’WPCŌ ˆŪŚ„E¹G‡–QäpvbæōN*šR+PvNŁ V·¤%KµÆā½q[ćh‹£"ł9ķ)£ą,ńēÖŻm¤Z8šS¬+°4ć¶ĆóƒExÉ+”IŲmOH%L$\ßx”dūęć}s?Š8ŸWÕZ]ģäµ./č;MUŖQ£]Ū6ąÄSˆ­)¬v¾–ÆR°]ĪĮ7žņōa…+”œS’¼i?‰|[·¤ z°ē@oĻØĒģé0ĆŃM’»G1‹w™—”ėiG‹zćńĄ;® ÷A›¦Œ]¼Œm§{‡µ `Œų)F757*V:ž~į?ŲåöÄÉb‹}§ßǹčŠéā'6į3 0{DŚYĪܗÖć"ˬ}s‘"€< „ĆIfhnüµ+†pģƒ!v i™’—™y“T[„„Pt9¹‘¤70Tć•oś“ŖŹÉezvÜN) OŅ7ܔ„Łń;qóśļLź¹ä;ŌŽŻ‡ģæ—;|q,}³Ė7­‹’¾˜KM焯&”²;CįhX’ ō„0Će$.p¬qH±€,«¹3&o“PóŽŅģóŻ&ü@/w ŅE¬Ō‘ ūo¹ń2J›ęū”*q~Ø*/³ŻÄ؋®({–pU.4UND %’ 0(˜4Ą'    0 ..10/18/9410/18/94   <ž6X9`(courier3|x—šÖĆ9 Z‹6Times New Roman RegularX($””B\;ierŻ ƒ!ŻŻ  ŻŃ  ŃŅܰŅŅܰŅņ ņÓ  ÓÓ33ÓŃ7€G^XXdģdČ7ŃĢĢą@~ ~ ģąTHE€RED€HERRING€AND€THE€PET€FISHó óˆŠ ę6 Šņņą@” ” ģąWņ ņhy€concepts€ņņstillóó€can't€be€prototypesˆŠ Q Šó óóóą@$$ģąJerry€Fodor€and€Ernest€LeporeˆŠ  p ŠĢą@||ģąDepartment€of€Philosophy€and€Center€for€Cognitive€Science,€Rutgers€UniversityˆĢĢĢņņIntroduction.óóņņ1óóŠ §÷  ŠĢThere€is€a€Standard€Objection€to€the€idea€that€concepts€might€be€prototypes€(or€exemplars,Ļor€stereotypes):€Because€they€are€productive,€concepts€must€be€compositional.€PrototypesĻaren't€compositional,€so€concepts€can't€be€prototypes€(see,€e.g.,€Margolis,€1994).ņņ2óóŠ c  ŠĢHowever,€two€recent€papers€(Osherson€and€Smith,€1988;€Kamp€and€Partee,€1995)€reconsiderĻthis€consensus.€They€suggest€that,€although€the€Standard€Objection€is€probably€right€in€theĻlong€run,€the€cases€where€prototypes€fail€to€exhibit€compositionality€are€relatively€exotic€andĻinvolve€phenomena€which€ņņanyóó€account€of€compositionality€is€likely€to€find€hard€to€deal€with;Š šź Šfor€example,€the€effects€of€quantifiers,€indexicals,€contextual€constraints,€etc.€KP€are€evenĻprepared€to€indulge€a€guarded€optimism:€"...€when€a€suitably€rich€compositional€theory...€isĻdeveloped,€prototypes€will€be€seen€...€as€one€property€among€many€which€only€when€takenĻaltogether€can€support€a€compositional€theory€of€combination"€(p.56).ĢĢIn€this€paper,€we€argue€that€the€Standard€Objection€to€prototype€theory€was€right€after€all:ĻThe€problems€about€compositionality€are€insuperable€in€even€the€most€trivial€sorts€ofĻexamples;€it€is€therefore€as€near€to€certain€as€anything€in€cognitive€science€ever€gets€that€theĻstructure€of€concepts€is€not€statistical.€Theories€of€categorization,€concept€acquisition,€lexicalĻmeaning€and€the€like,€which€assume€the€contrary€simply€don't€work.ĢĢWe€commence€with€a€general€discussion€of€the€constraints€that€an€account€of€concepts€mustĻmeet€if€their€compositionality€is€to€explain€their€productivity.€We'll€then€turn€to€a€criticism€ofĻproposals€that€OS2€and€KP€make€for€coping€with€some€specific€cases.€ĢĢĢŠ e+µ&# ŠņņPart€I:€Productivity€and€compositionality.Š ° ŠóóĢWithin€the€family€of€theories€that€identify€concepts€with€mental€representationsĻ(Representational€Theories€of€Mind€(RTMs)),€there€is€a€more€or€less€explicit€consensus€thatĻconcepts€are€productive,€and€that€their€productivity€is€explained€by€the€assumption€that€mentalĻrepresentations€(MRs)€are€compositional.ņņ3óó€We€will€assume,€as€do€the€authors€we'reŠ 7 ‡ Šdiscussing,€that€some€version€of€this€story€is€correct.€Our€first€aim€is€to€provide€a€minimalĻsketch€of€the€explanatory€architecture€that€it€presupposes.ĢĢHere's€the€general€idea:€A€compositional€theory€of€the€productivity€of€concepts€must,€at€aĻminimum,€specify€two€functions:ĢĢ„A€ņņComposition€Functionóó€(FC),€which€maps€a€finite€basis€of€simple€MRs€onto€an€infinity€ofŠ ōD  Šcomplex€MRs€together€with€their€structural€descriptions.€ĢĢ„An€ņņInterpretation€Functionóó€(FI),€which€maps€arbitrary€MRs,€simple€or€complex,€into€theirŠ E• Šsemantic€interpretations.€There€is,€alas,€no€general€consensus€on€what€sorts€of€things€semanticĻinterpretations€are;€but€examples€of€candidates€for€the€interpretations€of€general€conceptsĻinclude€properties,€sets,€intensions,€senses€and€functions.€In€any€case,€it's€assumed€that€theĻsemantic€interpretations€of€MRs€are€typically€`things€in€the€world'€and€not€themselves€mentalĻor€linguistic.ĢĢWe€want€to€make€clear€what€justifies€postulating€each€of€these€functions:€ĢĢņņWhy€you€need€the€composition€functionóó:€If€concepts€are€MRs€and€concepts€are€productive,Š 8ˆ Šthere€must€be€infinitely€many€€MRs.€Practically€without€exception,€people€who€accept€thisĻinference€conclude€that€infinitely€many€MRs€must€have€internal€structure;€specifically,€thatĻinfinitely€many€MRs€must€have€MRs€as€their€constituents.€ĢĢJust€what€the€argument€for€this€is€supposed€to€be€isn't,€after€all,€entirely€obvious.€What€logicalĻor€metaphysical€principle€prohibits€the€existence€of€infinitely€many€ņņunóóstructured€mentalŠ Ś%*! Šparticulars?€On€the€other€hand,€if€the€argument€from€the€productivity€of€concepts€to€theĻinternal€structure€of€MRs€isn't€demonstrative,€it€is€nevertheless€extremely€well€evidenced.€ItĻis€assumed€by€all€the€psychological€and€semantic€accounts€of€productivity€(and€systematicity;Ļā āsee€Fodor€and€Pylyshyn,€1988)€that€anybody€has€thus€far€been€able€to€imagine.€And€theŠ F*–%" Šintuitive€plausibility€of€the€idea€that,€for€example,€the€concept€BROWN€is€a€constituent€of€theĻconcept€BROWN€COW€appears€undeniable.Ģā āĢSo,€then,€according€to€this€line€of€theorizing,€concepts€are€productive€because€there€areĻinfinitely€many€MRs.€There€are€infinitely€many€MRs€because€new,€relatively€complex€MRsĻcan€be€constructed€by€using€old,€relatively€primitive€ones€as€their€constituents.€That€MRs€haveĻconstituent€structure€is€thus€essential€to€explaining€the€compositionality€of€concepts;Ļaccordingly,€FC€serves€to€specify€the€constituency€relations€that€MRs€enter€into.€ĢĢņņWhy€you€need€the€interpretation€functionóó€(FI):€What€RTMs€offer€to€reconstruct€theŠ £ó  Špretheoretic€concept€CONCEPT€is€ņņMRóó€ņņwith€its€semantic€interpretation.€óóAs€remarked€above,Š ¾  Šsemantic€interpretations€are€assumed€to€be€typically€nonmental.€Correspondingly,€theĻinterpretation€function€specifies€a€relation€between€MRs€and€things€in€the€world€(for€example,Ļbetween€MRs€and€their€extensions).€ĢĢThough€MR€theorists€are€lost€to€solipsism€or€idealism€from€time€to€time€(see,€e.g.,€Jackendoff,Ļ1992),€that€some€mind/world€relation€is€essential€to€concept€individuation€is,€in€fact,€prettyĻwidely€understood.€€In€particular,€it€is€common€ground€that€concepts€are€the€kinds€of€thingsĻthat€ņņapplyóó€to€things€(that€is,€they're€the€kinds€of€things€that€can€function€as€€ņņcategoriesóó)€andŠ –ę Šthat€it€is€constitutive€of€the€identity€of€a€concept€that€it€applies€to€the€things€that€it€does.ĻNothing€that€applies€to€bricks,€or€that€fails€to€apply€to€birds,€could€be€the€concept€BIRD.€ĢĢEssentially€all€philosophers€who€discuss€these€sorts€of€issues€take€this€for€granted.