CONCEPT CALCULUS: MUCH BETTER THAN
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CONCEPT CALCULUS: MUCH BETTER THAN

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1 CONCEPT CALCULUS: MUCH BETTER THAN Harvey M. Friedman* May 4, 2009 revised August 28, 2009 revised October 31, 2009 ABSTRACT. This is the initial publication on Concept Calculus, which establishes mutual interpretability between formal systems based on informal commonsense concepts and formal systems for mathematics through abstract set theory. Here we work with axioms for better than and much better than, and the Zermelo and Zermelo Frankel axioms for set theory.
  • basic facts about interpretation power
  • equivalence relation
  • t. interpretation
  • t1 definable
  • zf
  • theory of strict linear orderings
  • examples of incomparability between natural theories
  • logic of informal concepts from common sense thinking
  • proof

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WHATISIDENTITY(ASWENOWUSETHEWORD)?
JamesD.Fearon
DepartmentofPoliticalScience
StanfordUniversity
Stanford,CA94305
email:jfearon@stanford.edu
DRAFT{Commentsappreciated
November3,1999ABSTRACT
Thepaperundertakesanordinarylanguageanalysisofthecurrentmeaningsof\iden-
tity,"acomplicatedandunclearconceptthatnonethelessplaysacentralroleinongoing
debatesineverysub eld ofpoliticalscience(forexample,debatesaboutnational,ethnic,
gender,andstateidentities).\Identity"aswenowknowitderivesmainlytheworkofpsy-
chologistErikEriksoninthe1950s;dictionaryde nitions havenotcaughtup,failingto
capturetheword’scurrentmeaningsineverydayandsocialsciencecontexts. Theanalysis
yieldsthefollowingsummarystatement. Asweuseitnow,an\identity"refertoeither
(a)asocialcategory,de ned bymembershiprulesand(alleged)characteristicattributesor
expectedbehaviors,or(b)sociallydistinguishingfeaturesthatapersontakesaspecialpride
inorviewsasunchangeablebutsociallyconsequential(or(a)and(b)atonce).Inthelatter
sense,\identity"ismodernformulationofdignity,pride,orhonorthatimplicitlylinksthese
tosocialcategories.Thisstatementdi ers fromandismoreconcretethanstandardglosses
o ered bypoliticalscientists;Iargueinadditionthatitallowsustobetterunderstandhow
\identity"canhelpexplainpoliticalactions,andthemeaningofclaimssuchas\identities
aresociallyconstructed."Finally,Iarguethatordinarylanguageanalysisisavaluableand
perhapsessentialtoolintheclari cation ofsocialscienceconceptsthathavestrongrootsin
everdayspeech,averycommonoccurrence.1 Introduction
Inrecentyears,scholarsworkinginaremarkablearrayofsocialscienceandhumanities
disciplineshavetakenanintenseinterestinquestionsconcerningidentity. Withinpolitical
science,forexample,we nd theconceptof\identity"atthecenteroflivelydebatesin
everymajorsub eld. StudentsofAmericanpoliticshavedevotedmuchnewresearchtothe
\identitypolitics"ofrace,genderandsexuality.Incomparativepolitics,\identity"playsa
centralroleinworkonnationalismandethniccon ict (Horowitz1985;Smith1991;Deng
1995;Laitin1999). Ininternationalrelations,theideaof\stateidentity"isattheheart
ofconstructivistcritiquesofrealismandanalysesofstatesovereignty(Wendt1992;Wendt
1999;Katzenstein1996;LapidandKratochwil1996;BierstekerandWeber1996). And
inpoliticaltheory,questionsof\identity"marknumerousargumentsongender,sexuality,
nationality,ethnicity,andcultureinrelationtoliberalismanditsalternatives(Young1990;
Connolly1991;Kymlicka1995;Miller1995;Taylor1989)
Comparedtorecentscholarshipinhistoryandthehumanities,however,politicalsci-
entistsremainlaggardswhenitcomestoworkonidentities.Duetoin uences rangingfrom
MichelFoucaulttothedebateonmulticulturalism,thehistoricalandculturalconstructionof
identitiesofallsortshaslatelybeenapreoccupationforbothsocialhistoriansandstudents
1ofliteratureandculture.
