C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) T edition) of type theory lends itselftodi theory type considered. Forexample,theconcrete propositionalfunction“ propositional functionsofthattypeintheepistemicrealizationis a particularinterpretation,namelythesetofallconcrete type isassigned The BertrandRussellResearchCentre,McMasterU. russell: is toexaminelogicaltypesin aim My tance in the epistemic perspective atstake,andotherwisetoade perspective tance intheepistemic to beanobjectofacquain- out turns will correspondtoauniversalifit the formality of types as introduced in the Introduction (tothe Introduction the formalityoftypesasintroducedin it will be part of the interpretation of the type of predicative it willbepartoftheinterpretationtypepredicative description involvinghigher-orderquanti independent) perspectives.The lambda-calculus tobeused. would liketotakeup that question of modern sense.Buttypicalambiguityis type theoriesintheRusselliansenseof lambda-calculus. Thisopensthewayto type shows thatthetheoryoflogical speci SymbolicConventions”,inasmuchas of typical ambiguityasdescribedin Russe paradoxes, andtypesaslogicalprotot betw tobemade has that adistinction of logicaltypeasintroduced W c systems of moderntypedlambda-cal GENERALITY OFLOGICALTYPES The cipia Mathematica wo kindsofgeneralitycanbeattributedto logical types in h ora fBrrn usl tde n.s.31(summer2011):85–107 the JournalofBertrand RussellStudies Principia W rst one,whichIwillcall“externalgenerality”,pertainsto Philosophy /U.ParisOuest( . I claim indeed that the formal system oframi system . Iclaimindeedthattheformal bhalimi , and they ought to be clearlydistinguished. to , andtheyought 92001 Nanterre,France intheIntroduction(to T Brice Halimi erent “epistemic realizations”, inwhicheach W rst onepertainstothe ambiguity of the notion @ typical ambiguity, by extending the typed typical ambiguity,byextendingthe u-paris Principia Mathematica ll andWhitehead’s“PrefatoryStatement ypes. The second perspective bears on ypes. Thesecondperspectivebears een typesascalledforinthecontextof een s canbeformalizedinthewayofa not takenintoaccountinthepaper.I the word,andtypetheoriesin an interesting reconciliation between an interestingreconciliation it lends itself to a comparison with it lendsitselftoacomparison ireph culus. In particular,arecentpaper culus. 10 W cation. In the .fr ) & issn

sphere 0036-01631; online1913-8032 W rst edition).Iclaim z fromtwo(partly W rst caseonly, x W z isgreen” rst-order W Prin- W nite W rst ed C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) it canbeformalizedwithinspeci Referring to Quine andSommerville,NicholasGri Quine to Referring depending on the stock of individuals and universalswithwhichthesub- system thatgives rise to a multiplicity of epistemicanalyses,eachanalysis will be di for that function…. In general, sincedi general, In for thatfunction…. for others, with noacquaintance with calculus. Inparticular,arecentpaper(Kamareddine calculus. extending thetypedlambda-calculustobeused. take upthatquestionoftypicalambiguity,by paper. Iwouldliketo intoaccountinthe taken modern sense.Buttypicalambiguityisnot type theoriesintheRusselliansenseofword,and type theories in the toaninterestingreconciliationbetween way the calculus. Thisopens that thetheoryoflogicaltypescanbeformalizedin the way of a lambda- di Conventions” at the beginning of beginning Conventions” atthe described inRussellandWhitehead’s“Prefatory Statement of Symbolic bears ontypicalambiguity,as generality”, “internal and whichIwillcall of eachtypeiswhatsubstantiatesandexplainsitsformality. of possibleepistemiccounterparts propositional functions.Thatvariety bricehalimi 86 worth developing.Indeed,therami Ithink,ageneralandveryimportantpoint,whichis fact, in This is “ or be complex;andthustermslike“individual” inanother, turn outto may, betakentosimple may inonecontext, which, theory istosomeextent … Russell’stype duction of the lowing remark: Thus Socrates isapossiblevalueof complex hidden description tobeanalyzed byRussell’stheory ofdescriptions. tance; whereas,whenused when usedbySocrateshimselfdenotesa Anexampleoccursinconne contexts…. spooiinlfntoso oi.(Gri as propositionalfunctionsoflogic. T The secondkindofgeneralitythatcanbeattributed to logicaltypes, erent items, the rangeoftotalvariationforfunctionslike 1. w T formal theoryandepistemicrealizationsof erent fordi W rst editionof T erent people. Thus it is intole is erent people.Thusit by someone who has never met Socrates, it is a bysomeonewhohasnevermetSocrates,itis logical types Principia Mathematica z / z x ˆ z W isanindividual/onlyforSocrates himself, Principia c systemsofmoderntypedlambda- Socrates, Socratesisnotapossiblevalue W simple individualof Socrates’ acquain- contextsensitive:forexample,anitem ed typetheorysetoutintheIntro- ction withtheword T erent peopleareacquaintedwith , Volume W rst-truth” arenotstableacross rable totreatsuchfunctions z constitutes aformal ii U et al z . Myclaimisthat / n makesthefol- z x “Socrates”which ˆ z isanindividual/ U . 2002)shows n 1980, p. 138) p. n 1980, C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) versal tome,my in thatcase,supposingfurthermore order propositional function with one individualargument.Therefore, whereas inthe the individual other individual proposition expressingthatacertainindividual consideration isacquainted.Let’sconsider,forinstance,the ject under C C C (1) 3 ( (3) say: Having onlyaccesstogreenasin“the colour of grass”(“CoG”),Ishall On theotherhand,havingonlyaccessto guage of 2 ( (2) ti theless, itsoccurrencein any contextwillgenerateasecond-orderquan- 4 ( (4) Eventually, bycombiningthetwodescriptions,Ishallsay: W x hasthecolourofgrass “ individuals: Second-order propositionalfunctionsof Predicative propositionalfunctionsofindividuals: Individuals: In thethirdcase,“green”isnothingbut an incomplete symbol. Never- cation. In view of this fact, green may be identi be cation. Inviewofthisfact,greenmay Ga ' ' ' x f x z z Principia ) ) ) . : :. ( ( ( ' z y c z a ) f a z W andtheuniversal z ) . , z )

rst case, supposing furthermore that I understand the understand I that rst case,supposingfurthermore b epistemic diagram Fy : Being agreatgeneral is a great general. If I am in acquaintance withboth acquaintance in isagreatgeneral.IfIam z :. CoG( : (

.

c / z )

. :

c , (second-level truth,second-orderproposition) (second-level truth,second-orderproposition) y CoG( = z ) . x

(elementary truth, /

. (

W . c z &

rst-level truth, c Green z )

willbethefollowing: . = .

Gx

/ f

.

