Infinite vs unbounded.docx
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Infinite vs unbounded.docx
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Infinitevsunbounded
InfinityinEthics
RoutledgeEncyclopediaofPhilosophy,ElectronicSupplement,Forthcoming2001
Puzzlescanariseinethicaltheory(aswellasdecisiontheory)wheninfinityisinvolved.Thepuzzlesariseprimarilyintheories—suchasconsequentialisttheories—thatappealtothevalueofactionsorstatesofaffairs.Section1addressesthequestionofwhetheronesourceofvalue(suchasmajoraestheticpleasures)canbeinfinitelymorevaluablethananother(suchasminorgustatorypleasures).Anaffirmativeanswerisgivenbyappealingtothenotionoflexicographicpriority.Section2addressthequestionofwhatmoralityrequireswhenthereareaninfinitenumberoffeasibleoptionsandnooptionismaximallyvaluable?
Insuchcases,itissuggested,moralitycandemandnomorethanthatwe“almostmaximize”or(moreweakly)thatwe“satisfice”.Section3addressesapuzzlethatcanarisewhentimeisinfinitelylong.Isastateofaffairswithtwounitsofvalueateachtimemorevaluablethanastateofaffairswithoneunitateachtime(eventhoughbothproduceinfiniteamountsofvalue)?
Aplausibleprincipleisintroducedthatanswersaffirmatively,butitfacescertainproblems.Section4addressesapuzzlethatcanarisewhentimeisfinitebutinfinitelydivisible.
1.LexicographicPriority
Canthevalueofsomeeventsorstates(e.g.,thepleasureofhearingmusicthatoneabsolutelyloves)beinfinitelygreaterinrelativetermsthanthevalueofsomeothereventorstate(e.g.,thepleasureeatingacarrotthatoneisnotespeciallyexcitedabout)?
Isitpossible,thatis,thatthereisnofinitenumberofthelattereventssuchthatthevalueofallthoseeventstogetherisatleastasgreatasthevalueoftheformerevent?
Thisisimpossible,ifallstatesandeventshavesomestandardfinitevalue.Onecan,however,coherentlyrejectthisassumption.
First,valueneednotberepresentablebynumbers.Itmaysimplybeordinallyrepresentablebyarankingrelation(i.e.,asmore,less,orequallyvaluable;butnoassignmentofspecificnumbersforvalue).Allelsebeingequal,moreoftheinfinitelylessvaluablesourcesofvaluemakestheworldmorevaluable,butnofinitenumberofsuchsourcescanevercompensateforthelossofoneoftheformersourcesofvalue.Theinfinitelymorevaluablesourcesofvaluearesimplylexicographicallypriorinthegenerationofoverallvaluetotheinfinitelylessvaluablesourcesofvalue.Thus,wecanmakeperfectsenseoftheidea,ifwedonotrequirethatvaluebenumericallyrepresentable.
Second,evenifonerequiresthatnumbersbeassignedtothevalueofstatesoftheworld,thiscanbedoneusinginfinitesimalnumbersofnon-standardarithmetic.Instandardmathematics,therearenonumbersthatareinfinitesimallysmall.Inthe1960s,however,AbrahamRobinson,amathematician,provedthatonecanmakeperfectmathematicalsenseofsuchinfinitesimals,andthusthatitislegitimatetopositthem.Theadditionofapositiveinfinitesimaltoagivennumberproducesalargernumber,butthesumoffinitelymanyinfinitesimalsstillisstillinfinitesimallysmall,andhencesmallerthananyfinitenumber(althoughgreaterthaneachoftheoriginalinfinitesimals).Ifinfinitesimalsarerecognized,thensomesourcesofvaluemaygenerateonlyinfinitesimalvaluerelativetoothersourcesofvalue.
Thus,theideaofinfinite(orinfinitesimal)relativevalueisnotproblematic.
2.InfinitelyManyOptions
Supposethatanagenthasinfinitelymanyoptions(possiblechoices).Tosaythatthereareinfinitelymanyoptionsisjusttosaythattherearemoreoptionsthananyfinitenumber.Thepresenceofinfinitelymanyoptionsdoesnotautomaticallygenerateproblems,butitcanwhereagentsarerequiredtoperformanoptionthatismaximallygood(atleastasgoodasanyotheroption).Suppose,forexample,thattheoptionsarenumbered,thato1hasavalueof1/2,thato2hasavalueof2/3,andthatingeneralonhasavalueofn/(n+1).Inthiscase,thereisnooptionwithamaximalvalue.Thevaluesareallfiniteandlessthanone,butforanyoption,onsay,thereisanotheroptionwithgreatervalue(e.g.,on+1).Nooptionismaximallygood,andthusnooptionispermissibleaccordingtoavalueoptimizingtheory.Theresultthatnothingispermissibleispuzzling,butitcanbeavoidedbyreplacingtheoptimizationrequirementwitharequirementthatachosenoptionbeatleastasgoodas"triviallyless"(onsomespecifiedcriterion)thanthebestonecando.Forexample,ifonebillionthofaunitofgoodnessisthecutoffforbeingtrivial,then,intheaboveexample,thereareinfinitelymanyoptionsthatsatisfythisrequirement(andtheyareall"almost"maximal).
