Theory of Types

8/3/2019 Theory of Types

http://slidepdf.com/reader/full/theory-of-types 1/41

Mathematical Logic as basedon the Theory of Ty pes.

BY BERTRAND RUSSELL.

Thefollowing heoryof symbolicogic ecommendedtselfto mein thefirst

instanceby its ability to solvecertain contradictions,f which the one best

known o mathematicianss Burali-Forti's oncerninghe greatest rdinal.* Butthe theory in questionseemsnot wholly dependenton this indirect recom-

mendation; t hasalso,f I amnot mistaken, certainconsonanceithcommon

sensewhich makest inherentlycredible. This, however, s not a merit upon

whichmuchstress houldbe laid; for common enses far more allible than it

likesto believe. I shallthereforebeginby statingsomeof the contradictionso

be solved, nd shall then show how the theory of logical types effects heir

solution.

The Contradictions.

(1) The oldestcontradiction f the kind in questions the Epi'menides.

Epimenideshe Cretansaid hat all Cretanswere iars,and all other statements

madeby Cretanswere certainly ies. Wasthisa lie? Thesimplestorm of this

contradictions afforded y the manwhosays" I amlying;" if he is lying, he

is speakinghe truth, andvice versa.

(2) Let w be the classof all thoseclasses hicharenot membersof them-

selves. Then,whateverclass may be, " x is a w "is equivalent to "x is not

anx." Hence,givingto x thevaluew, " w is a w" is equivalent o "w is not

a w."

(3) Let T be the relationwhich subsists etween wo relations1 and S

wheneverB doesnothavethe relationR to S. Then,whatever elationsR and

S may be, " R has he relationT to S" is equivalento " R doesnot have the

*See below.

f Two propositionsare called equivalentwhenboth are true or both are false.



RUSSELL: Mathematicalogicas basedon the Theoryof Types. 223

relationR to S." Hence, giving the value T to bothR and S, " T has he rela-

tion Tto T" is equivalent o " T doesnothave the relationTto T."(4) The nurnberof syllablesn the English names of finite integers ends

to increase sthe integersgrow arger, and mustgradually ncreasendefinitely,sinceonly a finite numberof namescan be madewith a given finite number of

syllables. Hence he namesof somentegersmustconsist f at least nineteen

syllables, ndamonghese here mustbe a least. Hence" the least ntegernot

nameablen fewer than nineteensyllables"must denote a definite integer; in

fact, it denotes 11,777. But "the least integernot nameablen fewer than

nineteensyllables is itself a nameconsisting f eighteensyllables;hence he

least ntegernotnameablen fewer than nineteen syllablescan be named in

eighteensyllables,which s a contradiction.*(5) Amongtransfiniteordinalssome canbedefined,whileotherscannot;

forthe total numberof possibledefinitions s N, while the number of trans-

finite ordinals exceedsN. Hence there must be indefinableordinals,and

among hese heremustbea least. But this s definedas "the least ndefinable

ordinal,"which s a contradiction.t

(6) Richard'sparadox is akin to that of the least ndefinableordinal. It

isasfollows: Consider ll decimals hat canbe definedby meansof a finite

numberof words;let E bethe classof such decimals. Then E has N terms;

hencets members an be orderedas the 1st, 2nd, 3rd, .. .. Let N beanumber

defined sfollows:If thenth figure n the nth decimal s p, let the nth,figurein N be p + 1 (or 0, if p = 9). ThenN is different rom all the members f E,

since,whatever initevaluen mayhave,the nthfigure n N is different rom the

nth figure n the nth of the decimals omposing , andthereforeN is different

from the nth decimal. Neverthelesswe have definedN in a finite numberof

words,andthereforeN ought o bea memberof E. ThusN bothis andis nota memberof E.

(7) Burali-Forti'scontradictionmnay e statedas ollows: It canbeshown

*This contradiction wassuggestedo me by Mr. G. G. Berry of the BodleianLibrary.

f Cf. K6nig, " Ueber die Grundlagender Mengenlehre und das Kontinuumproblem," Xath. Annalen,Vol.

LXI (1905); A. C. Dixon, "On ' well-ordered aggregates," Proc.London Xath. Soc.,Series2, Vol. IV, Part I

(1906); and E. W. Hobson,"On the Arithmetic Continuum," ibid. The solution offered in the last of these

papers doesnot seemto me adequate.

I Cf. Poincar6, " Les mathematiqueset la logique," Revuede Y6taphysique t deMforale,Mai, 1906, especially

sectionsVII and IX; alsoPeano,RevistadeMathenzatica,ol. VIII, No. 5 (1906), p. 149 ff.

?" Una questione sui numeri transfiniti," Rendicontidelcircolomatematico i Palermo,Vol. XI (1897).



224 RUSSELL:MathematicalLogic as basedon the Theory of Types.

that everywell-ordered erieshas an ordinalnumber, hat the seriesof ordinals

up to and ncludingany givenordinalexceedshe given ordinalby one,and (on

certainvery.

natural assumptions)hat the seriesof all ordinals in order of

magnitude) s well-ordered. It follows that the seriesof all ordinalshasan

ordinalnumber,Q say. But in that casehe seriesof all ordinals ncludingQ2

hasthe ordinalnumberQ2 1, whichmustbe greater han l. HenceQl is not

the ordinal numberof all ordinals.

In all the above contradictionswhich are merely selections rom an

indefinitenumber) here s a common haracteristic, hichwe may describe s

self-referencer reflexiveness.The remarkof Epimenidesmust includeitself

in its ownscope If all classes, rovided hey are not membersof themselves,

are members f w, this mustalsoapplyto w; and similarly for the analogousrelational contradiction.In the casesof namesand definitions,he paradoxes

result romconsidering on-nameabilityndindefinabilityas elementsn names

and definitions. In thecaseof Burali-Forti'sparadox, heserieswhoseordinal

numbercauseshe difficultys theseries f all ordinal numbers. In each con-

tradictionsomethingssaidaboutall cases f some ind, andfromwhat is said

a new case eemso begenerated,whichboth s andis not of the samekind as

the cases f whichall were concernedn whatwassaid. Let usgo throughhe

contradictionsneby oneandseehowthisoccurs.

(1) Whena mansays" I am lying," we may interpret his statementas:

"There is a propositionwhich I am affirmingandwhich s false." All state-

ments that " there is" so-and-so aybe regardedas denying hat the opposite

is always rue; thus"I am lying" becomies: It is not true of all propositions

that either amnotaffirminghemor they are true;" in other words, It is

not true for all propositions that if I affirmp, p is true." The paradox

results rom regarding his statementas affirminga proposition,which must

therefore come within the scopeof the statement. This, however,makes it

evidenthat thenotionof " all propositions"s illegitimate; for otherwise,heremustbepropositionssuchas theabove)whichare aboutall propositions,nd

yet cannot, withoutcontradiction,eincluded monghe propositionshey are

about. Whateverwesupposeo bethetotalityof propositions,tatementsbout

thistotality generatenewpropositions hich,onpainof contradiction,must ie

outsidethe totality. It is uselesso enlargethe totality, for that equallyenlargeshescopeof statementsbout he totality. Hence there must be no

totality of propositions,nd" all propositions"mustbea meaninglesshrase.



RUSSELL: Mathematical Logic as basedon the Theory of Types. 225

(2) In thiscase, hieclassw is definedby reference o " all classes,"and then

turns out to be one amnongflasses. If we seekhelp by deciding that no class s

a memberof itself, then w becomes he classof all classes, nd we have to decide

that this is not a member of itself, i. e., is not a class. This is only possible fthere is no suchthing as the class of all classes n the sense required by the

paradox. That there is no suchclass esults from the fact that, if we suppose

there is, the suppositionmnmediately ives rise (as in the above contradiction)

to new classesying outside he supposed otal of all classes.

(3) This case is exactly analogousto (2), and shows that we can not

legitimately speakof " all relations."

(4) "The least integer not nameable in fewer tharn nineteen syllables"

involves the totality of names,for it is "the least integer such that all names

either do not apply to it or have more than nineteen syllables." Here we

assume, n obtainingthe contradiction,that a phrase containing" all names" is

itself a name, though t appearsfrom the contradiction that it can not be oneof

the names which were supposed o be all the names there are. Hence "all

names" is an illegitimate notion.

(5) This case,similarly, shows that "all definiitions" is an illegitimate

notion.

(6) This is solved, like (5), by remarking that " all definitions" is an

illegitimnate notion. Thus the number E is not defined in a finite number ofwords, being in fact not definedat all.*

(7) Burali-Forti's contradictionshows hat "all ordinals" is an illegitimate

notion; for if not, all ordinalsin order of magnitudeform a well-orderedseries,

which must have an ordinal number greater than all ordinals.

Thus all our contradictionshave in common the assunmptionf a totality

such that, if it were legitimate, it would at oncebe enlarged by new members

defined n terms of itself.

This leads us to the rule: "Whatever involves all of a collection mnustnot

be one of the collection;" or, conversely: " If, provided a certain collectionhada total, it would have membersonly definable in terms of that total, then thesaid collection hasno total." t

* Cf. " Les paradoxesdela logique," by the present author, RevuedeMetaphysiquet de Morale,Sept., 1906,p. 645.

t WhenI say that a collection has no total, I meanthat statements about all its members are nonsense.Furthermore, it will be found that the use of this principle requiresthe distinction of all and any considerednSection I.