, I shall say, usingtheformallan- shall I ,

: Generality ofLogicalTypes c

: z Being agreatgeneral ( & = a z z

y asto“the f z f ) a .

: f

W W a z & y rst-order proposition) rst-order proposition) is green and thatan- and green is

.

W / ed withasecond- x isagreatgeneral

F .

y y ”, Ishallsay: x = isgreen”,i.e., x

: z

isauni- z &

:

f z z 87 ! z x z C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) will moregenerallytherealizationofwholetheorylogicaltypes. so ofthesituationwillthusvaryfromsubjecttosubject,and structure sca leged. Onthecontrary,theory aparticularwayofanalyz otherwise of statements from(1)to(4),evenbetweenutility (3)and(4)—hencethe S predicates commontogreatgeneralsinhistory,itwillbe: ofbeingagreatgeneralonlyasthepropertyhavingall property bricehalimi 88 C C C volving propositionalfunctions other subject second-order propositionalfunction is apredicative that “beingapredicaterequiredingreatgeneral” guage. Thecontextualde usesapredicatethatsheborrowsfromanothersubject’slan- but only other words,theepistemiccounterpartoframi tions correspond,insomeway,tothe“same”stateof a 1.1 epistemic universetoanother. rulefroman great general”intheother,worksasaninterpretation a great general”, as uttered bysomesubject a greatgeneral”,as was “Napoleon example, pens tobeconsidered.TogetbackRussell’s hap- that ones dependsontheepistemicuniverse predicates—which by such andtype. will inhabit terms multiplicity ofrealizations,eachwhichselects express the fact that the the factthat express meant to And soforthfortheothercases.Thequotationmarksare z 2 .) Typesoframi w Principia Principia Second-order propositionalfunctionsof individuals: “ Predicative propositionalfunctionsofindividuals: Individuals: All terms turn out to be predicative, because theyallgetinterpreted All termsturnouttobepredicative, eral”, i.e., W T Types astranslationpatterns ne-grained types. Indeed, thesestatementsexpresspropositionsin- ne-grained types.Indeed, olding that every epistemic subjectimplementsinaspeci every that olding mentions only variables of propositional functions, because functions, mentionsonlyvariablesofpropositional z ’s rami x hasallthepredicatesthatmakeagreatgeneral S z 2 a as:“( , Green W W ed type theory allows ustodistinguishamongthe allows ed typetheory ed type theory keep track of properties thatsome properties ed typetheorykeeptrackof c z ) W : rst epistemicsubjectdoesnotspeakforherself, W

f nition of“green”inonecase,or“beinga

z ! z ( c z z ! of di z z ˆ z ) . ing reality would be wrongly privi- ing realitywouldbewrongly

of logical typesof remains aneutral T ' f

erent types z !

z . (

c c z z z ! z ! z z ˆ z (Napoleon)”. (Wesuppose z S ) commontoboth W z 1 , isunderstoodbysome ed typetheoryisanopen , eveniftheseproposi- x isgreen T airs. Thelogical x isagreatgen- . W c way.In S z 1 and C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) arguments ofthearguments,andsoforth,inaninductivemanner.So arguments argumentsofthe the in arguments atissuearethemselvesfunctions), guments, as well as (possibly) in theargumentsofthese(if anything involvedinthear- of diagram the all therealvariables,sothat and therespectivetypesof types; andnotonlythat,butalsothenumber variables so it is usually thought that they basically consist inranges. basically they variables soitisusuallythoughtthat without oversteppingthesphereofschematic.Typesareassignedto correspond toallpossible particular functional forms that onecanspecify ment below). alltheinstancesofsamestructure(seerulesfortypeassign- to Here thetranslationrule to her. available be would that subject cannotidentifywithanypredicate f W adds nonon-predicativetermsto verse. Here the complextype of the verse. Here a waytorenderpredicativequanti ing in fact(in ing be ableto quantify over such symbols, whichmeansusingvariablesrang- duced, butonlyinviewofasubstitutional reading of quanti predicative function.Accordingly,non-predicativevariablesareintro- a as of whatisaccessible,withinanotherepistemicperspective, spective, only thenominalequivalent,withinoneepistemicper- are terms tional but onlyin a substitutional way. Inotherwords,non-predicativefunc- symbol in order toaccountforsomeof in symbol from translation of termssuchas“ of any propositionalfunction any of ables only in a derivative way. Typesarenotdomains,butforms,whose derivative ables onlyina types ofpropositionalfunctions,andvari- types areprimarily playing not only the simpletypesofapparentvariablesoccurringin only playing not ne-grainedness oughttobemaximal. , but the number of such apparent variables, as well as their respective their but thenumberofsuchapparentvariables,aswell , Viewed astranslationpatternsbetweenepistemicuniverses,types This meansatleastthat,intherealmoframi 1 SeeChihara1973,p.43,foranexample. S z 2 ’s point of view, thatis,introducedin view, of ’s point S z 2 ’s perspective) over adomainofnon-predicativeterms, [ Gx z ] S z 1 =[( c Gx z ) f : z ” byprovidingitwith a pattern common

z x ˆ f

can be conceived of as a diagram dis- canbeconceivedofasadiagram S z ! S W z 2 z ( z ’s counterpartof cation in some other epistemic uni- epistemic cation insomeother 2 c ’s universe: “ z z ! z z ˆ S z ) Generality ofLogicalTypes z 1 . ’s sentences. Still, ’s sentences.

'

. W

c S ed typetheory,the z z z 2 G ! ’s languageasamere z x z x ˆ z ] y S ” isonlyemulated G z 2 z z ! z x ˆ underliesthe S W z 2 cation, as needsto 1 But 89 C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) ositional functions canthenbede ositional for exampleif may obtainafunctionwhosevaluesdoin for genuine propositional functionssuchas( for genuinepropositional fact, I think that that fact, Ithink two realvariables so theory oflogicaltypes,“ the gapbetweenthosetwoenvironments.Indeed,asaformaltermof upon relies environments, andthattheambiguityRussellbringsout i ofram- and theneedtosharplydistinguishbetweenformalsetting of ( the generalformofafunctionaltypeoughttobe: bricehalimi 90 At ... “ + +! internal structureof oaprn aibe. ( no apparentvariables.) following well-knownpassage: theotherhand,isshowninparticularby on temic realizationofit, occurrences. Thetypeofindividualsthenbecomes ofrealvariables,orderedaccordingtotheir are therespectivetypes 1.2 variable predicativewhenallitsinstanceshaveatype. functions, anda predicative of type the call atypepredicativewhenitis order is the least possible withrespecttotheirtype.Byextension, I shall where the that Russellsuggests.doesnotmeanthat,aslong that the orderofanindividualis can alwaysbedetermineddirectlyfromthetype: it because mentioned, want toseeit.Russellcouldmeanthat saying that containapparentvariables:thatwouldamountto cannot analyzed, it W t y ed type theory, on the one hand, andsuch-and-suchparticularepis- ed typetheory,ontheone a 1 ; The referencein w z f , …, W f x A Russellianconceptofmodel rst sight,itseemsdi o z z ) z z z ! ! z z z , x x . ,

