Necessary and Sufficient Conditions

Indeed, according to the standard theory, any truth will be a necessary condition for the truth of every statement whatsoever, and any falsehood ... StanfordEncyclopediaofPhilosophy Menu Browse TableofContents What'sNew RandomEntry Chronological Archives About EditorialInformation AbouttheSEP EditorialBoard HowtoCitetheSEP SpecialCharacters AdvancedTools Contact SupportSEP SupporttheSEP PDFsforSEPFriends MakeaDonation SEPIAforLibraries EntryNavigation EntryContents Bibliography AcademicTools FriendsPDFPreview AuthorandCitationInfo BacktoTop NecessaryandSufficientConditionsFirstpublishedFriAug15,2003;substantiverevisionWedJul6,2022 Ahandytoolinthesearchforprecisedefinitionsisthe specificationofnecessaryand/orsufficientconditionsforthe applicationofaterm,theuseofaconcept,ortheoccurrenceofsome phenomenonorevent.Forexample,withoutwaterandoxygen,there wouldbenohumanlife;hencethesethingsarenecessaryconditions fortheexistenceofhumanbeings.Cockneys,accordingtothe traditionaldefinition,areallandonlythosebornwithinthesound oftheBowBells.Hencebirthwithinthespecifiedareaisbotha necessaryandasufficientconditionforbeingaCockney. Likeotherfundamentalconcepts,theconceptsofnecessaryand sufficientconditionscannotbereadilyspecifiedinotherterms.This articleshowshowelusivethequestisforadefinitionoftheterms “necessary”and“sufficient”,indicatingthe existenceofsystematicambiguityintheconceptsofnecessaryand sufficientconditions.Italsodescribestheconnectionbetween puzzlesoverthisissueandtroublesomeissuessurroundingtheword “if”anditsuseinconditionalsentences. 1.PhilosophyandConditions 2.TheStandardTheory:Truth-functionsandReciprocity 3.ProblemsfortheStandardTheory 4.Inferences,ReasonsforThinking,andReasonsWhy 5.Conclusion Bibliography AcademicTools OtherInternetResources RelatedEntries 1.PhilosophyandConditions Anambitionoftwentieth-centuryphilosophywastoanalyseandrefine thedefinitionsofsignificantterms—andtheconceptsexpressed bythem—inthehopeofcastinglightonthetrickyproblemsof, forexample,truth,morality,knowledgeandexistencethatlaybeyond thereachofscientificresolution.Centraltothisgoalwas specifyingatleastinparttheconditionstobemetforcorrect applicationofterms,orunderwhichcertainphenomenacouldtrulybe saidtobepresent.Evennow,philosophy’suniquecontribution tointerdisciplinarystudiesofconsciousness,theevolutionof intelligence,themeaningofaltruism,thenatureofmoralobligation, thescopeofjustice,theconceptofpain,thetheoryofperception andsoonstillreliesonitscapacitytobringhighdegreesof conceptualexactnessandrigourtodiscussionsintheseareas. Ifmemoryisacapacityfortrackingourownpastexperiencesand witnessingsthenanecessaryconditionforPenelope rememberinggivingalectureisthatitoccurredinthepast. Contrariwise,thatPenelopenowremembersthelectureis sufficientforinferringthatitwasgiveninthepast.Ina well-knownattempttousetheterminologyofnecessaryandsufficient conditionstoilluminatewhatitisforonethingtobecauseof anotherthing,J.L.MackieproposesthatcausesareataminimumINUS conditions,thatis,“InsufficientbutNecessarypartsofa conditionwhichisitselfUnnecessarybutSufficient”fortheir effects(Mackie1965).AschemalikeMackie’sbecamethe foundationforthe“sufficientcausemodel”ofdiseasein epidemiology(seeRothman1976)andcontinuestohaveinfluenceon medicalthinkingaboutdiseasecausation,aswellasondefinitionsof causationinpsychologyandlaw(VanderWeele2017). What,then,isanecessary(orasufficient)condition?Thisarticle describesthedifficultiesinprovidingcompleteprecisionin answeringthisquestion.Althoughthenotionofsufficientcondition canbeusedindefiningwhatanecessaryconditionis(andvice versa),thereappearstobenostraightforwardwaytogivea comprehensiveandunambiguousaccountofthemeaningoftheterm “necessary(orsufficient)condition”itself. 2.TheStandardTheory:Truth-functionsandReciprocity Thefrontdoorislocked.Inordertoopenitinanormalwayandget intothehouse,Imustfirstusemykey.Anecessaryconditionof openingthedoor,withoutviolence,then,istousethekey.Soit seemstruethat IfIopenedthedoor,Iusedthekey. Canweusethetruth-functionalunderstandingof“if”to proposethattheconsequentofanyconditional(in(i),theconsequent is“Iusedthekey”)specifiesanecessaryconditionfor thetruthoftheantecedent(in(i),“Iopenedthedoor”)? Manylogicandcriticalthinkingtextsusejustsuchanapproach,and forconvenienceitcanbecalled“thestandardtheory” (seeBlumberg1976,pp.133–4,HintikkaandBachman1991,p. 328,MooreandParker2009,310–11,andSouthworthandSwoyer 