The Concept of Necessary Conditions and Sufficient Conditions

Definition: A condition A is said to be necessary for a condition B, if (and only if) the falsity (/nonexistence /non-occurrence) [as the case may be] of A ... Copyright©Norman Swartz,1997 DepartmentofPhilosophy SimonFraserUniversity TheConceptsof NecessaryConditionsandSufficientConditions Contents Introduction Definitionof"necessarycondition" Definitionof"sufficientcondition" Necessaryconditionsthatarenotjointlysufficient Sufficientconditionsthatarenotnecessary Theconceptofconverserelations "Isanecessaryconditionfor"and"isasufficientconditionfor"areconverserelations Fourpossiblecombinations Practiceexercise#1 Practiceexercise#2 Differentkinds(ormodes)ofnecessarycondition 1.Introduction Everyoneisfamiliarwiththeconceptofanecessarycondition.For example,weallknowthatairisnecessaryfor(human)life.Without air,thereisno(human)life.Similarly,amicroscope(orsomeother instrument)isnecessaryforhumanbeingstoseeviruses.(Viruses aretoosmalltobeseenbythenakedeye.) Similarly,everyoneisfamiliarwiththeconceptofasufficient condition.Forexample,itsuffices(i.e.itissufficientfor)an object'shavingfoursidesthatitisasquare.Or,again,itis sufficientforyourhavingsomethingtodrinkthatyouhaveaglassof Coca-Cola®. 2.Definitionof"necessarycondition" Definition:AconditionAissaidtobe necessaryforaconditionB,if(andonlyif) thefalsity(/nonexistence/non-occurrence)[asthecasemaybe] ofAguarantees(orbringsabout)thefalsity(/nonexistence /non-occurrence)ofB. Socommonisthisnotionofnecessary conditionthatthereare,notsurprisingly,agreatmanywaystoexpress thatsomethingisanecessarycondition.Hereareanumberofexamples,all -moreorless-sayingthesamething: "Airisnecessaryforhumanlife." "Humanbeingsmusthaveairtolive." "Withoutair,humanbeingsdie(i.e.donotlive)." "Ifahumanbeingisalive,thenthathumanbeinghasair(tobreathe)." Inan`if-then'statement(suchasthelastexampleimmediatelyabove),the clausethatfollowsthe"then"(i.e.theso-called`consequent')statesthe necessaryconditionfortheantecedent(i.e.theclauseimmediatelyfollowing the"if").Thusthatsomehumanbeinghasair(tobreathe)isa necessaryconditionforthathumanbeing'sbeingalive. Let'slookatsomefurtherexamples(again,allsayingprettymuchthesame thing).Noticetheuseof"if-then"(inthesixth[highlighted]example): "Havingamicroscope(orsomeotherinstrument)isanecessarycondition for(our)seeingviruses." "Amicroscope(orasimilarinstrument)isneededtoseeviruses." "Humanbeingsmustuse(have)amicroscopetoseeviruses." "Humanbeingscannotseeviruseswithoutamicroscope." "Anyonewhoseesviruseshas(uses)amicroscope." "Ifsomeoneseesviruses,thenthatpersonusesa microscope." NOTE "Withoutamicroscope,apersoncannotseeviruses." "Ifapersondoesnothave(theuseof)amicroscope,thenthatpersondoes notseeviruses." "Whoeverlacksamicroscopedoesnotseeviruses." "Onemusthaveamicroscopetoseeviruses." Butofcourse,aswewellknow,ingeneralanecessaryconditionisnota sufficientcondition.Allsortsofconditionsmaybenecessary forothers,butdonot-bythemselves-sufficefor,or guarantee,thoseothers. 