z z could be non-predicative, which Russell seems toexclude.In couldbenon-predicative,whichRussellseems ” is a function whichcontainsnoa ” isafunction c t o y m zz a z ( , z t ; x , thatis, y a f j t , y y ’s aretherespective types of apparent variables, andthe 1 r y ˆ z f z z , …, ! y ) is z x z z ! z f z ’s internal structure does not exist as long as we donot as long ’s internalstructuredoesnotexistas x z z is( f z ! + z z t ˆ Principia o z z z y ! and + f n r ; z ( x z z , y o z z is:max( z z and( ) ! o z z ); ( x . z , f x

z should mirror that of its possible instances, and possible its shouldmirrorthatof , thetypeof( U c . (Itshouldbeobservedthatwhen z z ! o zz ( cult tomakesenseoftheambiguity cult z x z x 0 ), z z z to an epistemic perspective on logical types, toanepistemicperspectiveonlogical y ” is but a ” , , and the order , andthe z ) y o z ). But so long as * . z ,

t , where ( , where c y a i W y zz * ( ned asusual,thefunctionswhose , x volve individualsasapparentvariables, , * W t y c y z rst-order predicativevariable.Its j r ) belongtotwototallydi y pparent variables,butcontainsthe f * z ) ) z o z . 1 1 !

z z z # # z x ) = f

* z j i isonlyaschematicmatrix z zz zz ! f # # z y z ( n z m f ) f z +! * w isvariable, ofafunction . z ! +

+ , z z c ˆ t z o y +! , zz a 1 1 ( z z , …, c , x . Predicativeprop- . Theorderisnot ; , z z ! !, y z z ˆ z f ). Butthenthe z , t = isassigned, we x y m a z f z ) is ; o f t z z ! . Thetype y z 1 r f x z z PM , …, z ! ++! contains z z x oftype T is not is z 1: 52) 1: erent ; f t t o y y z n r i z r ! y z z , , ’s z x ; , C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) in adrawingperspective. the frontalsides,andnon-predicativetypesto the dotted linesthatoccur section: predicativetypescorrespondto that of representation the logical for andtypescorrespond, ineachsection,toreferencemarks universe, 5 ( (5) be apropositionalfunctionwhoseschemeis( of thatpredicateinanotherepistemicuniversemay counterpart the but di Two whichisatcompleteoddswithTarskiansemantics. sentences, truth valuestheygiveto the in never and reality, their logicalanalysisof ian semantics.But,ontheotherhand,twodi which valuesareassignedtovariablesinTarsk- in way the analogous to entity variable, into x interpretations ofalogicallanguageandthedi interpretations assigned value in agivenepistemicuniversewillbespeci assigned value instantiate rami variable changing parameters.Thus,ontheonehand,Russellconsidersthat (amongwhicharetheavailabl guage “ here). There is no real comparison between the di here). Thereisnorealcomparisonbetweenthe same truthvalue(letusnotclarifythe“correspondence”whichisatstake of principle,two“corresponding”privatestatementshavethe matter a as corresponding feature)forgranted: the takes elementaryequivalence(or interpretation of notion the model-theoreticperspective,Russellian diagrams asabove,togetherwithtranslationrules between them. Unlike two di willusethesameterm,butanalyzeitin here universes areconsidered binary relation.Thetwocognitivesubjectswhoseepistemic some actual “ subject without beingabletoindicateanypossibleinstancefor c F , thatis,witheverything.Theadmissionof T y If asubject The “realizations” of the theory oftypesareallpossibleepistemic theory The “realizations”ofthe zz (the ( erent models oframi x W , ed typetheory,because,inthelattercase,resourcesoflan- ' T y z G x erent ways. )” isnotanothervariable,but S f z z ) 2 y )”, i.e.: z inacquaintancewith is “assigned”concretepropositionalfunctions,which is quite : x ( z . Nosubjectisacquaintedwithallthe possible instances of z y S z ) z 1 . knows

Gy S

. z

1 ’s epistemicdomain,hingeson / a

W . z only as“the

ed typetheoryareliketwosections of the same y = x

:

z & a z

, onthecontrary,wouldbeableto .

Fx e proper names) are precisely the e propernames)areprecisely G y a schematicletter standing for ”, she will assertsomethinglike she ”, Generality ofLogicalTypes T a erent realizationsdi , as a quasi-value for an for , asaquasi-value z y T z ) erent realizations of erent realizations .

S c T z 1 ’s believingthat zz erent Tarskian ( x x , W . Anyother c predicate, y z ) z — z where T er in 91 C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) case and the absenceofcomplexformsquanti epistemic transparency,atransparency in termsoffunctionsthatpossessacertain of knowingindividualsandclasses axioms ofacquaintanceandreducibility tensionally equivalentpredicativefunc and but iscomprised,fromthepointofvi of objectsacquaintance, domain players. Itdoesnotconsistinasingle subjects, orthesymmetricequilibriumhopefullyreachedbyallepistemic but also,foreachsubject ual variable. functions, thisin between mathematicalrealityandou means bywhichtheyareknown.Redu the occur inthehierarchy only under thegu entity S bricehalimi 92 is otherwiseunabletodemonstrate. Although our is how This trustworthy. be to from anyothersubjectwhomshedeems receives cipia 1.3 trusts of universal instantiation). ludes to here are all thelawsof the“t directly representobjects” allthelogicalpropertiesofsymbolswhich formally) “has (speaking itself toanepistemologicalinterpretation: The reducibilityaxiomlends environment. natural-language applied ina constructions. This means in particular that this de This meansinparticularthatthis constructions. asotherlogical status what Russellcallsaconstruction,withthesame totic resultoftheinformation z 2 ’s identi w Individual variables do not refertoactualobjectsofacquaintanceonly, 2 Epistemic realizationsareanaturalthing to bring up assoon