2020,ch.3.2forexamplesofthisapproach). Thestandardtheorymakesuseofthefactthatinclassicallogic,the truth-function“p⊃q”(“If p,q”)isfalseonlywhenpistrue andqisfalse.Therelationbetween“p” and“q”inthiscaseissometimesreferredtoas materialimplication(seetheentryon thelogicofconditionals). Onthisaccountof“ifp,q”,ifthe conditional“p⊃q”istrue,and pholds,thenqalsoholds;likewiseifq isfalse,thenpmustalsobefalse(iftheconditionalis true).Thestandardtheoryclaimsthatwhentheconditional “p⊃q”istruethetruthofthe consequent,“q”,isnecessaryforthe truthoftheantecedent,“p”,andthetruthof theantecedentisinturnsufficientforthetruthofthe consequent.Thisreciprocalrelationbetweennecessaryandsufficient conditionsmatchestheformalequivalencebetweenaconditional formulaanditscontrapositive(“~q⊃ ~p”isthecontrapositiveof“p ⊃q”).Descendingfromtalkoftruthofstatements tospeakingaboutstatesofaffairs,wecanequallycorrectlysay,on thestandardtheory,thatusingthekeywasanecessaryconditionof openingthedoor. Giventhestandardtheory,necessaryandsufficientconditionsare conversesofeachother:B’sbeinganecessary conditionofAisequivalenttoA’sbeinga sufficientconditionofB(andviceversa).Soitseemsthat anytruth-functionalconditionalsentencestatesbothasufficientand anecessaryconditionaswell.SupposethatifNellieisanelephant, thenshehasatrunk.Beinganelephantisasufficientconditionof herhavingatrunk;havingatrunkinturnisanecessaryconditionof Nellie’sbeinganelephant.Indeed,theclaimaboutthe necessaryconditionappearstobeanotherwayofputtingtheclaim aboutthesufficientcondition,justasthecontrapositiveofa formulaislogicallyequivalenttotheoriginalformula. Itisalsopossible—accordingthethestandardtheory—to use“onlyif”toidentifyanecessarycondition:wecan saythatJonahwasswallowedbyawhaleonlyifhewasswallowedbya mammal,forifacreatureisnotamammal,itisnotawhale. Equivalentto(i)above,onthisaccount,isthesentence“I openedthedooronlyifIusedthekey”—aperfectly naturalwayofindicatingthatuseofthekeywasnecessaryfor openingthedoor. Thisaccountofnecessaryandsufficientconditionsisparticularly appositeindealingwithlogicalinferences.Forexample,fromthe truthofaconjunction,itcanbeinferredthateachcomponentistrue (if“pandq”istrue,then “p”istrueand“q”is true).Suppose,then,thatitistruethatitisbothrainingand sunny.Thisisasufficientconditionfor“itisraining” tobetrue.Thatitisrainingis—contrariwise—anecessary conditionforitbeingtruethatitisbothrainingandsunny.A similaraccountseemstoworkforconceptualanddefinitional contexts.Soiftheconceptofmemoryisanalysedastheconceptofa facultyfortrackingactualpastevents,thefactthataneventisnow inthepastisanecessaryconditionofmypresentlyrecollectingit. Ifwaterischemicallydefinedasaliquidconstitutedmainlyof H2O,thenifaglasscontainswater,itcontainsmainly H2O.ThattheglasscontainsmostlyH2Oisa necessaryconditionofitscontainingwater. Despiteitsinitialappeal,objectionstothestandardtheoryhave beenmadebytheoristsfrombothphilosophyandlinguistics.In summary,theobjectionsbuildontheideathat“if”in Englishdoesnotalwaysexpressauniformkindofcondition.If differentkindsofconditionsareexpressedbytheword “if”,theobjectorsargue,thenitwouldbewiseto uncoverthesebeforeengaginginattemptstoformalizeandsystematize theconceptsofnecessaryandsufficient.Intrying toshowthatthereisanambiguityinfecting “if”-sentencesinEnglish,criticshavefocusedontwo doctrinestheyregardasmistaken:first,thatthereisareciprocity betweennecessaryandsufficientconditions,and,second,that “ifp,q”and“ponlyif q”areequivalent. 3.ProblemsfortheStandardTheory GivenanytwotruesentencesAandB,the conditional“IfA,thenB”istrue.For example,provideditistruethatthesunismadeofgasandalsotrue thatelephantshavefourlegs,thenthetruth-functionalconditional “Ifelephantshavefourlegs,thenthesunismadeofgas” isalsotrue.However,thegaseousnatureofthesunwouldnot normallyberegardedaseitheraconceptually,orevenacontingently, necessaryconditionofthequadripedalityofelephants.Indeed, accordingtothestandardtheory,anytruthwillbeanecessary conditionforthetruthofeverystatementwhatsoever,andany falsehoodwillbeasufficientconditionforthetruthofany statementwecaretoconsider. Theseoddresultsdonotariseinsomenon-classicallogicswhereit isrequiredthatpremissesberelevanttotheconclusionsdrawnfrom them,andthattheantecedentsoftrueconditionalsarelikewise relevanttotheconsequents.Buteveninthoseversionsofrelevance logicwhichavoidsomeoftheseoddresults,itisdifficulttoavoid alloftheso-called“paradoxesofimplication”(seethe entryon thelogicofconditionals and relevancelogic). Forexample,acontradiction(astatementoftheform “pandnotp”)willbeasufficient