Despitethis vastlyincreased andbroad-ranginginterest in\identity," theconcept
itselfremainssomethingofanenigma.WhatPhillipGleason(1983)observed15yearsago
remainstruetoday:Themeaningof\identity"aswecurrentlyuseitisnotwellcapturedby
1SeeBrubakerandCooper(1999)forsomecitationstothisvoluminousliterature. Forameasureof
thespreadof\identity"inacademicdiscourse,Ichartedtheprogressofthewordindissertationabstracts,
whichcannowbesearchedon-linegoingbackto1981.Thenumberofdissertationabstractscontainingthe
word\identity"almosttripledbetween1981and1995,risingfrom709to1,911.Thisincreasehasoccurred
entirelyinthelasttenyears. Theaverageincreasewasabout12%peryearfor1986to1995,whileitwas
roughly at at-2.3%for1981to1985.Someofthisincreasecouldbeduetoanincreaseinthetotalnumber
ofdissertationsabstracted.Ihavebeenunabletogetthese gures, butIdidtrysearchingyear-by-yearfor
aneutral\controlword"{Iused\study"{togetaroughestimate. Bythismeasure,thetotalnumber
ofabstractedincreasedbyanaverageof.64%peryearfor1981-1985,and4.4%peryearfor
1986-1995.Thusthenumberofdissertationsabstractsusingtheword\identity"hasbeengrowingalmost
threetimesfasterthantherateforallabstracteddissertations.
1dictionaryde nitions, whichre ect oldersensesoftheword.Ourpresentideaof\identity"is
afairlyrecentsocialconstruct,andarathercomplicatedoneatthat.Eventhougheveryone
knowshowtousethewordproperlyineverydaydiscourse,itprovesquitedi cult togivea
shortandadequatesummarystatementthatcapturestherangeofitspresentmeanings.
Giventhecentralityoftheconcepttosomuchrecentresearch{andespeciallyinsocial
sciencewherescholarstakeidentitiesbothasthingstobeexplainedandthingsthathave
explanatoryforce{thisamountsalmosttoascandal.Ataminimum,itwouldbeusefulto
haveaconcisestatementofthemeaningofthewordinsimplelanguagethatdoesjusticeto
itspresentintension.
Thisisthemainpurposeofthispaper,todistillastatementofthemeaningof\iden-
tity"fromananalysisofcurrentusageinordinarylanguageandsocialsciencediscourse.
Themainresultsareeasilystated,althoughafairamountofworkonalternativepossibili-
tieswillberequiredtoreachthem. Iarguethat\identity"ispresentlyusedintwolinked
senses,whichmaybetermed\social"and\personal."Intheformersense,an\identity"
referssimplytoasocialcategory,asetofpersonsmarkedbyalabelanddistinguishedby
rulesdecidingmembershipand(alleged)characteristicfeaturesorattributes.Inthesecond
senseofpersonalidentity,anidentityissomedistinguishingcharacteristic(orcharacteris-
tics)thatapersontakesaspecialprideinorviewsassociallyconsequentialbutmore-or-less
unchangeable.
Thus,\identity"initspresentincarnationhasadoublesense. Itrefersatthesame
timetosocialcategoriesandtothesourcesofanindividual’sself-respectordignity.There
isnonecessarylinkagebetweenthesethings. Inordinarylanguage,atleast,onecanuse
\identity"torefertopersonalcharacteristicsorattributesthatcannotnaturallybeexpressed
intermsofasocialcategory,andinsomecontextscertaincategoriescanbedescribedas
\identities"eventhoughnooneseesthemascentraltotheirpersonalidentity.Nonetheless,
\identity"initspresentincarnationre ects andevokestheideathatsocialcategoriesare
boundupwiththebasesofanindividual’sself-respect. Arguablymuchoftheforceand
22interestofthetermderivesitsimplicitlinkageofthesetwothings.