3.Definitionof"sufficientcondition" Definition:AconditionAissaidtobesufficient foraconditionB,if(andonlyif)the truth(/existence/occurrence)[asthecasemaybe]ofA guarantees(orbringsabout)thetruth(/existence/occurrence)ofB. Forexample,whileairisanecessaryconditionforhumanlife,itisby nomeansasufficientcondition,i.e.itdoesnot,byitself,i.e.alone, sufficeforhumanlife.Whilesomeonemayhaveairtobreathe,thatperson willstilldieifs/helackswater(foranumberofdays),hastakenpoison, isexposedtoextremesofcoldorheat,etc.Thereare,infact,averygreat manyconditionsthatarenecessaryforhumanlife,andnoone-orevenjust afewofthem-willsufficefor[orguarantee]humanlife. Or,further,considerthepropertyofhavingfoursides.Whilehaving foursidesisanecessaryconditionforsomething'sbeingasquare, thatsingleconditionisnot,byitself,sufficient(toguarantee) something'sbeingasquare,i.e.somefour-sidedthings(e.g.trapezoids) arenotsquares.Thereareseveralnecessaryconditionsforsomething's beingasquare,andallofthesemustbesatisfiedforsomething'sbeing asquare: xhas(exactly)foursides eachofx'ssidesisstraight xisaclosedfigure xliesinaplane eachofx'ssidesisequalinlengthtoeachoftheothers eachofx'sinterioranglesisequaltotheothers(theyare eachright[i.e.90o]angles) thesidesofxarejoinedattheirends Theforegoingisacompletesetofnecessaryconditions,i.e. thesetcomprisesasetofsufficientconditionforx'sbeingsquare. Frequentlytheterminologyof"individuallynecessary"and"jointlysufficient" isused.Onemightsay,forexample,"eachofthemembersoftheforegoing setisindividuallynecessaryand,takenalltogether,theyare jointlysufficientforx'sbeingasquare." Caution:Inthisexample,wehavebeenable,withease,to listasetofindividuallynecessaryconditionsthatisalso sufficientforsomething'sbeingasquare.However,wemustnot generalizefromthissimpleexampleandbelievethatitis usually,oroften,astraightforwardtasktospecifysetsof conditionsthatareindividuallynecessaryandjointly sufficient.Sometimesitismucheasiertospecify(some,or many,ofthe)necessaryconditionseventhoughweareunableto specifyasetthatisjointlysufficient.Othertimes,the converseistrue:forsomecasesitwillbeeasiertospecify sufficientconditionswithoutourbeingabletospecify individuallynecessaryones. 4.Necessaryconditionsthatarenotsufficient Example4.1-Asetofconditionsthatareindividuallynecessary withoutbeingjointlysufficient. ThomasWhite,theauthorofarecenttextbookinphilosophy, attemptedtouseas hisexamplethespecifyingofthenecessaryandsufficient conditionsforhearingmusicfromaWalkman®.Hereisthe listofnecessaryconditionsthatWhiteoffered(irrelevant conditionshavebeenhereremoved,andthelisthasbeen renumbered): TheWalkmanisingoodworkingorder. Thebatteriesaregood. [ Note 1 ] Theearphonesarepluggedin. Thetapehasmusiconitandisingoodworkingcondition. Youoperatethecontrolscorrectly. --DiscoveringPhilosophy,byThomasI.White,p.25 (EnglewoodCliffs,NJ:PrenticeHall),1991. Whitethengoesontomakeatoo-hastyclaim:"...takenall togethertheyaresufficient "(ibid.,p.25). Unfortunately,forhisillustrativepurposes,thelistisbyno meanssufficient.Herearejustafewofthemanyadditional necessaryconditionsthatmyownstudents,inpreviousyears, haveoffered: Thelistenermustnotbedeaf. Theambient(surrounding)soundmustnotdrownoutthe earphones. Thelistenermustbewearingtheearphones,ormustbeclose enoughtothem,tohearthemusic. Theremustbenothingblockingthesoundinthelistener'sears. Thetapemustbeinsertedcorrectly;thedooroftheWalkman mustbeclosed;andthetapemustnotbeattheendofthereel (morespecifically,itmustbepositionedsothatsomeofthe partsofthetapewhichcontainrecordedmusicwillpassoverthe playbackheads). Theearphonesareingoodworkingorder. Thelistenerdoesnotdieinthetimebetweenoperatingthecontrols correctlyandthemusic'semergingfromtheearphones. Eventhisexpandedlistisnotcomplete.Iimaginethatwill littleeffort,youyourselfcancomeupwithstillfurther necessaryconditions.Indeed,theredoesnotseemtobeany practicallimittothenumberofnecessaryconditions. Conclusion:Sometimes(asinthecaseofhearingmusicfrom aWalkman),itis(far)easiertospecifynecessary conditionsthansufficientones. 