€(However,Ļsee€Stich,€1983)€€But€so€too€do€practically€all€psychologists,€at€least€implicitly.€Most€of€theĻexperimental€data€on€which€the€discussion€of€prototypes€turns€use€categorization€tasks€toĻassess€concept€possession.€That€is,€they€assume€that€whether€one€has€the€concept€C€isĻrevealed,€at€least€in€part,€by€one's€capacity€to€distinguish€the€things€that€C€applies€to€from€theĻthings€that€it€doesn't.€This€research€strategy€would€be€incoherent€if€it€weren't€assumed€that€itsĻrelation€to€its€domain€of€application€is€among€a€concept's€essential€properties.ĻCorrespondingly,€the€interpretation€function€FI€is€required€to€specify€this€relation€for€each€ofĻthe€infinitely€many€MRs.€ĢĢThe€argument€so€far€is€that€everybody€who€wants€to€explain€the€productivity€of€concepts€byĻreference€to€the€compositionality€of€MRs€has€to€postulate€a€composition€function€and€anĻinterpretation€function.€Notice,€however,€that€assuming€the€mere€(as€it€were,€Platonic)Ļexistence€of€FC€and€FI€isn't€good€enough€for€the€task€at€hand.€The€productivity€problem€isŠ a+±&# Šnot€just€that€there€are€(Platonically)€infinitely€many€concepts;€it's€that€(given€the€usualĻidealizations)€there€are€infinitely€many€concepts€ņņthat€people€can€entertainóó.€But,€how€couldŠ Ė Šit€follow€from€the€mere€Platonic€existence€of€€FC€and€FI€that€people€can€entertain€infinitelyĻmany€concepts?€Compare:€The€facts€of€arithmetic€don't,€in€and€of€themselves,€explain€howĻpeople€are€able€to€add.€You€also€need€some€psychological€premises€about€what€they€knowĻand€what€is€going€on€in€their€heads.€Correspondingly,€explaining€the€productivity€of€people'sĻconcepts€requires€postulating€not€only€that€there€are€the€functions€FI€and€FC,€but€also€thatĻpeople€are€epistemically€related€to€these€functions€in€appropriate€ways;€that€people€can€ņņgraspóóŠ m ½ Šthese€functions.€What€explains€the€productivity€of€our€concepts€is€that€we€grasp€the€functionsĻFI€and€FC.ĢĢIt's€generally€assumed€that€you€can€only€grasp€a€function€that€is€finitely€specifiable.ņņ4óó€NoticeŠ Ł)  Šthat,€unless€this€ņņisóó€assumed,€it's€hard€to€see€why€productivity€is€a€problem.€One€generates€theŠ ōD  Šproductivity€problem€by€asking€how€a€finite€creature€could€have€an€infinite€epistemic€capacity:Ļhow€there€could€be€infinitely€many€concepts€that€it€can€entertain.€Clearly,€either€the€existenceĻof€such€infinite€capacities€is€unproblematic,€in€which€case€the€productivity€problem€doesn'tĻarise;€or,€if€there€really€is€a€productivity€problem,€the€solution€must€not€itself€presupposeĻepistemic€relations€to€infinite€sets.€ĢĢSo,€then,€if€the€story€is€that€we€can€entertain€an€infinity€of€concepts€because€we€can€grasp€FIĻand€FC,€these€functions€must€themselves€be€finite€objects.€This€doesn't€mean,€of€course,€thatĻthey€must€have€finite€ņņextensionsóó.€Rather,€the€idea€is€that€a€finite€creature€can€get€into€anŠ ē7 Šepistemic€relation€to€an€infinite€set€only€by€being€in€some€epistemic€relation€to€a€finite€objectĻthat€specifies€the€set.€Similar€considerations€suggest€that€each€mental€representation€mustĻitself€€be€finitely€specifiable,€that€the€primitive€basis€from€which€complex€MRs€are€constructedĻmust€be€finite,€and€so€forth.€As€remarked€above,€the€consensus€on€this€sort€of€point€appearsĻto€be€general€among€MR€theorists€who€accept€the€productivity€of€concepts€as€a€realĻphenomenon.ĢĢOur€present€concern€is€with€a€further€consequence€that€an€account€of€their€graspability€placesĻon€FI€and€FC.€Principle€P€provides€a€rough€formulation:ĢĢP:€The€interpretation€that€FI€assigns€to€a€certain€MR€must€be€computed€from€the€structuralĻdescription€that€FC€assigns€to€that€MR.Ģā āŠ F*–%" ŠLet's€continue€to€assume,€for€purposes€of€exposition,€that€semantic€interpretations€are€setsĻā ā(specifically,€extensions).€Then,€presumably,€FI€assigns€to€the€mental€representation€BROWNĻCOW€the€intersection€of€the€set€of€brown€things€with€the€set€of€cows.€However,€P€furtherĻrequires€that€FI€does€so€ņņbecauseóó€FC€assigns€to€BROWN€COW€a€structure€which€includes€theŠ Q Šconstituent€representations€BROWN€and€COW€(in,€of€course,€the€appropriate€configuration).ĻTo€put€it€slightly€differently,€the€operations€FI€performs€must€be€sensitive€to€the€structuralĻdescriptions€that€FC€enumerates,€so€that€the€structure€of€the€interpretation€that€FI€assignsĻderives€from€the€structure€that€FC€assigns.ĢĢTo€see€what€is€at€issue,€consider€what€would€happen€if€we€had€a€semantical€theory€for€MRsĻthat€failed€to€satisfy€P.€One€could€perhaps€imagine€that€such€a€theory€somehow€succeeds€inĻgetting€the€`right'€extensions€assigned€to€each€of€the€infinitely€many€MRs.€So,€the€set€of€cowsĻgets€assigned€to€COW,€the€set€of€brown€cows€gets€assigned€to€BROWN€COW,€etc.ĻTechnically,€such€a€theory€would€succeed€in€representing€the€productivity€of€the€MRs;€itĻwould€represent€them€as€an€infinite€set€of€interpreted€objects.€But€it€would€nonetheless€leaveĻcertain€glaring€explanatory€gaps€in€the€resulting€explanation€of€why€concepts€are€productive.€ĢĢFor€one€thing,€we€would€be€at€a€loss€to€explain€what€the€constituent€structure€of€MRs€is€ņņforóó.Š {Ė ŠConstituent€structure€would€be€idle€in€the€general€case€in€just€the€way€that€it€really€is€idle€inĻthe€case€of€idioms.€(Compare€the€semantically€irrelevant€constituent€structure€of€`kick€theĻbucket'€(=€ņņdieóó)€with€the€semantically€relevant€constituent€structure€of€`brown€cow'€andŠ Ģ ŠBROWN€COW.)€Second,€it€would€fail€to€explain€why€a€given€MR€has€the€interpretation€thatĻit€does€(or,€equivalently,€why€the€isomorphism€between€the€structure€of€MRs€and€theĻstructure€of€their€interpretations€is€reliable).€By€contrast,€when€principle€P€is€enforced,€we€canĻsee€straight€off€why€an€MR€that€has€COW€as€a€constituent€has€as€its€interpretation€a€set€allĻthe€members€of€which€are€cows€(as€in€the€case€of€BROWN€COW)€or€members€of€theĻcomplement€of€the€set€of€cows€(as€in€the€case€NOT€COW),€etc.€€Failing€P,€a€theory€representsĻthis€sort€€of€parallelism€as€accidental.ĢĢSo€much€for€what€we€take€to€be€common€assumptions;€we're€about€to€see€that€even€MRĻtheorists€who€talk€as€though€perhaps€they€don't€accept€one€or€another€of€them€actually€do€soĻin€practice.€We€turn€now€to€our€main€topic,€which€is€how€theories€of€the€productivity€ofĻconcepts€fare€when€they€take€MRs€to€be€prototypes,€and€hence€take€prototypes€to€constituteĻboth€the€domain€and€the€range€of€FC,€and€the€domain€over€which€the€operations€of€FI€areĻdefined.ĢŠ a+±&# ŠWe€will€consider€two€kinds€of€objections€that€an€account€of€the€compositionality€ofĻprototypes€appears€to€face€if€it€is€to€take€the€general€form€we've€just€been€considering.