conditionforthetruthofanystatementunlessthesemanticsforthe logicinquestionallowtheinclusionofinconsistentworlds. Theseodditiesmightbedismissedasmereanomalieswereitnotfor thefactthatwritershaveidentifiedanumberofotherproblems associatedwiththeideasofreciprocityandequivalencementionedat theendoftheprevioussection.Accordingtothestandardtheory, thereisakindofreciprocitybetweennecessaryandsufficient conditions,and“ifp,q”sentencescan alwaysbeparaphrasedby“ponlyifq” ones.However,aswritersinlinguisticshaveobserved,neitherof theseclaimsmatcheseitherthemostnaturalunderstandingof necessary(orsufficient)conditions,orthebehaviourof “if”(and“onlyif”)inEnglish.Consider,for example,thefollowingcase(drawnfromMcCawley1993,p.317): Ifyoutouchme,I’llscream. Whileinthecaseof(i)above,usingthekeywasnecessaryfor openingthedoor,noparallelclaimseemstoworkfor(ii):inthe naturalreadingofthisstatement,myscreamingisnotaconditionfor yourtouchingme.JamesMcCawleyclaimsthatthe “if”-clauseinastandardEnglishstatementgivesthe condition—whetherepistemic,temporalorcausal—forthe truthofthe“then”-clause.Thenaturalinterpretationof (ii)isthatmyscreamingdependsonyourtouchingme.Totakemy screamingasanecessaryconditionforyourtouchingmeseemstoget thedependenciesbacktofront.Asimilarconcernarisesifitis maintainedthat(ii)entailsthatyouwilltouchmeonlyifI scream. Asimilarfailureofreciprocityormirroringarisesinthecaseof thedoorexample((i)above).Whileopeningthedoordepended, temporallyandcausally,onusingthekeyfirst,itwouldbewrongto thinkthatusingthekeydepended,eithertemporallyorcausally,on openingthedoor.Sowhatkindofconditiondoestheantecedentstate? Itmaybehelpfultoconsiderthefollowingpuzzlingpairof conditionalsentences(amodificationofSanford1989, 175–6): Ifhelearnstoplay,I’llbuyLambertacello. LambertlearnstoplayonlyifIbuyhimacello. Noticethatthesetwostatementsarenotequivalentinmeaning,even thoughthestandardtheorytreats“ifp, q”asjustanotherwayofsaying“ponly ifq”.While(iii)statesaconditionunderwhichIbuy Lambertacello(presumablyhefirstlearnsbyusingaborrowedone, ormaybehehiresone),(iv)statesa(necessary)conditionofLambert learningtoplaytheinstrumentinthefirstplace(theremaybe otherstoo).Understoodthisway,thestatementstakentogetherleave pooroldLambertwithnoprospectofevergettingthecellofromme. Butif(iv)werejustequivalentto(iii),combiningthetwo statementswouldnotleadtoanimpasselikethis. Buthowelsecanweformulate(iii)intermsof“onlyif”? AnaturalEnglishequivalentissurprisinglyhardtoformulate. Perhapsitwouldbesomethinglike: LamberthaslearnedtoplaythecelloonlyifIhaveboughthim one. wheretheauxiliary(“has”/“have”)hasbeen introducedtotrytokeepdependenciesinorder.Yet(v)isnotquite right,foritcanbereadasimplyingthatLambert’ssuccessis dependentonmyhavingfirstboughthimacello—somethingthat iscertainlynotimpliedin(iii).Astillbetter(butnotcompletely satisfactory)versionrequiresfurtheradjustmentoftheauxiliary, say: LambertwillhavelearnedtoplaythecelloonlyifIhavebought himone. Thistime,itisnotsoeasytoread(vi)asimplyingthatIbought Lambertacellobeforehelearnedtoplay.Thesechangesinthe auxiliary(sometimesdescribedaschangesin“tense”)have ledsomewriterstoarguethatconditionalsinEnglishinvolve implicitquantificationacrosstimes(see,forexample,vonFintel 1998). Analternativeviewisthatdifferentkindsofdependencyare expressedbyuseoftheconditionalconstruction:(iv)isnot equivalentto(iii)becausetheconsequentof(iii)provideswhat mightbecalledareasonforthinkingthatLamberthas learnedtoplaythecello.Bycontrast,theverysame condition—thatIbuyLambertacello—appearstofulfila differentfunctionin(iv)(namelythatIfirsthavetobuyhima cellobeforehelearnstoplay).Inthefollowingsection,the possibilityofdistinguishingbetweendifferentkindsofconditionsis discussed.Theexistenceofsuchdistinctionswouldseemtoprovide someevidenceforasystematicambiguityaboutthemeaningof “if”andintheconceptsofnecessary(and sufficient)condition. Thepossibilityofambiguityintheseconceptsraisesafurther problemforthestandardtheory.Accordingtoit—asGeorgHenrik vonWrightpointsout(vonWright1974,7)—thenotionsof necessaryconditionandsufficientconditionarethemselves interdefinable: AisasufficientconditionofB =dftheabsenceofAisanecessary conditionoftheabsenceofB BisanecessaryconditionofA =dftheabsenceofBisasufficient conditionoftheabsenceofA Ambiguitywouldthreatenthisneatinterdefinability.Inthefollowing section,wewillexplorewhetherthereisanissueofconcernhere. ThepossibilityofsuchambiguityfiguresinworkbyPeterDowningon so-called“subjunctive”conditionals(Downing1959)andis exploredsubsequentlyasamoregeneralprobleminWilson(1979), Goldsteinetal.