C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) T The BertrandRussellResearchCentre,McMasterU. russell: W Mathematica playsanimportantroleinthelogicof reducibility The axiomof clothing”. fact(inQuine’swords)“settheoryinsheep’s in is and cal principles a second-order propositionwillcontainquanti a second-order 66–7). a propositioninvolvesthelevelofquanti type. Theorderof that of function equivalent predicativepropositional of whateverorderagiventypeargument,thereisanextensionally reducibility, theaxiomwhichstates that forany propositional function there arealsoappeartobedi pretations itisnoteasytoseetheax along thelinesofLinskyandMares.Iar nominalist interpretationalong the line claim. Ilookattwootherwaysofun that theaxiomisclearlynotalogical If weunderstandpropositionalfunctionsas no smallpartontheun will dependin of reducibility.Whethertheaxiomcanpl yield alltherequiredtheorems. In this therami because was added which simply rst-order propositionwillcontainquanti 1 This is Quine’s of characterization seco mathematics to logic fails because the theory demands non-logi- demands theory mathematics tologicfailsbecausethe Mathematica he mostcitedreasonfortheviewthatlogicismof THE AXIOMOFREDUCIBILITY h ora fBrrn usl tde n.s.31(summer2011): 45–62 the JournalofBertrand RussellStudies 1 The non-logical principle mostobjectedtoistheaxiomof Thenon-logical , buthasgenerallybeencondemnedasan was a failure isthepositionthatreductionof failure a was Pocatello, Philosophy /IdahoStateU. U wahlruss Russell Wahl culties inacceptingitasapurelylogicalaxiom. id iom asanon-logicalclaimabout the world, 83209-8056, 83209-8056, axiom and in fact makes an implausible axiom andinfactmakesan paper I examine the status of the axiom paper Iexaminethestatusof derstanding ofpropositionalfunctions. derstanding propositional functions,a derstanding propositional @ s of Landini and a realist interpretation realist s ofLandinianda nd-order logicingeneral(Quine1970,pp. gue thatwhile on eitheroftheseinter- ausibly be included as alogicalaxiom isu.edu W ed typetheorywithoutitwouldnot constructionsofthemind,itisclear W ers rangingoverindividuals, usa W ers in the proposition. A ers intheproposition. issn W ad hoc ers rangingover 0036-01631; online1913-8032 z non-logical axiom non-logical Principia Principia W rst- C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) of theformalsystemanddi and onefortwo-placepropositionalfunctions: axiom isgivenintwoforms,oneforone-placepropositionalfunctions bound, or“apparent”,variable. restricted topredicativefunctions,doesnot treat circum more austere Landiniformulationhasallpropositional functionvariables types andorderstreatscircum i theram- incorporates cause theformalsystemisnotmadeexplicitand propositional functions( ( ( calls it the axiomofclasses fact places theassumptionofclassesandin because itgivesforpropositionalfunctionsotherwisede argument. Theaxiomisimportant its with compatible is thelowestorder functionwillbeonewhich by avariable.A“predicative”propositional aconstantorfunction either derived fromapropositionbyreplacing be level propositionalfunctions,etc.Afunctionwill russellwahl 46 (1976). Church’s formulation includes bound variablesofalldi (1976). Church’sformulationincludesbound thatofChurch Probably themostcommonlyacceptedformulationis f   pecially with regard to the ontological commitments ofthetheory. pecially withregardtotheontologicalcommitments quanti and de thus servesasacomprehensionprinciple. classescanbeconstructed.Theaxiom which ositional function,from and theancestralrelation),andinde and quently, there havebeenmanydi andWhitehead.Conse- Russell type indices,thesearenotsuppliedby tional function. ment, andthe W PM PM 12.11 12.1 is a propositional functionofanyorderwhichcantake isapropositional ed theory oftypes.Whilethetypetheorycriesoutforvariables with theory ed The logicof 2 See,forexample,Hatcher1968,Chihar

1: 167).In  W 13) andintheaccountofmathematicalinduction(through W | | cation over predicative functions anequivalentpredicativeprop- nes purported predicates of classesintermsofpredicates of nes purportedpredicates

: : ( ( ' ' f f yzz zyz f Principia Mathematica Principia z with the exclamation after itisapredicativeproposi- after withtheexclamation ) ) f : : is a real variable inRussell’sterminologyand variable real a is

f f z x ( x

. ,

/ y PM z the axiom occurs in the discussion ofidentity the the axiomoccursin ) x

. .

f

/  T

z ! x erent accompanying interpretations, es- erent accompanyinginterpretations, 20.01). Russellhimselfsaysthe axiom re- z , x y

T .