ĻAccording€to€the€first€objection,€prototype€theory€can't€€account€for€certain€relations€of€logicalĻequivalence€among€concepts.€According€to€the€second,€prototype€theory€can't€€predict€theĻsemantic€relations€between€complex€concepts€and€their€constituents.€We€claim€that€both€theseĻobjections€are€warranted.ĢĢņņPart€II:€Boolean€concept.Š m ½ ŠĢóóWe€start€by€considering€the€sort€of€complex€concepts€that€are€built€up€from€their€basicŠ £ó  Šconstituents€by€the€use€of€such€Boolean€operators€as€AND,€IFTHEN,€OR,€and€NOT.€ThereĻare€two€sorts€of€problems€such€concepts€raise€for€prototype€theories.€We'll€argue€that€the€firstĻis€a€sort€of€red€herring€in€that€neither€the€problem€itself,€nor€the€sort€of€solution€KP€propose,Ļare€specific€to€prototype€theories.€The€second€is€more€serious€since€it€appears€to€jeopardizeĻprinciple€P.€€ĢĢņņThe€red€herring:€truth€value€gaps.Š `° ŠóóĢWhether€or€not€concepts€are€prototypes,€it's€clear€that€many€concepts€are€vague;€in€manyĻcases,€no€definite€truth€value€attaches€to€the€judgment€that€a€certain€concept€applies.€ForĻexample:€It's€surely€just€true€that€tables€and€chairs€are€furniture,€and€it's€surely€just€false€thatĻfish€are€furniture;€but€what€about€wall„to„wall€carpets?€It's€not€mandatory,€but€it's€natural,€toĻsay€that€there's€no€fact€of€the€matter€here;€viz.,€that€the€judgment€that€wall„to„wall€carpets€areĻfurniture€is€neither€true€nor€false.ņņ5óóŠ m ŠĢIf€concepts€are€prototypes,€then€whether€something€falls€under€a€concept€is€a€matter€of€howĻsimilar€to€the€prototype€it€is.€Since€SIMILARITY€is€itself€presumably€a€vague€concept,€so€tooĻare€whatever€concepts€are€defined€in€terms€of€it.€So,€if€vague€concepts€(inter€alia)€areĻprototypes,€it€is€intelligible€that€judgments€in€which€they€occur€will€often€exhibit€`truth€valueĻgaps€(tv„gaps).€€Notice€that€the€suggested€account€of€tv„gaps€is€metaphysical,€not€epistemic.ĻAn€epistemic€treatment€of€vagueness€would€claim€that€there€ņņisóó€a€fact€about€whether€wall„to„Š Ś%*! Šń–ńŠ Ś%*! Šń–ńwall€carpeting€is€furniture,€though€it's€a€fact€we€don't€happen€to€be€apprised€of;€maybe€futureĻresearch€will€decide.€We€don't€find€it€easy€to€take€the€epistemic€view€of€vagueness€seriously,Ļand€we€won't€discuss€it€in€what€follows.Ģā āŠ F*–%" ŠNow,€if€a€concept€exhibits€a€tv„gap,€then€it€will€contribute€that€gap€to€infinitely€many€complexĻā āconcepts€of€which€it€is€a€constituent.€So,€for€example,€if€there€is€no€fact€of€the€matter€aboutĻwhether€wall„to„wall€carpets€are€furniture,€then€there€is€likewise€no€fact€of€the€matter€aboutĻwhether€wall„to„wall€carpets€are€expensive€furniture€(viz.,€whether€they€are€both€furniture€andĻexpensive).€And€there€is€also€no€matter€of€fact€about€whether€wall„to„wall€carpets€areĻfurniture€or€not€furniture,€and€so€forth.€But€this€sort€of€consequence€is€arguably€not€tolerable.ĻExcluded€middle€says€thatņņ€everythingóó€is€either€furniture€or€not€furniture,€and€excluded€middleŠ R ¢ Šis€a€law€of€logic,€and€laws€of€logic€are€necessarily€true.€So,€it€looks€as€though,€if€you€assumeĻthat€concepts€are€prototypes,€you€will€be€forced€to€deny€a€necessary€truth.ĢĢWhether€or€not€you€find€this€line€of€thought€convincing,€it€bears€emphasis€that€the€problemĻbeing€raised€really€has€nothing€in€particular€to€do€with€prototypes€as€such€or€even€withĻvagueness€as€such.€It€comes€up€wherever€you€suppose€that€tv„gaps€can€occur;€and,€primaĻfacie,€tv„gaps€can€have€all€sorts€of€sources:€failed€presuppositions,€empty€names,€vacuousĻpredicates,€the€truth€paradoxes€and€so€on.€Correspondingly,€we're€about€to€see€that€theĻmechanisms€that€KP€propose€for€dealing€with€predicates€like€`is€wall„to„wall„carpet€that€eitherĻis€furniture€or€is€not€furniture'€do€not,€in€any€way,€exploit€the€characteristic€properties€ofĻprototypes.€ĢĢWe€propose€to€scant€the€details.€In€effect,€supervaluation€assigns€arbitrary,€stipulative€truthĻvalues€to€those€base€clauses€of€Boolean€concepts€which€lack€them€(see€below.)€€Given€thatĻsupervaluations€allow€such€tv„gaps€to€be€filled,€the€logical€truths€can€then€be€identified€in€theĻusual€way;€viz.,€as€the€sentences€that€remain€true€however€tvs€are€assigned€to€theirĻconstituents.€The€logical€forms€that€concepts€are€ascribed,€according€to€the€supervaluationĻtreatment,€are€exactly€what€one€would€suppose€on€classical€assumptions€about€what€conceptsĻare.€(For€example€CARPET€THAT€EITHER€IS€FURNITURE€OR€IS€NOT€FURNITUREĻis€assumed€to€have€a€mental€representation€of€the€form€ņņc€which€is€F€or€not„F.)óó€Since€theŠ n!¾ Šclassical€notion€of€logical€form€is€respected€by€supervaluation€theory,€all€the€necessary€truthsĻof€classical€Boolean€logic€can€be€preserved.ĢĢHere€is€the€essential€point:€The€supervaluation€treatment€allows€the€recovery€of€logical€truthsĻbecause€it€assigns€classical€logical€forms€to€Boolean€concepts.€For€exactly€that€reason,€it€isĻindifferent€whether€Boolean€concepts€are€vague€because€their€constituents€are€prototypes€orĻwhether€it's€something€else€that€makes€them€vague.€In€consequence,€assuming€thatĻā āsupervaluation€works€as€a€treatment€of€the€vagueness€of€Boolean€concepts€if€they€areŠ F*–%" Šconstructed€from€€prototypes,€it€will€work€equally€well,€and€in€exactly€the€same€way,€ifĻBoolean€concepts€are€vague€but€ņņnotóó€constructed€from€prototypes.€Š Ė Šā āĢFor€example,€they€might€be€vague€because€they€are€definitions€couched€in€a€vagueĻmetalanguage.€(BROWN€COW€=€BROWN€&€COW,€and€BROWN€is€vague.)€IfĻsupervaluation€cures€the€sort€of€vagueness€that€prototypes€cause,€it€also€cures€the€sort€ofĻvagueness€that€vague€definitions€cause,€and€does€so€equally€well€and€in€exactly€the€same€wayĻ(viz.,€by€`precisifying';€see€below).€The€moral€is€that€vagueness€is€ņņeverybody'sóó€problem.Š m ½ ŠSupervaluation€may€solve€it,€but€the€way€it€solves€it€doesn't€favor€prototype€theory€over€anyĻaccount€of€concepts€which€allows€that€they€are€often€vague.ĢĢHaving€stressed€the€neutrality€of€the€supervaluation€treatment€of€vagueness€in€respect€of€theĻnature€of€concepts,€we€can't€resist€adding€that€we€have€pretty€severe€doubts€whether€theĻsupervaluation€treatment€of€vagueness€actually€works;€in€particular,€whether€assumingĻsupervaluation€as€a€semantics€for€tv„gaps€will€lead€to€the€desired€result€that€`John€is€bald€orĻJohn€is€not€bald'€and€the€like€will€get€appropriate€truth€values.€The€problems€are€technical€andĻa€full€treatment€would€go€beyond€the€scope€of€the€present€discussion.€But€a€quick€sketch€mayĻserve€to€supply€the€intuition.ĢĢFor€present€purposes,€the€basic€idea€employed€in€applying€supervaluation€theory€to€theĻanalysis€of€vagueness€is€that€of€a€ņņprecisificationóó€of€a€vague€language€L.