z Epistemic modeltheory PM ofquasi-valuesconveyedbyothers. z ’s rami S a

S z  2 isreliable.Grantedthat the z . The all-inclusive range of “ . Theall-inclusiverangeof 1 14.18, 1: 180. See 14.18, 1:180. getsthemissing quasi-value of W cation of(what W access ed type theory is understood as a formal schemeto be no wayprecludesthe toafunction  20.71 aboutclasses.TheformalpropertiesthatRussellal- S 2 z : it can, by extension, instantiateanyindivid- , to the virtual objects ofacquaintancethat , tothevirtual S z 1 heory ofapparentvariables”,suchas conceivesofas)the f X z is mediated byquanti mediated is r knowledgeofitthat the rami ow shared by all truthful epistemic all ow sharedby tions or functions oforder1.…The or tions embodied byacquaintanceintheone G x postulate (respectively) the possibility postulate (respectively)the ew of each subject, of proper values each subject,ofproper of ew ise ofpredicativefunctions,whichare (Demopoulos andClark2005,p.158) existence z ” becomesinthat way the asymp- cibility thuspostulatesaconcordance actuallyexists,“the x z in(5) , withinthehierarchy,ofex- W cation intheother.Classes z — W z to theextentthatshe nite descriptionthen G with some actual withsome W cation overother G  y ” becomes 9.2 (therule W ed theory Prin- S z C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) along with( predicative counterpart. Supposenowthat predicative term. In the case ofindividuals,anyobjectintroducedbyade term. Inthecase function,asasymbol,isinterpretedbygenuine that tion inwhich that, for any propositional function, there is alwaysanepistemicrealiza- some attention, thereducibilityschematacanbeunderstoodas meaning universe, only subsequentconnectionswithepistemologyproperlyspeaking. subjects. So, in fact,aformalsemanticsbasedon epistemic diagramshas and besidesremainquiteremotefromactualconcretecognitive loaded, in whichrealitycould be carved up. Hence,theyarenotmetaphysically mean nothingelseherebutcomplete tional functions in in ( despite thefactthat one never understand theirsyntacticpossibility icative terms—andinparticularto thestatusofnon-pred- understand from theformerisonlywayto thelatter erpart ofthetheorylogicaltypes,andthatdistinguishing tion aside. view, thatobjectsdoexistwithout belonging epistemic universe.Then,foranyparticular consider, forexample,atranslationpatternsuchas an actualpredicate,goinginfactwithawholelistofconditions.Let’s non-empty functionaltermcorresponds,insome epistemic universe,to actual individual.Thisisakindofquasi-acquaintanceprinciple. France”), mustcorrespond,uptoashiftinepistemiccontext,an to exist(asopposed“theactualkingof description andassumed validity of This equivalence holds even though, among all thepredicates among This equivalenceholdseventhough, Once theepistemologicalbackgroundof 3 according to type, single The arrayofallpossibleinterpretationsa Once again,Ithinkthatepistemicrealizationsarethesemanticcount- In the case ofpropositionalfunctions,theaxiomschemasaysthatany In the On that score, there is an issue about whet about issue an score,thereis Onthat ’ ) that Fx S Fx z ’ 1

has access to, there are some for which for hasaccessto,therearesome .

).

.

' ' x x

. . Principia

f f yz yz [( ( ( G G f z z ! ! z ) z z z z ˆ ˆ . z z , ,

f . It should be addedthatepistemicsubjects x x

z z ! z ) is part of the list of conditions that goes W ) hastobe validin z ( nds variablesofnon-predicativeproposi- f z ! z z ˆ z , to anyepistemicuniverse.Ileavethatques- x her it is possible or impossible, in Russell’s in orimpossible, possible is her it epistemic diagrams,thatis,ways epistemic z )] S Generality ofLogicalTypes 1

/ f [ yz z ! Fx z ( Principia G f y ˆ z ] z , also belonging tothat alsobelonging S z ˆ 2 z ) itselfbelongsto S S z 2 z 2 ’s realization.The does nothaveany z ’s typesisgiven f z atstake W 3 ( nite S ’ z 93 2 ). ’s C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) duction. Typical ambiguity has been deemed tobean been has duction. Typicalambiguity Intro- the merely tackedontotheconceptionoftypesasdevelopedin type,andnotafeature logical ity isanintrinsicpartofthenotion the system. kind ofshortcoming matic treatment,and,tothatextent,itinvolvesa as aninformalconventionthatisnotreallyamenable to anyintra-syste- may be,itiscommonlyconsidered it innocuous and Though powerful having suchauniversaltype. of itself opentothelogicalinconsistency allows thesystemof than asanexternalone.Itisofte types, typical ambiguity can be describedasaninternalgenerality, rather logical types,asjusti universe. The resulting system Theresulting universe. brought upcanbesaidtoexternal.Onthe contrary, the of ordinarylanguageandknowledgeconditions,thegeneralitythus ment”, is a totally di ment”, isatotally as many th thefact (p.138)despite face” of saving inde Principia di bricehalimi 94 guity, andthatthelatterisonlypossi theory, byshowingthattheformerisrich dicate both type theory,intheformofmo X egories. Feferman considers that and in particular toaccountforcasesofse of many nested“re not consider does by HarperandPollack(1991),butit particular in explored, been has is neededtoprovideforitsformalregimentation.Suchaformalization the system.Allthat of derstood asafeaturethatisonparwiththerest Such adiagnosiscanbecorrected:typicalambiguityindeedun- it is based onactualapplicationsoframi is it icativity reasons.Russellwastryingto“havehiscakeandeatittoo”. icativity for impred- discharge ageneralitythatwasatthesametimeprecluded exible expression ofmathematicalproperties T ZF On the contrary, I would like todefendtheviewthattypicalambigu- would On thecontrary,I 4 See Feferman 2004. According toFeferman According 2004. See Feferman erent epistemicperspectives,isthetrue content of its W z , is intended primarilytoprovidefo a nitely many others, within thesamehierarchyofpurelylogical nitely manyothers,within 0 ’s as there are types.” Feferman proposes . Since it consists in the abilitytousetypes asrepresentativesof Principia X ective universes” Principia W z T as anything other than a historical landmark, and landmark, as anythingotherthanahistorical ed in Russell andWhitehead’s“PrefatoryState- in ed erent kindofgeneralitybestowedontypesin ZF typetheory,onthecontrary, z / U U n to have a type of alltypeswithoutlaying of tohaveatype , each of which is an elementary substructure elementary of an is the ofwhich , each < v z , whichis proved tobe n conceived of as a useful device that n conceivedofasauseful at, as Russell himself acknowledges, “there are ble semantical framework forthosesystems. framework ble semantical dern systemsoflambda lf-membership suchasthecategoryof all cat- enough toformalizeRusselliantypicalambi- undational frameworkforcategorytheory, ” (p. 140). In what follows, I follows,I wish what to In vin- (p.140). ” , “theuseoftypicalambiguityisaway asystemofsettheorywithcountably- W ed typetheorytotheanalysis “doesn’t lenditselftothe a conservative extension a conservative -calculus, and category ad hoc formality ambiguity z solutionto . Since z z of 4 C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) these valuesonlyinpropositionalpositions. values withargumentswhichareconstants 1907, andentitled“Types”: 1907, works.Thisisborneoutbyamanuscriptdatingbackto sell’s logical Typical ambiguity is actuallyathemethatshowsupquiteearlyinRus- parent variablesofanytype,howeverhigh?… matter atall.Thewaytoexplainthingsisasfollows: than individuals: then than individuals: got fromlowertypesalone. as argumentbeyondwhatcanbe in put suppose anyknowledgeofwhatistobe di ment”, as “symbolicforms”.Asymbolicformisnothing other than what strued afewyearslater,inRussellandWhitehead’s“PrefatoryState- involves thedi value of same functionalskeleton,fordi mains, butthemselvespropositionalschemes: complicated iables. Thus typical ambiguity con iables. Thustypicalambiguity calculus canbesu following, Ihopetoshowthat a moresimpleextensionoftypedlambda- and soon.Inthis way,the theory ofalltypescan bedone atonce. T (3). … Wecanconstruct (1). The pointisthat (2). 5 ofconstruction” towhichRussellrefersin1907arecon- “ways The We mustwrite There issomeobscurityaboutthe BertrandRussellArchives,ma erent typeshaveincommon,totheextentthattheyarisefrom 2. w w w f F f