30




The above principle is, however, purely negative in its scope. It suffices o

show that many theoriesare wrong, but it doesnot showhow the errors are to

be rectified. We can not say: "When I speak of all propositions, mean all

except those n which 'all propositions'are mentioned;" for in this explanation

we havementioned he propositionsn which all propositions re mentioned,which we can not do significantly. It is impossible o avoid mentioninga thing

by mentioningthat we won't mention it. One might as well, in talking to a

man with a long nose, say: " When I speak of noses, except such as are inor-

dinately long," which would not be a very successfuleffort to avoid a painful

topic. Thus it is necessary, f we are not to sin against the above negative

principle, to constructour logic without mentioning such things as "all propo-

sitions" or " all properties," and without even having to say that we areexcludingsuchthings The exclusionmust result naturally and inevitably from

our positivedoctrines,which must make it plain that "all propositions" and

"all properties" are meaninglessphrases.

The first difficulty that confrontsus is as to the fundamental principles of

logic known under the quaint name of "laws of thought." "All propositions

are either true or false," for example, has becomemieaningless. If it were

significant, it would be a proposition, and would come under its own scope.

Nevertheless,somesubstitutenmust efound,

or allgeneral

accounts fdeductionbecome mpossible.

Another more special difficulty is illustrated by the particular case of

mathematical induction. We want to be able to say: " If n is a finite integer,

n has all properties possessedy 0 and by the successorsf all numbers possess-

ing them." But here "all properties" must be replaced by soine other phrase

not open to the samnebjections. It miglht be thought that " all properties pos-

sessedby 0 and by the successors f all numbers possessinghem" might be

legitimate even if "all properties" were not. But in fact this is not so. We

shall find that phrasesof the form "all propertieswhich etc." involve all prop

erties of which the "etc." can be significantlyeither affirmedor denied,and not

only those which in fact have whatever characteristic s in question; for, in the

absence of a catalogue of properties having this characteristic, a statement

about all those that have the characteristicmust be hypothetical, and of the

form: " It is always true that, if a property has the said characteristic, hen

etc." Thus mathematicalinduction s prima facie incapableof beingsignificantly



RuSSELL: MathematicalLogic as basedon the Theory of Types. 227

enunciated,f " all properties isa phrasedestituteofmeaning. Thisdifficulty,

as we shallsee ater, canbe avoided; for the presentwemustconsiderhe laws

of logic,since heseare far more undamental.

II.

All and Any.

Givena statementcontaining variablex, say" x = x," we mayaffirm hat

thisholds n all instances, r we may affirm any one of the instanceswithout

deciding s to which nstancewe are affirming. The distinction s roughlythe

sameas that betweenthe generaland particularenunciationn Euclid. The

generalenunciation ells us somethingabout say)all triangles,whilethe par-

ticular enunciationakes one triangle, and asserts he samething of this onetriangle. But the triangle takenis any triangle,not someone special riangle;

and thus although,hroughouthe proof,only one riangleis dealt with,yet the

proofretains ts generality. If wesay: "Let ABC be a triangle, henthe sides

AB, AC are togethergreater han the sideBC," we are sayingsomethingabout

one riangle,not aboutall triangles; but theonetriangleconcerneds absolutely

ambiguous,nd ourstatement onsequentlys alsoabsolutelyambiguous.We

do not affirm any one definite proposition,utan undetermined neof all the

propositionsesulting rom supposing BC to be this or that triangle. This

notionof ambiguousssertions very important,and it is vital not to confoundan ambiguous ssertionwith thedefiniteassertioinhat the same hingholdsn

all cases.

The distinctionbetween 1) asserting nyvalueof a propositionalunction,

and(2) assertinghat the functions always rue, is present throughoutmathe-

matics,asit is in Euclid'sdistinctionof generalandparticularenunciationls.n

any chain of mathematicaleasoning,he objectswhosepropertiesare being

investigatedare the arguments o any value of somepropositionalunction.

Take as an illustration hefollowingdefinition:"We call f(x) continuousorx = a if, for everypositive umbera, different

from 0, there exists a positivenumberE,differentfrom 0, suchthat, for all

valuesof z whicharenumericallyess thanE,the difference(a + ^)-f(a) is

numericallyess hana."

Here the function is any functionor which the abovestatementhas a

meaning; the statements about , and varies asf varies. But the statemeint

is not about or Eor S, because ll possible aluesof theseare concerned, ot



228 RUSSELL: Mathematical Logic as basedon the Theory of Types.

one undetermined value. (In regard to e, the statement" there exists a positive

number Esuch that etc." is the denial that the denial of "etc." is true of all

positive numbers.) For this reason,when any value of a propositional function

is asserted, he argument (e.g., f in the above) is called a real variable; whereas,

when a functionis said to be always true, or to be not always true, the argument

is called an apparent variable.* Thus in the above definition, f is a real

variable, and o, S,z are apparent variables.

When we assertany value of a propositionalfunction, we shall say simply

that we assert the propositional unction. Thus if we enunciate the law of

identity in the form " x = x," we are asserting he function " x = x" i. e., we

are asserting any value of this function. Similarly we may be said to deny a

propositional unction when we deny any instance of it. We can only trulyassert a propositional function if, whatever value we choose, hat value is true;

similarly we can only truly deny it if, whatever value we choose, hat value is

false. Hence in the general case, n which somevalues are true and some alse,

we can neither assert nor deny a propositional function.t

If px is a propositional function, we will denote by " (x) . mx" the propo-

sition "qx is always trule." Similarly " (x, y) . (p x, y)" will mean " q (x, y) is

always true," and so on. Then the distinction between the assertion of all

valuesand the assertionof any is the distinction between (1) asserting (x) . qpxand (2) assertingCpxwhere x is undetermined. The latter differs from the

former in that it can not be treated as one determinate proposition.

The distinctionbetween asserting qx and asserting (x) . cpxwas, I believe,

first emphasizedby Frege4I His reason for introducing the distinction explicitly

was the samewhichhad caused t to bepresentin the practiceof mathematicians;

namely, that deduction can only be effectedwith real variables,not with apparent

variables. In the case of Euclid's proofs, this is evident: we need (say) some

one triangle ABC to reasonabout, though it doesnot matter what triangle it is.

The triangle ABC is a real variable; and althoughit is any triangle, it remainsthe same triangle throughout the argument. But in the general enunciationi,

* These two terms are due to Peano, who uses hem approximately in the abovesense. Cf., e.g., Formuz(aire

Mathdmatique, ol. IV, p. 5 (Turin, 1903).

t Mr. MacColl speaksof "spropositions" as divided into the three classes of certain, variable, anu im-

possible. We may accept this division as applying to propositional functions. A function which can be

asserted s certain, one whichcan be denied s impossible, and all others are (in Mr. MacColl's sense) variable.

t See his Grundgesetzeer Arithmetik,Vol. I (Jena, 1893), ?17, p. 31.




the triangle s anapparentvariable. If we adhere o theapparentvariable,we

cannot perform any deductions,nd this is why in all proofs, eal variables

have to be used. Suppose,o take the simplestcase, hat we know "'px is

always rue," i. e."(x) . px,"andweknow"px always mpliesAx," i. e. "(x) . (Pximplies4x ." How shallwe infer "+4x s always true," i.e. "(x) . Ax?" We

knowit is always rue that if px s true, andif cpxmpliesAx, then Ax is true.

But wehavenopremiseso theeffect hat 'pxs true and cpxmplies 4x; what

wehave is: qpxs always rue,andcpx lwaysmplies x. In orderto makeour

inference,we mustgofrom "qx isalways rue" to (px,and from "?pxalways

impliesAx" to "+x impliesAx," where the x, while rermainingny possible

argument,s to bethe samien both. Then,from " +px"and" px impliesAx,"

we infer "+x;" thusAx is true for any possibleargument, nd therefore salways rue. Thus n orderto infer "(x) . Ax from "((x) . cpx" and "(x) . +pximplies?x ," we haveto passromthe apparent o the realvariable,and then

backagain o the apparent ariable. Thisprocesssrequiredn all mathematical

reasoningwhichproceedsromnheassertion f all valuesof oneor morepropo-

sitional functions o the assertionof all valtuesof soine other propositional

function, s,e.g., from" all isoscelesriangleshaveequalanglesat the base to

" all triangleshavirngqualanglesat the baseareisosceles." In particular,his

processs requiredn provingBarbara and the other moodsof the syllogism.

In a word,all deductionperatesith realvariablesor with constants).It inight be supposedhat we could dispensewith apparentvariables

altogether,contenting urselves ith anyas a substituteor all. This,however,

is notthe case. Take,for example,hedefinitionof a continuousuinctiotnuoted

above: in this definitiona, , and A must be apparentvariables. Apparent

variables are constantly equiredfor definitions. Take, e.g., the following:

"An integer scalledaprimewhen t hasno ntegral actors xcept1 and tself."

Thisdefinitionunavoidablynvolves n apparentvariable n theform: " If n is

an integerother than1 or thegiven nteger,n is nota factorofthegiven nteger,for all possiblealtuesf n"

The distinction etweenall and any is, therefore,necessaryo deductive

reasoning,and occurshroughoutmathematics; hough,so far as I know,its

importanceemainedunnoticed ntil Fregepointedt out.

For our purposest hasa differentutility, which svery great. In the caseof suchvariablesaspropositionsr properties, anyvalue" is egitimate,hough"all values"is not. Thuswe may say: "p is true or false,wherep is any



230 RUSSELL: MathematicalLogic as basedon the Theory of Types.

proposition," though we can not say " all propositionsare true or false." The

reason s that, in the former, we merely affirm an undetermined one of the

propositionsof the form " p is true or false," whereas n the latter we affirm (if

anything) a new proposition,different from all the propositionsof the form

"C_ps true or false." Thus we may admit " any value" of a variable in cases

where " all values" would lead to reflexive fallacies; for the admissionof " any

value" doesnot in the same way create new values. Hence the fundamental

laws of logic can be stated concerningany proposition, though we can not

significantly say that they hold of all propositions. These laws have, so to

speak,a particular enunciation but no general enunciation. There is no one

propositionwhich is the law of contradiction(say); there are only the various

instances of the law. Of aniy propositionp. we can say: "p and not-p cannot both be true;" but there is no such proposition as: " Every propositionp

is suchthat p and not-p can not both be true."