erent formulationsorreconstructions f

z X ! a 1973,Church1976,andLandini1998. z ( exion aslambdaabstraction.The x is di , y z )Pp Principia W U nition ofrealnumbers.The cult tocapture. This isbe- has ano-classtheory X W exion as a term- exion asa ned in termsof x asanargu- T f erent z isa  Pp 90 2 C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) di ought tobeuniversals. functionswould beor the non-linguisticcorrelatestopropositional be naturaltothinkthat ist aboutuniversalsduringthistime,anditmay Landini (1998). On the otherhand,Russellcertainlyappearstobeareal- semanticsforpredicatevariablesasadvocatedby nominalist suggest a non-linguistic correlatestothem.Thislinewould no are and sothere thatpropositionalfunctionsaremerelyexpressions sometimes suggests takentowardpropositionalfunctions.Russell are avarietyofattitudes way asetisde all types bythevicious-circle principle, which states that “whateverinvolves propositional functions,foritisthesethat are rami only comprehensionprinciple. well asthe as schema letters, andsotreatstheaxiomofreducibilityasa forming operator,andtreatsmanyinstances of realvariablesasschematic propositional functions which are entitiesin which propositional functions and nopropositions onesensethereare eventhoughin that wouldsay B. Linsky his accountofpropositionsandpropositional as types. Russellrejectspropositions it variables. Inthatpresentation propositionsassingleentitiesand treated W 40). Much work has been done on the vicious-circle principle, vicious-circle Much workhasbeendoneonthe 40). would phrase his position as position his would phrase at all,givenRussell’sviewthatthey,like tions areconstructionsofthemind. As Gödelpointed outin1944,this func- (1983). Theconceptualistview,simplyput,isthatpropositional ciated with predicative logicsasstudiedbyFeferman(1964)and Hazen easy to see that the principle suggestsaconceptualismofthekind asso- cannot containtermsonlyde as thepositionthat“thevaluesofafunction of propositionalfunctions principle interms the rephrase tional functions,RussellandWhitehead come in(rami rst explicationoftheprincipleisinterms of setsandinparticularthe T 6 5 4 3 In alltheinterpretationsisstatusof for issue At theheartof erent reasons,bybothLa of a collection must notbeoneofthecollection”( must of acollection Theidenti Landinidoesnotwanttotalkaboutprop In the1908article,“MathematicalLogic SeeGödel1944,Chihara1973,Hazen19 Principia W W ed) types. cation ofuniversalsandpropositiona W , Russell andWhiteheadmotivatedtherami , ned. Given the “no classtheory”andtheroleofproposi- 5 the view that all predicate variables are predicative. are variables predicate all that theview ndini 1998andLinsky1999. was propositions whichcame indiwas entitiesintheIntroduction to W 3 nable in terms of the function” ( nable intermsofthefunction” propositions, are “incomplete symbols.” He propositions,are“incompletesymbols.” Principia as entitieswhichcouldbesubstitutedfor functions as certain kinds of constructions, kinds functions ascertain 83 andGoldfarb1989,forexample. as Based on the Theory of Types”, Russell Types”, of ontheTheory Based as ositional functionsorfunctionvariables ositional The AxiomofReducibility , theyarebothconstructionswhich l functions l functions is rejected, thoughfor W ed andtyped. Principia PM T erent ordersand W 1: 37). Their ed theoryof . Because of . Because 6 anditis 4 There PM 47 1: C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) donment ofpropositionsasentities,whichisevidentinRusselland shift fromthesubstitutiontheoriesof1905–06isRussell’saban- biggest in whichtheaxiomofreducibilityisfalse. argued thattherearelogicallypossibleworlds Ramsey and Wittgenstein functions”,both ositional functions”and“predicativepropositional are taking“prop- grounds forbelief.Withoutbeingexplicitonhowthey no something inwhichtheconceptualistnotionofpropertyprovides 374), p. (1988, it a logicalprinciple,but,asHazenrightlyputs not only mental activity, then it appears clear thattheaxiomofreducibilityisnot we think that propositionalfunctionscomeintobeingasa result of some willcertainlyseemtobeasubstantialnon-logicalclaim.If reducibility frompredicativefunctions,thentheaxiomof distinct tions areentities the type theory of the type to manuscripts inwhichRussellmovedfromthistheory the and theory in Chapter we think of propositional functionsasentitieswhichcomeindi we thinkofpropositional logical types, and thesetypesare rami takes veryseriouslytheoriginsof view, theotherarealist. nominalist predicative propositionalfunctions.Oneoftheseviewsisa as themselvesbeing functions conjunctions ofpredicativepropositional structure which lends supporttotherationaleof the rami whicharenotconstructions,buthavea entities tions ashigher-order conception doesnot russellwahl 48 I want to look at two interpretations of interpretations two I wanttolookat plausibly be seenasalogical principle. The justi leadtotheviewthataxiomofreducibilitycan naturally they more withinthem.Ipicktheseinterpretationsbecause fares of reducibility both oftheseviews will involve understandingin accounts. The nominalist position is articulated in Landini (1998). Thisposition Landini The nominalistpositionisarticulatedin