Intheabovecasethereareinfinitelymanyoptions,eachoptionhasafinitevalue,andthevaluesoftheoptionsarebounded(i.e.,thereissomefinitevalue—forexample,1inthiscase—suchthatnooptionhasagreatervalue).Thingsarenotsosimplewhenthevaluesarenotbounded.Suppose,forexample,thatthevalueofo1is1,o2is2,andingeneralonisn.Giventhatthereareinfinitelymanyoptions,thereisnofinitelimitonhowhighthevaluescanbe(eventhougheachoptionhasafinitevalue).Inthiscase,optimizingand"almostoptimizing"theoriessaythatnooptionispermissible.Absolutesatisficingtheories—thatis,theoriesthatjudgeanoptionpermissiblejustincaseitsvalueis"goodenough"onsomespecifiedabsolutesense—havenoproblemwiththiscase.Whateverthecriterionforbeinggoodenough,thereareinfinitelymanyoptionsthatarepermissible.Peoplewhoareinclinedtodefendanoptimizing,oralmostoptimizing,theoryinthefinitecasethuseitherhavetoacceptthatnothingispermissibleinsuchinfinitecases(astrangeclaim)ortoexplainwhysatisficingisacceptableintheinfinitecasebutnotinthefinitecase.(Onepossibilityistoholdthatoneshouldmaximizewhenpossible,that,whenthisisnotpossible,oneshouldalmost-maximize,andthat,whenthisisnotpossible,oneshouldsatisfice.)
3.InfiniteTime
Theabovepuzzlesinvolvedinfinitelymanyoptionsforanagentinagivenchoicesituation.Relatedpuzzlescanarisewhenthereareonlyafinitenumberofoptionsatagiventime,butthereareinfinitelymanyfuturechoicesituationsbecausetimeextendsinfinitelyintothefuture.Hereletussupposeforsimplicitythatthevalueofanactionisthevalue(e.g.,happiness)thatitproducesintheworld,andthattimeisdiscrete(i.e.,foreachtimethereiswelldefined"next"time).Furthermore,supposeforsimplicitythatthereisafirsttime,andthatthatateachtimethereisexactlyoneagent(eitheroneagentexistsforever,orwhenanagentdiesareplacementagentcomesintobeing).Ateachtime,theagenthastwooptions.Oneoptionistoproduceacertainamountofvalueimmediately,inwhichcasenofurthervaluewillbeproducedintheworldatlatertimes.Theotheroptionistoproducenovalueimmediately,inwhichcaseatthenexttimetheagentwillhaveachoicebetween
(1)producingevenmorevalueimmediatelyandnothingthereafterand
(2)producingnovalueimmediatelybuthavingasimilarlystructuredchoiceatthenexttime.Forexample,thesequencesofpossiblechoicesmightlooklikethis:
<1unitimmediatelyvs.postpone>,<2unitsimmediatelyvs.postpone>,….
Asatisficingtheoryusinganabsolutecriterionofbeinggoodenough,however,willjudgeitpermissibleatsomepointtoproduceimmediatevalue.
Evensatisficingtheories,however,confrontapuzzlehere.Fortheyalsoclaimthatitispermissibleateachpointintimetopostponetheproductionofvalue.For,ifatsomepointproducingnunitsofvalueissatisfactory,thenpostponingtheproductionofvalueisalsosatisfactory(sincethetotalamountofvaluethatcanbeproducedatwillisevengreater).Onewayofavoidingthisproblem(probablytheonlyway)istoholdthatmoralityisrule-basedratheract-based.Theideaisthatatagiventimetheagentfacesinfinitelymanyrulesorstrategiesthatshecouldadoptandthencomplywithinthefuture.Intheproblemsituation,thepossiblestrategiesare:
neverproduceimmediatevalue(i.e.,postponeateachstep),produceimmediatevalueattime1,produceimmediatevalueattime2,etc.Ifnunitsaretheminimalsatisfactorylevel,thenallstrategiesthatproduceatleastnunitsofvaluearepermissible.Thestrategyofneverproducingimmediatevalueisclearlynotsatisfactoryandthusnotpermissible.Thissolvesthepuzzle,butisalsoraisesquestionsaboutwhetherthemoralpermissibilityofactionsisindeedbasedontheconsequencesofcompliancewithrulesratherthandirectlyontheirconsequences.
Theabovepuzzlesarisewhentryingtodeterminewhatismorallypermissible.Puzzlescanalsoarisewhentryingtodeterminewhatismorallybetterthanwhat.Supposeagainthattimeextendsinfinitelyintothefutureandthatanagenthasachoicebetweenproducingtwounitsofvalueateachtimeoroneunitofvalueateachtime.Intuitively,itwouldseemthattheformeroutcomeisbetterthanthelatteroutcome.Thetotalvalueproduced,however,isthesameinfinityineachcase.Thus,ifoverallvalueissimplythesumofthevaluesateachtime,thenitwouldseemthatneitherisbetterthantheother.Ofcourse,sometheoriesofvaluerejectthesummativeview(e.g.,egalitarianviews),buttho
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