A similar explanation appliesto properties. We canspeakof any property

of x, but not of all properties, because new properties would be thereby

generated. Thus we can say: " If n is a finite integer, and if 0 has the prop-

erty qp,and m + 1 has the property (p provided m has it, it follows that n has

the property p." Here we need not specify qp;q stands for "any property."

But we can not say: "A finite integer is defined asonewhich haseveryproperty

(ppossessedy 0 and by the successorsf possessors."For here it is essential

to considereveryproperty,* not any property; anid n usingsuch a definition we

assume hat it embodies a property distinctive of finite integers, wlhichis just

the kind of assumption from which, as we saw, the reflexive contradictions

spring.

In the above nstance, t is necessaryo avoid the suggestionis f ordinary

language,which is not suitable for expressingthe distinction required. The

point may be illustrated further asfollows: If induction s to be used or defining

finite integers,induction must state a definite property of finite integers, not anambiguous property. But if p is a real variable, the statement "n has the

property p provided this property is possessedy 0 and by the successors f

possessors assignso n a property which varies as pvaries,andsucha property

can not be usedto define the classof finite integers. We wish to say: "'n is a

finite integer' means: 'Whatever property qpmay be, n has the property cppro-

* This is indistinguishablefrom 4 all properties."



RuSSELL: Mathematicalogicas basedon the Theoryof Types. 231

videdcps possessedy 0 andby the successorsf possessors.'"But herecphas

become napparent ariable. To keepit a real variable,weshould aveto say:

"Whatever property pmay be, 'n is a finite nteger' means:n has heproperty

qp rovidedqps possessedy 0 andby the successorsf possessors.'" But here

the meaniingf 'n is a finite integer' variesascp aries,and hussuch definition

is impossible. This case illustrates ani important point, namely the following:

"The scope* of a real variable can never be less thanl the whole propositional

function in the assertion of which the said variable occurs." That is, if our

propositional unction is (say) "cpx implies p," the assertionof this function will

mean " any value of 'cpx mplies p' is true," not "'ainy valuieof qpxs true' im-

plies p." In the latter, we have really " all valuesof cpxare true," and the x is

an apparent variable.III.

The Meaning and Range of GeneralizedPropositions.

In this section we have to consider first the meaning of propositions n

which the word all occurs, and then the kind of collections wlhichadnmitof

propositionsabout all their inembers.

It is convenientto give the nan-meener lizedpropositions ot only to suchas

contain all, but also to such as containso?neuindefined). The proposition "cpx

is sometimes true" is equivalent to the denial of "not-qx is always true;""some A is B" is equivalent to the denial of "all A is not B;" i. e., of "no A

is B." Whether it is possibleo find interpretations which distinguish "qPx is

sometimes true" from the denial of "not-?px is always true," it is unneces-

sary to inquire; for our purposeswe may define"?bx is sometimiesrue" as the

denial of " not-?px s always true." In aniy case, he two kindsof propositions

require the samekind of interpretation, and are subjectto the same limitations.

In each there is an apparent variable; and it is the presence of an apparent

variable which constitutes what I mean by a generalizedproposition. (Note

that there can not be a real variable in any proposition;for what contains a realvariable is a propositional unction, not a proposition.)

The first questionwe have to ask in this section s: How arewe to interpret

the word all in suchpropositionsas "all men are mortal ?" At first sight, it

might be thought that there could be nodifficulty, that " all men" is a perfectly

* The scope of a real variable is the whole function of which " any value" is in question. Thus in

implies p " the scopeof x is not *px,but " ox implies p."



232 RUSSELL: Mathemrtnical ogicas basedon the Tlheoryof Types.

clear idea, and that we say of all men that they are mortal. But to this view

there are rnany objections.

(1) If this view were right, it would seemthat " all mienare mortal" could

not be true if there were no men. Yet, as Mr. Bradley hasurged,* " Trespasserswill be prosecuted may be perfectly true even if no onetrespasses;andhence,

as he further argues,we are driven to interpret suchpropositions s hypotheticals,

meaning "if anyone trespasses, e will be prosecuted;" i. e., "if x trespasses,

x will be prosecuted,"where the range of valueswhich x may have, whatever it

is' is certainly not confined o those who really trespass. Similarly "all men

are mortal " will mean " if x is a man, x is mortal, where x may have any value

within a certain range." What this rangeis, remains to be determined; but in

any case t is wider than "mnen,"for the above hypotheticalis certainly often

true when x is not a man.

(2) "All men" is a denoting phrase;and it would appear,for reasonswhich

I have set forth elsewhere,t that denoting phrasesnever have any rneaning in

isolation, but only enter as constituentsinto the verbal expressionof proposi-

tions which contain no constituent correspondingto the denoting phrases n

question. That is to say, a denoting phraseis definedby meansof the propo-

sitions in whoseverbal expression t occurs. Hence it is impossiblethat these

propositionsshould acquire their mieaning through the denoting phrases; we

must find an independent interpretation of the propositionscontaining suchphrases,and mustnot use these phrasesin explaining what such propositions

mean. Hence we can not regard "all men are mnortal"as a statementabout

"all men."

(3) Even if there were suchan objectas "all men," it is plain that it is

not this object to which we attribute mortality when we say "all men are

mortal." If we were attributing mortality to this object,we shouldhave to say

"all men s mortal." Thus the supposition hat there is such an object as "all

mnen will not help usto interpret " all men are mortal."

(4) It seemsobviotushat, if we meet somethingwhich may be a manormnay

be an angel in disguise, t comnes ithin the scopeof " all men are mortal " to

assert " if this is a man, it is mortal." Thus again, as in the case of the tres-

passers,t seemsplain that we are really saying " if anything is a man, it is

mortal," and that the questioniwhether this or that is a man doesnot fall within

the scopeof our assertion,as it would do if the all really referred to "all men."

* Logic,Part I, Chapter II. ti" On Denoting,". Mind, October, 1905.



RUSSELL: Mathematical ogicas based n the Theoryof Types. 233

(5) We thusarrive at theview that what ismeantby "all menaremortal"

may be more explicitly statedn some uch ormas" it is always rue that if x

is a man, x is mortal." Here wehave to inquireas to the scopeof the word

always.(6) It is obvioushat alwaysncludes ome asesn whichx is not a man,

as we sawin the caseof the disguised ngel. If x were limited to the case

when x is a man,we couldinfer that x is a mortal,since f x isa man,x is a

mortal. Hence,with the samemeaning f always,we shouldind" it is always

truethat x is mortal." But it is plain that, without altering the meaningof

always, his newpropositions false, hough he otherwas rue.

(7) One might hopethat "always" would mean "for all valuesof x."

But " all valuesof x," if legitimate,would nclude as parts" all propositionsand "all functions,"and such llegitimatetotalities. Hence the values of x

must be soinehowestrictedwithinsomeegitimate otality. Thisseemso lead

us to the traditionaldoctrineof a " universeof discourse"withinwhichx must

besupposedo lie.

(8) Yet it is quiteessentialhat we shouldhave somemeaningof always

which doesnot haveto beexpressedn a restrictivehypothesis s to x. For

suppose always means 'whenever belongso theclass." Then " all men

are mortal" becomeswheneverx belongso the class , if x is a man, x is

mortal;". e., " it is always rue that if x belongsotheclass , then,if x is aman,x is mortal." But whatis ournewalways o mean?There seems omore

reasonor restrictingx, in this new proposition,o the class , than there was

beforeor restrictingt to the classman. Thusweshallbe edon o a newwider

universe,andsoon ad infinitum,unlesswe can discoverome aturalrestriction

upon hepossiblealuesof (i. e.,some estrictiongivenwith) thefunction"if x

is a man,x ismortal," andnotneedingo beimposedromwithout.

(9) It seems bvioushat, sinceall men are mortal,there can not be any

false propositionwhich is a valueof the function" if x is a man,x is mortal."

For if thisis a propositiont all, thehypothesisx is a man" mustbe a propo-

sition,andsomust he conclusionx is mortal." But if the hypothesiss false,

the hypotheticalstrue; and f thehypothesiss true, the hypothetical s true.

Hencethere can be no falsepropositionsf the form "if x is a man, x ismortal."

(10) It follows hat, if anyvaluesof x are to beexcluded, heycanonly bevaluesor whichthere snopropositionf the form" if x is a man,x ismortal;"

31




i. e., for which this phrase s meaningless. Since,as we sawin (7), there must be

excludedvalues of x, it follows that the function "if x is a man, x is mortal"

must have a certain range of significance,*which falls short of all imaginable

valuesof x, thoughit exceeds he valueswhich are men. The restriction on x

is therefore a restriction to the range of significanceof the function " if x is a

man, x is mortal."

(11) We thus reachthe conclusion hat "all men are mortal" means" if x

is a man, x is mortal, always," where always means "for all values of the

function 'if x is a man, x is mortal."' This is an internal limitation upon x,

given by the nature of the function; and it is a limitation which doesnot require

explicit statement,sinceit is impossible or a function to be true moregenerally

than for all its values. Moreover, if the range of significanceof the functionis i, the function " if x is an i, then if x is a man, x is mortal " has the same

range of significance,since it can not be significantunless ts constituent"if x

is a man, x is mortal " is significant. But here the range of significances again

implicit, as it wasin 'if x is a man, x is mortal;' thus we cannot make ranges

of significanceexplicit, since the attempt to do so only gives rise to a new

proposition n which the samerange of significances implicit.