Insection2belowIjustifytheenterpriseatgreaterlength,arguingthatforcontested,
complicated,orunclearsocialscienceconceptswithstrongrootsinordinarylanguage(i.e.,
mostofthem),acarefulanalysisofordinarylanguagemeaningsshouldprecedee orts to
legislateade nition forparticularresearchpurposes.Section3considerstheinadequacyof
dictionaryde nitions of\identity"andverybrie y tracesthehistoricalevolutionofitsnew
3setofmeanings. Section4beginstoaskaboutthecurrentmeaningof\identity"bytesting
possiblede nitions againstexamplesfromusage.Thetrailleads rst totheformulationof
aidentityasasocialcategory,and,insection6,toidentityasdistinguishingfeaturesofa
personthatformthebasisofhisorherself-respectordignity(andmore).Inbetween,section
5developsapotentiallyvaluabledistinctionbetweenroleand\type"identities. Sections7
and8drawoutsomeimplicationsoftheanalysisfortwoissuesofconcerntosocialscience
usersofconcept.Insection7Iusetheresultsoftheordinarylanguageanalysistoconsider
howidentitiesbearontheexplanationofactions(politicalandotherwise). Insection8I
brie y extendtheanalysisof\identity"appliedtoindividualstocorporateactorssuchas
statesand rms. Acentralargumentinrecentinternationalrelationstheoryholdsthatstate
interestsaredeterminedby\stateidentities."Themeaningofthisclaimobviouslydepends
onthemeaning\stateidentities,"whichIarguemightrefertoanyofseveraldi eren tthings.
Section9concludes.
2 Whybother?
Giventheintenseinterestinidentityandidentitiesacrossabroadspectrumofdisciplines,
onemightinitiallyexpectiteasyto nd simpleandclearstatementsofwhatpeoplemean
2Theaddedvalueofthisstatementofthecurrentmeaningof\identity"isnotthedistinctionbetween
\social"and\personal"sidesperse.Thereisalongtraditionofscholarsdrawingaofthissort,
contrastingvariousformulationsofindividualorpersonalidentity,ontheonehand,andsocialorgroupor
collectiveidentityontheother.Whatisnovelintheformulationderivedhereisthespeci c contentofthe
twosidesofthedistinction(whichcanbeandhasbeen lled inmanyways).
3Foranexcellentandmoredetailedsemantichistoriesof\identity,"seeGleason(1983)andMackenzie
(1978).
3whentheyusetheseconcepts.WhileIhavenotdoneanexhaustivesearch,Ihavenotfound
thistobethecase. Overwhelmingly,academicusersoftheword\identity"feelnoneedto
explainitsmeaningtoreaders.Thereaders’understandingissimplytakenforgranted,even
4when\identity"istheauthor’sprimarydependentorindependentvariable.
Thisisperhapsnotsosurprising. Inthe rst place,whiletheoriginsofourpresent
understandingof\identity"lieintheacademy,theconceptisnowquitecommoninpopular
discourse.Sinceweallknowhowtoemploythewordandweunderstanditinotherpeoples’
sentences,whybotherwithde nitions orexplanations?Second,inpopulardiscourseidentity
isoftentreatedassomethingine able andevensacred,whileintheacademyidentityisoften
5treatedassomethingcomplexandevenine able. Onehesitatestotrytode ne thesacred,
theine able, orthecomplex.
Ofcourse,onecan nd briefde nitions andclari cations inmanyplaces. Theserun
thegamut,fromsuggestiveglossestosomefairlycomplicatedandopaqueformulations.Here
aresomeexamples,culledmainlybutnotexclusivelyfromtheareasIreadmostin(political
science,internationalrelations):
1.Identityis\people’sconceptsofwhotheyare,ofwhatsortofpeopletheyare,andhow
theyrelatetoothers"(HoggandAbrams1988,2).
2.\Identityisusedinthisbooktodescribethewayindividualsandgroupsde ne them-
selvesandarede ned byothersonthebasisofrace,ethnicity,religion,language,and
culture"(Deng1995,1).
3.Identity\referstothewaysinwhichindividualsandcollectivitiesaredistinguishedin
theirsocialrelationswithotherindividualsandcollectivities"(Jenkins1996,4).