€A€precisification€isŠ Ģ Šan€assignment€of€€T€and€F€to€the€atomic€sentences€of€€L€such€that:€definitely€false€sentencesĻare€assigned€F,€definitely€true€sentences€are€assigned€T,€and€vague€sentences€are€ascribed€TĻor€F€arbitrarily.ņņ6óó€€Then€a€complex€sentence€ņņsóó€is€`definitely€true'€if€it€receives€T€under€everyŠ m Šprecisification,€and€it€is€false€if€it€receives€F€under€every€precisification.€Just€as€you'd€expect,Ļa€ņņlogicalóó€truth€is€a€sentence€that€comes€out€true€on€ņņeveryóó€assignment€of€tvs€to€its€atomicŠ S £ Šsentences,€the€assignments€made€by€precisification€included.€€So,€€as€KP€put€it,€supervaluationĻtheory€may€provide€"a€sound€logical€framework€in€which€prototype€theory€and€classical€logicĻcan€peacefully€coexist."€(ms€p.22)€€€Consider€`John€is€bald€or€John€is€not€bald,'€and€supposeĻthere€is€no€fact€of€the€matter€whether€the€atomic€sentence€`John€is€bald'€is€true.€There€are€twoĻways€of€precisifying:€assign€`John€is€bald'€T€or€assign€it€F.€`John€is€bald€or€John€is€not€bald'Ļcomes€out€true€on€either€of€these€assignments,€so€the€usual€truth€functional€construal€of€`or'Ļand€`not'€make€`John€is€bald€or€John€is€not€bald'€a€logical€truth.€Analogous€treatment€appliesĻto€(e.g.,)€`John€is€bald€or€not€bald'.Ģā ā€Š F*–%" ŠHere,€€however,€is€the€problem:€supervaluation€theory€involves€stipulating€assignments€ofĻā ātruth€values€to€sentences€which,€by€assumption,€have€no€truth€value€in€the€actual€world€(forĻexample,€to€atomic€sentences€with€vague€predicates).€This€seems€perfectly€all€right€when€itĻis€ņņcontingentóó€that€a€sentence€has€a€tv„gap.€For€example,€`John€is€bald'€may€have€no€truth€valueŠ Q Šas€a€matter€of€fact,€but€we€can€perfectly€well€make€sense€of€counterfactual€hypotheticalsĻwhich€assume€that€it€is€true:€e.g.,€`If€John€weren't€bald,€he€would€be€happier€than€he€is'.€ButĻnow,€there€are€lots€of€cases€where€if€a€sentence€has€a€truth€value€gap€at€all,€then€it€has€itĻņņnecessarilyóó.€Consider,€for€example,€`The€man€who€is€taller€than€he€is,€is€a€Russian.'€SinceŠ m ½ Šthere€can't€be€a€man€who€is€taller€than€he€is,€the€presupposition€of€this€sentence€is€necessarilyĻfalse;€so€if€failure€of€presupposition€entails€a€tv„gap€(as€we€may€suppose),€then€this€sentenceĻis€necessarily€without€truth€value.€But€the€question€now€arises:€What€could€it€mean€toĻstipulate€a€truth€value€for€a€sentence€which€lacks€a€truth€value€ņņnecessarilyóó?€What,€forŠ Ł)  Šexample,€is€the€force€of€counterfactuals€with€antecedents€like€`If€the€man€who€is€taller€thanĻhimself€were€Russian,€then....'?€We€doubt€that€there's€any€sense€to€be€made€of€suchĻhypotheticals.ĢĢBut€if€one€can't€assign€a€truth€value€to€`The€man€who€is€taller€than€himself€is€Russian,'€we€can'tĻassign€T€(or,€of€course,€any€other€tv)€to€`Either€the€man€who€is€taller€than€he€is€is€Russian€orĻthe€man€who€is€taller€than€he€is€is€not€Russian'.€This€latter,€however,€appears€to€have€theĻlogical€form€ņņa€is€F€or€a€is€not€Fóó,€so€it€has€to€come€out€true€if€supervaluation€theory€is€toŠ ± Šrecover€the€Law€of€Excluded€Middle.€Something€appears€to€have€gone€wrong€with€theĻtheory.ĢĢPrecisely€the€same€sort€of€difficulties€are€produced€by€vagueness€assuming,€as€seemsĻextremely€plausible,€that€some€vague€sentences€lack€truth€values€necessarily.€So,€suppose€thatĻthere's€no€fact€of€the€matter€about€whether€wall„to„wall€carpeting€is€furniture.€Then,€surely,Ļthe€fact€that€there€is€none€is€not€itself€contingent;€if€it's€true€in€our€world,€then€it's€true€in€everyĻworld.€If€you're€inclined€to€think€that€it's€merely€contingent€that€there's€no€fact€of€the€matterĻwhether€wall„to„wall€carpets€are€furniture,€ask€yourself€what€discovery€about€the€world€(orĻabout€English,€for€that€matter)€would€convince€you€that€you€that,€by€gosh,€you€were€wrongĻand€that€it€really€is€(or€really€isn't)€furniture€after€all.€Similarly€with€examples€like€`SomeoneĻwho€has€precisely€38€hairs€is€bald';€if€it€has€a€truth€value€gap€at€all,€then€it€has€its€truth€valueĻgap€necessarily.€ĢĢBut€now€the€previous€argument€applies:€If€a€sentence€lacks€a€truth€value€necessarily,€then€itĻcan't€be€precisified;€if€it€can't€be€precisified,€then€no€truth€value€can€be€assigned€to€complexŠ a+±&# Šsentences€in€which€it€occurs,€including€ones€which€have€the€form€of€logical€truths.€So€theĻprogram€of€using€supervaluation€theory€to€construct€a€semantics€of€vagueness€that€preservesĻthe€logical€truths€apparently€fails.Ģń—ńĢń—ńņņń—ńĢń—ńĢBoolean€concepts€continued:€The€status€of€Principle€Póó.Š  l ŠĢOur€second€point€is€that,€whether€or€not€prototypes€generate€tv„gaps€in€Boolean€concepts,Ļthere's€another,€and€quite€different€problem€about€assimilating€the€semantics€of€such€conceptsĻto€prototype€theory.€Namely,€that€for€indefinitely€many€Boolean€concepts,€there€ņņisn't€anyóóŠ ˆ Ų Šprototype€even€though:€ĢĢ(i)€the€primitive€constituent€concepts€all€have€prototypes,€and€ĢĢ(ii)€the€complex€concept€itself€has€definite€truth€conditions.€ĢĢSo,€for€example,€consider€the€predicate€`isn't€a€cat';€and€let's€suppose€(probably€contrary€toĻfact)€that€`cat'€is€NOT€vague;€i.e.,€`is€a€cat'€has€either€the€value€T€or€the€value€F€for€everyĻobject€in€the€relevant€universe€of€discourse.€Then,€clearly,€there€is€a€definite€semanticĻinterpretation€for€`is€not€a€cat';€i.e.,€it€expresses€the€property€ofņņ€notóó€ņņbeing€a€catóó,€a€propertyŠ –ę Šwhich€all€and€only€objects€in€the€extension€of€the€complement€of€the€set€of€cats€instantiate.ĻHowever,€although€`isn't€a€cat'€is€entirely€well€behaved€on€these€assumptions,€it€pretty€clearlyĻhas€no€stereotype;€and€nor€do€indefinitely€many€other€Boolean€complex€concepts.€There€isn'tĻany€stereotypic€nonprime€number,€and€there€isn't€anything€that€is€stereotypically€pink€if€it'sĻsquare.€And€so€on.€This€is€a€point€that€KP€recognize€explicitly€(cf.€circa€p.€48).ĢĢWe€remark,€in€passing,€that€this€difficulty€does€not€depend€on€a€proprietary€reading€ofĻ`prototype';€for€example,€it€holds€whether€you€think€that€prototypes€are€something€like€featureĻsets,€as€OS2€explicitly€does,€or€whether€you€think€that€prototypes€are€something€likeĻexemplars.€KP€aren't€entirely€explicit€about€which€of€these€notions€of€prototype€they€have€inĻmind,€but€on€balance€it€seems€to€be€the€second.ĢĢTo€return€to€the€main€theme:€there€are€indefinitely€many€cases€in€which€there€is€no€prototypeĻcorresponding€to€a€complex€Boolean€concept;€a€fortiori,€the€MR€corresponding€to€such€aĻconcept€isn't€a€prototype.€€Faced€with€this€problem,€a€theorist€might€just€give€up€and€admitĻthat€in€at€least€€indefinitely€many€cases,€what€a€primitive€concept€transmits€to€its€complexĻhosts€is€not€its€prototype;€and€that,€in€such€cases,€the€identification€of€MRs€with€prototypesŠ a+±&# Šis€simply€false.