Some morerecentrealistinterpretations treatedpropositionalfunc- Some 9 8 7 See,forexampleGoldfarb1989an SeeRamsey1925,p.57,and 1983. Hazen SeeGödel1944;also ii i. oftheIntroduction to w two interpretationsof Principia W t wellwithRussell’srealistviewofuniversals. . ItalsotakesveryseriouslyRussell’sremarks Wittgenstein 1922,6.1232,6.1233. d Linsky 1998 and 2006 for two di two d Linsky1998and2006for Principia Principia W ed, sothatnon-predicativefunc- Principia 9 z ’s logicinthesubstitution . What Landiniseesasthe “ principia W z cation fortheaxiomon and seehowtheaxiom W nite disjunctionsand W ” ed theory. T erent such T erent 7 8 If C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) ized. Quanti theory oftypesandordersisathe symbolism, not thesymbol- assert such complexes ( assert suchcomplexes self apredicate.Russellputs it isnotit- predicates, of predicates ofagreatgeneral”referstototality as a predicative function which is trueofthatobject.Since“ equivalent to this function. Russell here de equivalent tothisfunction.Russellhere reducibility, asthataxiomwillstatethereisapredicativefunction ( “second-order truth”( have one complex,buttomany.Thesejudgmentswill to correspond not every individual,butstatethatthesejudgmentsdo of ositional function the-relation judgment. Theybeginwithabasicontologyofsuch complexes as“ andthemultiplerelationtheoryof theory endorse theno-propositions Whitehead’s accountoftruthandfalsehood( substitution theory,though(see Theory ofTypes”(1908).Propositionalfuncti isin distinct theThis theory orders. pres initially adopted aricher ontologywhichin discussion ofthistheory.Herejectedtheth into entity variables andsubstitutingone symbols for propositionalfunctionsasinco truth”. Theythende down ontologyofthattime. theystillseemtosubscribethesamecut- toSeptember1906, prior substitutiontheorywhichRussellheld the of variables inthemanner function propositional of account no longerattempttogroundthe head language suchas“property”, “quality”, etc. So while Russell andWhite- functions ofagivenorder( ( functions ally; thetheoryoftypesisastructuredvariables. of individualsisthefunction“ often repeated)Russellgivesofanon-predicativefunction example (now ofnon-predicativefunctions.The treatment own Russell’s by considering great general, and then expresses the claim that this second-order pred- this that claim great general,andthenexpressesthe PM 10 no di Onare thisnominalistview, there The di Forawhileafterhisadoptionofthetheo 1: 56). Russell uses the exampletoexplainforceofaxiom 1:56).Russelluses T erence betweenthispositionand realist onescanbeillustrated PM W R cation overpredicatesshouldbeunderstoodsubstitution- -to- 1:48–55)makesreferencetomakingassertionsaboutall b z ” andde W ne ageneraljudgmentasonewhich asserts a prop- PM PM LK z 1: 45). The discussionofthehigher-order The 1:45). 1:44).Thesejudgmentshave“ W 10 f PM y , p.77). ne elementaryjudgmentsasthosewhich ( x f z hasallthepredicatesofagreatgeneral” z 1: 51–2) but does notuseontological does but 1:51–2) ! z z ˆ ent in“MathematicalLogicasBasedonthe y ) for another.SeeLandini1998forathorough eory inthe “Paradoxof(1906) and the Liar” mplete symbolsexplainedintermsof entity volves propositions asentitieswhichcome volves propositions on variables werestilltreatedaswiththe variables on ry ofdescriptionsRu f The AxiomofReducibility W z ! z z nes a“predicate”ofanobject ˆ T z isapredicaterequiredin erent typesofentities;the PM 1: 41–7), where they where 1:41–7), ssell soughttotreat x hasall the W rst-order a -in- 49 C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) Now howshouldweunderstand In this case we shouldnotethe generals Russell’s exampleslightly,that modify assigned a well-formed formula which contains thefunction( which assigned awell-formedformula 1 ( (1) icate appliestoNapoleonas russellwahl 50 predicative functions (Landini 1998,p.264). (Landini functions predicative predicative” (1: 165). Landini’s interpretation requires that there be novariables for non- possible …withoutlossofgene However, stitutionally, notobjectually. sub- semanticsisthatthefunctionvariablesbeunderstood nominalist thatitisnot.Whatimportant forthe reveals but afurtheranalysis but only in apredicateposition.Itappears(1)asifsubject position, example, ( pia context ofhisquestioningRussell’sclaimthatvariablesin in the propositional functionhold? higher-order the does that group.But,onecouldask,invirtueofwhat function heldofexactly predicative no but individuals, held ofagroup propositionalfunction false? Itwouldbeaworldwherehigher-order common togreatgenerals,suchthat ofbeingonetheproperties property that otherpropertieshave,a 2 ( (2) then, justis On thisnominalistsemantics, 11 On a realist interpretation,thesemanticswouldassignto We cangetclearonthisbytakinganexample mentioned by Hylton wheretheaxiomofreducibilityis one be What sortofworldwould Hylton 1990, p. 308n. The referenceistotheclaimin The 308n. p. Hylton1990, range only over predicative propositional functions. propositional range onlyoverpredicative istrue of Fred and also true of Mary, although Fred has this in f x ( zz )( f Gx z ! z z ˆ z f ) would notbeidenticaltotheproperty (