4.\Nationalidentitydescribesthatconditioninwhichamassofpeoplehavemadethe
sameidenti cation withnationalsymbols{haveinternalisedthesymbolsofthenation
..."(Bloom1990,52).
5.Identitiesare\relativelystable, role-speci c understandingsandexpectationsabout
self"(Wendt1992,397).
4See,forinstance,Calhoun(1991)orFox(1985),thoughanynumberofsimilarexamplescanbegiven.
5Forastrikingexampleofthelatter,seeJamesCli ord’s (1988)essay\IdentityinMashpee."Likewise,
CharlesTaylor,afterspendingseveralpagesof Sources of the Self: The Making of the Modern Identity
explainingwhathemeansby\identity,"writes: \Butinfactouridentityisdeeperandmoremany-sided
thananyofourpossiblearticulationsofit"(Taylor1989,29).
46.\Socialidentitiesaresetsofmeaningsthatanactorattributestoitselfwhiletaking
theperspectiveofothers,thatis,asasocialobject. ... [Socialidentitiesare]atonce
cognitiveschemasthatenableanactortodetermine‘whoIam/weare’inasituation
andpositionsinasocialrolestructureofsharedunderstandingsandexpectations"
(Wendt1994,395).
7.\Bysocialidentity,Imeanthedesireforgroupdistinction,dignity,andplacewithinhis-
toricallyspeci c discourses(orframesofunderstanding)aboutthecharacter,structure,
andboundariesofthepolityandtheeconomy"(Herrigel1993,371).
8.\Theterm[identity](byconvention)referencesmutuallyconstructedandevolvingim-
agesofselfandother"(Katzenstein1996,59).
9.\Identitiesare...prescriptiverepresentationsofpoliticalactorsthemselvesandoftheir
relationshipstoeachother"(KowertandLegro1996,453).
10.\Myidentityisde ned bythecommitmentsandidenti cations whichprovidetheframe
orhorizonwithinwhichIcantrytodeterminefromcasetocasewhatisgood,or
valuable,orwhatoughttobedone,orwhatIendorseoroppose"(Taylor1989,27).
11.\Yetwhatifidentityisconceivednotasaboundarytobemaintainedbutasanexus
ofrelationsandtransactionsactivelyengagingasubject?"(Cli ord 1988,344).
12.\Identityisanysourceofactionnotexplicablefrombiophysicalregularities,andto
whichobserverscanattributemeaning"(White1992,6).
13.\Indeed, identity is objectively de ned as location in a certain world and can be
subjectively appropriatedonlyalongwiththatworld. ... [A]coherentidentityin-
corporateswithinitselfallthevariousinternalizedrolesandattitudes."(Bergerand
Luckmann1966,132).
14.\Identityemergesasakindofunsettledspace,oranunresolvedquestioninthatspace,
betweenanumberofintersectingdiscourses. ... [Untilrecently,wehaveincorrectly
thoughtthatidentityis]akindof xed pointofthoughtandbeing,agroundofaction
...thelogicofsomethinglikea‘trueself.’...[But]Identityisaprocess,identityissplit.
Identityisnota xed pointbutanambivalentpoint. Identityisalsotherelationship
6oftheOthertooneself"(Hall1989).
Therange,complexity,anddi erences amongthesevariousformulationsareremark-
able. Inpart,thedi erences re ect themultiplelineagesthat\identity"haswithinthe
6ExceptingthequotefromCli ord andanessaybyHandler(1994),Ihavehadlittleluck nding de nitions
orglossesof\identity"o ered byanthropologists,eventhough(orperhapsbecause)theytendtorelyvery
heavilyontheterm(forexample,inFox(1985)\identity"appearsnumeroustimesonpracticallyeverypage
ofthebook,butisneverde ned). Thissimplyindicatesthatanthropologiststendtotaketheconceptfor
granted,whichisappropriateiftheymainlyshareacommonunderstandingofwhatitdesignates.Handler
claimsthatthedictionaryde nition \approximately"(p.28)capturesthewaythewordisnowused;Iargue
againstthisbelow.