Thus generally: " (x) . px" is to mean " px always." This may be inter-

preted, thoughwith lessexactitude, as "qx is always true," or, more explicitly:

"All propositionsof the form cpxare true," or "All values of the function cpxare true."t Thus the fundamental all is "all values of a propositionalunction,"

and every other all is derivative froin this. And every propositional function

has a certain rangeof significance,within which lie the argunmentsor which the

function has values. Within this range of arguments,the function is true or

false; outside this range, it is nonsense.

The aboveargumentationmay be summedup asfollows:

The difficulty which besets attempts to restrict the variable is, that

restrictionsnaturally expressthemselvesas hypotheses that the variable is ofsuchor sucha kind, and that, when so expressed, he resulting hypothetical is

free from the intended restriction. For example, let us attempt to restrict the

*A function is said to be significant or the argumentx if it hasa value for this argument. Thus we may

sayshortly 4 oxis significant," meaning " the function l has a value for the argument x." The range ofsignificanceof a function consists of all the argumentsfor which the function is trule, together with all theargumentsfor which it is false.

tA linguistically convenientexpression or this idea is: "px is true for all possible alues of x," a possiblevalue being understood o be onefor whichoxis significant.



RUSSELL. Mathematicalogicas basedon the Theoryof Types. 235

variableto men,and assertthat, subject o this restriction,";x is mortal" isalwaystrue. Then what is always rue is that if x is a man,x is mortal; andthishypotheticals trueevenwhenx is not a man. Thus a variable canneverberestrictedwithin a certain ange f the propositional unction n which thevariableoccursemains ignificantwhenthevariable s outside hat range. Butif the functionceaseso besignificantwhenthevariablegoesoutside certainrange, hen the variable s ipso actoconfinedo that range,without he needofanyexplicitstatement o that effect. Thisprinciple sto be bornein mind inthedevelopment f logicaltypes, o whichwe shallshortlyproceed.

We cannowbegin o seehow it comeshat " all so-and-so's"s sometimesa legitimatephraseandsometimes ot. Supposewe say" all termswhichhave

the property(phave the property 4." That means,accordingo the aboveinterpretation,"(pxalways mplies4x." Provided he range of significancef

Cpxs the same as that of 4x, this statements significant; hus, given anydefinite unctionpx,therearepropositionsbout" all the termssatisfying Px."But it sometimesappensas we shallseemore ully lateron)thatwhatappearsverballyasone functions really many analogousunctions ithdifferent angesof significance.Thisapplies,or example,o "p is true," which,we shall ind,isnotreallyone functionof p, but is ditferent unctionsaccording o the kindof propositionhat p is. In sucha case,he phraseexpressinghe ambiguousfunctionmay, owing o theambiguity,besignificanthroughout set of valuesof the argumentexceeding he rangeof significancef anyonefunction. Insucha case,all isnot legitimate. Thus f wetry to say " all true propositionshave theproperty p," i. e.," ' p is true' always mpliespp," he possibleargu-ments o 'p is true' necessarilyxceedhepossible rgumentso (p, and there-forethe attemptedgeneralstatement s impossible.For this reason,genuinegeneralstatementsboutall true propositionsannotbemade. It mayhappen,however, hat the supposedunction pis reallyambiguousike 'p is true;' and

if it happenso havean ambiguitypreciselyof the samneind as that of 'p istrue,' we maybe ablealways o giveaninterpretationo theproposition 'p istrue' implies(pp." This will occur,e.g., if (ppis " not-p is false." Thus weget an appearance,n suchcases, f a generalpropositiononcerningll propo-sitions;but thisappearances dueto a systematicmbiguityaboutsuchwordsas true and false. (This systematic mbiguityresultsfrom the hierarchyofpropositions hichwill beexplained ater on). We may, n all suchcases,makeour statement boutany propositioni,ince hemeaninigf the ambiguousords



236 RUSSELL: MathematicalLogtc as basedon the Theory of 7ypes.

will adapt tselfto any proposition. But if we turn our proposition nto anapparentvariable,and saysomething boutall, we mustsupposehe ambiguouswords ixed to thisor that possiblemeaning,hough t may be quite irrelevantwhichof their possiblemeaningshey are to have. This is how it happens oththat all has imitationswhichexclude" all propositions,"nd that there never-theless eemo be true statementsbout" all propositions."Both these points

will become lainerwhenthe theory of typeshasbeenexplained.It hasoftenbeensuggested*hat what is required n order that it may be

legitimate o speakof all of a collections that the collectionshouldbe finite.Thus "all men areinortal" will be legitimatebecausemen form a finite class.But that is not really the reasonwhy we can speak of "all men." What is

essential, sappearsromtheabovediscussion,s not finitude,but what may becalled ogical omogeneity. hisproperty is to belong o any collectionwhosetermsareall containedwithin the range of significancef someone function.It would alwaysbe obviousat a glanicewhether a collectionpossessedhispropertyor not, if it werenot for the concealed mbiguity in common ogicaltermssuchas trueand alse, whichgivesan appearancef beingasingleunctionto what is really a conglomerationf many functionswith different angesof

significance.

The conclusionsf thissectionare asfollows:Every proposition ontaining

all assertshat somepropositionalunction salways rue; and this meansthatall valuesof the said unctionare true, not that the functions true for all argu-ments,since hereare argumentsor which any given function s meaningless,i. e.,has no value. Hence we can speakof all of a collectionwhen andonly

when the collection orms part or the wholeof the range of significanceofsomepropositional unction,the range of significance eing definedas the

collection f thoseargumentsor which the function n questions significant,i. e., hasa value.

IV.The Hierarchyof Types.

A type s definedas the rangeof significancef a propositionalunction,i. e., as the collectionof argumentsor which the said functionhas values.Wheneveranapparentvariableoccursn a proposition,herajrge f values f theapparentvariable s a type, thetypebeing ixedby the functionof which" all

*E. g., by M. Poincar6, RevuedeAUtaphysique t deMorale, Mal, 1906.



RUSSELL: Mathematical ogicas basedon the Theoryof Types. 237

values"areconcerned.The divisionof objectsntotypes snecessitated y the

reflexive fallacies which otherwisearise. These fallacies, s we saw, are to

be avoidedby what may be called the " vicious-circleprinciple;" i. e., "no

totalitycancontainmembers efinedn termsof itself." This principle, n our

technical anguage, ecomes:" Whatevercontainsan apparent variable must

not be a possible alue of that variable." Thuswhatevercontains n apparent

variablemust be of a different ype from the possible aluesof that variable;

we will say that it is of a higher ype. Thius he apparentvariables ontained

in an expressionre what determinests type. This is the guiding principle n

what follows.

Propositions hichcontainapparentvariablesare generated rom such as

do notcontainiheseapparentvariablesby processesf which one is always heprocess f generalization,. e., the substitutionf a variable or one of the terms

of a proposition, nd the assertionof the resultingfunction for all possible

valuesof the variable. Hencea propositions called a generalized roposition

whenit contains n apparentvariable. A proposition ontainingno apparent

variable we will call an elementary roposition. It is plainthat a proposition

containingan apparentvariable presupposesthers from which it can be

obtained by generalization;hence all generalizedpropositions resuppose

elementarypropositions. n an elementary ropositionwe can distinguishone

or more ermsromoneor moreconcepts;he terms rewhatevercanberegarded

asthe subjectfthe proposition, hiletheconceptsre the predicatesr relations

asserted f theseterms.* The terms of elementary propositionswe will call

individuals; these orm thefirst or lowest ype.It isunnecessary,n practice,o know what objectsbelongto the lowest

type, or even whether he lowest ypeof variableoccurringn a given context

is that of individualsor someother. For in practiceonly the relative ypesof

variables re relevant; thus he lowest ype occurringn a given contextmay

becalled hat of individuals) o ar as that contexts concerned. It follows hattheaboveaccount f individualss not essentialo the truth of what follows;

all that is essentials the way in which other typesare generatedrom indi-

viduals,however hetype of individualsmaybeconstituted.

By applying the processof generalizationo individualsoccurring n

elementarypropositions, e obtain nlewpropositions.The legitimacyof this

* SeePrinciplesof MathematieB,48.



238 RUSSELL: Mathematical Logic as based on the Thleory of Types.

process equires only that no individuals should be propositions. That this is

so,is to be securedby the meaningwe give to the word individual. We may

definean individual as somethingdestitute of complexity; it is then obviously

not a proposition,since propositionsare essentiallycomplex. Hencein applying

the processof generalization o individuals we run no risk of incurring reflexive

fallacies.

Elementary propositions ogether with such as contain only individuals as

apparentvariables we will call first-orderpropositions. These form the second

logical type.

We have thus a new totality, that of first-orderpropositions. We can thus

form new propositions in which first-order propositions occur as apparent

variables. These we will call second-orderropositions; these form the thirdlogical type. Thus, e. g., if Epimenides asserts"all first-order propositions

affirmed by me are false," he assertsa second-orderproposition;he may assert

this truly, without asserting truly any first-order proposition,and thus no con-

tradiction arises.

The aboveprocesscan be continued ndefinitely. The n + Ith logicaltype

will consistof propositionsof order n, which will be suchas containipropositions

of order n - 1, but of no higher order, as apparent variables. The types so

obtainedare mutually exclusive, and thus no reflexive fallacies are possibleso

long as we reinemberthat an apparent variable must always be confined within

someone type.