(2005),ch.6,andlaterwriters.Thegeneral argument,inbrief,isthatinsomecontextsthereisalackof reciprocitybetweennecessaryandsufficientconditionsunderstoodin acertainway,whileinothersituationstheconditionsdorelate reciprocallytoeachotherinthewayrequiredbythestandardtheory. Ifambiguityispresent,thenthereisnogeneralconclusionthatcan safelybedrawnaboutreciprocity,orlackofit,betweennecessary andsufficientconditions.Insteadtherewillbeaneedtodistinguish thesenseofconditionthatisbeinginvokedinaparticularcontext. Withoutspecificationofmeaningandcontext,itwouldalsobewrong tomakethegeneralclaimthatsentenceslike“ifp, q”aregenerallyparaphrasableas“p onlyifq”.Thephilosophicalliteraturecontains numerouswaystomakesenseofthelackofreciprocitybetweenthetwo kindsofconditions.Usingasemi-formalargument,CarstenHeld,for example,hassuggestedawayofexplainingwhynecessaryand sufficientconditionsarenotconversesbymakingappealtoaversion oftruthmakertheory(Held2016–seeOtherInternetResources). Inwhatfollows,theentryfocusesonaratherdifferentandpossibly morewidelyacceptedapproach:theattempttomakesenseofthelack ofreciprocityintermsofthedifferencebetweeninferential, evidentialandexplanatoryusesofconditionals. 4.Inferences,ReasonsforThinking,andReasonsWhy Arethefollowingtwostatementsequivalent?(Wertheimer1968, 363–4): Theoccurrenceofaseabattletomorrowisanecessaryand sufficientconditionforthetruth,today,of“Therewillbea seabattletomorrow.” Thetruth,today,of“Therewillbeaseabattle tomorrow”isanecessaryandsufficientconditionforthe occurrenceofaseabattletomorrow. DavidSanfordarguesthatwhile(vii)issensible,(viii)“has thingsbackward”(Sanford1989,176–7).Hewrites: “thestatementaboutthebattle,iftrue,istruebecauseofthe occurrenceofthebattle.Thebattledoesnotoccurbecauseofthe truthofthestatement”(ibid.)Whathemaymeanis thattheoccurrenceofthebattlegivesareasonwhythe statementistrue,butitisnotconverselythecasethatthetruthof thestatementprovidesanyreasonwhythebattleoccurred.Ofcourse, peoplesometimesdoundertakeactionsjusttomaketruewhattheyhad formerlysaid;sotherewillbeunusualcaseswherethetruthofa statementisareasonwhyaneventoccurred.Butthisseemsan unlikelyreadingoftheseabattlecase. NowletSbethesentence“Therewillbeaseabattle tomorrow”.IfSistruetoday,itiscorrecttoinfer thataseabattlewilloccurtomorrow.Thatis,eventhoughthetruth ofthesentencedoesnotexplaintheoccurrenceofthebattle,the factthatitistruelicensestheinferencetotheoccurrenceofthe event.Ascendingtothepurelyformalmode(seesection4oftheentry on RudolfCarnap), wecanmakethepointbyexplicitlylimitinginferencerelationsto onesthatholdamongsentencesorotherbearersoftruthvalues.Itis perfectlypropertoinferfromthetruthofStodaythatsome othersentenceistruetomorrow,suchas“thereisaseabattle today”.Since“thereisaseabattletoday”istrue tomorrowifandonlyifthereisaseabattletomorrow,thenwecan inferfromthefactthatSistruetodaythatitistruethat aseabattlewilloccurtomorrow. Fromthisobservation,itwouldappearthatthereisagapbetween whatistrueofinferences,andwhatistrueof“reason why”relations.Thereisaninferentialsenseinwhichthetruth ofSisbothanecessaryandsufficientconditionforthe occurrenceoftheseabattle.However,thereisanexplanatorysense inwhichtheoccurrenceoftheseabattleisnecessaryandsufficient forthetruthofS,butnotviceversa.Itwould appearthatincaseslike(vii)and(viii)theinferencesruninboth directions,whileexplanatoryreasonsrunonlyoneway.Whetherwe read(vii)asequivalentto(viii)seemstodependonthesensein whichthenotionsofnecessaryandsufficientconditionsarebeing deployed. Isitpossibletogeneralizethisfinding?Thedoorexample((i) above)seemstobeacaseinpoint.ThefactIusedthekeygivesa reasonwhyIwasabletoopenthedoorwithoutforce.ThatIopened thedoorwithoutforcegivesagroundforinferringthatIusedthe key.HereisafurtherexamplefromMcCawley: IfAudreywinstherace,wewillcelebrate. Audrey’swinningtheraceisasufficientconditionforus havingacelebration,andherwinningtheraceisthereasonwhywe willbecelebrating.Ourcelebration,however,isnotlikelytobethe reasonwhyshewinstherace.Inwhatsensethenisthecelebrationa necessaryconditionofherwinningtherace?Again,thereisaground forinferring:thatwedon’tcelebrateisagroundforinferring thatAudreydidn’twintherace.Englishtimereferenceappears sensitivetotheasymmetryuncoveredhere,inthewaynotedinthe previoussection.Thenaturalwayofwritingthecontrapositiveof (ix)isnottheliteral“Ifwewillnotcelebrate,thenAudrey doesnotwintherace”,butrathersomethinglike: Ifwedon’tcelebrate,Audreydidn’twinthe race. or Ifwearen’tcelebrating,Audreyhasn’twonthe race. oreven Ifwedon’tcelebrate,Audreycan’thavewonthe race. GilbertoGomes(Gomes2019)discussesindetailarangeofcasesin whichsometimeselaborateparaphrasingofthis,andother,kindscan beusedtopreservereciprocitybetweenconditionalsandtheir contrapositives. IanWilson(inWilson1979)proposesthattherearefivepossible relationssymbolisedby“if,…then…” includingthereasonwhyreading.Subsequentauthorshavenot adoptedthisproposal,preferringtofocusmainlyonthreeofthese relations,eachofwhichbearsonquestionsofnecessityand sufficiency.Firstistheimplicationrelationsymbolisedbythehook operator,“⊃”orperhapssomerelevantimplication operator.Suchanoperatorcapturessomeinferentialrelationsas alreadynoted.Forexample,wesawthatfromthetruthofa conjunction,itcanbeinferredthateachcomponentistrue(from “pandq”,wecaninferthat “p”istrueandthat“q”is true).Hook,orarelevantimplicationoperator,seemstocaptureone oftherelationsencounteredintheseabattlecase,arelationwhich canbethoughtofasholdingparadigmaticallybetweenbearersoftruth values,butcanalsobethoughtofintermsofstatesofaffairs.For thisrelation,weareabletomaintainthestandardtheory’s reciprocitythesiswiththelimitationsalreadynoted. Twofurtherrelations,however,areoftenimplicatedinreflectionson necessaryandsufficientconditions.Toidentifythese,considerthe differentthingsthatcanbemeantbysaying IfSolangewaspresent,itwasagoodseminar. Onescenariofor(xiii)isthesituationwhereSolangeisinvariablya livelycontributortoanyseminarsheattends.Moreover,the contributionsshemakesarealwaysinsightful,henceguaranteeingan interestingtimeforallwhoattend.Inthiscase,Solange’s presenceatleastinpartexplainsorisareasonwhythe seminarwasgood.Notethatsomewritersdistinguishexplanatory reasonsofthissortfromfull-bodiedexplanations(Nebel2018).A differentscenariodepictsSolangeassomeonewhohasanalmost unerringknackforspottingwhichseminarsaregoingtobegood,even thoughsheisnotalwaysactiveinthediscussion.Herattendanceata seminar,accordingtothisstory,providesareasonfor thinking—orisevidence—thattheseminaris good.Wemightsaythataccordingtothefirststory,theseminaris goodbecauseSolangeisatit.Inthesecondcase,Solangeisatit becauseitisgood.Examplesofthiskindwerefirstintroducedin Wilson(1979)followingonfromtheattackonthevalidityof contrapositioninDowning(1975).Noticethatthehook(asunderstood inclassicallogic)doesnotcapturethereasonforthinking relation,foritpermitsanytrueconsequenttobeinferredfromany otherstatementwhatever.Wheretheconditionalisareasonfor thinkingrelation,thentheantecedentmustprovidesomesupport fortheconsequent—hencehastoprovidesomesupportingevidence foracceptingtheconsequent.Aformalexplorationofonereason forthinkingrelationisgiveninVincenzoCrupiandAndrea Iacona’sstudyoftheevidentialconditional(Crupiand Iacona2020,andseethefullydevelopedlogicoftheevidential conditionalinRaidl,IaconaandCrupi2021).Thesewriterstreatthe reasonforthinkingorevidentiaryrelationasoneinwhich theantecedentprovidesa“reasonforaccepting”the consequent.SeealsoIgorDouvenonthe“evidentialsupport thesis”(Douven2008,2016). Thereasonwhyandreasonforthinkingthat conditionsmayhelptoshedlightonthepeculiaritiesencountered earlier.ThatIopenedthedoorisareasonforthinkingthatIused thekey,notareasonwhy.Incase(iii)above,thathelearnstoplay theinstrumentisthereasonwhyIwillbuyLambertacello,andthat Ibuyhimacellois(inthesamecase)areasonforthinking that—butnotareasonwhy—hehaslearnedtoplaythe instrument.OurcelebratingisareasonforthinkingthatAudreyhas wontheraceincase(ix),butisunlikelytobeareasonwhy. Althoughthereissometimesacorrelationbetweenreasonswhy,onthe onehand,andevidentiaryrelations,ontheother,fewgeneralisations aboutthiscanbesafelymade.IfAisareasonwhy Bhasoccurred(andsoperhapsalsoisevidencethat Bhasoccurred),thentheoccurrenceofBwill sometimesbeareasonforthinking—butnota guarantee—thatAhasoccurred.IfAisnomore thanareasonforthinkingthatBhasoccurred,then Bwillsometimesbeareasonwhy—butnotaguarantee that—Ahasoccurred.Goingbacktoexample(i)above, myopeningthedoor(withoutviolence)wasareasonforthinking,that istosayevidence,thatIhadusedthekey.ThatIusedthekey, however,wasnotjustareasonforthinkingthatIhadopenedthe door,butoneofthereasonswhyIwasabletoopenthedoor.Whatis importantisthatthe“if”clauseofaconditionalmaydo anyofthreethingsdescribedinthepresentsection.Oneoftheseis wellcapturedbyclassicaltruth-functionallogic,namely(i) introduceasentencefromwhichtheconsequentfollowsintheway modelledbyanoperatorsuchashook.Buttherearetwootherjobs