w x

z x ! z . ! typical ambiguityandpolymorphismoftypes z standsforanythingcontaining f z f

z x z ˆ ! f f z z stands for anything containing values of standsforanythingcontaining y ˆ . The fact that y x ˆ z standsforanythingcontainingvaluesof U f

cult tools of the calculus of constructions. Inthe cult toolsofthecalculusconstructions. U z ! f z a z cient toformalizetypicalambiguityinaneatway. and for a function whose argumentsareofanytypeother whose function fora f

z ! z f F 5 a e b e a z z x stand for ways of construction, anddonotpre- ofconstruction, forways stand ˆ f z with any nuscript 230.030890,fos.32–3.

may ’ y ( T z f erent typical determinationsofthe var- containan

z ! Primitive ideaf z b W z ) rms thattypesarenotlevelsordo- f . or apparentvariables,andcontaining = welike,andthensubstituteamore x , andbeingapropositionforevery Generality ofLogicalTypes .

f

z ! f z z a as apparent variable does not apparentvariabledoes as

z ! z f a . Maythiscontainap- . , andcontainingthese f z inasimilarway. Df 95 C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) from that of individuals, andwe“see”thatthisprocesscanbeinde umerated. Wehaveinfactenumeratedaverylimitednumberoftypesstarting arising forthesetofpossibletypical iudb nlg.( tinued byanalogy. it representsdi var t Begin a “ “permanent truth.”… same proofholds inanyother assigned 2.1 stronger “polymorphic”type-theoreticcalculi. canbecaptured,ononecondition:shiftingto ambiguity that typical ambiguity isleftaside.Iwould like to extend the suggestionbyshowing bymeans ofalambda-calculus.Buttypical types canbeformalized A recentpaper(Kamareddine on drawing qualify thatpredicament, ambiguity has often been understood as a merecorrective.Myaimisto overtypes.Thatiswhytypical circularity, itisimpossibletoquantify of reasons and Whiteheadherealludeto.But,forobvious Russell what beneededtoexpressformally would of embarrassment.Typevariables bricehalimi 96 swap to be true in the lowest signi to betrueinthelowest biguous symbolsaredi variables rangingoverintegers: iour is the programme whichperformstheswappingofvaluestwo the generalityof “generic datatypes”areintroducedthatallowprogrammerstoexpress z : = symbolic form It is convenienttocallthesymbolic To “assert a symbolicform”istoasse To “assert When a proposition containing typically ambiguoussymbols When apropositioncontaining 7 6 kind a Admittedly, RussellandWhitehead’sresortto“seeing”betrays Polymorphism is a feature of programming languagesinwhich Polymorphism isafeatureofprogramming w Crole1993,p.202. SeeScedrov1990. t Typical ambiguityasparametricity m z : Int (var m , T n z ”. Thus,ifasymbolicformcontainssymbolsofambiguoustype erent propositionalfunctionsacco z : Int) uniform T erently adjusted.… z algorithms. W cant type, andwecan“see”thatsymbolicallythe cant et al 7 . 2002) shows that the theoryoflogical . 2002)showsthat determinations whicharesomewhereen- determinations form ofapropositionalfunctionsimply moderntype-theoreticframeworks. 6 type, we say that the propositionhas that type, wesay rt eachofthepropositionalfunctions An example of that kind of behav- of kind that of example An rding asthetypesofitsam- can beproved W nitely con- PM z 2:xii) C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) This ismadepossiblebytheintroductionof regard programme, a generalswapping can beintroducedthroughaquanti why Thatis giving riseto“polymorphictypes”andterms”. t Begin straction orquanti types. Thecharacteristicfeatureof whichincludevariablesof and tax forwritingdownfunctionalterms lambda-calculus whichprovideaformalsyn- typed of of formalsystems var other type. It is, ofcourse,muchmore e It isclearthattheprogrammeworksexactlysame for variables of any End where thetypevariable End n m m general swap that is, as a uniform procedure toprovepropositionswhichdi procedure that is,asauniform own type), giving termsasoutputs,andcanbeviewed as a generic proof, Consequently, anypolymorphictermtakestypesas inputs (includingits tion codedbyitstype:thisisthe“propositions-as-types”interpretation. types). n 2.2 substantiate theideathatcertainprocedureshaveagenericvalue. theoriesto variables havebeenintroducedintherealmofmoderntype dure. z : = z z : = : = z = : = Polymorphic second-ordertypetheory( In ordinary lambda-calculus eachtermcodesaproofoftheproposi- In ordinarylambda-calculus This is exactlythekindofsituationthatRussellhadinmind.Type This is w t Polymorphic second-ordertypetheory m z n : t t pso n X is also called “impredicative type theory”(inasmuchasatype type isalsocalled“impredicative (type: W cation on type variables, bothintypesandterms, cation ontypevariables, X X zz ; var zz isassignedavalueateachcalloftheproce- m , n z : X pso W y ) cation over the collection ofall cation overthecollection less ofthetypevariables. U Generality ofLogicalTypes is that it allows explicitab- isthatit pso cient to beableimplement cient type variables ) is actuallyawholefamily z : T er only 97 C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) through construction construction rule through abstraction type variables blocks fortheconstructionofwell-formed(or“proved”)types. non-zero arity The rules to generate well-formed types referto well-formed The rulestogenerate swapping ofvalues. logous proofs. Thisholdsforproofsaswell operations such as the buttherepresentationofpatterncommontoana- nothing term is “s are spective proofs,consequently, pertain, andwhosere- they with respecttothekindofthingswhich bricehalimi 98 z C C C D summed upinthefollowinglist: On topofthat,ifa type variable binary functionaltypes). type (thinkhereofpropositionalconnectivesas F of distinct type variables that aresupposedtoserveaselementary that ofdistincttypevariables Proved types: 8 Universal types: Functions: Variables: Let me

IfollowhereRoyCrole’sexpositionof : = X z

W *

rst set outthemainlinesof ; F X y ( n z X F auniversaltype to givesrise , andif , , …, Y , Z F , …. Types are then constructed inductively , ….Typesarethenconstructed F ) (forinstance D 1 s. Inparticular,if“ , …, D | w z | F

D D F 1 D ww , z | F y ( X X , 8 F ; n … X z , occurs in a type a occursin are tructurally” thesame.Apolymorphic X 1 D

, … pso | ww

N . | F

F n D pso F giveninCrole1993. X z types,then F ww z × | n . Thebasictypesymbolsare F z ) F ; n F and X y type contexts ” isanytypesymbolof

.

