’srstclause,andsotheremustbeacherry:bananacherryWhenmodelslackanyinformationaboutaproposition,individualstendtointerpretthislackasequivalenttothenegationofthecorre-spondingproposition.Thismodellacksanapple,andsoindividualsinferthatthereisnotanapple,butthemodelhasacherry,andsoitisconsistentwiththethirdproposition.However,incontrasttoProblem1,whichcanbeevaluatedbygeneratingasinglementalmodelofthepropositions,Problem2requiresndingasecond,alternativemodel.Hence,theevaluationofProblem2shouldbeharderthantheeval-uationofProblem1.LIFEISEASIERWHENTHEFIRSTMODELSUFFICESOurrstexperimenttestedwhethertheevaluationofconsistencyiseasierwhentherstmodelsufces,asinProblem1,thanwhenitdoesnot,asinProblem2(Johnson-Lairdetal.,2000).Itexploitedthefactthatadisjunctionimpliesaconditional.Forexample,thedisjunctionThereisanappleorthereisabanana.impliestheconditional:Ifthereisnotanapplethenthereisabanana.Individualsrecognizethisimplication(Ormerod&Richardson,2003),thoughtheimplicationcanbeblockedwhentheinterpretationofadisjunctionorconditionalismodiedbyknowledge(Johnson-Laird&Byrne,2002).Ourexperimentcontrastedeightsortsofconditionalproblems(includingProblem1)witheightsortsofdisjunctiveprob-lems(includingProblem2).Wetested522ofthebesthighschoolgraduatesinItaly,askingthemtodecideforeachproblemwhetheritwaspossibleforthethreesentencestobetrueatthesametime.Theresultsshowedthattheproblemsbasedonconditionalshadarobustadvantageinaccuracy(of15%)overthosebasedondisjunctions,especiallywhentheparticipantscorrectlyjudgedthatasetwasconsistent.Therewasasmallerbutsignicantadvantageforcon-sistentproblemsoverinconsistentproblems.Couldtheresultsreectsomedifferencebetweenconditionalsanddisjunctionsotherthanthenatureoftheirmentalmodels?Asubsequentexperimenttestedthemodeltheorymorestringently(Legrenzi,Girotto,&Johnson-Laird,2003).Iftheparticipantsjudgedthatasetwasconsistent,theyalsohadtodescribethepropertiesofthecorrespondingentity.Considerthesepropositions:3.Thechairissaleableifandonlyifitiselegant.Thechairiselegantifandonlyifitisstable.Thechairissaleableoritisstable,orboth.OnepossibilityaccordingtothersttwopropositionsissaleableelegantstableThethirdpropositionistrueinthispossibility,andsoindividualsshoulddescribethechairassaleable,elegant,andstableIncontrast,considerthefollowingdescription:4.Thechairisunsaleableifandonlyifitisinelegant.Thechairisinelegantifandonlyifitisunstable.Thechairissaleableoritisstable,orboth.OnepossibilityaccordingtothersttwopropositionsisunsaleableinelegantunstableThethirdpropositionisfalseinthispossibility,andsoindividualshavetosearchforanalternativemodel.Therstpropositionisalsocompatiblewithachairthathastheseproperties:saleableelegantThesecondpropositioncanbeusedtoupdatethispossibility:saleableelegantstableThethirdpropositionisconsistentwiththismodel.ButtheevaluationofconsistencyshouldbeharderforProblem4thanforProblem3becauseoftheneedtorejecttheinitialmodel.Thisparticularprob-lemmightbeharderbecauseitcontainsnegatives,andsotheiroccurrencewascounterbalancedinthetwosortsofproblemsillus-tratedin3and4.Theresultsshowedthatparticipantsweremuchmoreaccuratewhentherstmodelsufced(97%correct)thanwhenitdidnot(39%correct).ILLUSIONSOFCONSISTENCYANDINCONSISTENCYTheprincipleoftruthiscentraltothemodeltheory.Eachsimpleproposition,afrmativeornegative,isrepresentedinamentalmodelonlyifitistrueinthepossibilitythatthemodelrepresents.Forin-stance,considerandisjunction,whichallowsforonepos-sibilityortheother,butnotboth:Volume13—Number2 