that“if”maydoaswell,namely:(ii)stateareasonwhy whatisstatedintheconsequentisthecase;(iii)stateareasonfor thinkingthatwhatisstatedintheconsequentisthecaseevenwhen thisisnotareasonwhy. Ingeneral,ifexplanationisdirectional,itmaynotseemsurprising thatwhenAexplainsB,itisnotusuallythecase thatB,oritsnegation,isinturnanexplanationof A(oritsnegation).Audrey’swinningtheraceexplains ourcelebration,butourfailuretocelebrateisnot(normally)a plausibleexplanationofherfailuretowin.Solange’spresence mayexplainwhytheseminarwassuchagreatsuccess,butaboring seminarisnot—inanynormalsetofcircumstances—areason whySolangeisnotatit.Thisresultseemstounderminetheusual understandingthatifAisasufficientconditionof B,itwilltypicallybethecasethatBisa necessaryconditionforA,andthefalsityofBa sufficientconditionforthefalsityofA. Indefenceofcontraposition,itmightbearguedthatinthecaseof causalclaimsthereisatleastaweakformofcontrapositionthatis valid.Gomesproposes(Gomes2009)thatwhere‘A’ isclaimedtobeacausallysufficientconditionfor ‘B’,or‘B’acausally necessaryconditionof‘A’,thensomeformof reciprocitybetweenthetwokindsofconditionsholds,andsosome versionofcontrapositionwillbevalid.Goingbacktoexample(ii), supposewereadthisasstatingacausalcondition—thatyour touchingmewouldcausemetoscream.Gomessuggeststhat ‘A’denotesasufficientcauseofB, providedthat(1)‘A’specifiestheoccurrenceof aneventthatwouldcauseanotherevent‘B’,and doesthisby(2)statingaconditionthetruthofwhichissufficient forinferringthetruthof‘B’.Insuchacase, wecouldfurthermaintainthat‘B’,inturn, denotesanecessaryeffectof‘A’,meaningthat thetruthofBprovidesanecessaryconditionforthetruth ofA(Gomes2009,377–9).ItisnotablethatCrupiand Iacona’sevidentialconditionalalsopreservescontraposition, whichtheyclaimasadistinctivefeatureofoftheiranalysis,based ontheirunderstandingthat“providesevidencefor”is anotherwayofsaying“supports”(section6ofCrupiand Iacona2020).Forthem,ifSolange’spresenceprovidesevidence fortheseminarbeinggood(see(xiii)above),thenthe seminar’snotbeinggoodisevidenceofSolange’sabsence. They—likeGomes(2019)—thusrejectDowning’s(1975) generalscepticismaboutcontraposition.Wilson,bycontrast,argues thattheseminarnotbeinggoodmightalsobeareasonwhySolange doesnotattend.Butthis,heclaims,“savesonlytheappearance ofcontraposition”(Wilson1979,274–5). Whileitispossibletodistinguishthesedifferentrolesthe “if”clausemayplay(theremaybeotherstoo),itisnot alwayseasytoisolatethemineverycase.Theappealto “reasonswhy”and“reasonsforthinking” enablesustoidentifywhatseemtobeambiguitiesbothintheword “if”andintheterminologyofnecessaryandsufficient conditions.Unfortunately,theconceptofwhatisexplanatory itselfmaybetoovaguetobeveryhelpfulhere,forwecanexplaina phenomenonbycitingareasonforthinkingitisthecase,orby citingareasonwhyitisthecase.Asimilarvaguenessinfeststhe word“because”,asweseebelow.Consider,forexample, caseswheremathematical,physicalorotherlawsareinvolved(one locusclassicusforthisissueisSellars1948).Thetruthof “thatfigureisapolygon”issufficientforinferring “thesumofthatfigure’sexterioranglesis360 degrees”.Likewise,from“thesumofthefigure’s exterioranglesisnot360degrees”wecaninfer“the figureisnotapolygon”.Suchinferencesarenottrivial. Rathertheydependongeometricaldefinitionsandmathematical principles,andsothisisacaseofmathematicallynecessaryand sufficientconditions.Butitappearsquiteplausiblethat mathematicalresultsalsogiveusatleastareasonforthinkingthat becauseafigureisapolygonitsexteriorangleswillsumto360 degrees.Wemayevenbeabletothinkofcontextsinwhichsomeone claimsthatafigure’sbeingapolygonisareasonwhyits exterioranglessumto360degrees.Anditmightnotbeunnaturalfor someonetoremarkthatacertainfigureisapolygonbecauseits exterioranglessumto360degrees. Asimilarpointholdsforthetheoryofknowledgewhereitis generallyheldthatifIknowthatp,thenpistrue (seetheentryon theanalysisofknowledge). Thetruthofpisanecessaryconditionofknowingthat p,accordingtosuchaccounts.Insayingthiswedonotrule outclaimsstrongerthansimplysayingthatthetruthofp followsfromthefactthatweknowthatp.Thatabeliefis true—forexample—maybe(partof)areasonforthinkingit constitutesknowledge.Othercasesinvolveinferenceslicensedby physics,biologyandthenaturalsciences—inferencesthatwill involvecausalornomicconditions.Againthereisneedforcarein determiningwhetherreasonwhyorreasonfor thinkingrelationsarebeingstated.Theincreaseofmeankinetic energyofitsmoleculesdoesnotjustimplythatthetemperatureofa gasisrisingbutalsoprovidesareasonwhythetemperatureis