5.Sufficientconditionsthatarenotnecessary Example5.1-Asetofconditionsthatare(jointly)sufficient withoutbeingindividuallynecessary. AsufficientconditionfortravellingfromCalgaryto Vancouverwouldbeyourtakinganuneventfultripasapassenger onaregularlyscheduledairflight.Butwhilethatmethodof gettingfromtheonecitytotheotherwouldsuffice,itisbyno meansnecessary.Thereareallsortsofotherconditions thatwouldalsosufficeforyourgettingfromCalgaryto Vancouver. YoucouldtakeVIArail;or Youcouldtravelbycar;or Youcouldhike;or Youcouldrideabicycle;or Youcouldtravelonhorseback;or ...,... Example5.2-A(nother)setofconditionsthatare(jointly)sufficient withoutbeingindividuallynecessary. Ifyou'llforgivethemorbidexample,thinkofalltheways apersonmightdie:havinghis/herheadchoppedoff;beingat `Ground-Zero'whenanuclearbombisdetonated;tearingagaping holeinone'sspacesuitwhileona`space-walk'ontheMoon; etc.,etc.Butnoneoftheseconditionsisanecessary conditionforaperson'sdying.Indeedalmosteveryonedies withouthavingsatisfiedoneoftheseunusualsufficient conditions. Conclusion:Sometimes,itiseasiertospecifysufficient conditionsthannecessaryones. 6.Theconceptofconverserelations Herearesomeexamplesoftwo-placerelations: ...islargerthan.. ...istallerthan... ...isabove... ...isaparentof... ...isachildof... ...loves... ...employs... ...detests... Examplesofthree-placerelations: ...isbetween...and... ...travelsto...by... ...sends...to... Certaintwo-placerelationsareconversesofoneanother.For example,eachmemberofthefollowingpairsisaconverseoftheother member: ...isaparentof... ...isachildof... ...istallerthan... ...isshorterthan... ...isabove... ...isbelow... ...loves... ...islovedby... Definition:Two(two-place)relations,R1 andR2,areconversesofoneanother, ifandonlyif,(1)xR1y(e.g.Sandraistaller thanLouise)guaranteesyR2x(e.g.Louise isshorterthanSandra),and(2)yR2x guaranteesxR1y. Thesetwo-placerelationsarenotconverses ofoneanother: ...isadaughterof... ...isaparentof... The`trouble'withthiscaseisthatalthough"xisadaughterofy" doesindeedguarantee"yisaparentofx",thesecondcondition stipulatedinthedefinitionfor"converserelation"doesnothold. Foritisfalsethat"yisaparentofx"guarantees"xisa daughterofy". Or,considerthiscase: ...isnottallerthan... ...istallerthan... Again,thesetworelationsarenotconversesofoneanother. Inthisinstancethefirstconditionofthedefinitionof"converse relation"doesnothold.From"xisnottallerthany"itdoesnot followthat"yisshorterthanx".Perhapsxandyareexactlythe sameheight.Ifso,"xisnottallerthany"willbetrue,but"y istallerthanx"willbefalse. 7."Isanecessaryconditionfor"and"isasufficientconditionfor" areconverserelations Ifxisanecessaryconditionfory,thenyisa sufficientconditionforx. And,equivalently, Ifyisasufficientconditionforx,thenxisa necessaryconditionfory. Let'slookatsomeexamplesthatillustratetheclaimintheboxabove: "Sincehavingamicroscopeisnecessaryforseeingviruses,thenseeingviruses guaranteesthatonehasamicroscope(i.e.sufficesforhavingamicroscope)." Similarly,"Sincehavingairtobreatheisnecessaryforhumanlife,iffollows thattheexistenceofhumanlife(demonstrates,assures,guarantees,i.e.) sufficesfortheexistenceofair." "Sinceeverysquaremusthavefoursides(i.e.sincehavingfour sidesisanecessaryconditionforbeingasquare),beingasquare isasufficientcondition(butnotanecessaryone)forsomething's havingfoursides."Putanotherway:"Allsquares(must)havefour sides;butnotallfour-sidedthings[e.g.trapezoids]aresquares." "Beingafatherisasufficientconditionforbeingmale,andbeingmale isanecessaryconditionforbeingafather."