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Thereisanappleorelsethereisabanana.Thisdisjunctionhastwomentalmodels:Therstmodeldoesnotrepresentexplicitlythatitisfalsethatthereisabananainthispossibility,andthesecondmodeldoesnotrepresentexplicitlythatitisfalsethatthereisanappleinthispossibility.Reasonersmake‘‘mentalfootnotes’’aboutwhatisfalse,butsoonforgetthem.Withthesefootnotes,however,theycanconvertmentalmodelsintofullyexplicitapplebananaHence,asindividualsunderstand,‘‘orelse’’meansthatwhenonepropositionistrue,theotherisfalse.Acomputerimplementationofthetheoryshowedthatinsomecasesmentalmodelsarewrongaboutwhatispossible(Johnson-Laird&Savary,1999).Ifreasonersrelyonmentalmodels,theyshouldthereforesuccumbtoillusionsincasesinwhichfalsitymatters.Theyshouldevaluatesomesetsofpropositionsasconsistentwheninfacttheyareinconsistent,andviceversa.Considerthisdescription:Thetrayisportableorelsenotbothbeautifulandheavy.Thetrayisportableandnotbeautiful.ThementalmodelsofthedisjunctioninthisdescriptionarebeautifulheavyTherstmodelincludesbutlacks,andsoin-dividualsshouldjudgethatthesecondassertionisconsistentwithit.Theywouldbewrong.Peopletake‘‘orelse’’tomeanthatwhenonepropositionistrue,theotherisfalse.Soifitistruethatthetrayisportable,thenfromthedisjunctiveassertionitisfalsethatthetrayisnotbothbeautifulandheavy;thatis,itbothbeautifulandheavy.Thatisinconsistentwiththesecondassertion.Andifitisfalsethatthetrayisportable,thenthattooisinconsistentwiththesecondas-sertion.Hence,thetwoassertionsareinconsistent.Incontrast,supposethatthesamedisjunctiveassertionoccurswithadifferentsecondassertion:Thetrayisnotbeautifulandnotheavy.Onceagain,individualsshouldjudgethatthetwoassertionsareconsistent(seethefourthofthefourpossibilitiesforthedisjunction).Thistime,however,theyarecorrect.Anexperimentcomparedcontrolproblemsthatthetheorypredictsshouldyieldcorrectevaluationswithexperimentalproblemsthatthetheorypredictsshouldyieldillusions,eitherofconsistencyorofin-consistency.Theresultscorroboratedthetheory’spredictions:Theparticipantsrespondedmoreaccuratelytosixsortsofcontrolprob-lems(86%correct)thantosixsortsofillusions(27%correct).Only11ofthe459participantswentagainstthistrend(Legrenzietal.,2003).Couldtheparticipantshavemisunderstood‘‘orelse,’’failingtorealizethatwhenoneofitsconstituentpropositionsistrue,itsotherconstituentpropositionisfalse?Afurtherexperimentconveyeditsmeaningusinganunambiguousrubric:‘‘Onlyoneofthefollowingassertionsistrue.’’Onceagain,theparticipantssuccumbedtoillu-sionsofconsistencyandtoillusionsofinconsistency,butrespondedcorrectlytothecontrolproblems(Legrenzietal.,2003).Socialpsychologistshaveknownformanyyearsthatindividualstrytoadjusttheirbeliefstoaccommodateinconsistencies.Inafamouscase,membersofacultwhoseleaderhadpredictedtheendoftheworldreasonedthatthepredictionhadfailedasaresultoftheirpiouslabors(Festinger,Riecken,&Schachter,1964).ButthepioneeringworkofKahnemanandTversky(e.g.,2000)showedthatindividualsarein-consistentintheirchoicesandintheirjudgmentsofprobabilities.Evenexpertsmakethesamemistakeswhentheirmemoryistaxed.Yet,untilrecently,therewerefewinvestigationsofhowindividualsdetectinconsistencies.Themodeltheoryproposesthattheydosowhentheyareunabletoaccommodateaproposition—anobservationoranassertion—intotheirexistingmentalmodels.Twostrandsofevidencehavecorroboratedthisidea.First,inevaluatingasetofassertionsasconsistent,participantsaremoreaccuratewhentherstmodelofadescriptionsufcesthanwhentheymustlookforamodelofanalternativepossibility.Second,whenfalsitymatters,theysuc-cumbtoillusionsofconsistencyandofinconsistency.SupposethatyouarewaitingforPaolo,asoneofusoncewas.Youbelievethathehasgonetogetthecar,andthatifhehasgonetogetthecarhewillbebackshortly.Youinfercorrectlythathewillbebackshortly.Buthedoesnotreturnevenafteraquarterofanhour.Youdetecttheinconsistencywiththeconsequenceofyourbeliefs.Thatisonlytherststep,thoughtheonethatwehavetriedtoexplainhere.Thenextstepistomodifyyourbeliefs.Psychologistshavebeguntostudythisprocess(e.g.,Elio&Pelletier,1997),buttheyhavenocomprehensiveaccountofit.Indeed,youdonotjustchangeyourbeliefs,youtrytoformulateanexplanationthatresolvestheincon-sistency.Perhaps,forexample,thecarwouldnotstart.Howyoucreatetheseexplanationsis,atpresent,amystery.RecommendedReadingJohnson-Laird,P.N.(2001).Mentalmodelsanddeduction.TrendsinCognitive,434–442.Kahneman,D.,&Tversky,A.(Eds.).(2000).(SeeReferences)Legrenzi,P.,Girotto,V.,&Johnson-Laird,P.N.(2003).(SeeReferences)ThisresearchwassupportedbyagrantfromtheNationalScienceFoundation(Grant0076287)totherstauthortostudystrategiesinreasoning,andbygrantsfromMIUR(ItalianMinistryofUniversitiesandResearch)andCNR(ItalianNationalCouncilforScienticResearch)totheotherauthors.Black,A.,Freeman,P.,&Johnson-Laird,P.N.(1986).Plausibilityandthecomprehensionoftext.BritishJournalofPsychology,51–62.Elio,R.,&Pelletier,F.J.(1997).Beliefchangeaspropositionalupdate.CognitiveScience,419–460.Volume13—Number2 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Festinger,L.,Riecken,H.W.,&Schachter,S.(1964).Whenprophecyfails.NewYork:Harper&Row.Jeffrey,R.C.(1981).Formallogic,itsscopeandlimits(2nded.).NewYork:McGraw-Hill.Johnson-Laird,P.N.,&Byrne,R.M.J.(1991).Deduction.Hillsdale,NJ:Erlbaum.Johnson-Laird,P.N.,&Byrne,R.M.J.(2002).Conditionals:Atheoryofmeaning,pragmatics,andinference.PsychologicalReview,646–678.Johnson-Laird,P.N.,Legrenzi,P.,Girotto,V.,Legrenzi,M.,&Caverni,J.-P.(1999).Naveprobability:Amentalmodeltheoryofextensionalrea-PsychologicalReview,62–88.Johnson-Laird,P.N.,Legrenzi,P.,Girotto,V.,&Legrenzi,M.S.(2000).Illusionsinreasoningaboutconsistency.,531–532.Johnson-Laird,P.N.,&Savary,F.(1999).Illusoryinferences:Anovelclassoferroneousdeductions.,191–229.Kahneman,D.,&Tversky,A.(Eds.).(2000).Choices,values,andframesCambridge,England:CambridgeUniversityPress.Legrenzi,P.,Girotto,V.,&Johnson-Laird,P.N.(2003).Modelsofconsistency.PsychologicalScience,131–137.Ormerod,T.C.,&Richardson,J.(2003).Onthegenerationandevaluationofinferencesfromsinglepremises.Memory&CognitionPerrow,C.(1984).Normalaccidents:Livingwithhigh-risktechnologies.NewYork:BasicBooks.Rips,L.(1994).Thepsychologyofproof.Cambridge,MA:MITPress.Volume13—Number2 P.N.Johnson-Laird,PaoloLegrenzi,andVittorioGirotto