Logic, Models, and Paradoxical Inferences

Logic,Models,andParadoxicalInferencesISABELORENESANDP.N.JOHNSON-LAIRDAbstract:Peoplereject‘paradoxical’inferences,suchas:Luisadidn’tplaymusic;therefore,ifLuisaplayedsoccer,thenshedidn’tplaymusic.Forsometheorists,theyareinvalidforeverydayconditionals,butvalidinlogic.Thetheoryofmentalmodelsimpliesthattheyarevalid,butunacceptablebecausetheconclusionreferstoapossibilityinconsistentwiththepremise.Hence,individualsshouldacceptthemiftheconclusionsreferonlytopossibilitiesconsistentwiththepremises:Luisadidn’tplaysoccer;therefore,ifLuisaplayedagamethenshedidn’tplaysoccer.Twoexperimentscorroboratedthispredictionforthreesortsof‘paradox’,includingadisjunctiveparadox.1.IntroductionSomesententialconnectiveshaveinterpretationsthataretruthfunctionalandotherinterpretationsthatarenottruthfunctional.Forexample,theconjunction:PatistallandVivisshort,isnormallyinterpretedastruthfunctional,i.e.itistrueifandonlyifbothofitsclausesaretrue,andsoitissynonymouswith:VivisshortandPatistall.Incontrast,theassertion:PatpushedVivandVivfellover,isnormallyinterpreted,notastruthfunctional,butasimplyingatemporalorderbetweenthetwoevents:thefirsteventoccurredbeforethesecondevent,andperhapscausedit.Hence,theassertionisnotsynonymouswith:VivfelloverandPatpushedher.ThisresearchwassupportedbygrantSEJ2006-14714andConsolider-Ingenio2010CSD2008-00048fromtheSpanishGovernmenttothefirstauthor,andbyNationalScienceFoundationGrantSES0844851tothesecondauthortostudydeductiveandprobabilisticreasoning.WethankDavidBeltr´an,NickChater,JonathanEvans,VittorioGirotto,SangeetKhemlani,KlausOberauer,CarlosSantamaría,andWalterSchroyens,fortheirthoughtsaboutthepresentresearch.WealsothankMikeOaksfordandtwoanonymousreviewers,fortheircriticismsofpreviousdraftsofthearticle.Addressforcorrespondence:IsabelOrenes,UniversidaddeLaLaguna,FacultaddePsicología,DepartamentodePsicologíaCognitiva,Social,yOrganizacional,CampusGuajara,sn.38205LaLaguna,Tenerife,Spain.Email:[e-mail]&Language,Vol.27,No.4September2012,pp.357–377.©2012BlackwellPublishingLtd358I.OrenesandP.Johnson-LairdOnewaytoaccountforthesedifferencesistopostulatethat‘and’isambiguous,andhasbothatruth-functionalandanon-truth-functionalinterpretation.Grice(1989)madeananalogousargument,andheextendedhisanalysistoconditionals,suchas:IfPatpushedVivthenshefellover.Hearguedthatthemeaningofconditionalsistruthfunctional.Theyaretrueiftheirif-clausesarefalseortheirthen-clausesaretrue,orboth,andsotheyaretrueineverycaseexceptwhentheirif-clausesaretrueandtheirthen-clausesarefalse.Thissemanticscorrespondstotheconnectiveof‘materialimplication’inlogic.Table1showsitstruthtable.But,Gricealsoarguedthatpragmaticfactorscouldoverruletheseinterpretations.Notalltheorists,however,acceptthisview(e.g.Stalnaker,1968;EvansandOver,2004).Andtheyhaveanapparentlycogentreasonforrejectingit,whichconcernstheso-called‘paradoxes’ofmaterialimplication—thetopicofthepresentpaper.Theparadoxesofmaterialimplicationarenotrealparadoxes,buttheyaresufficientlyperplexingthatwewillcontinuetousethistermtodescribethem(seePfeiferandKleiter,2011;Schroyens,2010).Tounderstandtheparadoxes,wemustfirstclarifythemeaningof‘validity’,whichisopentomorethanoneinterpretation.Theusagewefollowisstandardinmodernlogicthatrecognizesthedistinctionbetweenprooftheoryandmodeltheory:aninferenceis‘valid’ifitsconclusionmustbetruegiventhetruthofitspremises,i.e.theconclusionistrueineverycaseinwhichthepremiseistrue(Jeffrey,1981,p.1).Asacorollary,aninferenceisinvalidifitsconclusioncouldbefalseeventhoughitspremisesaretrue.Anyinferenceisaccordinglyeithervalidornotvalid(invalid).Akeyquestionabouttheparadoxesiswhetherornottheyarevalid.Therearetwosortsofparadox,andthefirstsortisillustratedhere:Luciadidn’tweartheshoes.(Not-B)Therefore,IfLuciaworejewelrythenshedidn’tweartheshoes.(∴IfAthennot-B)Thecategoricalpremiseassertsthetruthoftheconditional’sthen-clause.AsTable1shows,amaterialimplicationistrueinthiscaseregardlessofwhetheritsif-clauseistrueorfalse.Theinferenceisaccordinglyvalidformaterialimplication,becausethetruthofthepremisesufficesforthetruthoftheconclusion.PatishereVivishereIfPatisherethenVivishereTrueTrueTrueTrueFalseFalseFalseTrueTrueFalseFalseTrueTable1Anillustrationofthetruthtableformaterialimplicationinlogic.©2012BlackwellPublishingLtdLogic,Models,andParadoxicalInferences359Thesecondsortofparadoxisillustratedhere:Itwon’traintoday.(Not-A)Therefore,ifitrainstodaythentheforecastisright.(∴IfAthenB)Thecategoricalpremisedeniestheconditional’sif-clause,and,asTable1alsoshows,theinferenceisvalidformaterialimplication.Yet,bothsortsofparadoxareimplausible,andindividualstendtorejectthem.Onthisaccount,ifconditionalsareinterpretedasmaterialimplicationsthentheparadoxesarevalidinsemanticsensethatwehaveadoptedfrommodernlogic:theypreservetruth.But,theirunacceptabilityhasledsometheoriststoarguethatconditionalsindailylifearenotinterpretedasmaterialimplications.Stalnaker(1968)arguedthatthefalsityoftheantecedentisneversufficientreasontoaffirmaconditional,evenanindicativeconditional,andsotheparadoxesareinvalid(cf.Stalnaker,19