F F F y y ( F (

X v X y ( yz zy F ). Thesyntaxis z F ), second-order 1 ) , …, , thatis,lists *

; F ( X X n

z ) isa . e ª

F

D . ) C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) using construction rules:functionsymbols using construction sorting abstraction andapplication.Thisgivesthefollowingtermsyntax: a judgment conclusion is whose built be can therulesabove,derivationtrees Using C C C D X contexts. A term context is a sequence sequence a contexts. Atermcontextis the context, dom distinct typed variables (the variablesaresaidtomakeup the swapping ofvalues, exampleofthe former trized” proofsoroperations.Toreturntothe canthenbeconsideredas“parame- constituents variables amongtheir iables, tobuildthem.Termshaving(possiblybesidestermvariables)type asvar- As onecansee,typesarenotonlyusedtotypeterms,butalso,

| . Functional abstraction: Binary Products: Term Variables: As isthecasefortypes,rulesfo Terms are de Proved terms

l F f y = i

z foreach :

s X x zz

( z

. a

* + D z ) = [ z

y z z a , | (typingrules): x s F W F z z ( , F z ( ned startingwithvariablesofdi a . i G ) z , calleda in ( 1 , …, z )). Onewrites f D l , …, x z G D 2 D

: . D G | 2 D Int F

z

2 G G f z | D 2 zyz

n

G

wwww ) , proof 2 | G f z .

F

G | , : * l l z

F x ], termabstractionandapplication,type , x

y z l +

z x

D

z wwww : ofthetype z : r generatingwell-typedtermsmention f : x z D zz

, Int zz F | :

F : f zz | F

F , z

F N, . G G z D

z wwww . | z f

(

N z . : 2 + |

D z

Generality ofLogicalTypes f z :

f y G z z F G

zz zz , F

x z being a typecontext)tomean : =[

x z z * ×

C D |

: z

, v a zz F | generalizesinto F f f C

withspeci x y z inthecontext G N C z y 1 z

:

N T z : C *

erent types, and then erent types,and

F l 1 X , …,

.

f y W

ed arityand x *

m domain

f L D y :

F X . F

z . . m

l z ] of of x 99

: C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) vicious circularity.Indeed,apolymorphictypesuchas judgments constructderivationtreesfor to one The above-mentionedrulesallow proved terms: lambda-calculus properly speaking, context C brice halimi 100 C C C C up with vicious-circle principle. How could it be possible,onthatbasis,tocatch Thisclearlyviolatesthe own. its takes asinputanytype,including itself. Besides,apolymorphicterm be appliedtoanytype,including As wecan It remainstosettheconversionrules Application toaterm: Polymorphic equalities: Functional equalities: Application toatype: Type abstraction: D Principia , G D . 2

W D G nally see,

2 D |

z D ’s theoryoflogicaltypes? D G 2 t 2 2

y G

|

: D G G D ,

( F

2

x | , | D L

D z

G , calleda X : ( 2 f X 2 pso

z

F |

l 2 : G

D G . z

F x D

G z f

z

, f | | z z

| allows atype-theoreticformalizationof allows yzz 2

z : X : v | )

zz F L zz f F z G

; C y 2 z f

z X C

z C :

X

proof | = G : . z C

z . wwww F

F .

which consist in equalities between which consistinequalities f

: | z y f f F wwww y g

z

wwww

) F z z z [ f

f : wwww z z : oftheterm C =

zz y ;

z : [ D C

C / X F F D X 2 D z —

y

/ y [ 2 z D G . ] X

z

F f | G /

z | : y that is,therulesof that | ] zz x z F C z | C z g ] [

f : : z

C

zz C : F t z F

/ : X zz F y ] L z inthedouble X z

. z T y ( ( ( X X X y z z

) can e e ª ª zy zy G G z z ) ) C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) (i) cated earlier,therulesfortypesare: polymorphic lambda-calculussoastoformalizetypicalambiguity. implement therulesfor to stead ofpropositions-as-types.Butitremains 2.3 C C C stead of lambda-calculus terms, and types in thesenseof stead oflambda-calculusterms,andtypes in- functions propositional of logicaltypesasatypetheorywith theory iables (inkeepingwiththeirrespectivetypes).Themainrulesare: i)if (ii) here areconsideredupto“ (As alreadystated,thementionofordersisinfactredundant.) D (merging thetypescorrespondingtoidenticalvariablesorconstants) In compliancewiththe Following Kamareddine Connectives: Functional variables: Individuals: Let’s move on now to the typing rules. All propositional functions Let’s moveonnowtothetypingrules.Allpropositional w 2

Typical ambiguityinpolymorphicclothing G o u D

y z = | b 1 t 2 ,…, 1 y

a 1 f G ,…, +! 1

: | ( ; f t u !,

z 1 , …, y t  n b y w o m n a

isarami , m g , g

isarami : u ( t y z D m b 1 t ,…, z z 1 ; 1 z , …, | u D z z G 1 , …, W 2 W a wwww

ne-grained presentation of typesIhaveadvo- ed type(theofindividuals); G u -equivalence”, that is, up to achangeofvar- up -equivalence”, thatis, et al D t W z y m , n D ed type,with b D 2 ww z n , f

u 2 zz G 2

t D

z . (2002),onecanthinkof : n z

are rami N G

G z y 1 ) z | F , …, | w

z | | D

, z F ~ f a G 2

wwww

f N : : G t

Generality ofLogicalTypes |

z N o t

: k W | z f

; t z ed types,then

u z g g : D

y z 1 =max( F : , …, z | ( t z G z N 1 N , …, u z n a z , i u z t , z N N b k 1 , …, z + ; j t z Principia ) + u y a 1 ,…, N 1 Principia 1 , …, 1 u . N l z ) t y u m a z in- N m

101 l y z ) ; z ’s C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) functions isneeded: is builtintothetheory. type, asintheTermVariablesruleof duced attheonset,becauseallbound variables are recorded within each Such aruleisnotneededin C brice halimi 102 G C C C at formalizing G

: In Kamareddine Abstraction fromparameters: Type Application: Polymorphism: Quanti The resulting system The resultingsystem , y y {

z :