increasing.However,iftemperatureisjustonewayofmeasuringmean molecularkineticenergy,thenachangeintemperaturewillbea reasonforthinkingthatmeankineticenergyofmoleculeshaschanged, notareasonwhyithaschanged. Asmentionedatthestartofthearticle,thespecificationof necessaryandsufficientconditionshastraditionallybeenpartofthe philosopher’sbusinessofanalysisofterms,conceptsand phenomena.Philosophicalinvestigationsofknowledge,truth, causality,consciousness,memory,justice,altruismandahostof othermattersdonotaimatstatingevidentialorexplanatory relations,butratheratidentifyinganddevelopingconceptualones (seeJackson1998foradetailedaccountofconceptualanalysis,and thesupplementaryentryon conceptionsofanalysisinanalyticphilosophy foranoverview).Butevenhere,thetemptationtolookfor reasonswhyorreasonsforthinkingthatisnotfar away.Whileconceptualanalysis,likedictionarydefinition,eschews evidentialandexplanatoryconditions,evidentialconditionsseemto benaturalconsequencesofdefinitionandanalysis.ThatNellieisan elephantmaynotbea(orthe)reasonwhysheisananimal,anymore thanthatafigureisasquareisareasonwhyithasfoursides.But someevidentialclaimsseemtomakesenseeveninsuchcontexts:being anelephantapparentlygivesareasonforthinkingthatNellieisan animal,andacertainfiguremaybesaidtohavefoursidesbecauseit isasquare,inanevidentialsenseof“because”. Tospecifythenecessaryconditionsforthetruthofthesentence “thatfigureisasquare”istospecifyanumberof conditionsincluding“thatfigurehasfoursides”, “thatfigureisonaplane”,and“thatfigureis closed”.Ifanyoneoftheselatterconditionsisfalse,then thesentence“thatfigureisasquare”isalsofalse. Conversely,thetruthof“thatfigureisasquare”isa sufficientconditionforthetruthof“thatfigureis closed”.Theinferentialrelationsinthiscaseare modelled—evenifinadequately—byanoperatorsuchas hook. Nowconsiderapreviousexample—thatofmemory.ThatPenelope rememberssomething—accordingtoastandardaccountof memory—means(amongotherthings)thatthethingrememberedwas inthepast,andthatsomepreviousepisodeinvolvingPenelopeplays anappropriatecausalroleinherpresentrecallofthethingin question.Itwouldbeamistaketoinferfromthecausalroleofsome pastepisodeinPenelope’scurrentremembering,thatthe definitionofmemoryitselfinvolvesconditionsthatareexplanatory inthereasonwhysense.ThatPenelopenowrememberssome eventisnotareasonwhyitisinthepast.Norisitareasonfor thinkingthatitisinthepast.Rather,philosophicaltreatmentsof memoryseekforconditionsthatareaprioriconstitutiveof thetruthofsuchsentencesas“Peneloperemembersdoing X”.Theuncoveringofsuchconditionssimplyprovides insightintowhether,andhow,“remember”istobe defined.Reasonwhyandreasonforthinkingthat conditionsseemnottoplayaroleinthispartofthe philosopher’senterprise. Finally,itshouldbenotedthatnotallconditionalsentences primarilyaimatgivingeithernecessaryorsufficientconditions.A commoncaseinvolveswhatmightbecalledjocular conditionals.AfriendofOctavia’smistakenlyrefersto “Plato’sCritiqueofPureReason”and Octaviasays,“IfPlatowrotetheCritiqueofPure Reason,thenI’mAristotle”.Ratherthanspecifying conditions,Octaviaisengaginginaformofreductio argument.SinceitisobviousthatsheisnotAristotle,herjoke invitesthelistenertoinfer(bycontraposition)thatPlatodidnot writetheCritiqueofPureReason.Anothercaseisthe so-calledconcessiveconditional,wheretheantecedentdoes notappeartobeaconditionontheconsequentofevenaninferential kind.Supposeweplanonhavingpicnicandhopeitwillbesunny.But evenifraincomes,wewillstillgo.Insuchacaseitdoes notseemplausibletomaintainthatthethreatofrainprovidesany conditionatallonacceptingtheconsequent.Suchconcessive conditionalsdonotadmitofcontraposition(CrupiandIacona2022, andcompareGomes2020).Othershavearguedthatincaseswherea conditionalconstructiondoesnotappeartobeputtingforwardany genuineconditions,theseare“nonconditional” conditionals(GeisandLycan1993)—inotherwordstheymayjust bedisguisedaffirmations. 