5academy. Di eren tresearchtraditions{in uenced variouslybysymbolicinteractionism,
roletheory,Eriksonianpsychology,socialidentitytheory,andpostmodernism,tonamea
few{haveevolvedsomewhatdi eren tconventionsregardingtheterm. Further,perhaps
someoftheseauthorsintendmerelytostipulateade nition of\identity"appropriateor
usefulfortheirspeci c purposes,sosomevariationmightbeexpectedwithvaryingpur-
poses.
Nonetheless,itisalsostrikingthatthede nitions seemtorefertoacommonunderlying
concept.Almosteveryoneevokesasenseofrecognition,sothatnoneseemsobviouslywrong,
despitethediversity.Thisisalsotobeexpected,because\identity"hasforsometimenow
beenastapleofordinarylanguage.Regardlessofparticularresearchtraditionsorpurposes,
itwouldbeverystrangetoo er ade nition of\identity"thatborenorelationtowhatwe
alreadyintuitivelyunderstandbytheconcept.
Thereisanimportantandmoregeneralpointtobemadehereaboutthede nition
ofsocialscience concepts. Incontrast tomany areasinthenaturalsciences, insocial
7sciencemostofourkeyconceptseitherderivefromorenterintoordinarylanguage. Power,
rationality,democracy,ethnicity,race,thestate,andevenpoliticsareexamples.Whenoneis
naminganentityinphysicsorbiochemistry,orde ning forthe rst timeatechnicaltermor
neologismlike\subgameperfection,"\bureaucraticauthoritarianism,"or\postmodernism,"
8itmakes perfectsense tostipulatethemeaningafterthemannerofHumpty Dumpty.
Indeed,thereisnoalternativeinthiscase. Butwhenatermhasstrongrootsinordinary
language,itispotentiallyveryconfusingtostipulateade nition withoutpayinganyexplicit
attentiontotheprior,ordinarylanguagemeaningofterm.
SupposeIstipulatethat,henceforth,by\table"Imean\chair,"andvice-versa. In
additiontobeingunnecessary,thiswouldrightlybeconsideredaninvitationtoconfusion.
Thereisastrongercaseforstipulatingade nition forsocialscienceconceptssuchaspower
oridentity,whereitislessinitiallyclearwhattheordinarylanguageversionmeans. But
7Typically,theymovebackandforth;seethediscussionof\identity"’shistorybelow.
8\WhenIuseaword,itmeansjustwhatIchooseittomean{neithermorenorless"(Carroll1992,124).
6doingsostillrisksseriousconfusiontotheextentthatthestipulatedde nition divergesfrom
thereaders’unarticulatedpriorunderstanding. Andthereisnowaytoguagethiswithout
rst explicatingthemeaningincurrentusage.Intheend,socialscientistsmayoften nd it
necessarytore ne andrede ne ordinarylanguagemeanings.Butwithoutaclearstatement
9ofthepriormeaning,eventhestipulatorwillnotknowwhatsheisdoingwiththeconcept.
Anotherargumentforexplicatingcurrentusageisthatthemethodcanyieldadeeper
understandingofcontestedandunclearconceptslike\identity."Theintuitionsbehindordi-
narylanguagemeaningsoftenhavemuchinterestingstructure,whichislikelytobemissedif
wejumptostipulatingde nitions. Intheiranalysesoftheconceptof\identity,"bothGlea-
son(1983)andBrubakerandCooper(1999)concludethatthewholesale,chaoticspreadof
10\identitytalk"inpopularandacademiclanguagehasdepriveditofanymeaningatall.
QuotingA.O.Lovejoyontheword\romantic",Gleasonsaysthat\identity"has\cometo
meansomanythingsthat,byitself,itmeansnothing.Ithasceasedtoperformthefunction
ofaverbalsign"(p. 914). BrubakerandCooperbelievethatthetermhasacquiredso
manycontradictorymeaningsandusesinsociologythatitshouldbepurgedinfavorofmore
speci c terms. IwillargueBrubakerandCooperandGleasonaregivinguptoosoonon
bothpopularand\popularacademic"usage.
