Infinite vs unbounded.docx

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>,….,etc.Assumeherethatanyrelevantdiscountingofvaluefortemporaldelaysisalreadyreflectedinthenumbers.Anoptimizing,oralmostoptimizing,theorysaysthatateachtimethechoiceshouldbetopostpone,butthiswillhavetheresultthatnovalueiseverproduced!

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