€The€obvious€elaboration€of€this€view€is€that€the€MRs€corresponding€to€suchĻcomplex€concepts€specify€ņņnotóó€their€prototypes€but€their€logical€forms,€and€that€theirŠ Ė Šinterpretations€are€computed€from€their€logical€forms€in€the€standard€Classical€way.€So,€theĻfunction€FC€assigns€to€€`isn't€a€cat'€the€logical€form€€ņņnot(F)óó,€and€the€rule€of€interpretation€forŠ Q Šan€MR€of€that€form€assigns€as€its€extension€the€complement€of€the€set€of€Fs.€This€is,€ofĻcourse,€to€abandon€the€project€of€using€prototype€structure€to€account€for€the€productivityĻof€complex€Boolean€predicates.€So€be€it.ĢĢWe€are€seriously€unclear€whether,€or€to€what€extent,€that€is€the€course€of€action€that€KPĻendorse;€their€text€often€suggests€that€they€have€a€rather€different€treatment€in€mind.€SupposeĻyou€were€to€give€up€not€the€idea€that€MRs€are€prototypes€but€rather€Principle€P,€accordingĻto€which€the€semantic€value€of€a€concept€is€computed€from€the€corresponding€MR.€You€couldĻthen€grant€that€the€interpretation€of€`not(F)'€isn't€computed€from€a€prototype,€but€argue€thatĻis€compatible€with€prototype€theory's€account€of€MRs.€That's€because€the€interpretation€ofĻ`(not)F'€ņņisn't€computed€from€its€MRóó.€In€fact,€as€far€as€the€process€of€interpreting€it€isŠ *z Šconcerned,€it€needn't€be€assumed€that€`not(Fņņ)'€even€hasóó€an€MR.€"Consider€[the€concept]€ņņredóó.Š E• ŠThe€concept€ņņ(is)€not€redóó€does€not€appear€to€have€a€prototype;€for€how€might€one€resolve€theŠ `° Šchoice€among€ņņwhite,€green,€black,€yellowóó€and€all€the€other€colors€that€ņņredóó€excludes?Š {Ė ŠNevertheless,€the€degree€of€membership€in€the€conceptņņ€not€redóó€is€[sic]€a€matter€ofŠ –ę Šprototypicality.€Only,€the€relevant€prototype€is€not€some€prototype€for€ņņnot€red€óóbut€theŠ ± Šprototype€for€ņņred,óó€and€the€degree€to€which€something€is€not€red€is€a€matter€of€how€ņņlittleóó€ratherŠ Ģ Šthan€how€much,€it€resembles€that€prototype"€(p.48).€(Similarly€for€conjunction,€disjunctionĻand€the€rest;€cf.€p.49.)€ĢĢNotice€that,€on€this€account,€if€the€prototype€for€`red'€is€a€fire„engine€in€`That's€red',€then€it'sĻa€fire„engine€`That's€not€red'€too,€and€`red'€contributes€the€ņņsameóó€extension€in€both€cases;€viz.,Š S £ Šit€contributes€the€set€of€things€that€are€sufficiently€similar€to€fire€engines.€However,€`red'€doesĻnot€contribute€its€extension€to€`not€red'€ņņbyóó€ņņcontributing€the€prototype€of€`red'€to€the€prototypeŠ ‰"Ł Šof€`not€redóó'.€That's€the€sense€in€which€P€is€violated.Š ¤#ō ŠĢWe€remarked€above€that€there€seem€to€us€to€be€decisive€reasons€for€holding€onto€P.€We€won'tĻrepeat€them€here.€The€point€we€want€to€emphasize€is€that€it's€unclear€that€giving€up€P€is€evenĻa€ņņcoherentóó€alternative€to€giving€up€`MRs€are€prototypes'.€KP€tell€us€that,€in€computing€anŠ (`#  Šinterpretation€for€`not€red',€"the€relevant€prototype€is€not€some€prototype€for€ņņnot€red€óóbut€theŠ +){$! Šprototype€for€ņņred,óó€and€the€degree€to€which€something€is€not€red€is€a€matter€of€how€ņņlittleóó€ratherŠ F*–%" Šthan€how€much,€it€resembles€that€prototype."€But€the€question€arises€how€they€know€that€thisŠ a+±&# Šis€so.€More€precisely:€How€does€the€computation€that€assigns€an€interpretation€to€the€formulaĻ"not€red"€know€that€it's€the€prototype€for€"red"€(and€not,€say,€the€prototype€for€"green"€orĻ"soup"€or€"transcendental")€that€it€should€consult€when€it€does€so?€The€answer€surely€mustĻbe€that€the€MR€it€computes€over€is€not€the€prototype€for€"red"€but€rather€a€representation€ofĻthe€logical€form€of€"not€red".€But€this€means€(a)€that€P€is€still€in€force;€i.e.,€the€interpretationĻof€"not€red"€is€computed€from€the€MR€of€"not€red,"€ņņnotóó€from€the€MR€of€"red";€and€(b)€theŠ 7 ‡ ŠMR€of€"not€red"€is€a€logical€form,€ņņnotóó€a€prototype.€In€effect,€though€their€text€rather€suggestsŠ R ¢ Šthe€contrary,€what€KP€have€really€opted€for€is€the€alternative€that€makes€all€complex€BooleanĻpredicates€ņņcounterexamplesóó€to€prototype€theory.€Š ˆ Ų ŠĢņņPart€III:€Pet€Fish.Š ¾  ŠóóĢThere€is€another€kind€of€case,€discussed€in€both€SO2€and€KP,€in€which€it€is€apparently€notĻpossible€to€provide€the€correct€interpretation€of€a€complex€predicate€given€just€its€structureĻand€the€prototypes€of€its€primitive€constituents.€The€problem€here€is€not€that€the€complexĻconcept€fails€to€have€a€prototype,€as€in€many€of€the€Boolean€cases,€but€rather€that€an€object'sĻsimilarity€to€the€prototype€for€a€complex€concept€seems€not€to€vary€systematically€as€aĻfunction€of€its€similarity€to€the€prototypes€of€the€constituents€concepts.€So,€for€example,€aĻgoldfish€is€a€poorish€example€of€a€fish,€and€a€poorish€example€of€a€pet,€but€it's€quite€a€goodĻexample€of€a€pet€fish.€ĢĢNow,€according€to€prototype€theory,€to€have€a€concept€is€to€have€its€prototypeņņ€óótogether€withŠ ē7 Ša€measure€of€the€distance€between€the€prototype€and€an€€arbitrary€object€in€the€domain€ofĻdiscourse;€in€effect,€this€distance€measure€is€the€form€that€FI€takes€in€prototype€versions€ofĻcomputational€theories€of€mind.€Prima€facie,€however,€the€distance€of€an€arbitrary€object€fromĻthe€prototypic€pet€fish€isņņ€notóó€a€function€of€its€distance€from€the€prototypic€pet€and€its€distanceŠ S £ Šfrom€the€prototypic€fish.€In€consequence,€knowing€that€PET€and€FISH€have€the€prototypesĻthat€they€do€does€not€permit€one€to€predict€that€the€prototypical€pet€fish€is€more€like€aĻgoldfish€than€like€a€trout€or€a€herring,€on€the€one€hand,€or€a€dog€or€a€cat,€on€the€other.€ButĻif€prototypes€aren't€compositional,€then,€to€put€it€mildly,€the€identification€of€concepts€withĻprototypes€can't€explain€why€concepts€are€€productive.ĢĢBoth€OS2€and€KP€offer€solutions€for€this€problem,€but€it€seems€to€us€that€neither€is€even€closeĻto€satisfactory.€We'll€review€them€very€briefly.Ģā āŠ F*–%" ŠAs€we€remarked€above,€OS2€takes€prototypes€to€be€matrices€of€weighted€features€(rather€thanĻā āas€exemplars).ņņ7óó€So,€for€example,€the€prototype€for€`apple'€might€specify€a€typical€shape,€color,Š Ė Štaste,€size,€ripeness....€etc.€Let's€suppose,€in€particular,€that€the€prototypical€apple€is€red,€andĻconsider€the€problem€of€constructing€a€prototype€for€`purple€apple'.€The€basic€idea€is€that€youĻform€a€derived€matrix€that's€just€like€the€one€for€apple,€except€that€the€featureņņ€purple€óóreplacesŠ  l Šthe€feature€ņņredóó€and€the€weight€of€the€new€feature€is€increased.€Pet€fish€presumably€work€theŠ 7 ‡ Šsame€way.ņņ8óóŠ R ¢ ŠĢIt's€pretty€clear,€however,€that€this€treatment€is€flawed.€To€see€this,€ask€yourself€ņņhow€muchóóŠ ˆ Ų Šthe€feature€purple€weighs€in€the€feature€matrix€for€PURPLE€APPLE.