3 Theconstructionof\identity"
Ifinneedofade nition, onelooks rst todictionaries.Hereisthemostrelevantentryfor
\identity"intheOED(2ndedition,1989):\Thesamenessofapersonorthingatalltimesor
inallcircumstances;theconditionorfactthatapersonorthingisitselfandnotsomething
else;individuality,personality."Notethatthisdoesnoteasilycapturewhatweseemtomean
9Oneexampleoftheconfusionthatcanresultfrominattentiontoordinarylanguagemeaningscomesfrom
theuseof\rational"inrational-choice-in uenced politicalscienceapplications,whereithasbeenpopular
toarguethatcontrarytoconventionalwisdom,phenomenonX(war,genocide,ethnicviolence,etc.)canbe
explainedastheproductofrationalactorsmakingchoices.Butthemeaningof\rational"inrationalchoice
theoryconcernsprimarilythee ciency ofmeansforattainingdesiredends,whereasinordinarylanguage
\rational"alsoreferstowhetheraperson’sendsarecomprehensibleorevenmorallydefensible.
10SeealsoMackenzie(1978),whoseinitialdismayattheproliferationofidentitytalkinthe1970sBritain
leadshimtospeakofthe\murder"oftheconcept.
7whenwereferto\nationalidentity"or\ethnicidentity,"forexample. Isnationalidentity
thesamenessofanationinalltimesandplaces,ortheconditionofbeingthisnationand
notanother?Certainlytheideaofnationalidentityentailsanideaoftemporalandspatial
continuityofanation,butthisisn’twhatanessayonthenationalidentityoftheRussians
(forexample)wouldbefocusedon. Norisnationalidentitythefactorconditionofbeing
di eren tfromothernations,butrathersomethingaboutthecontentofthedi erences.
Thedictionaryde nition alsofailstocapturewhatweintendbydeclarationsofthe
form\myidentityis[suchandsuch]...",although\individuality"maycomeclosehere
(\personality"isclearlywayo ).MosttellingisthecomparisonbetweentheOEDde nition
andthesocialscientist’sde nitions listedabove.Whilethereisconsiderableoverlapamong
socialscientist’sde nitions, thereisalmostnonewiththedictionarymeaning.
Animportantpointfollows: Ourpresentconceptof\identity"isrecent,oratleast
recentenoughthatdictionarieshavenotcaughtupwithcurrentusage.TheOEDde nition
isreportinganoldermeaningofthewordthatisstillusedquitefrequentlyineveryday
speechbutisnonethelessnarrowerthanourpresentconceptofidentity.Inthisoldersense,
\identity"referstothe(oftenlegal)associationofaparticularnametoaparticularperson{
thequalityofbeingaparticularperson,orthesamepersonasbefore,asin\sherevealedthe
identityofthemurderer"or\acaseofmistakenidentity."Thisusageisstillverymuchwith
11us. Forexample,thereisaminorgenreofnewspaperarticlesaboutthetheftofcreditand
12otheridenti cation cardsthatrefersto\stolenidentities." Butnotethatthisisaquite
di eren tsensefromwhatwemeanwhenwesay\Ican’tdothatbecauseitisinconsistent
withmyidentity"orclaimthat\Ethniccon icts areparticularlypronetoviolencebecause
theyinvolvemattersofidentity."
11Mackenzie(1978,25)callsthisthe\bureaucraticusage."
12Laitin(1998)identi es thisgenreinananalysisofusagebasedonaNexis-Lexissearch.Hereportsthe
followinginstance,from USA Today: \AuthoritieshavechargedJanetzke,40,ofStreamwoodwithwhat
amountstothetheftofanotherperson’sidentity. Policesayheusedthenameandcredithistoryofa
35-year-oldtruckerfromWoodDale... andeventookoutatelephonenumberinhisname. ‘Hejusttook
awaymyhusband’sidentity,’thetruckdriver’swifesaid.‘It’sjustabigmess.’"Inmyownrandomsample
of40usesof\identity"inmajorEnglishnewspapersidenti ed byNexis-Lexis,Ifoundthatabout40% t
thedictionary,\mistakenidentity"sense.
8

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