5.Conclusion Giventhedifferentrolesfor“if”justidentified,itis hardlysurprisingthatgeneralisationsaboutnecessaryand/or sufficientconditionsarehardtoformulate.Suppose,forexample, someonetriestostateasufficientconditionforaseminarbeinggood inacontextwherethespeakerandallthelistenerssharetheview thatSolange’spresenceisareasonwhyseminarswouldbegood. Inthiscase,Solange’spresencemightbesaidtobea sufficientconditionoftheseminarbeinggoodinthesensethather presenceisareasonwhyitisgood.Now,isthereasimilarsensein whichthegoodnessoftheseminarisanecessaryconditionof Solange’spresence?Thenegativeanswertothisquestionis alreadyevidentfromtheearlierdiscussion.Ifwefollowvon Wright’sproposal,mentionedabove,wegetthefollowingresult: thattheseminarisnotgoodisasufficientconditionofSolangenot beingpresent.Butthiscannotplausiblybereadasasufficient conditioninanythinglikethesenseofareasonwhy.Atmost,the factoftheseminarnotbeingagoodonemaybeareasonforthinking thatSolangewasnotatit.Sohowcanwetell,ingeneral,whatkind ofconditionisbeingexpressedinan“if”sentence?As notedinthecaseoftheseabattle,whenrewritingintheformalmode capturesthesenseofwhatisbeingsaid,andwhentheformulations “ifp,q”and“ponlyif q”seemidiomaticallyequivalent,thenaninferential interpretationwillbeinorder,vonWright’sequivalenceswill hold,andthematerialconditionalgivesareasonableaccountofsuch cases.Asindicatedabove,therearelimitationstosuchanapproach anditprovidesatbestapartialaccountofthecircumstancesunder whichconditionalsentencesexpressnecessaryorsufficient conditions. Asalreadynoted,eventheinferentialuseof“if”isnot alwaysassociatedprimarilywiththebusinessofstatingnecessaryand sufficientconditions.Thisobservation,togetherwiththecasesand distinctionsmentionedinthepresentarticle,showstheneedfor cautionwhenwemovefromnaturallanguageconditionalstoanalysisof themintermsofnecessaryandsufficientconditions,andalsothe needforcautioninmodellingthelatterconditionsbymeansofformal operators.Itappearsthatthereareseveralkindsofconditionals, andcorrespondinglyseveralkindsofconditions.Asthedevelopments outlinedabovehaveshown,thereareasaresultseveralformal schemesfortranslatingandmakingsenseofthevarietyof conditionalsusedinnaturallanguageandtheconditions,ifany,that theyexpress. Bibliography Blumberg,AlbertE.,1976.Logic:AFirstCourse,New York:AlfredE.Knopf. 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Gomes,Gilberto,2009.“AreNecessaryandSufficient ConditionsConverseRelations?”,AustralasianJournalof Philosophy,87:375–87. –––,2019.“MeaningPreserving ContrapositionofNaturalLanguageConditionals”,Journalof Pragmatics152:46–60. –––,2020.“Concessiveconditionals withoutEvenifandnon-concessiveconditionalswithEven if”,ActaAnalytica,35:1–2. doi:10.1007/s12136-019-00396-y Goldstein,Laurence,Brennan,Andrew,Deutsch,MaxandLau,Joe Y.F.,2005.Logic:KeyConceptsinPhilosophy,London: Continuum. Hintikka,JaakkoandBachman,James,1991.WhatIf…? TowardExcellenceinReasoning,London:Mayfield. Jackson,Frank,1998.FromMetaphysicstoEthics:ADefenceof ConceptualAnalysis,Oxford:OxfordUniversityPress. Mackie,J.L.,1965.“CausesandConditions”, AmericanPhilosophicalQuarterly,12:245–65. McCawley,James,1993.EverythingthatLinguistshaveAlways WantedtoKnowaboutLogic*(Subtitle:*ButWereAshamedto Ask),Chicago:ChicagoUniversityPress. Moore,BrookeN.,andParker,Richard2009.Critical Thinking,ninthedition,NewYork:McGrawHill. Nebel,Jacob2019.“NormativeReasonsasReasonsWhyWe Ought”,Mind128(510):459–484. doi:10.1093/mind/fzy013 Raidl,Eric,Iacona,Andrea,andCrupi,Vincenzo2021.“The LogicoftheEvidentialConditional”,TheReviewofSymbolic Logic,FirstView:1–13.doi:10.1017/S1755020321000071 Rothman,K.J.1976.“Causes”,AmericanJournalof Epidemiology,104:587–92 Sanford,DavidH.,1989.IfP,thenQ:Conditionalsandthe FoundationsofReasoning,London:Routledge. Sellars,Wilfrid,1948.“Conceptsasinvolvinglawsand inconceivablewithoutthem”,PhilosophyofScience,15: 289–315. Southworth,JasonandSwoyer,Chris,2020.“Critical Reasoning:AUser'sManual,v.4.0”.PhilosophyOpen EducationalResources,2. [SouthworthandSwoyer2020availableonline] VanderWeele,TylerJ.,2017.“InvitedCommentary:The ContinuingNeedfortheSufficientCauseModelToday”, AmericanJournalofEpidemiology,185,11:1041–1043. doi:10.1093/aje/kwx083 VonFintel,Kai,1997.“BarePlurals,BareConditionalsand Only”,JournalofSemantics,14:1–56. VonWright,GeorgHenrik,1974.Causalityand Determinism,NewYork:ColumbiaUniversityPress. Wertheimer,Roger,1968.“Conditions”,Journalof Philosophy,65:355–64. Wilson,IanR.,1979.“ExplanatoryandInferential Conditionals”,PhilosophicalStudies,35: 269–78. Woods,M.,Wiggins,D.andEdgingtonD.(eds.),1997. Conditionals,Oxford:ClarendonPress. AcademicTools Howtocitethisentry. PreviewthePDFversionofthisentryatthe FriendsoftheSEPSociety. Lookuptopicsandthinkersrelatedtothisentry attheInternetPhilosophyOntologyProject(InPhO). Enhancedbibliographyforthisentry atPhilPapers,withlinkstoitsdatabase. OtherInternetResources Held,Carsten,2016. Conditions, downloadfromphilsci-archive.pitt.edu/12164. TheConceptsofNecessaryandSufficientConditions, maintainedbyNormanSwartz,Philosophy,SimonFraser University. 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