: t D i ( a i 2

t W | 1 D a G cation: z 1 D 2

, …,

f |

2 G Principia wwwwwwww D

: D ( G

( | 2 x 2 ,

t t

j G D 1 g z n x G ) a a

et al z 1 j n

,

: , …, .

| z : )

|

X f

t a u

y wwww f y } L [ 2 . (2002) aruleforabstraction from propositional z j : ’s theory of logical typeswhentypicalambiguity of ’s theory D

prtt

| ( | X G D z t 2 t n

z

f t y a z

2 . 1 z z |

/ G n , …, D

y g z f ) G : (

f z a 2 ] z ( | y (polymorphicrami prtt

:

ww 1 | : t G : , …,

( z ( ; 1

, …,

t t L z f | t f wwww z X m z z zyz ispredicative 1 X , …, z ) ,

: variablesofanytypecanbeintro- f . t

y u

z . N t : k

t j

wwww z ( z f pso ) ; : z f m

z

ispredicative t ( u : z t t ; : z x 1 ( y z z m

, …, [ 1 u z ; , …, j t z (where t . z ; isthe z z 1 1 N D a X , …, u / z 1 , …, X z

z | 1 . , …, y t

] u t t z m wwwwwww z w N j z z u ; !

j

W : g 1 -th freevariablein w j u t z z Type , , …, z n ed typetheory)aims isaparameterin u FV( a z 1 u n , …, z n z , ( j z +1 , z t , …, t z f z u ) 1 zyz a z ) z 1 u n , …, ( z ) z # z n z

) u e dom ª z dom n z ) t n ( a X n z ) z (

a z e G ( ª z ) G

a z f f G ) +1 yzz zyz )) ) ) ) C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) the type fully generalproof,i.e.,togettheuniversal fully “Axiom C”: by addingthefollowing: chart have suggested accounting for the“generic”valueoftypes in (1983, p.51),GiuseppeLongoandThomasFru- gestion duetoCarnap the type(as one bears no information asinput.Soif 2.4 Axiom CisconsistentwithGirard’ssystem erality, as opposed to quanti erality, asopposedto ambiguity inthe“PrefatoryStatement”isanexampleofparametricgen- Indeed,typical ity whilestickingmorecloselytoRussell’sconception. pens, thetype applies stitution of the inputtype:insomecases,aparticularinstance on depend not does or computationcodedbyapolymorphicterm proof ametric in one can reduce “In otherwords,thebehaviour of polymorphic terms is so ‘uniform’that is broughtintoplayonlyinthespeci other words,thatithas“permanenttruth”.Typicalambiguity type or,in symbolscanbeprovedtotrueinanyassigned typically ambiguous wise anyterm Let M and Nhavetype Axiom Ccanprovethefollowing“GenericityTheorem”: Axiom Cintuitivelymeans that an input( There isawaytoextendtype-theoreticallytheideaoftypicalambigu- M But Russellpreciselydoesnotclaimthateverypropositioncontaining w Typical ambiguityandtype-theoreticgenericity single t = M s M to z (i.e. type s t F F t N z If M does notdependon does

z , can be described as being obtained by the uniform sub- , canbedescribedasbeingobtainedbytheuniform or foratypevariable for alltypes s s F t Fc z isnotafunctionof

t issaidtobe (nomatterwhichone!)”( :

N :

equalityon F andonemayconsiderbothresultstobeequal.

; z couldleadtoaproofoftheabsurdum X z

. ;

s

X t

, z and Xisnotamongthefreevariablesof .

t s N W . . generic every cational generality.Pickinguponasug- If M X X y M ) ofthecorresponding output value ( , and thusmaysu , X z possibletypes[ hasthetype t y . Thisisnotalwaysthecase,other- = ), then it does not matterwhetherone ), thenitdoes W Fc Generality ofLogicalTypes c realm of“formalarithmetic”. realm c

N (Fruchart and Longo 1999, p. 46) p. (Fruchart andLongo1999, t t z ), whichisnotusedtoestablish

ibid for onetype F , and thesystem and , ; ; .). Thatmeansthatthe X X

U

sic . . s

s

s ce to determine the ce todetermine z z [ ] to z and . When thishap- F

t / , X Fc then M= X y ], being par- isnot free in equalityon ; X Fc = F

. s y X , . M Fc then pso 103 N t + z ), . C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) maps [ types are considered themselvesasmembersofaset considered types are dependent productsoveritself,exceptforaset whose typeis More speci there isonlyoneindividual(oneelementin 2.5 arithmetical truths,particularly,havea“stabletruth-value”. Type-theoretic genericitycanbeaway brice halimi 104 W C C C C over and isclosedunderexponentsaswelldependentproducts uals is tointerpreteachtype functions betweenthesetsin X proved thatthereisnot set of sets modelof be anysuchset-theoretic type theory is notset-theoretic,butpertainsto“ not practicableasanoption. nite productsofsomedistinguished object

e 9 If If [ If [ [ Let’s nowlookatanaturalset-theoretic semantics for To take stock: Actually, one of the mainavailableinterpretationsofpolymorphic the Actually, oneof w SeeReynolds1984andJacobs1999,8.3.3. o

Semantical issues  z  L ] = B [ [( z z zz f f L hasatmostoneelement. z X zy ( =[ (forany X x ] X z z x j

1 I z ) , . e z

)]

e .

X t f . A

z

] W f z  f e 2 yz : =[( zy , … e cally, proved types ]

] zy A ; ; e  u e =[( X

F z

X . Becauseofthequanti . z w y , then[ w t X



z P 1 e . e e P , …, z z [ n F

zz

 containsa(presumably)non-emptyset

t  A | z 1 w z and , …, z j z ] F 

[ F , y t z ( z X f z f w u m zz X zy z ] : (

] z P z e ; as a set, namely as the set [ asaset,namelytheset F y t

z ) y e z  u m z =[ U z )];

e z w z 1 ; B , …,

z F  n u  z A

y X v z z e 1 canserveasinterpretationsofterms: ( z . , …, X

t X 9 

This would implyinparticularthat This v y U  z u 1 ) zy , z z ; inacategory

e n

z X F z that isclosedunderexponentsand )]

prrt u  z to describeaccuratelythefactthat 2 z zz n , … ( z e ). It turns out that there cannot ).Itturnsoutthatthere z W )] X

 cation bindingtypevariables, y )] e . Indeed,JohnC.Reynolds zz and

X  e U z

n zz forany 

z . A proved typeisthusa |  W A  zz bred categorytheory”.