' z f )

( : z f ) f ( : z z ! x )

z x f z )( : ( z ), where“

Gx f f ( rality, to use no apparent variables exceptsuchasare variables rality, tousenoapparent z ! f z

z ˆ ' z z ! ) f z z ˆ

implies z f ) z ! z z ˆ / z f ! ( z Gx z doesnotappearinasubjectposition, x ( f f z ) x is a free variablethatshouldbe free is a z z z ' ” justis “ ! )(

z z f having apropertycommontogreat ˆ Gx z

)? z f ! z (Napoleon). z

! ' z (Napoleon).

f x z ! z isagreatgeneral”.(1), z x z ). Principia x 11 z )( Suppose,to

f Gx  z a property a f 12 that“itis z !

' z z ˆ z ) Princi-

z f — z ! z z for x z ). C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) over functionswherewhatwasinvolvedonlya true ofFredandMary,butnopredicativefunctionwouldbe. to it.Thedi such functions,therewouldalwaysbea lower-level function equivalent case the second-levelfunction,( tions, then the disjunctionofall then tions, onpredicativefunctionsyieldfurtherfunc- ean operations that Bool- i.e. is alwaystheassertionthatoneorothersofthesehold, ti valent to conjunction of those functions would yieldapredicativefunctionequi- second-level function tothe ious propertieswouldbeitselfapredicativefunctionequivalent junction.” So Russell himselfallowsthatany So junction.” is apredicate,becausenopredicateoccursasapparentvariable in the dis- ( virtue of havingproperty virtue where the quanti ity ofsuchdisjunctionsorconjunctions. (linguistic) propositionalfunctionsofin functions,astherearenosentencesor or conjunctionofpropositional predicative function ispresentbutnotcapturedbyany predicative ity wouldbejusti that order. It is easy to see that at the at that see that order.Itiseasyto a givenorderisitselffunctionof of junction ofpropositionalfunctions falsi ( thought itwaslegitimate, for inafootnote( clearly also wonderwhetherthisstepislegitimate.Russell,though, might we exist, to it that somemindhadtothinkthedisjunctionfor thought wonder whetherthisstepislegitimate. Similarly, if as conceptualists we each other,thenwemight of thought eachoftheselogicallyindependent Fred and Mary do nothaveanygenuine propertiesin common. erty in virtue of having property x W W z )( nite) set of predicates is given by actual enumeration,theirdisjunction by given is nite) setofpredicates If one held that whenever there are the predicates the are If oneheldthatwheneverthere 12 If wefollowthenominalistsemanticsandunderstandfunction ed predicative propositional functionswithgenuineuniversalsand ed predicativepropositional Gx Hyltonattributesthe example toThomas W cation of the axiom of reducibility, weneedacasewherenon- cation oftheaxiomreducibility,