€Clearly,€it€must€weighĻmore€than€the€feature€red€does€in€the€feature€matrix€for€APPLE€since,€though€there€can€beĻapples€that€aren't€red,€there€can't€be€purple€apples€that€aren't€purple€(any€more€than€there€canĻbe€red€apples€that€aren't€red€or€purple€apples€that€aren't€apples.)€In€effect,€purple€has€to€weighĻņņinfinitelyóó€much€in€the€feature€matrix€for€PURPLE€APPLE€because€`purple€apple€„„„>€purple'Š _  Šis€a€ņņlogicalóó€truth.€So€the€theory€faces€a€dilemma:€either€treat€the€logical€truths€as€(merely)Š *z Šextreme€cases€of€statistically€reliable€truth,€or€admit€that€the€ņņweightsóó€assigned€to€the€featuresŠ E• Šin€derived€matrices€aren't€compositionally€determined€even€if€the€features€themselves€are.ĻNeither€horn€of€this€dilemma€seems€happy.€What€really€sets€the€weight€of€the€PURPLE€inĻPURPLE€APPLE€isn't€its€prototype;€it's€its€logical€form.€€€Ģ€ĢAnd,€even€if€the€treatment€weren't€flawed,€it€is€clearly€not€general.€The€problem€is€that€theĻ`features'€associated€with€the€Ns€in€AN€constructions€are€not,€in€the€general€case,Ļindependent.€So,€suppose€that€the€prototype€for€NURSE€includes€the€featureņņ€femaleóó.€€YouŠ R Šcan't€derive€the€prototype€for€MALE€NURSE€by€just€replacing€ņņfemaleóó€with€ņņmale;óó€all€sortsŠ m Šof€other€things€have€to€change€too.€Notice€that€this€is€true€even€though€`male€nurse'€is€(in€KP'sĻterm)€`intersective';€i.e.,€even€though€the€set€of€male€nurses€is€the€overlap€of€the€set€of€malesĻwith€the€set€of€nurses.€Things€go€even€worse€for€the€OS2€proposal€when€one€considersĻnonintersective€concepts€like€STONE€LION,€DECOY€DUCK,€FAKE€DIAMOND,ĻPOTENTIAL€PROVOST€and€so€forth.ĢĢKP€offer€a€more€complicated€analysis€of€the€pet€fish€case,€but€it€doesn't€work€well€either.ĻScanting€the€details€once€again,€the€basic€idea€is€that€the€failure€of€pet€fish€to€be€goodĻexamples€of€pets€is€a€kind€of€context€effect,€analogous€to€the€failure€of€big€ants€to€be€goodĻexamples€of€big€things.€KP€think,€plausibly€enough,€that€the€meaning€of€`big€ant'€is€somethingĻlike€ņņbig€for€anóó€ņņantóó,€so€that€a€really€good€example€of€a€big€ant€would€be€something€that's€as„Š F*–%" Šń—ńŠ F*–%" Šń—ńgood„an„example„of„something„big€as€an€ant€can€be.€Similarly,€a€really€good€example€of€aŠ a+±&# Šmale€nurse€would€be€something€that's€as€good€an€example€of€a€nurse€as€something€male€canĻbe;€a€really€good€example€of€a€pet€fish€would€be€something€that's€as€good€an€example€of€a€petĻas€a€fish€can€be,€and€so€on.Ģ€ĢIt€seems€to€us,€however,€that€the€assimilation€of€PET€FISH€to€BIG€ANT€is€clearly€ill€advised:Ļ`pet€fish'€entailsņņ€petóó€but€`big€ant'€does€ņņnotóó€entail€ņņbigóó,€and€KP's€proposal€leaves€this€asymmetryŠ 7 ‡ Šentirely€unexplained.€ĢĢWe're€claiming,€in€effect,€that€if€AN€„„„>€A,€that's€a€very€strong€reason€to€suppose€that€theĻinterpretation€of€the€A€in€AN€is€ņņnotóó€affected€by€the€interpretation€of€the€N.€For,€suppose€theŠ £ó  Šcontrary;€suppose€that€the€meaning€of€A€were€some€how€converted€from€ņņAóó€to€ņņA'óó€when€A€isŠ ¾  Ša€constituent€of€AN;€then€AN€shouldn't€entail€A€but€A'.€What€better€argument€could€there€beĻthat€`male'€meansņņ€maleóó€in€`male€nurse'€than€the€necessity€of€`Male€nurses€are€male'.Š ōD  Š(Analogously,€what€better€argument€could€there€be€that€the€content€of€MALE€is€ņņmale€óóthanŠ _  Šthat€`Males€are€male'€is€necessary?€Indeed,€what€other€argument€have€we€got?)ĢĢIt€follows€from€this€proposal€that€AN€„„„>€A€is€a€necessary€condition€for€an€adjective€beingĻintersective,€hence€that€`big'€is€not€intersective€in€`big€ant'.€KP,€however,€use€a€substitutivityĻtest€to€decide€on€intersectivity:€€A€is€intersective€in€AN1€only€if€(`a€is€AN1'€and€`a€is€N2')€€„„„>Ļ`a€is€AN2.'€So,€for€example,€`skillful'€is€not€intersective€according€to€KP€since€(`a€is€a€skillfulĻviolinist'€and€`a€is€a€plumber')€does€not€entail€`a€is€a€skillful€plumber.€ĢĢYou€might€think€that€the€two€tests€for€intersectivity€should€be€coextensive€and€that€`big'Ļwould€fail€to€be€intersective€by€either€one.ņņ9óó€However,€KP€argue€that€`big€ant'€ņņisóó€intersectiveŠ m Šbut€that€the€fact€that€it€is€is€obscured€by€the€(putative)€context€effect€of€`ant'€on€theĻinterpretation€of€`big'.€We€find€this€doctrine€hard€to€construe.€If€`big€ant'€is€in€the€intersectionĻof€the€big€things€with€the€ants,€then€`big€ant€„„„>€big'€must€be€valid,€which,€however,€intuitionĻdenies.€Indeed,€on€KP's€own€analysis,€a€big€ant€ought€to€be€(not€in€the€set€of€big€things€but)Ļin€the€set€of€things€that€are€bigņņ€for€antsóó.€But€the€set€of€things€that€are€big€for€ants€isn't€includedŠ ¤#ō Šin€the€set€of€things€that€are€bigņņ€tout€courtóó.€In€fact,€`big€ant'€doesn't€look€to€be€intersective€onŠ æ$  Šany€interpretation€that€we€can€think€of.ņņ10óóŠ Ś%*! ŠĢThe€bottom€line,€then,€is€that€PET€FISH€is€a€counterexample€to€the€compositionality€ofĻprototypes,€and€there€is€no€reason€at€all€to€suppose€that€the€problem€it€raises€would€be€solvedĻby€whatever€mechanism€it€is€that€the€semantics€employs€to€cope€with€BIG€ANT.€We€want€toĻemphasize€that,€quite€aside€from€the€technical€issues,€this€conclusion€really€does€seem€quiteŠ a+±&# Šplausible.€The€reason€you€can't€derive€the€PET€FISH€prototype€given€the€PET€prototype€andĻthe€FISH€prototype,€is€simply€that€what€ņņkinds€of€fish€people€keep€as€pets€is€a€fact€about€theŠ Ė Šworld,€not€a€fact€about€concepts€or€language.óó€It€is€therefore€possible€to€be€perfectly€clearŠ ę6 Šwhat€`pet€fish'€means,€and€yet€have€no€idea€which€pet€fish€are€prototypical.€Which€pet€fish€areĻprototypical€is€something€you€ņņjust€have€to€go€out€and€learnóó.€The€ņņlanguageóó€(/concept)€assuresŠ  l Šyou€that€the€prototypical€pet€fish€is€a€pet€and€a€fish,€just€as€the€language€assures€you€that€theĻprototypical€big€ant€is€big€for€an€ant.€After€that,€you're€on€your€own.ĢĢņņConclusion:€prototypes€are€fish€out€of€water.Š ˆ Ų ŠĢóóThe€prototype€for€PET€FISH€is,€as€it€were,€an€idiom;€a€merely€linguistic€(/conceptual)€inquiryŠ ¾  Šwill€tell€you€that€pet€fish€are€fish,€but€no€merely€linguistic€(/conceptual)€inquiry€will€tell€youĻwhich€pet€fish€are€prototypical.€Putting€it€that€way€might,€however,€suggest€that€there€is€Ļsome€hope€for€the€prototype€theorist€after€all.€Why€shouldn't€he€just€admit€that€his€story€aboutĻthe€compositionality€of€concepts€doesn't€work€for€PET€FISH,€but€argue€that€the€reason€itĻdoesn't€is€precisely€that€PET€FISH€ņņisóó€an€idiom.€On€anybody's€story,€idioms€are€expressionsŠ E• Šwhere€the€interpretation€that€the€compositional€semantics€predicts€is,€as€it€were,€over„riddenĻby€special€conventions€that€must€simply€be€acquired€case€by€case.