In practice,a hierarchy of functions s more convenientthan one of propo-

sitions. Functions of various orders may be obtained from propositions of

various ordersby the methodof substitution. If p is a proposition,and a a,con-

stituent of p, let "p/a;x" denote the propositionwhich results from substi-

tuting x for a wherever a occurs n p. Then p/a, which we will call a matrix,

may take the placeof a function; its value for the argument x is p/a; x, and its

value for the argumenta isp. Similarly, if "p/(a, b); (x, y) " denotesthe resultof first substitutingx for a and then substituting y for b, we may usethe double

matrix p/(a, b) to represent a double function. In this way we can avoid

apparent variables other than individuals and propositionsof various orders.

The orderof a matrix will be definedas being the order of the proposition in

which the substitutionis effected,which propositionwe will call the prototype.

The order of a matrix doesnot determine its type: in the first placebecause t

doesnot determine the number of argumentsfor which othersare to be substi-



RUSSELL: MathematicalLogic as basedon the Theory of Types. 239

tuted (i. e., whether the matrix is of the form p/a or p/ta, b) or p/(a, b, c)

etc.); in the secondplace because, f the prototype is of more than the first

order, the arguments may be either propositionsor individuals. But it is plain

that the tvpe of a matrix is definable always by means of the hierarchy ofpropositions.

Although it is possibleo replace functions by matrices, and although this

procedure introduces a certain simplicity into the explanation of types, it is

technically inconvenient. Technically, it is convenient o replacethe prototype

p by ma, and to replace p/a; x by (px; thus where, if miatriceswere being em-

ployed, p and a would appear as apparent variables, we now have cpas our

apparent variable. In order that (pmay be legitimate as an apparent variable,

it is necessary hat its values shouldbe confined o propositions f someonetype.

Hence we proceed as follows.

A functionwhose argument is an individual and whose value is always a

first-order propositionwill be called a first-order function. A functioninvolving

a first-order function or propositionas apparent variable will be called a second-

order function, and soon. A function of one variable which is of the order

next abovethat of its argument will be called a predicativefunction; the same

name will be given to a function of severalvariablesif there is one among these

variables in respectof which the functionbecomespredicative when values are

assigned o all the other variables. Then the type of a function is determinedby the type of its values and the number and type of its arguments.

The hierarchy of functionsmay be further explained as follows. A first-

order function of an individual x will be denoted by (p!x (the letters i, X, 0,

f, g, F, G will also be used for functions). No first-order function contains a

function as apparentvariable; hencesuch unctionsform a well-definedtotality,

and the (pin (p! x canbe turned into an apparent variable. Any proposition in

which (pappearsasapparentvariable, and there isnoapparentvariable of higher

type than (p, is a second-orderproposition. If suicha propositioncontains an

individual x, it is not a predicative function of x; but if it containsa first-order

function (p, it is a predicativefunction of cp,and will be written f! (4O!%). Then

is a second-orderredicative unction; the possiblevalues of f agrainform a

well-defined totality, and we can turn f into an apparent variable. We can

thus define third-orderpredicative unctions,which will be such as have third-

order propositionsfor their values and second-orderpredicative functions for

their arguments. And in this way we canl proceed ndefinitely. A precisely

similar developmentappliesto functions of several variables.



240 RUSSELL: MathematicalLogic as basedon the Theoryof Types.

We will adopt the following conventions. Variables of the lowest type

occurring n any context will be denoted by small Latin letters (excluding f

and g, which are reservedfor functions); a predicativefunction of an argument

x (where x may be of any type) will be denotedby p!x (where 4',Xy, f, g, F

or G may replace(p); similarly a predicativefunction of two arguments x and y

will be denotedby (p (x, y); a general functionof x will be denotedby (px,and

a general functionof x and y by cp(x,y). In (px, cpcan not be made into an

apparent variable, since its type is indeterminate; but in (p!x, where (p is a

predicative unction whoseargument is of somegiven type, (p can be made into

an apparent variable.

It is important to observe hat sincethere are various typesof propositions

and functions,and sincegeneralizationcan only beappliedwithin somnene type,all phrasescontaining the words "all propositions" or "all functions" are

prima facie meaningless, houghin certain casesthey are capable of an unob-

jectionable interpretation. The contradictionsarise fronmhe useof suchphrases

in caseswhere no innocentmeaningcan be found.

If we now revert to the contradictions,we seeat once that some of them

are solvedby the theory of types. Wherever " all propositions are mentioned,

we must substitute "all propositionsof order n," where it is indifferent what

value we give to n, but it is essentialthat n shouldhavesomevalue. Thuswhenl

a man says "I am lying," we must interpret him as meaning: "There is a

propositionof order n, which I affirm,and which is false." This is a proposition

of order n + 1; hence the man is not affirming any propositionof order n;

hencehis statement is false,and yet its falsehood does not imply, as that of

" I am lying " appearedto do, that he is making a true statement. This solves

the liar.

Consider next "the least integer not nameable in fewer than nineteen

syllables." It is to be observed, n the first place, that nameablemust mean

((nameable by means of such-and-suchssignednames," and that the number ofassignednamesnlust be finite. For if it is not finite, there is no reason why

there shouldbe any integer not nameable n fewer than nineteen syllables,and

the paradox collapses. We may next supposehat " nameablein termsof names

of the classN" means"being the only term satisfying somefunction composed

wholly of namesof the classN." The solutionof this paradox lies, I think, in

the simple observationthat "nameable in termnsof names of the classN" is

never itself nameable in terms of names of that class. If we enlarge N by



RUSSELL: MiathematicalLogic as basedon the Theory of Types. 241

addingthe name " nameable in terms of namesof the classN," our fundamental

apparatusof names s enlarged; calling the new apparatus N', "nameable in

terms of names of the classN"' remnainsot nameablein terms of namesof the

class N'. If we try to enlarge N till it embracesall names,"nameable" be-comes by what was said above) "being the only term satisfying some function

composedwholly of names." But here there is a function as apparentvariable;hence we are confined o predicative functionsof someone type (for non-predi-

cative functions can not be apparent variables). Hence we have only to observe

that nameability in terms of such unctions s non-predicative n order to escape

the paradox.

The caseof " the least indefinable ordinal" is closelyanalogous o the case

we have just discussed. Here, asbefore,

"definable must be relative to somegiven apparatus of fundamental ideas; and there is reason to suppose hat

" definable in terms of ideasof the classN" is not definable in terms of ideas

of the classN. It will be true that there is somedefinite segment of the series

of ordinalsconsistingwholly of definable ordinals, and having the least inde-

finable ordinal asits limit. This least indefinableordinal will be definable by a

slight enlargement of our fundamental apparatus; but there will then be a new

ordinal which will be the least that is indefinable with the new apparatus. If

we enlargeour apparatusso as to include all possibleideas,there is no longer

any reasonto believe that there is any indefinableordinal. The apparent forceof the paradox lies largely, I think, in the suppositionthat if all the ordinals

of a certainiclass are definable, the classmust be definable,in which case its

successors of coursealsodefinable; but there is no reason for accepting this

supposition.

The other contradictions,hat of Burali-Forti in particular, require somne

further developmrrentsor their solution.

V.

The Axiom of Reducibility.

A propositionalfunctionof x may, as we lhaveseen,be of any order; hence

any statement about " all properties of x " is mneaningless. A " property of x"

is the samething as a "propositional function which holds of x.") But it is

absolutelynecessary, f mathematics s to be possible, hat we shouldhiavesome

methodof mnaking tatementswhich will usually be equivalent to what we havein mind wheniwe (inaccurately) speakof " all propertiesof x." This necessity

32




appears n many cases,but especially n connectionwith mathematicalinduction.

We can say, by the use of any instead of all, "Any property possessed y 0,

and by the successors f all numbers possessingt, is possessed y all finite

numbers." But we can not go on to: "A finite number is one which possesses

all properties possessed y 0 and by the successors f all numbers possessing

them." If we confinethis statement to all first-orderpropertiesof numbers,we

can not infer that it holds of second-orderproperties. For example, we shall

be unableto prove that if m, n are finite numbers, hen m + n is a finite number.

For, with the above definition, " m is a finite number " isa second-order roperty

of in; hence the fact that n + 0 is a finite number, and that, if m + n is a

finite number, so is m + n + 1, doesnot allow us to concludeby inductionthat

m + n is a finite number. It is olvious that such a stateof things rendersmuchof elementary mathematics mpossible.

The other definition of finitude, by the non-similarity of whole and part,

faresno better. For this definition is: "A class s said to be finite wh)en every

one-one relation whose domain is the class and whose converse domain is

contained in the class has the whole classfor its conversedomain." Here a

variable relation appears, i. e., a variable function of two variables; we

have to take all valuesof this function, which requiresthat it shouldbe of some

assignedorder; but any assignedorder will not enable us to deduce many of

the propositionsof elementary mathematics.

Hence we must find, if possible,some method of reducing the order of a

propositional function without affecting the truth or falsehood of its values.

This seems o be what commoni-senseffectsby the admissionof classes. Given

any propositional unction px,of whatever order, this is assumed o be equivalent,

for all valuesof x, to a statementof the form "x belongs o the classa." Now

this statement is of the first order, since t makesno allusion to "all fiunctions

of such-and-such type." Indeed its only practical advantageover the original

statement px is that it is of the first order. There is no advantagein assumingthat there really are such things as classes,and the contradiction about the

classeswhich are not members of themselvesshowsthat, if there are classes,

they must be somethingradically different from individuals. I believe the chief

purposewhich classes erve,and the chief reasonwhich makesthem linguistically

convenient,s that they provide a miethodof reducingthe orderof a propositional

function. I shall, therefore, not assumeanything of what may seem to be

involved in the common-senseadmissionof classes,except this: that every



RUSSELL: Mathematical ogicas basedon the T'heory f Types. 243

propositional unction is equivalent, or all its values, to somepredicative

function.