' W

cation substitutionally, then it appears thatto come upwitha appears it cation substitutionally,then Ax c having all thepropertiesofagreatgeneral z mustbeanon-predicativefunction. z ) and( U culty occurs in the casewherewearedealingwithan in x z )( W Gx ed: for every higher-order functionthatquanti ed: foreveryhigher-order

' having a property of a great general having apropertyofgreat

Dx F z z ) andfrom (saybeingarrogant)andMaryhasthisprop- ' G f (say being deceitful), but otherwise but (saybeingdeceitful), W z ) A rst-level functions assertingthevar- W : z f and f and ( nite level,theaxiomofreducibil- Ricketts.Hylton’sconcernisthatfrom W x nite length.Itisn’tclearwheth- z )( The AxiomofReducibility D Gx PM z m wecandeduce

W ' nite disjunctionorcon- z 1:56)hesays,“Whena

. Ifasrealistsweiden- f z ! z F x W z z ) and nite collectionof W . nite disjunction

f . Similarlythe z ! G ' z z ˆ c z z thenthere , wouldbe y ( 12 c y Inthat f .

c W W y m), ed n- 51 C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) type restrictionsonpropositionalfu of apropositionalfunctiondeterminesth while Landini does (Cocchiarella 1980, pp. 105–8, Landini 1998, Chap. 10). 1998,Chap. Landini pp.105–8, 1980, does(Cocchiarella Landini while be amodeloftheaxiomreducibility. he assertedthatourinability to write thepropositionsofin sell’s realism. selves predicative functions, then every intended model of model intended every selves predicativefunctions,then rami is “logicallyamereaccident”(Ramsey the possibilitythatthereisapossiblein mightleaveopen develops he er Russell’s“assertions”inthesemantics russellwahl 52 advocate typingsetsaccordingtotheirpr accused of advocatingwithth W and relations whichholdofanindividual.This“in and position it is easy to imagine thattheaxiomofreducibilityisfalsein to position itiseasy Onthiskindof of thevariousboundpropositionalfunctionvariables. particular, butalsopropositions.These intentional entities arethevalues the constructivisminvolvedinrami Goldfarb, inparticular,defendsthis whether overindividualsorfunctions,arein allows thatallgeneralizations, which system shall see,Ramseyproposeda give thepredicativefunctionrequiredbyaxiom of reducibility. Aswe di tion ofanin which isin necessary, becausewedonothavealanguage is universal generalization that all entities, therefore, are of thesametype. that allentities,therefore,are the unrestrictedvariableandview of doctrine philosophy oflogic:the Nino Cocchiarellahaspointedout)what drove much ofRussell’searly function and so is notn sois and function function in this way, the axiomis true, and is of reducibilityshouldtheref conjunctions anddisjunctions(Ramsey1925, in these toinclude function” predicative theconceptofa he doesby“widening nite conjunctions and disjunctions ofpredicativefunctionsasthem- disjunctions and nite conjunctions T Peter HyltonandWarrenGoldfarbhavesuggestedthatdespitethe 15 14 13 So farwehavebeendiscussingthenominalistline which preserves (as erent ontological types of abstract entities: propositionalfunctionsin abstract of erent ontologicaltypes SeeGoldfarb1979,especiallypp. 30–4. In fact, while Ramsey suggests that hereje suggeststhat Ramsey while Infact, Cocchiarella, though, does not think not Cocchiarella,though,does W ed hierarchy, Russell wasarealistaboutpropositionalfunctions. ed hierarchy, W nite. W 15 nitely longsentence,theaxiomwillbedubious. His position is that Russell was committed to various to HispositionisthatRussellwascommitted eeded as aseparate axiom. ore beunderstoodthatonce e vicious-circleprincipleas nctions withoutconstructivism(p.31). esentation. Goldfarb arguesthataspeci esentation. e function presented in a way that justi e functionpresentedinawaythat Russell maintainedthispositionin interpretation against thosewhosee showntobesuchbytheconceptofatruth Goldfarb sees theconstructivismRussellis pp. 38–9).Ramsey’scr cts the axiomofreducibility,this is what 1925, p.41).If we accept thesein- W 13 W ed theory as being alien to Rus- to alien being ed theoryas The axiom is necessary, just as axiom isnecessary,just The nite assertionof the properties W nite truth-functions,and appropriate toonewhowould 14 we understand a predicative we understand a WithoutRamsey’sno- W nite assertion”would iticism of the axiom Principia W nite length Principia W W cation es the z W will nite , C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) they are not“logical but constructions, are they not bethoughtofasindependententities and propositionalfunctionsshould propositions both On Linsky’sview, individuals and the totalityofelementarypropositionsisconstructedout that sense not adopting a nominalistattitudetowardthese. the no-propositionsandno-propositionalfunctionspositionwhile thepassagesin Linsky isthusabletoaddress andso“notsingleentities”. 1999, p.29).Theyarestructuredentities, (Linsky these as“creationsofthehumanmind” see not does and entities Linskyseestheconstructionasametaphysicalrelationamong way. the same in sumably becausethereisnomethodofeliminatingthem of for Linskycomeindi he does notidentifyuniversalswith propositional functions. Universals Edwin Mares(2007). These related viewshavebeenadvocatedbyBernardLinsky(1999)and possibleworlds. the axiomofreducibilitydoesappeartobetrueinall tured entities. Theyarenot tured his constructions man”above); average which canbeeliminatedinanalysis(suchas“the constructions functionsalikeas view. Rouilhanseespropositionsandpropositional constructivism. Rouilhan’sconstructions constructivism. of kinds three on thisviewbyPhilippedeRouilhan,whodistinguished is a falls under them,butthesetypesarenotidenticaltotherami falls some possible world. eralization overentities,etc.Hisconstructions propositions are thoseconstructedoftheseandsuchoperations as gen- tion betweenuniversals,whicharenotconstructedentities,andproposi- p. 34)rejectthesepassagesasno these passages in in these passages Linsky, Rouilhanthinksconstructions and favoured byFefermanandHazen(Rouilhan1996,p. 217). Like Goldfarb Linsky makes a distinction in type between entities anduniversals,but Linsky makesadistinctionintypebetween 17 16 Principia So Linsky, even with his realist view which is akin to that of Goldfarb, can accept can to ofGoldfarb, that akin is which view realist his evenwith SoLinsky, Goldfarbseestheaxiomasshowingth W ction in the proposition “the average man is ction intheproposition“theaveragemanis , whichapplytopropositionsandpropositionalfunctions. 