€Having€a€green€thumb€turnsĻout€not€to€be€having€a€thumb€that€is€green,€compositional€semantics€to€the€contraryĻnotwithstanding.€Why,€then,€shouldn't€the€semantics€say€that€a€paradigm€pet€fish€ought€to€beĻa€paradigm€pet€and€a€paradigm€fish?€If€that€prediction€is€wrong,€that€only€shows€that€`pet€fish'Ļisn't€compositional.ĢĢSo,€then,€here's€the€proposal:€The€prototypical€ANs€are€the€intersection€of€the€prototypicalĻA€things€with€the€prototypical€N€things€ņņin€the€unmarked€caseóó.€But€you€default€to€theŠ 8ˆ Šunmarked€case€only€if€you€do€not€have€specific€information€to€the€contrary.€Just€as€theĻsemantics€of€English€supports€the€inference€that€green€thumbs€are€green€and€thumbs,€so€theĻsemantics€of€concepts€supports€the€inference€that€prototypical€pet€fish€are€prototypical€petsĻand€prototypical€fish.€It's€just€that€when€you've€learned€which€fish€it€is€that€people€actually€doĻkeep€as€pets,€you€learn€to€override€the€inference€that€the€semantics€supports.ĢĢThe€idea€that€composition€works€on€prototypes€to€deliver€default€values€is€maybe€a€littleĻcounterintuitive;€first€blush,€neither€`pet€fish'€nor€`big€ant'€seem€plausible€€candidates€forĻidioms.€But€it€might€nonetheless€be€a€hard€theory€to€refute.€That's€because€it€is€required€toĻpredict€default€to€the€compositional€prototype€only€in€cases€where€no€real€world€beliefs€areĻover„riding.€Failures€to€default€(PET€FISH,€BIG€ANTS,€and€so€forth)€can€thus€be€viewed€asŠ a+±&# Šprima€facie€indications€that€such€over„riding€has€indeed€occurred.€Correspondingly,€a€clearĻtest€of€the€theory€would€require€examining€cases€which€there€is€independent€reason€toĻsuppose€that€ņņonly€óólinguistic€(/conceptual)€knowledge€is€employed€in€drawing€an€inference.€ItŠ ę6 Šis,€however,€notoriously€difficult€to€construct€such€cases,€the€vagaries€of€the€analytic/syntheticĻdistinction€being€what€there€are.ĢĢSo€we€can't€prove€that€defaulting€to€the€compositional€prototype€isn't€the€strategy€that€peopleĻactually€do€follow.€But€it's€easy€to€see€that€it€shouldn't€be€because€it's€an€irrational€strategy.ĻThat€it€is€irrational€follows€from€two€considerations,€rough€formulations€of€which€go€asĻfollows:ĢĢi.€All€else€equal,€the€more€complexly€modified€a€concept€is,€the€ņņlessóó€you€are€likely€to€haveŠ Ł)  Šspecial€knowledge€about€the€things€in€its€extension.ĢĢSo,€for€example,€I€know€a€little€about€cows€qua€cows.€But€I€know€next€to€nothing€aboutĻbrown€cows€qua€brown€cows€(except,€of€course,€that€they€are€brown€and€cows);€and€I€knowĻliterally€nothing€about€brown€cows€owned€by€people€whose€last€names€start€with€`W'€quaĻbrown€cows€owned€by€people€whose€last€names€start€with€`W'€(except,€of€course,€that€theyĻare€brown,€and€cows€and€owned€by€people€whose€last€names€start€with€`W').€ĢĢThe€upshot€is€that€if€my€strategy€is€to€default€to€the€compositional€prototype€when€I€have€noĻspecial€information€to€the€contrary,€then€the€more€heavily€modified€a€concept€is,€the€moreĻlikely€I€am€to€default€to€its€compositional€prototype.ĢĢNotice,€however,€that€(ii)€is€also€true:ĢĢii.€All€else€equal,€the€more€heavily€modified€a€concept€is,€the€ņņlessóó€likely€that€its€prototype€isŠ n!¾ Špredicted€by€the€prototypes€of€€its€constituents.€€€€ĢĢSo,€for€example,€pet€fish€aren't€good€bets€for€satisfying€the€pet€prototype;€but€still€less€so€areĻpet€fish€that€live€in€Armenia;€and€still€less€so€are€pet€fish€who€live€in€Armenia€and€haveĻrecently€swallowed€their€owners....€And€so€forth.€This€is€actually€quite€close€to€a€point€weĻmade€above:€the€`features'€that€prototypical€Ns€exhibit€are€not,€in€the€general€case,Ļindependent€of€one€another.€In€consequence,€the€more€A's€you€put€on€to€modify€the€N,€theĻmore€likely€it€is€the€prototypical€ANs€will€ņņnotóó€exhibit€the€features€that€the€€prototypical€Ns€do.€Š F*–%" ŠŠ a+±&# ŠIf€you€now€put€(i)€and€(ii)€together,€it's€clear€why€defaulting€to€the€compositional€prototypeĻis€an€irrational€strategy:€on€the€one€hand,€the€more€complexly€modified€a€concept€is,€the€moreĻlikely€it€is€that€(i)€will€require€you€to€default€to€the€compositional€prototype.€On€the€otherĻhand,€the€more€complexly€modified€a€concept€is,€the€more€(ii)€makes€it€likely€that€defaultingĻto€the€compositional€prototype€will€give€you€the€wrong€interpretation.€It€is,€however,Ļirrational€to€employ€a€strategy€if€the€more€likely€you€are€to€use€it,€the€more€likely€it€is€to€fail.ĢĢņņThe€Bottom€line.€Š m ½ ŠóóĢPrototypes€aren't€compositional;€they€work€like€idioms.€Concepts,€however,€ņņmustóó€beŠ £ó  Šcompositional;€nothing€else€could€explain€why€they€are€productive.€So€concepts€aren'tĻprototypes.€This€is€too€sad€for€words.€A€theory€of€concepts€has€two€things€to€explain:€howĻconcepts€function€as€categories,€and€how€a€finite€mind€can€have€an€infinite€conceptualĻcapacity.€Prototypes€do€a€not„bad€job€of€explaining€the€first€(though,€notoriously,€they're€notĻso€good€at€penguins€being€birds;€see€also€Armstrong€et€al.,€1983).€Anyhow,€they€doĻnoticeably€better€than€definitions.€But€they€are€ņņhopelessóó€at€the€second€job.€In€fact,€what€aŠ E• Šconstituent€concept€contributes€to€its€host€appears€to€be€precisely€ņņnecessary€conditionsóó€andŠ `° Šnot€ņņstatistical€correlatesóó;€`pet'€contributesņņ€petóó€to€`pet€fish',€and€not,€for€example,€ņņfurry€andŠ {Ė Šcuddlyóó;€`bachelor'€contributes€ņņhasn't€a€spouseóó€to€`elderly€bachelor,'€and€notņņ,€óóforņņ€óóexample,ņņ€hasŠ –ę Ša€live„in€girl€friendóó;€`big'€contributes€`big€for€an€N'€to€`big€N',€and€not,€for€example,€ņņheavyóó€orŠ ± Šņņhard€to€kill€by€stepping€onóó.€Etc.€The€penultimate€line€is:€ņņit€is€not€in€virtue€of€their€statisticalŠ Ģ Šproperties€that€concepts€are€compositional.ĢóóĢAnd€the€bottom€line€is:€ņņnobody€knows€what€makes€concepts€compositional,€so€nobody€knowsŠ m Šwhat€concepts€are.ĢĢAfterthought.ĢĢóóWhile€we€were€writing€this€paper,€an€article€appeared€by€(Huttenlocher€and€Hedges,€1994)Š ¤#ō Šthat€proposes€a€statistical€model€for€the€formation€of€complex€prototypes,€one€that€relies€onĻthe€assumption€of€independence€of€features.€"In€constructing€a€conjoint€category....€theĻlistener€must€make€some€assumption€about€the€form€of€the€relation€between€values€of€theĻconstituent€categories.€The€assumption€that€the€categories€are€independent€is€an€obviousĻone....€If€the€assumption€of€independence€holds,€the€formal,€mechanism€we€have€proposed€isĻapplicable"€(p.163).€However,€as€we've€just€been€seeing,€the€more€complex€a€concept€is,€theĻmore€likely€the€`listener'€must€depend€on€its€internal€structure€(rather€than€his€backgroundŠ a+±&# Šknowledge)€to€decode€it;€and€the€more€complex€the€concept€is,€the€more€the€assumption€thatĻthe€features€its€constituents€contribute€are€independent€is€likely€ņņnotóó€to€be€true€(so€long€asŠ Ė Šfeature€assignments€express€statistical€correlates€rather€than€necessary€conditions).