Thisassumption ith regard o functionss to be made whatevermay bethe type of their arguments. Let px be a function,of any order,of an argu-ment x, whichmay itself be either an individual or a function of any order.If cp s of the order next abovex, we write the function n the form p x; in

sucha casewewill call (pa predicativeunction. Thusa predicativeunctionof

an individual is a first-order unctioni;and for higher types of arguments,predicativeunctionsake the place hat first-order unctionsake in respectofindividuals. We assume, hen, that every function s equivalent, or all itsvalues, o somepredicative unctionof the same argument. This assumption

seems o be the essence f the usualassumption f classes; at any rate, itretainsas muchof classes s we have any use or, and little enougho avoid he

contradictionshich a lessgrudging dmission f classess apt to entail. We

will call thisassumptionhe axiomof classes,r the axiomof reducibility.

We shallassumeimilarlythat every unctionoftwovariablessequivalent,for all its values,o a predicative unction f thosevariables,whereapredicativefunctionof two variables s one such hat there is oneof the variablesn respect

of which thefunctionbecomes redicative in our previoussense)whena valueis assignedo the othervariable. Thisassumptions what seems o be meant

by saying that any statementabout two variablesdefinesa relation betweenthem. We will call this assumptionhe axiomof relations r the axiomofreducibility.

In dealingwith relationsbetweenmorethantwo terms,similarassumptionswouldbeneededor three, four, ... variables. But these assumptionsre notindispensableor our purpose, nd are thereforenot made n this paper.

By the help of theaxiom of reducibility,statements bout"all first-orderfunctions f x" or "all predicative unctionsof a" yield most of the results

whichotherwisewould equire" all functions." Theessential oint s that suchresultsare obtainedn all caseswhereonly thetruth or falsehood f valuesof

the functionsconcerned re relevant,as is invariablythe casen mathematics.

Thus mathematicalnduction, or example,neednow only be stated for allpredicativefunctionsof numbers; it then follows rom the axiom of classesthat it holdsof anyfunctionof whateverorder. It might be thought that theparadoxesor the sakeof which we invented he hierarchyof types wouldnowreappear. But this is notthe case,because,n suchparadoxes,ithersomething



244 RUSSELL: Mathemnaticalogic as basedon the Theory of Types.

beyond the truth or falsehoodof values of ftunctionss relevant, or expressions

occurwhich are unmeaniingeven after the introduction of the axiom of reduci-

bility. For example, such a statement as "Epimenides asserts 4x" is nlot

equivalent to "Epimenides assertscp x," even though 4'x and q)!x are equiv-

alent. Thus "I am lying" remains unmeaning if we attempt to include all

propositionsamong those which I may be falsely affirming, and is uinaffectedby

the axiom of classesf we conlfinet to propositionsof order n. The hierarchy

of propositionsand functions,therefore, remainsrelevant in just those cases n

which there is a paradox to be avoided.

VI.

Primitive Ideas and Propositionsof SymbolicLogic.

The primitive ideas required in symbolic ogic appear to be the following

seven:

(1) Any propositional function of a variable x or of several variables

x, y, z, . . . This will be denotedby cpxor ?p(x,y, z, . .)

(2) The negation of a proposition. If p is the proposition, its negation

will be denotedby -vp.

(3) The disjunction or logical sum of two propositions; . e., " this or that."

If p, q are the twyopropositions,heir disjunctionwill be denotedby p Vq.(4) The truth of any value of a propositional unction; i. e., of cpxwhere x

is not specified.

(5) The truth of all valuesof a propositional unction. This is denotedby

(x) . cpxor (x) :px or whatever larger number of dots may be necessaryto

bracket off the proposition.t In (x) . cpx, x is called an apparent variable,

whereaswhen cpx s asserted,where x is not specified,x is called a real variable.

(6) Any predicative function of an argument of any type; this will be

represented by p ! x or cp a or p ! R, accordingto circumstances. A predica-

tive functionof x is one whosevalues are propositionsof the type next above

that of x, if x is an individual or a proposition,or that of values of x if x is a

*In a previous article in this journal, I took implication as indefinable, nstead of disjunction. The

choice betweenthe two is a matter of taste; I now choosedisjunction, becauset enables us to diminish the

number of primitive propositions.

t The use of dotsfollows Peano's usage. It is fully explained by Mr. Whitehead, "On Cardinal Num-

bers," AMERICAN JOURNAL OF MATHEMATICS, Vol. XXIV, and "1On Mathematical Conceptsof the Material

World," Phil. Traits. A., Vol. CCV, p. 472.




function. It may be described s one n which he apparentvariables,f any,

are all of the same ype asx or of lowertype; and a variable is of lower type

thanix if it can significantly ccuras argument o x, or as argument o an argu-

mentto x, etc.

(7) Assertion;. e., theassertionhat somepropositions true, or that any

value of somepropositionalunictions true. This is requiredto distinguisha

propositionctuallyassertedrom onemerely considered, r from one adduced

ashypothesiso someother. It will be indicatedby the sign '1-" prefixedto

whatis asserted, ith enoughdots o bracketoff whatis asserted.*

Beforeproceedingo the primitivepropositions,eneedcertaindefinitions.

In thefollowingdefinitions, s well as in the primitive propositions,he letters

p, q, r areusedo denotepropositions.

p3q. =* -Vq Df.

This definition tateshat "p ) q" (which s read"p impliesq ") is tomnean

"p is falseor q is true." I do notmeanto affirmthat " implies can not have

any othermeaning,but onlythat this meanings the one which t is most con-

venient o give to "implies" in symbolic ogic. In a definition,he sign of

equalityand he letters" Df" are to be regardedasonesymbol,meaningointly

"is definedo mean." The signof equality without the letters "Df" has a

differentmneaning,o be definedshortly.

PaV. ~(~SV~q) Df.

Thisdefineshelogicalproductof two propositions andq, i. e., "p andq

are both true." The abovedefinitionstateshat this is to mean: " It is false

that either p is falseor q isfalse." Hereagain, he definition oesnot give the

only meaningwhichcan begiven to "p and q are both true," but gives the

meaningwhich smostconvenientor ourpurposes.

p_q. =.pq.q)p Df.

That is, "p-q," which s read"p isequivalent oq," means p impliesq

and q impliesp;" whence,of course,t follows that p andq areboth true or

bothfalse.f9x.(px om= J4x).opx D)f

* This sign, as well as the introduction of the idea which it expresses,are due to Frege. See his Begriffs-

schrift (Halle, 1879), p. 1, and Grundgesetzeer Arithmetik, Vol. I (Jena, 1893), p. 9.



246 RUSSELL:Mathematical Logic as basedon the Theory of Types.

This defines"there is at least one value of x for which cpxis true." We

define it as meaning" it is falsethat cpxs always false."

x=y -=:(cp):cp!x.).cp!y Df.This is the definition of identity. It states that x and y are to be called

identical when every predicative function satisfied by x is satisfied by y. It

follows from the axiom of reducibility that if x satisfiesAx, where 4 is any

function, predicativeor non-predicative,then y satisfies4py.

The following definitionsare less mportant, and are introduced solely for

the purposeof abbreviation.

(x, y) . (x, y). =: (x): (y) .p (x, y) Df.

(lZ,y).cp(x,y). -:(Jx): (y). p(x,y) Df.+R; ) * (P(X Y) (Z + )APX, Df.(px x $ Ax: =:(x): px .-.4 Df.

'k : . kLx ~~~~~~Df.

+(Px, ) . 3x v *1;(X,y):= (x, y) : +(p, y) . 3 . i(x, y) Df.,

and so on for any number of variables.

The priinitive propositionsequired are as follows. (In 2, 3, 4, 5, 6, and 10,

p, q, r stand for propositions.)

(1) A proposition mplied by a true premise s true.

(2) :pVp.).p.(3) :g.).pVq.

(4) :pVq.).qVp.(5) I-:pv(q'Vr). ). qvVp )(6) :.q)r. ):pVq.)opVr.(7) I-:(x).px.).4py;

i. e., " if all valuesof px'are true, then mpys true, where cpys any value." *

(8) If py is true, where mpys any value of px5,hen (x). px is true. This

cannot be expressedn our symbols;for if we write "'py. ). (x). cpx,"hat means

"i y implies that all values of cpxare true, where y may have any value of the

appropriate type," which is not in generalthe case. What we mean to assert s:

"If, however y is chosen,py is true, then (x) . px is true;" whereas what is

expressedby " mpy. . (x). px" is: " However y is chosen, f mPys true, then

(x). pxis true," which is quite a differeintstatement,and in general a false one.

e It is convenient o use the notation ox to denote the function itself, as opposed o this or that value of

the function.



RuSSELL: Mathematical ogicas basedon the Theoryof Types. 247

(9) 1: (x) . px . (pa,wherea is any definiteconstant.This principle s really as many different principles stherearepossible

valuesof a. I. e., it statesthat, e.g., whateverholdsof all individualsholds

of Socrates;also hat it holdsof Plato; andsoon. It is the principlethat a

general ule maybeapplied o particularcases;but in orderto give it scope,t

is lnecessaryo mentionheparticularcases, inceotherwise e need heprinciple

itself to assure s that the generalrule that generalrulesmay be appliedto

particular casesmay be applied (say)to the particularcaseof Socrates. It is

thusthat thisprinciplediffers rom(7); ourpresentprinciplemakes statenment

aboutSocrates,or about Plato, or someother definiteconstant,whereas 7)madea statement bouta variable.