2 Principia , as does Linsky.OnLinsky’sviewthereisasharpdistinc- W 2 rst-level conceptswhichapplyto them, and are the structured entities that are constructed in the in arethestructuredentitiesthatconstructed W 16 despite his realism.Church(1976,p.748)andGoldfarb(1989, ctions” assomeconstructionsare,butrather struc- T But on the two variants of realism I will discuss, will I realism Butonthetwovariantsof erent types, dependingerent on t consistentwiththelogicof W ctions inthesensethat“theaverageman” e failureofRussell’slogicism(p.38). The AxiomofReducibility 1 arethelogicalconstructions 3 arenotpertinenttoRussell’s 3 Principia are mentalconstructions 17 Linskywasanticipated whichtypeofentity Principia W ve foot nine”,pre- foot ve which advocate . W W rst-order ed types 53 C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) predicates isequivalenttoasinglepredicate’ equivalent totheassumptionthat‘anycombinationordisjunctionof axiomofreducibilityis “The said, ity onallmodels.AsRussellhimself the AxiomofReducibil- of validation the for same type.Thisthenallows anddisjunctivepropertiesforanysetofthe conjunctive function variables(i.e.thenon-predicativefunctionsof predicative typesisobjectual,andthesemanticsfornon-hereditary be natural to think that, onthisview,the axiom ofreducibilitywould that, be naturaltothink of universalandindividuals.Itmight predicative ones,asconstructions tionally. Linskythinksof all propositionalfunctions,includingthenon- ti ontological counterpartsfornon-pred involve frames which are sets of facts which include, forany involve frameswhicharesetsoffactsinclude, functions andnoontological“variables”.Themodelsforthesemantics stitutional. Therearenoontologicalcounterpartstothenon-predicative Linsky’s ontology for Linsky’s ontology di various typesofuniversalsarenotre tional functions,whichare. russellwahl 54 of thatsametype.Using this thesis, Mares of agiventypeyieldsproperty conjunction ordisjunctionofproperties the “Booleancombinationthesis”,thatany accepts Mares entities stand. allandonlythose which in quence ofentitiesinthemodel,arelation types. Thesemanticsforquanti p. 237).Onhis reconstruction, universals comeinahierarchyofsimple (Mares 2007, hereditary predicativefunctionsymbolsdenoteuniversals thiswaytheysharplydisagreewithLandini.Mares’ functions, andin He thencontinuestheseconstructionsforuniversalsofdi would apply to any set of entitiesthesametype(Mares2007,p.239). that these cepts ofindividualsandthenformsdisjunctiveproperties for functions withuniversals.MaresandLinskybothacceptaformal system this assumption into his models and then demonstrates Russell’sclaim. then and this assumptionintohismodels objectually, andquanti Mares strengthenstheBooleancombinationthesistoincludein Mares W T Edwin Mares (2007) has developed a semantics somewhat basedon Edwin Mares(2007)hasdevelopedasemanticssomewhat 18 The majordi cation over predicative propositional functions should betreated cation overpredicativepropositionalfunctions erences between individuals erences betweenindividuals Principia Theremaybeaconcernthatthetypedi z whichincludesfreeandboundvariablesfornon-predicative T erence betweenMares and LinskyisthatMareshasno Principia W cation overnon-predicativefunctions substitu- and propositionalfunctions. 18 X ected inthelanguage,whichonlyshowstype . Mares,though,identi W cation usingvariablesofhereditary icative functions. For Mares, quan- T erences between individuals andthe betweenindividuals erences W z ” ( rst displaysLeibniziancon- PM z 1: 58–9). Mares builds Mares 1:58–9). Principia W es predicative T erent levels. W z ) is sub- nite se- W nite C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) he suggests thatatleastwithanin he suggests structivism an illusion (Gödel 1944, p. 143) and bringing in an bringing structivism anillusion(Gödel1944,p.143)and oftheaxiomasmakingcon- sell, inparticularGödel’scriticism Rus- on in hiscontributiontotheSchilppvolume made Gödel remarks contradiction” (Rouilhan 1996, p. 273). Rouilhan ispickinguponsome as“hardlyworthmore…thana ity, heseestheaxiomofreducibility rather thanseeingthisinterpretationas validating the axiom ofreducibil- ofthekindLinskysuggests,but constructivism also discussesarealist Mares’ strengthened Boolean thesis whichmark ofalogicaltruthisitsextremegenerality. have the stage, this to at least at be Russell, faced. for that is position Linsky’s of Rouilhan Part ple. viewed asalogicalprinci- ing ofhowtheaxiomreducibilitycouldbe better understand- a predicative functionswithuniversals,andsogivesus versals” (p. 106).Mares’positionmakesthisexplicit, by identifying the buttheydonotintroduceanynewrealuni- them, ing oforclassifying duce newpropertiesofthings.They may characterize new ways of think- typepropositionalfunctionsdonotreallyintro- He remarks:“Higher (p. 107). ositional functionsinheritsomeofthecharacteruniversals” (Linsky 1999,p.108),and generality” principle…ofgreat claim. Linskydoesinfactcallita“metaphysical certainly notlooklikealogicalprinciple,butasgrandmetaphysical W atory Statement” toVolume 16). RussellandWhiteheadthemselvesmentionthispointinthe“Pref- functionsatthatlevel(Black1933,pp.115– predicative a continuumof there mustbeatleast thus and number, real each be distinctproxiesfor that givenatleastthetypelevelofclasses ratios, there need to (p.152).Infact,MaxBlackhadalreadypointedout “occult qualities” “occult qualities”?GödelblendedRouilhan’sconstructions ofperfectlyrespectablepredicativefunctionsaresomehow junctions Rouilhan attributestoGödel’sremarkabout“occultqualities”. functionsateachtypeleveliswhat predicative tions. Thisplethoraof by the methodssuggestedin co interpretation,since the of spirit is contradictory,butratherthattheaxiomof reducibility contradictsthe than acontradictionisnotthatthesystem more ducibility isnotworth tions ctions (Gödel1944,p.143),andcalledintoquestionthe claim that these There are, however, some consequences of the assumption involved in involved are, however,someconsequencesoftheassumption There But isitreallycorrectto think that these in 2 , andconstructions 3 orLinsky’sstructuredentitieswithhislogical Principia ii ( PM nstructing thepropositionalfunctions W z nity ofindividualstheaxiomre- 2: vii).WhatRouilhanmeanswhen he suggests that “predicative prop- he suggeststhat“predicative z wouldnotleadtothismanyfunc- The AxiomofReducibility W nite disjunctionsandcon- 1 , construc- W nity of 55 C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) de betweentheaxiomofreducibilityand There isacloseconnection marks that without the axiom we would be led to identitiesofdi led be marks thatwithouttheaxiomwewould Perhaps abetterwayofputtingitisthat,without the axiom ofreducibil- degrees, so thatstrictidentitywouldhavetobetaken as a primitive idea. which is theprinciplewhichjusti which class wehave. toallandonlythemembersofwhatever new universalwhichapplies less “occult”thansome words, other predicative functions.Theyseem,in toawholenewkindofentitybyadmittingthese are notjumping junctions and disjunctions of these alsoasconstructedentities,thenwe predicative functionsassomewhatfamiliarstructured entities, and con- of couldbetreatedasincompletesymbols.Butifwethink functions russellwahl 56    From this, given the axiom of reducibility, the generaltheoremfollows: reducibility, From this,giventheaxiomof trary propositionalfunction”(p.101). but ratherentity, just of apredicative prop pointing outthat“itdoes not assert the ex hand other (Linsky 1999,p.106n.).Onthe ofclearl objects has a“nature”,is then it And fromthisweget: sell requiresreducibilitytoprove also recognizedthiscloselink(1999,pp.104–9). important forthepurportedcounterexamplestoaxiom.Linskyhas 13.12 13.101 13.01 W 19 Russell de nition ofidentityin Linsky himself suggests thatif the axiomis taken as saying that any arbitrary class ii. counterexamples totheaxiomofreducibility w the identityofindiscernibles,ramsey,and x | | =