€The€waryĻlistener€will€therefore€either€avoid€the€strategy€that€Huttenlocher€and€Hedges€commend,€orĻavoid€decoding€complex€concepts€by€assigning€them€prototypes.ĢĢHuttenlocher€and€Hedges€refer€to€data€that€suggest€that€subjects€do€in€fact€reliably€default€toĻthe€independence€assumption.€We€suspect€that€if€this€is€so,€it€shows€only€that€subjects€whoĻare€required€to€make€guesses€in€experimental€environments€don't€care€much€whether€theirĻguesses€are€true.€It€would€be€interesting€to€know€what€happens€in€situations€where€theĻoutcomes€matter€more.€How€much€are€ņņyouóó€prepared€to€bet€that€Mongolian€Grey€Geese€satisfyŠ ¾  Šthe€Goose€prototype?€(Compare:€How€much€are€you€prepared€to€bet€that€Mongolian€GreyĻGeese€are€Grey,€Geese,€and€Mongolian?)Š  ōD  Šņ ņĢÓ  ÓNotesĢĢÓ  Ó1.ó ó€A€word€on€notation:€We€use€caps€for€names€of€concepts€(as€in€`the€concept€COW'.€WeŠ Q Šuse€italics€for€names€of€semantic€interpretations€(as€in€`the€concept€COW€expresses€theĻproperty€of€ņņbeing€a€cowóó')€and,€occasionally,€for€names€of€semantic€features.€Since€weŠ ; ‹ Šassume€that€words€and€phrases€express€concepts€(e.g.,€that€the€English€phrase€`brownĻcow'€expresses€the€concept€BROWN€COW),€and€that€the€productivity€of€language€isĻparasitic€on€the€productivity€of€thought.€we€are€thus€explicit€about€drawing€theĻlanguage/thought€distinction€only€when€it€matters€to€the€discussion.€Often€we'll€go€backĻand€forth€between€concepts€and€terms€as€convenience€of€exposition€dictates.€ĢĢ2.€For€further€discussion€of€the€Standard€Objection,€see€Fodor,€1981;€Osherson€and€Smith,Ļ1981.€For€a€survey€of€the€literature€on€prototype€effects€in€categorization€tasks,€see€SmithĻand€Medin,€1981.€€ĢĢ3.€Some€Connectionists€apparently€hold€that€conceptual€repertoires€are€intrinsicallyĻņņnonóóproductive€(viz.,€finite).€For€purposes€of€this€discussion,€they€are€beyond€the€pale.Š Ļ ŠĢ4.€Clearly,€this€condition€must€be€satisfied€whenever€grasping€a€function€requires€itsĻexplicit€internal€representation.ĢĢ5.€An€alternative€("fuzzy€logic")€treatment€assumes€that€there€are€infinitely€many€truthĻvalues€(tv's)€between€T€and€F,€and€that€`wall„to„wall€carpets€are€furniture'€has€one€of€them.ĻHowever,€Osherson€and€Smith€(1981)€have€shown€that€this€approach€leads€to€its€ownĻkinds€of€difficulties€in€the€case€of€concepts€that€are€built€out€of€the€Boolean€connectives.€ĢĢ6.€In€fact,€some€further€"penumbral"€requirements€on€the€coherence€of€these€`arbitrary'Ļassignments€are€in€force,€but€they€needn't€concern€us€here.€For€details,€see€Fine,€1975.€ĢĢ7.€Like€every€other€cognitive€scientist€we've€encountered€who€uses€the€notion,€OS€don'tĻsay€what€a€feature€is,€or€why€the€concept€FISH€has€more€than€the€one€feature€ņņ+fishóó.€(It'sŠ ł&I" Štrue,€of€course,€that€if€you€ask€subjects€to€list€some€typical€properties€of€fish€they€willĻhardly€ever€include€ņņbeing€fishóó.€But€familiar€Griceian€considerations€explain€their€not€doingŠ /)$! Šso.€Notice€that€they€usually€don't€listņņ€existingóó,€or€ņņbeing€thingsóó€either,€though€both€areŠ J*š%" Šfeatures€that€prototypical€fish€exhibit.)€Since,€however,€we€have€long€abandoned€hope€ofŠ e+µ&# Šfiguring€out€what€cognitive€scientists€might€mean€by€`feature',€we'll€just€take€the€notion€forĻgranted€in€what€follows.ĢĢ8.€Though€it's€not€entirely€clear€what€one€is€supposed€to€do€if€the€prototype€for€FISHĻdoesn't€have€a€parameter€for€degree€of€domesticatedness.€Maybe€the€pet€fish€features€areĻjust€the€union€of€the€pet€features€and€the€fish€features.ĢĢ9.€It's€not€possible€to€prove€this,€of€course,€since€what€rules€of€inference€are€valid€for€ANĻconstruction€is€moot.€But€the€following€arguments€seem€to€us€pretty€persuasive.€ĢĢFor€any€X,€if€€N€and€X€are€substitutable,€then€if€€'a€is€AN'€is€true,€then€'a€is€AX'€is€true;€thatĻis,€'a€is€A'€is€trueņņ€whatever€else€is€true€of€aóó;€€that€is,€€`a€is€A'€is€true€tout€court.€€So€if€KP'sŠ Ł)  Štest€is€satisfied,€so€too€is€ours.€ĢĢLikewise€in€the€other€direction:€If€`a€is€AN1'€entails€`a€is€A',€then€`a€is€AN1€&€a€is€N2'Ļentails€`a€is€AN2'€(assuming€that€conjunction€introduction€holds).€So€if€our€test€is€satisfied,Ļso€too€is€KPs.€So€it€appears€that€the€two€tests€are€equivalent.ņ ņŠ `° ŠĢó ó10.€Our€own€view€is€that€`big'€and€the€like€are€best€analyzed€as€"defined€in€use"€(seeŠ –ę ŠRussell€1956.€In€effect,€the€relevant€semantic€rule€interprets€`big€N'€for€variable€N,€to€theĻinstances€of€which€it€assigns€all€and€only€the€Ns€that€are€big€for€Ns.€This€seems€compatibleĻwith€everything€KP€say€about€the€case,€except€for€their€claim€that€`big€N'€is€intersective;Ļwhich€we€find€puzzling,€as€previously€remarked.€In€any€case,€KP€apparently€concede€thatĻ`big€ant'€doesn't€entail€`big'€tout€court,€which€is€all€that€our€argument€in€the€text€requires.ĢĢņ ņŠ  S £ ŠĢÓ  ÓĢReferencesĢĢó óĢÓ  ÓArmstrong,€S.L.,€Gleitman,€L.R.€and€Gleitman,€H.€(1983).€"What€some€concepts€mightĻą 4 ąnot€be."€ņņCognition,€óó13,€263„308.Š R ¢ ŠFine,€K.€(1975)€"Vagueness,€truth€and€logic,"€ņņSyntheseóó€30,€265„300.Š m ½ ŠFodor,€J.€(1981).€"The€present€status€of€the€innateness€controversy,"€in€ą L ąŠ ˆ Ų ŠREPRESENTATIONS,€MIT€Press,€Cambridge,€MA.ĢFodor,€J.€and€Pylyshyn,€Z.€(1988).€"Connectionism€and€cognitive€architecture:€A€criticalĻą 4 ąanalysis,"€ņņCognition,óó28,€3„71.Š Ł)  ŠHuttenlocher,€J.€and€Hedges,€L.€(1994)€"Combining€graded€categories,"€Psych.€Rev.€volĻą 4 ą101€no€1,€157„165.ĢJackendoff,€R.€(1992).€"The€problem€of€reality,"€in€LANGUAGES€OF€THE€MIND,€MITĻą 4 ąPress,€Cambridge,€Mass.ĢKamp,€H.€and€Partee,€B.€(1995)€(=KP).€"Prototype€theory€and€compositionality,"€ą ¤ ąŠ `° Šń—ńŠ `° Šń—ńą 4 ąforthcoming,€ņņ€Cognitionóó.Š {Ė ŠMargolis,€E.€(1994)€"A€reassessment€of€the€shift€from€the€Classical€Theory€of€concepts€toĻą 4 ąPrototype€theory,"€ņņCognition,€51,€73„89.Š ± ŠóóOsherson,€D.N.€and€Smith,€E.E.€(1981).€"On€the€adequacy€of€prototype€theory€as€a€theoryŠ Ģ Šą 4 ąof€concepts,"€ņņCognition,€óó9,€35„58.Š ē7 ŠOsherson,€D.N.€and€Smith,€E.E.€(1988)€(=OS2).€"Conceptual€combination€with€prototypeĻą 4 ąconcepts,"€€in€Collins€A.€and€Smith€E.€(eds.)€READINGS€IN€COGNITIVEĻą 4 ąSCIENCE,€Morgan€€Kaufman€Publishers,€Inc.€San€Mateo,€CA.ĢRussell,€B.€(1956)€"On€denoting,"€in€R.€Marsh€(ed.)€LOGIC€AND€KNOWLEDGE,€39„56,Ļą 4 ąUnwin€Hyman,€London.ĢSmith,€E.E.€and€Medin,€D.L.€(1981).€CATEGORIES€AND€CONCEPTS,€HarvardĻą 4 ąUniversity€Press,€Cambridge,€MA.ĢStich,€S.€(1983)€FROM€FOLK€PSYCHOLOGY€TO€COGNITGIVE€SCIENCE,€MITĻą 4 ąPress,€Cambridge,€Mass.ĢÓ  Óņ ņĢÓ  ÓĢÓ  Óó óŠ  +){$! ŠŃ7€=²XXdČXXdģ7ŃÓ  ÓĢ