The aboveprinciplesnever usedn symbolicogicor in puremathematics,sinceall ourpropositionsregeneral,andevenwhen(asin " one s a number )

weseem o have a strictly particular case,his turns out not to be so whenl

closelyexamined. In fact, the use of the aboveprinciple is the distinguisIling

mark of appliedmathematics. Thus,strictlyspeaking,we miighthave omitted

it fromour list.

(10) F: *(x).pv(P. ):p.V.(x) .(Px;i. e., " if 'p or (px' isalways rue, theneitherp is true, or (px s always rue."

(11) When f(+px)s truewhateverargumentx may be,and F(py) is true

whateverpossibleargumenty may be,then jf((px). F(cpx)t s true whateverpossible rgumentx maybe.

Thisis the axiom of the " identification f variables." It is neededwhentwo separate ropositionalunctions re eachknownto be always true, and we

wish o inferthat their logicalproduct s always true. This inference s only

legitimate if the two functions ake argumentsof the same type, for otherwise

their logicalproducts meaningless. n the aboveaxiom,x and y must be ofthe same ype, becauseothoccurasargumentso (p.

(12) If (px (px $x is true for any possible , then Ax is true for anypossible .

Thisaxiom is required n order to assure sthat the rangeof significanceof Ax, in the casesupposed,s the sameas that of (px (px x . ) . Ax; botharein fact thesame s hat of (px. Weknow, n the case upposed,hat Ax is true

whenever px (px ?xand (px (px) Ax. ) . A&x re bothsignificant, utwe donotknow, withoutanaxiom, hat 4x is true wheneverAx is significant. Hencetheneedof the axiom.




Axioms (11) anld 12) are required, e.g., in proving

(x) .q x: (x) .x )x: ). (x) .x.

By (7) and (Ii),

I-: . (x) .+cx: (x) . cx) ? :) py.py ) Aywhenceby (12),

F :. (x) . px: (x) .Px) : ):y,

whence the result follows by (8) and (10).

(13) *:.(Yf):.(x):qpx.=.f!x.

This is the axiom of reducibility. It states that, given any function qX",

there is a predicativefunction f! x such that f !x is always equivalent to ox.Note that, since a propositionbeginningwith "Qf)" is, by definition,the nega-

tion of one beginningwith "(f)," the aboveaxiomi involves the possibility ofconsidering" all predicativefunctionsof x." If ox is any functionof x, we can

not make propositionsbeginningwith " (P) " or " (Yp)," sincewe can not con-

sider "all functions," but only "any functionl" or " all predicative unctions."

(14) F : . (Yf ): . (x,y): +p(x,) .-.= f! (xyy).

This is the axiom of reducibility for doublefunctions.

In the abovepropositions,our x and y may be of any type whatever. The

only way in which the theory of typesis relevant is that (11) only allows us to

identify real variablesoccurringin different contentswhen they are shown o beof the sametype by both occurringas argumentsto the samefunction, and that,

in (7) and (9), y and a must respectively be of the appropriate type for argu-

ments to pz. Thus, for example, supposewe have a proposition of the form

(op)f ! (op! , x), which is a second-order unction of x. Then by (7),

F:(o) f! (0! z,,x) 3).f !(4!z, x),

where 4! z is any first-order function. But it will not do to treat (cp) f! (p !z,x)

as if it were a first-order function of x, and take this fiunctionas a possible

value of 4 !z in the above. It is such confusionsof typesthat give rise to theparadox of the liar.

Again, considerthe classeswhich are not members of themselves. It is

plain that, since we have identified classeswith functions,* no classcan be

significantlysaidto be or not to be a member of itself; for the members of a

classare arguments o it, and argumentsto a function are always of lower type

* This identification s subjectto a modification o be explained shortly.



RUSSELL: Mathematical Logic as basedon the Theory of Types. 249

than the function. And if we ask: "But how about the class of all classes?

Is not that a class,and so a member of itself?", the answer is twofold. First,

if " the classof all classes means" the classof all classesof whatever type,"

then there is no such notion. Secondly, if " the class of all classes means" the classof all classesof type t," then this is a classof the next type abovet,

and is therefore again not a member of itself.

Thus althoughthe above primitive propositionsapply equally to all types,

they do not enable us to elicit contradictions. Hence in the course of any

deduction t is never necessary o consider he absolute ype of a variable; it is

only necessary o see that the different variablesoccurring n onepropositionare

of the proper relative types. This excludessuch functionsas that from which

our fourth contradictionwas obtained, namely: "The relation R holdsbetweenR and S." For a relation between R and S is necessarily of higher type than

either of them, sothat the proposed unction is meaningless.

VII.

Elementary Theory of Classesand Relations.

Propositionsin which a functioncpoccursmay depend, or their truth-value,

uponthe particular function p, or they mnay epend only upon the extensionof

q, i. e., upon the argunments hich satisfy cp. A function of the latter sort wewill call extensional. Thus, e.g., " I believe that all men are mortal " may not

be equivalent to " I believe that all featherlessbipedsare mortal," even if men

are coextensive with featherless bipeds; for I inay not know that they are

coextensive. But "all men are mortal" must be equivalent to "all featherless

bipeds are mortal" if men are coextensivewith featherlessbipeds. Thus "all

men are mortal" is an extensional functionof the function "x is a man," while

"I believe all men are mortal " is a function which is not extensional; we will

call functions intensionalwhen they are not extensional. The functions of

functions with which mathematics is specially concernedare all extensional.

The mark of an extensionalfunction f of a function qp!z is

mp!X$.=XA+Z),:f+!) .(!).

From any function f of a function cp z we can derive an associated xten-

sionalfunction as follows. Put

z z Df.33



250 RUSSELL:1MathematicalLogic as basedon the Theory of Types.

The functionf Iz 4z) t is in reality a function of 4A,though not the same

function sf(4z), supposinghis latter to besignificant. But it is conveniento

treat f tz(4z) technicallyas though t had an argumenit (4'z),which we call

"the classdefinedby 4." We have

F :+2 _ Y.l4 :%0

fiA

((p) )A (

whence, pplyingo the fictitious bjects (pz)andz(4z) the definition f identity

givenabove,we find

F Sqx. +$'x) * (qpZ)=A(Z).

This, with its converse which can also be proved), is the distinctive

propertyof classes.Hencewearejustified n treatingz(pz)as the classdefinedby c. In the sameway weput

X4YV( t Y) } ( +) : (p! X

Y) * S.Y * x( ) (}y) Df

A few words are necessary ere as to the distinctionbetween+p!(Y,y) andc ! (y, x). We will adopt hefollowingconvention:Whenafunction asopposed

to its values)srepresentedn a fornmnvolvingx andyi,or anyother wo etters

of the alphabet, he value of this function or the argumentsa and b is to be

foundby substitutingfor Z andb for i; i.

e., the argumentmentioned irst isto besubstitutedor the letter which comnesarlier in the alphabet,and the

argumentrimentionedecond orthe later letter. This sufficiently istinguishes

between Y!(w,) and p!(y, x); e.g.:

Thevalueof p! (x, y) for argumentsa, b is (p! a, b)." "' "' ' " 6,a " ! (b, a).