W : : 19

ned identityin y x x

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' ' f z

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: c c

f x x

z ! / ' z x Principia z

c

. c

y ' W y y y ahighlycontentiousmetaphysicalprinciple , y es the substitution of identicals. Rus- es thesubstitutionofidenticals.  .

istence ofsomenew, . ositional functioncoextensivewithanarbi- 13.101 and therefore andtherefore 13.101

, hedefendstheaxiomofreducibilityby f z ! as follows: z y , andthisconnectionis non-logical categoryof  13.12. Here- T erent Df. C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) properties. tion whichcanincludeanin by disjunctionorconjunc- constructed predicative functions,functions icative properties,butdi icative since theremightbedistinctindividualswhichagreedonall their pred- ity, thede rejection ofRussell’stheoryorders,butadi nota is This isnotquitewhatRamseydoes.Instead,heproposes types. the wholesystemofordersandjustadoptingasimpletheory but empiricalterms.”NowitiseasytoreadRamseyasthereforerejecting some referencetothought,language andtheothersemanticalones,orthosewhich“contain tradictions Onegroupinvolvesthemathematicalandlogicalcon- p.20). (1925, (Ramsey 1925,p.28). theaxiomisnotatautology whole system.Hismajorconcernisthat om of reducibility in strong language as illegitimatedefect of anda the Introduction toMathematicalPhilosophy and also Russell’s own qualms about theaxiomasheexpressedthemin helpful inunderstandingRamsey’s criticism of theaxiomreducibility, tions formed by in tions formedby contains thequanti ascribed toNapoleonabove?Ramseywouldsaythatwhiletheexpression p.41).Whataboutthenon-predicativefunctionwe (1925, functions sey’s understandingof“predicativefunction”. be Russell’s.Theaxiom of reducibility will therefore betrue,withRam- whatRamseytakesto with not will beco-extensivewithRamsey’s,but formed fromin functions Russell’s byjustthispermissionofincludingpropositional takes tobe he what from functions distinguishes hisaccountofpredicate what apredicativefunctionis. a “real” characteristic (Ramsey 1925, p. 47). The linkbetweentheaxiomofreducibilityandde 20 Ramsey is famous for dividing the contradictions intotwogroups contradictions the Ramsey isfamousfordividing Ramsey criticizedtheaxi- Mathematics”, of “Foundations In his1925 As Mares’ reconstruction includes as predicative functionsthosefunc- as includes As Mares’reconstruction Ramsey says that on his view, therefore, allfunctionsarepredicative Ramsey saysthatonhisview,therefore, Ramsey points outthat,on W nition W nite truthfunctions(p.39). W  W nite Booleanoperations,Mares’predicative functions 13.01 wouldbedefectiveasade cation over functions, this shouldnotbeunderstood cation overfunctions, T ered withrespecttosomenon-predicative his view, order is acharacteristic of asymbol andnot W nite numberofarguments(pp.38–9). He 20 In fact, Ramsey explicitly includesas explicitly Infact,Ramsey , or symbolism, which are not formal or symbolism,whicharenot , . The AxiomofReducibility T erent understandingof erent W nition ofidentity W nition  13.01 is 57 C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) stead, he takes the axiomtogetherwithde directly comeupwithapossibleworldinwhichtheaxiomisfalse.In- hedoesnot But world. that theaxiommaybefalseinsomepossible ( is a“defect” Philosophy Mathematical evident axiom” ( evident axiom” more an axiombecauseitmightfollowfroma“morefundamentaland as the axiom (or its logicalstatus),butwithwhetherit should be included to anyelementaryfunctionof to theaxiomofreducibility: he functions”) from his“predicative position con itismerelyoursymbolwhichinvolvesit”(p.41).This functions; involve thetotalityof not does put it,“thefactitassertstobethecase Ramsey functions fromwhichitwasdeveloped.As predicative the from isinsomewaydistinct as givingusanew(ontological)functionwhich russellwahl 58 There will be no other individual but There willbenootherindividual ofthefunctiontaken.Then( respect agreeingwith individual an individual be anin should there that For itisclearlypossible W exactly thesamepredicates,innarrowsensewhich wehavebeen well, as a matter of abstract logicalpossibility,betwothingswhichhad might tity ofIndiscernibleswhichhethencallsintoquestion.“There form ofLeibniz’slaw”( the lattertherearecorrelatesonlytopredicativefunctions. correlatestopropositionalfunctionsatall,whileon are noontological with thatadvocatedbyMaresandLinsky,foronthe former viewthere duction to this regard.WhenhementionsqualmsabouttheaxiominIntro- atomic functions. respectto his counterexamplefromarejectionofLeibniz’slawwith to beRussell’ssense)that but inRamsey’sworldthereis no predicativefunction (inwhathetakes rst level, at least, he calls “elementary functions” to distinguish them distinguish rst level,atleast,hecalls“elementaryfunctions”to Ramsey’s insistence that Russell’s predicative functions (whichatthe predicative Russell’s Ramsey’s insistencethat Russell’s ownobjectionstotheaxiomofreducibility are interesting in Principia X a IMP icts neither withthepositionadvocatedbyLandininor neither icts such thatwhicheveratomicfunc PM , p.193).Hisreasonforholdingthisisthatit is possible , his concern doesnotseemtobewiththetruthof concern , his 1:60).Heismorecriticalinhis IMP a . Here he says that the admission of the axiom z inrespectofalltheotherfunctions,butnot individuates, sotospeak, x , p.192).Itisthe necessary truthoftheiden- (1925, p.57) . f lps usunderstandhis counterexample z ) a .

z towhichthisfunctionwillapply, f z ! W z x nition

/ W

tion we take there isanother tion wetakethere f nity ofatomicfunctions,and z ! z a  z couldnotbeequivalent 13.01 asa“generalized a . Ramsey hasbuilt Introduction to C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) speci … exac believing and properties “It wouldseempossiblefortheretobetwo being terialism andpropositionalattitudesas kind isnotamatterforlogictodecide. is alsocleartomethatwhetherornotth assumptions abouttheexistenceofpredicativefunctions: theidentityofindiscernibles,butmakesverystrong mention does not not showthattheaxiomofreducibilityis using theword‘predicate’ independent fromthequestion example. notrespectedinthis is by theBooleanoperators(includingnegation) arealsothepredicativepropositionalfunctionswhichformed there The principle that whenever thereisapredicatepropositionalfunction … imaginewelivedinaworld the axiomofreducibilitywouldcertainly other andfurtherneverheldbetweena the thingsandinsuchawaythatit di predicative if This particularcountere did not given anaccount of notgiven did he gaveanaccountofthe truth of proposit rejected understanding belief if Leibniz’s law is rejected as a de if Leibniz’slawisrejectedasa identity ofindiscernibles.But the of thesis into hismodelsaverystrong theaxiom ofreducibility,builds giving areconstructionthatvalidates over andabovethem does notsupplyuswiththatpossibility. propositionalfunction,andRussell distinguished byanon-predicative theseitemswereinfact unless the axiomofreducibilityintoquestion 23 22 21 The counterexample given by Wittgenstein ina1913letterto Russell Hazen (1983, p. 364) hassuppliedapossi 364) p. Hazen(1983, See Wittgenstein 1974,p.39. SeeWittgenstein Ramseyrejectsde W ed level,butdi p is apropositionwhichcontainsnon-predicativefunction.Russellhad T , ering in their attitudes toward so ering intheirattitudestoward W only xample requires that thepropertyofbelieving iii. nition beliefsinvolvingthese. w as arelationbetweenperson a concluding remarks of the validity oftheaxiom the of tly thesamepropositionsateverylevellowerthansome z  ” ( single 13.01, but sees the rejection of13.01, Leibniz’s butsees lawas beingrejection the The letterinquestionisR.23. ibid which nothingexisted except z relation holdingbetween in W ., p.192).Thisbyitselfwouldnotcall nition of identity, thisbyitselfwould of nition d not hold betweeneachthingandevery ions involvingnon-predicative functions, he 23 e world inwhichweliveis really of this W individuated bythe propositions involved: spirits sharing all their non-psychological their all sharing spirits nite numberofthings.Itisclearthat not bility whichinvolvesarejectionofma- z Butit hold good aworld. in such The AxiomofReducibility 21 X As we have seen, Mares, in awed. me propositionofthatlevel.” s and propositions, butwhile of reducibility(1925,p.30). 22 W p nitely manyof  has tobe non- 0