(Butbeingafatherisnot anecessaryconditionforbeingamale[e.g.JamesDeanwasnotafather, butwasmale];andbeingamaleisnotasufficientconditionforbeinga father[againthecaseofJamesDean].) 8.Fourpossiblecombinations Pickanytwoconditionswhatsoever.Therelationship betweenthetwoconditionsmustbeexactlyoneofthe followingfourpossibilities: Thefirstisanecessary,butnotasufficient, conditionforthesecond;or Thefirstisasufficient,butnotanecessary, conditionforthesecond;or Thefirstisbothanecessaryandasufficient conditionforthesecond;or Thefirstisneitheranecessarynora sufficientconditionforthesecond. Examples8.1-Thefirstisanecessary,butnotasufficient,conditionforthe second. "Sam'sbeingamaleisanecessary,butnotasufficientcondition,forbeing afather." "Atable'shavingfoursidesisanecessary,butnotasufficient,condition foritsbeingsquare." "Beingmorethan6feet(183centimeters)tallisanecessary,butnota sufficientconditionforbeing6feet3inches(190.5centimeters) tall." "Owninga1996ChevroletCavalierisanecessary,butnota sufficientcondition,forowningared,6-cylinder,1996 ChevroletCavalier." "Havingaticketinalotteryisanecessary,butnotasufficientcondition, forwinningthatlottery." Examples8.2-Thefirstisasufficient,butnotanecessary,conditionforthe second. "Sam'sbeingafatherisasufficient,butnotanecessarycondition,for beingamale." "Atable'sbeingsquareisasufficient,butnotanecessarycondition,for itshavingfoursides." "Being6feet3inchestall(190.5centimeters)tallisasufficient, butnotanecessaryconditionforbeingmorethan6feet(180 centimeters)tall." "Owningared,6-cylinder,1996ChevroletCavalierisasufficient, butnotanecessarycondition,forowninga1996 ChevroletCavalier." "Winningalotteryisasufficient,butnotanecessarycondition,forhaving aticket." Examples8.3-Thefirstisbothanecessaryandasufficient conditionforthesecond. "Sam'sbeingafatherisbothanecessaryanda sufficientconditionforhisbeingamaleparent." "Frankie'sbeingolderthanJohnnyisbothanecessary andasufficientcondition forJohnny'sbeingyoungerthanFrankie." "Today'sbeingneitherSaturdaynorSundayisbotha necessaryandasufficient conditionfortoday'sbeingaweekday." [TRICKY,buttrue.] "X's-being-a-necessary-condition-for-Yisbothanecessary andasufficientconditionfor Y's-being-a-sufficient-condition-for-X." Examples8.4-Thefirstisneitheranecessarynorasufficient conditionforthesecond. "John'slovingPamelaisneitheranecessarynora sufficientconditionforPamela'slovingJohn." "Havingamarriedbrotherisneitheranecessarynora sufficientconditionforbeinganaunt." "Beingthesmarteststudentinaclassisneithera necessarynorasufficient conditionforachievingthehighestgradeinthatclass." "Aziz'splayingbaseballwellisneitheranecessary norasufficientcondition forhishavingadriver'slicense." "Wantingtosucceedisneitheranecessarynora sufficientconditionforsuccess." PracticeExercise#1 FirstSetofPracticeExercises onNecessaryConditionsandSufficientConditions. PracticeExercise#2 SecondSetofPracticeExercises onNecessaryConditionsandSufficientConditions. 9.Differentkinds(ormodes)ofnecessarycondition (Tobewritten....) NOTES Strictly,althoughthebatteries'beinggoodisusuallya necessaryconditionforaWalkman'sworking,itisnotabsolutely necessary.ForitispossibletooperateaWalkmanwithout batteries,e.g.byusingaplug-inAC-to-DCadapter.What isnecessary,alongtheselines,isthattherebeaproper powersourcefortheWalkman.[ Resume ] Return/transfertoNormanSwartz'sPhilosophical Notes Return/transfertoNormanSwartz's HomePage