C whoseeverymember , then I z j F =[ y ). Thisis, of course, z consistingofallthe z

u z are interpretedas  u y x ] ofalltheterms z z z j

ofsets,sothat z prtt ] :

e t z

with I  z ofindivid- z , then . Theidea C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) Principia context scribed by Russell himself as aformofcontextrelativity ( ambiguity or astopgap of as ashortcoming not speaking, properly has todowithtypicalambiguity cording totheepistemicperspectivethatisconsidered.Thesecond one realizations, so that each type willbeinhabitedbydi realizations, sothateach the typing rules, as well as for the functional and polymorphic equalities. and the typingrules,aswellforfunctional course, necessary tothrowinfurtherconstraintsorderaccountfor consisting ofallthe is interpretedastheprojection member ofsome W 2.6 prtt z f | each object icativity, it would be interesting to see how to adapt it to the case of case the icativity, itwouldbeinterestingtoseehowadapt type foravariableamountstoreindexing. proved typesbecomemapsbetweensuch proved and whose morphisms are corresponding provedterms.Thejudgment and whosemorphismsarecorresponding logic describingwhathappensinthatcontext, two kindsof generality are independentandshouldnotbeconfused.The [ morphism [ binary productconsistsinafamilyofoperations U U U D X rst one has to do with thefactthattypetheorygivesriseto a variety of x y y y

2 z = ), that is, a category whoseobjectsareprovedtypesinthatcontext, category a ), thatis, ) ) f 1 To sumupbrie toconsiderabasecategory is Still, theunderlyingideaisclear:it 10 Since categorical semantics is calledforasawaytohandleimpred- semantics Since categorical

w , …,

x SeeJacobs1999,p.173. v v G Conclusion [ : X . Ileaveitforfurtherexamination.

F

C | z C 1 , …, , suppose that eachprovedtype D zz zz

( X ( z : t U , which is an internal one.Eventhoughtypicalambiguityisde- , whichisan U z |

formality z n z

z : U z n z n f z

| G , , X yz F z ] : U n U . Hence, each such type context becomes anindexfora . Hence,eachsuchtypecontextbecomes z

n in

F z becomesalogicalrelationinthe y y | ). Finally, foreachprovedterm Principia ), andabstractionontypesinafamily f j C z C X

] z 1 F ×… , which is an external generalitypeculiartotypes,and zz y, therearetwokindsofgenerality of logical typesin z ( — e U W ]

nite productsofsomeobject C e z z thought ofasatypecontext n

, zz C ( U U z ’s system,norasaslackeningintheoriginal zz ( y z f U ). Forexample,theprovedtype n , z m z n U ,

U v U y ), andthat y

2 y f

). Itisthennaturaltointerpret v inthecategory

U z . Then, the rulesofderivation Then, . Generality ofLogicalTypes F j z hasbeeninterpreted as C F zz ( hasbeeninterpretedas U 10 B X z and substitution ofa substitution and n n z , 1 : z , … — U U C W C bre overthetype z y zz the “ zz , and to assign to and , )’s. Forexample, ( ( T U U X ; PM erent termsac- z n z z n n n

, , : 2 W U

U z 1: 65), these 65), 1: X bre” C F y z y 1 ) × zz ). It). is, of , ( 1 U × , …, X C C z 2

f zz zz | z ( ( asa f U U 105 F

j X = z z z C n n n m z , , , 1 z C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) Feferman, S. 2004. “Typical Ambiguity: S. 2004.“Typical Feferman, andW., P.Clark.2005.Demopoulos, “T cal perspective.Inthe a logicalway. in for accounted seriousness, butasapositivefeaturethatcanbe logical brice halimi 106 Chihara, C. “The LogicistFoundat Carnap, R.1983. Reynolds, J. Gri provide theresultingpolymorphicrami ambiguitypolymorphically,andto possibility ofconstruingtypical right a“model”ofthattheory.Inthesecondcase,Ihavebroughtup Jacobs, B. 1999. 1999. B. Jacobs, Kamareddine, F., T. Laan, and R. Nede Kamareddine,R. F.,T.Laan,and nections between These twoperspectivesareenoughtosuggestsubstantialcon- calculus. along thelinesofcategoricalinterpretation of second-order lambda- ization oftheformalsystemrami Crole, R. Fruchart, T., and G. Longo. 1999. “Carnap’s Remarks onImpredicative “Carnap’s 1999. Longo. andG. T., Fruchart, Harper, R., and R.Pollack.Harper, 1991. and Logic Too”. InG.Link,ed. Russell”. InS.Shapiro, ed. Cambridge: Cambridge U.P. and H.Putnam,eds. Science, Florence, August1995 Florence, Science, TenthInternationalCongressof the of eds. Minari, Each ofthetwothreadsthatI have justsetoutgivesrisetoasemanti- Mathematics before1940”. don: CornellU.P. De de Gruyter. Computer Science 117–88. U n, N. 1980. “Russell on the Nature of Logic (1903–1913)”. (1903–1913)”. Logic “Russellon theNatureof 1980. N. n, W nitions and the Genericity Theorem”. Genericity nitions andthe y L. 1993. 1993. L. y . Oxford: Oxford U.P. y S. 1973. 1973. S. C. 1984. “Polymorphism Is Not Set-Theoretic”. In G. Kahn, D. Kahn, G. C. 1984.“PolymorphismIsNotSet-Theoretic”.In Logic andFoundationsof Mathematics; Selected ContributedPapers Categorical LogicandTypeTheory Categories forTypes Principia z 89: 107–36. Ontology andtheVicious-CirclePrinciple Philosophy of Mathematics: SelectedReadings Philosophy ofMathematics: W One Hundred Years of Russell’s Paradox ofRussell’s Years One Hundred rst case, I have argued that each epistemic real- rst case,Ihavearguedthateachepistemic The Oxford Handbook of Ph The OxfordHandbookof z ’s logicandmodernsemantics. The BulletinofSymbolicLogic . Dordrecht:Kluwer. references “Type CheckingwithUniverses”. . Cambridge: Cambridge U.P. W ed typetheoryconstitutesinitsown ions ofMathematics”.InP.Benacerraf Trying to Have Your Cake and Eat It Eat and Cake Your Have to Trying Logic, MethodologyandPhilosophyof Logic, he Logicism of Frege, Dedekind and Dedekind ofFrege, he Logicism rpelt. 2002.“TypesinLogicand W ed type theorywithamodel, ed In A.Cantini,E.Casari,andP. . Amsterdam: Elsevier. ilosophy ofMathematics z 8: 185–245. 8: . Ithaca. and Lon- . Berlin: Walter Synthese Theoretical . 2nd edn. 45: C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) Whitehead, A. Poly to Guide “A 1990. A. Scedrov, and ComputerScience vols. Cambridge: Cambridge U.P.(1st edn., 1910–13.) MacQueen, andG.Plotkin,eds. y N., andB.Russell.1925–27. . London: Academic Press. morphic Types”.InP.Odifreddi,ed. Semantics ofDataTypes Generality ofLogicalTypes Principia Mathematica . Berlin: Springer. . 2nd3 edn. Logic 107