things and, 59 C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) stracted fromfacts. which arepredicativepropositionalfunctionsthemselvesab- by Mares,theaxiomistruebecauseofatruthaboutconstructions to formulatethese.Finally,onLinsky’sview,especiallyasdeveloped able propositional functions,werewe predicative of junction orconjunction whether hisdi this last view infactdoescapturethelogic of whether wondering the axiommayperhapsbeseenasareasonfor about a logical principle than apurelymetaphysical one. Russell’s own qualms isatleastmoreof that to theunderstandingofaxiomassomething attractive asitlendssupport is view last This which isequivalenttothese. function axiom ofreducibilitytogiveusthepredicativepropositional disjunctions ofpredicative propositional functions,wewouldn’tneedthe to those which canbeformedby which to those Russell’s constructionsofpredicatepropositionalfunctions to be limited becausehetook Ramsey The axiomofreducibilityappearedfalseto russellwahl 60 W W in constructivism, this posit is unjusti posit in constructivism,this propositional functions.Gödelthought that if rami itpositstheexistenceofsomanypredicative later Rouilhan)because functions.TheaxiomappearedsuspecttoGödel(and atomic tions to it as somehow capturing whatwewould intendtosay by an in seems falseunlesswesee axiom the then interpretation, give anominalist claims.Ifwe entities, theaxiomappearstomakenon-logicalexistence of constructionsasmental.Ifwethinkpropositionalfunctions as real evenifwedon’tthink quired forthetruthofaxiomreducibility in allthepropositionalfunctionsre- result won’t construction ations of that theoper- thought still Gödel’s understandingofconstructivism,he need neitherrami aboutsets,andsowe His ownviewwasthatweshouldbePlatonists ositions, as Ramsey suggested, we wouldn’t need to have separate quanti- have ositions, asRamseysuggested,wewouldn’tneedto Landini for manyhelpful criticisms and clari University in2010. Thanksmembersto of cational axioms.Similarly,ifwe could express in axioms play.Ifwecouldexpressin cational On this view, the axiomsimilar to plays therole a role that quanti- 24 Anearlierversionofth U culty issolelywiththeaxiomorLeibniz’slaw. W cation northeaxiom.WhileRouilhanrejected is paper was readatthe paper is references W nite applicationsofBooleanopera- W myaudienceandespeciallytoGregory W cations. ed asistheaxiomofreducibility. PM W Principia nite conjunctions of prop- nite conjunctionsof y @ z 100 conference at McMaster 100 conferenceat W nite conjunctionsand W cation hasitsroots , butitisnotclear W nite dis- 24 C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) Hylton, Peter.1990. Linsky, Bernard. 1999. 1999. Bernard. Linsky, Landini, Greg.1998. Russell, Bertrand. 1906. “The Paradox of the Liar”, dated September 1906. Russell, Bertrand.1906.“TheParadoxoftheLiar”,datedSeptember Cocchiarella, Nino.1980. “The Developm Feferman, Solomon. 1964. “Systems of Predicative Analysis”. Predicative of Feferman, Solomon.1964.“Systems Goldfarb, Warren. 1989. “Russell’s Reasons forRami Goldfarb, Warren.1989.“Russell’sReasons Gödel, Kurt.1944.“Russell’s Mathematical Logic”. InSchilpp,ed. Rouilhan, Philippe de. 1996. Foun “The Ramsey, Frank.1925. Chihara, Charles.1973. Black, Max.1933. z z z Church, Alonzo. 1976. “A Comparisonof Quine, W.V.O.1970. Hazen, Allen. 1983. “PredicativeLogics Hatcher, William. 1968. Mares, Edwin.2007.“TheFactSemanticsforRami Gri — — — P. Oxford: OxfordU.P. Manuscript in the Russell Archives, McMasterU. Manuscript intheRussellArchives, Symbolic Logic 71–115. 45: and the NotionofaLogicalSubject in Russell’s Early Philosophy”. Knowledge phy of Bertrand Russellphy of Mathematics Paul. U. P. Antinomies withThatofTarski”. 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In z suppl.16:395–417. Languageand36: Mind), Journal ofSymbolicLogic sed ontheTheoryofTypes”. ”. Chapter1.5inGabbayandGuenthner, . Dordrecht:D.Reidel. Russell’sResolutionofthe Semantical . Englewood Cli . Englewood Rereading Russell ent of the Theory ofLogicalTypes Theory the ent of The AxiomofReducibility . London: RoutledgeandKegan London: . . London:AllenandUnwin. . Stanford: W ed Type Theory andthe Theory ed Type Principia Mathematica y C. Marsh, ed. W cation”. InC.Wade T . Oxford: Oxford U. . Minneapolis: U.of s, . Paris: nj csli z The Foundations 41: 747–60. 41: . Ithaca: Cornell : PrenticeHall. The Journalof 48: 237–51. . Oxford: Per- . The Philoso- puf Logic and American Synthese Synthese . 61 z ”. . C:\Users\Milt\WP data\TYPE3101\russellC:\Users\Milt\WP 31,1078red 002.wpd May 14, 2011 (10:00 am) Wittgenstein, L. 1922. Whitehead, A.N.,andB.Russell.1910–13. russellwahl 62 z — bridge: Cambridge U.P.(2nd edn., 1925–27.) z B. don: RoutledgeandKeganPaul. . 1974. y F. McGuinness.Ithaca:CornellU.P. Letters toRussell,Keynes,andMoore Tractatus Logico-Philosophicus. Principia Mathematica. . Ed. G. y Trans. C. H. vonWright assisted by y K. Ogden.Lon- 3 vols.Cam-