" " p) (y, 2;) " " a, b " qA(b, a).{; { { { ;v b,

~~~~ap!(a,b)We put

xEp !z. =.p! x Df.,whence

I : X.zx'z) *=: (Y'p) : 'p!y. _ . 4'y: !Xx.

Alsoby thereducibility-axiom,e have

(p) : ' !Y -Y --vwhence

I : XEZ Z) *- ASx.



RUSSELL: Mathematicalogtcas based n the Theoryof Types. 251

This holds whatever x may be. Supposenow we want to consider

z(iz) eCf {z !z) . We have, by the above,

F: _*z(,+z)A

A(?) ! =:I(~)) t *-. z(z t : (Ly+): ( - yp 0

whenceF

(,Z) A(XZ) A(,iZ),

A-X* (Z)?0

wherex is written for anyexpressionf theform f/{z(q) z)}.

We putcla=_(2p). a=z( p!z)} Df.

Here clshasa meaningwhichdepends pon he type ofthe apparentvariable(P.Thus, e.g., the proposition cls Ecls," which is a consequencef the abovedefinition, equireshat " cls" shouldhavea differentmeaningn the two places

where t occurs. Thesymbol" ci " canonlybe usedwhere t is unnecessaryo

knowthe type; it has an ambiguitywhich adjusts tself to circumstances.f

we introduceas an indefinable he function "1ndiv!x," meaning"x is an

individual,"we may put

Kl a ( . z !z.Indiv!z)t Df.

ThenKi is anunambiguousymbolmeaning"classesof individuals."We will use small Greek letters (other than e, (, A),X 0) to represent

classesf whatever ype; i. e.,to stand or symbols f the form zp(P!) or z(cz).

The theory of classes roceeds,rom this point on, much as in Peano's

system;^(qz) replaces (Qpz). AlsoI put

oaICf.:xexa.).xsg Df.

X !ac.. (9X;)E Do Df.

V xz( x) Df.

A = (x x) Df.

whereA, as with Peano,s the null-class. ThesymbolsY, A, V, like clsand E,

areambiguous,ndonly acquirea definitemeaningwhen he type concerneds

otherwisendicated.

We treat relations n exactlythe sameway, putting



252 RUSSELL:MathematicalLogic a-sbasedon the Theory of Types.

(the orderbeingdetermined y the alphabeticalorderof x andy and thetypo-

graphicalorderof a and b); whence

[ : . aIx y(x, y)}b._: (Yp) :+s(x,y) . y. (p! (x, y) :q !(a, b),

whence,by the reducibility-axiom,

f:a X '̂,(x, y) t b .-* (a, b)

We useLatin capital letters as abbreviationsfor suchsymbolsas 'xy^'(x,y),

and we findf :.R=8.=_:xRyO -,s,xsy,

whereR=S.=:f!R.bf.f!S Df.

We putRel=R{(9p) .R=-xy$qp!(x,y) Df.,

and wefindthat everythingproved orclasses as ts analogueordualrelations.

FollowingPeano,weput

a f 3 x(xea . xe) Df.,

defininghe product,or common art, of two classes;

a WdX3 x(xsa.v. xet3) Df.,

defininghe sumof two classes; nd- a c(xEa)} Df.,

defininghenegationof a class. Similarlyfor relationswe put

R S ~ (xRy.xSy) Df.

RQ S= iy (xRy V. xSy) Df.

*R-$y (XRY)? Df.

VIII.

Descriptive Functions.

The functionshitherto consideredhave been propositional functions,with

the exceptionof a few particularfunctionssuchR t, S. But the ordinary

functionsof mathematics,suchas X2, sin x, log x, are not propositional.

Functions f this kind alwaysmean"the term havingsuch-and-such relation

to x." For this reasonthey may be called descriptiveunctions,because hey

describecertain ermbymeans f its relatiorto theirargument.Thus"sin7z/2"




describeshe number1; yetpropositionsnwhichsin7t/2 occursre not hesameasthey wouldbe if 1 weresubstituted.This appears, .g., fromthe proposition"sin 7x/2 = 1," which conveys valuable information, whereas "1 = 1" is

trivial. Descriptiveunctions avenomeaningby themselves,ut only ascon-stituentsofpropositions;ndthisappliesgenerally o phrases f the form"thetermhavingsuch-and-suchproperty." Hence in dealingwith suchphrases,we mustdefineanypropositionn which heyoccur,notthephraseshemselves.*We are thus ed to thefollowingdefinition,n which " (7x) px)" is to be read"thetermx whichsatisfiesx."

4.{ (ix) (ox).= :(Yb) :OX.=X. X= b 4b Df.

This definitionstates

hat"the term which satisfiesq satisfies4" is tomean: "There isa term b such hat cpxs true when and only whenx is b,

and 4b is true." Thus all propositions bout"the so-and-so"will befalse fthere areno so-and-so'sr severalso-and-so's.

Thegeneraldefinitionof a descriptiveunctions

R'y- (ix) (xRy) Df.;

that is, "Rcy" is to mean"the termwhichhas herelation R to y." If thereare several ermsornonehaving he relationR to y, all propositionsbout R'y

will befalse. We putE! (?X) ((m) =:(X ): X x - b Df.

Here "E! (1x) qx)" mayberead"there is sucha term asthe x whichsatisfies

px," or " the x whichsatisfiespx xists." We have

-: .E! Ry .-: (=IXb): Ry.-.x = b.

The inverted comman R'y maybe readof. Thus f R is the relationof fatherto son," R'y" is " thefather of y." If R is the relation of sonto father, allpropositionsboutR'y will befalseunless hasonesonandnomore.

From the aboveit appears hat descriptiveunctionsare obtained romrelations. The relationsnowto be definedarechiefly mportant on accountofthedescriptiveunctionso whichtheygiverise.

Cnv = QP 1xQy , y t Df.

* Seethe above-mentioned rticle ' On Denoting," where the reasons or this view are given at length.



254 RUSSELL: Mathematical Logic as basedon the Theory of Types.

Here Cnv sshortfor " converse." t is the relationof a relationtoitsconverse;e.g., of greatero less, f parentage o sonship, f precedingo following,etc.

We haveCnv P = (7Q) JxQy.* Xv *Y-9-Wt

For a shorternotation,often moreconvenient,weput

P = CnveP Df.

We want next a notation or the classof termswhichhave therelationR to y.

For thispurpose,we put

R=ayJa=( (xRy)} Df.,whence

.R'y = ; (xRy)

Similarlyweput

RP={3XJ='(xRy) Df.,whence

.R' (xRy)

We want next the domainof R (i. e.,the classof terms which have therelation R to something),he converseomainof R (i. e.,theclassof terms towhichsomethingas he relationR), and the ieldof R, which sthe sumof thedomainandthe converse omain. For thispurposewe definethe relationsofthedomnain,onverseomain,andfield,to R. The definitions re:

D = aRl?a= ((y) . xRy)} Df.

a ,R { y((9x) .xRy) Df.

C R { xRy -V*yRx) Df.

Notethat the third of thesedefinitionss only significantwhenR is what wemaycalla homogeneouselation; . e.,one n which, f xRy holds,x andy areof

the same ype. For otherwise, oweverwemaychoose andy, either xRy oryRx will be meaningless.This observations importantin connectionwithBurali-Forti'scontradiction.

We have, n virtue of the abovedefinitions,

[ . D)'R-= {(gy). xRy-aR = A(x). XRyt,

.C'R = x (9y): xRy V yRx




the last of thesebeing significantonly whenR is homogeneous. DR" is

read "the domainof R;" "aR" is read "the converse omain of R" and

" C'R" is read" the field of R." The letter C is chosen s the initial of the

word " campus."

We wantnext a notationfor the relation,to a classa containedn the

domainofR, oftheclassof terms o whichsomememberof a hasthe relation

R, andalso or the relation,to a class3 containedn the converseomainofR,

of the classof termswhichhave he relationR to somememberof . For the

secondf theseweput

Re= ag a:=- ((Jy) . ycg . xRy) t Df.So that

[ .Re I (gy) Y XRy.

Thus f R is the relationof fatherto son,and 3 is the classof Etonians,Re.C3will be the class fathersof Etonians;" if R is the relation "less than," and

t is the classof proper ractionsof the form 1 - 2' for integral valuesof

n Rc3 will be the class of fractions less than some fraction of the form

1 -2; i. e., R$c3will be the classof properfractions. The other relation

mentioned boves (R1)e.

We put, asan alternativenotationoftenmoreconvenient,

Rccg= R,c/3 Df.

The relativeproductof two relationsR, ?S s the relation which holds

between and z wheneverhere s a termy such hat xRy and yRz both hold.

The relativeproducts denotedbyR S. Thus

RJ S = 2[Z{(y) . Ry . yRz D .We put also

R2 =PRjP Df.

The productand sumof a classof classes re often required. They aredefined s follows:

s c 'x (ELa*scw Dx= ax)eax . xesa Df.pxx{cex)xe4 Df.

Similarlyfor relationswe put

s '= 2xy {(R).PRe.xPRy Df.

PCX= z P )R7.x yR Df.



256 RUSSELL: Mathematical Logic as based on the Theory of Types.

We needa notation or the classwhoseonly members x. Peanouses X,

hencewe shalluse x. Peanoshowedwhat Frege also had emphasized)hat

this class annot be identifiedwith x. With the usualview of classes,he need

for sucha distinctionemainsa miystery;but with the view set forth above,t

becomesbvious.

We putX {oXyQ (y ) } 2Df.,

whence= y(y =x),

andF E!t"a) -t'a= (7x) (xma);

i.e., if a is a classwhiclh asonlyonemember, henl a is thatonemember.*

For the class f classesontainedn a givenclass,we put

Cll'a=,($ a) Df.

We cannowproceedo theconsiderationf cardinaland ordinalnumbers,

andof how theyareaffected y the doctrineof types.

Ix.

Cardinal Numbers.Thecardinalnumberof a class is definedas the classof all classesimilar

to a, two classes eingsimilarwhen here is a one-one elationbetweenthem.

The classof one-oneelationss denoted y I I anddefined sfollows:

1-> =R I Ry. X'Ry. XRy. 3Xl, x,y. x_= xt.y = yf Df.

Similarityis denoted y &im; its definitions

Sim=a- (gR).RRE11.DI'R=a.(l'R=3 Df.

ThenSim 'a is, by definition,he cardinalnumberof a; thiswe will denoteby

NcCa; hence we put

Nc Sim Df.,whence

. Ncca= Sim 'ca.

*Thus 'ca is what Peano calls ia.



RUSSELL: Mathematicalogicas basedon the Theoryof TEypes. 257

The classof cardinalswewill denoteby NC; thus

NC= Nc"cls Df.

0 is definedastheclasswhoseonlymembers the null-class,A, sothatO= tA Df.

The definitionof 1 is1 =a 4(c): X'c.- ct Df.

It iseasy o provethat 0 and 1 arecardinals ccordingo the definition.It is to be observed,owever,hat 0 and 1 and all the other cardinals,

accordingo the abovedefinitions, re ambiguousymbols,ike cls,andhave asmanymeaningss thereare types. To beginwith 0: the

meaningf 0

dependsuponthat of A, and the meaningof A is differentaccordingo the type ofwhich it is the null-class. Thus hereare asmanyO'sas thereare types; andthe sameapplieso all the other cardinals. Nevertheless,f two classes , d

are ofdifferent ypes,we canspeakof themashavingthe samecardinal,or ofoneashavinga greatercardinal han the other,because one-oneelationmayholdbetweenhe members f a andthemembers f fi, evenwhena and areof different ypes. For example,et fi bet7a; i. e., the classwhosemembers rethe classesonsistingf singlemembers fa. Thentea is ofhigher ypethana,

but similar to a, beingcorrelatedwith a by the one-oneelationt.The hierarchy of types has important results in regard to addition.

Suppose e havea classof a termsand a classof /3 terms,wherea and /3 arecardinals; t maybequite impossibleo add them together o get a classofa and d3 terms,since, f the classesrenot of the same ype, their logicalsumis meaningless.Where only a finitenumberof classesre concerned, e canobviate he practicalconsequencesf this, owing o thefactthat we canalwaysapplyoperationso a classwhich raise ts type to any requiredextent withoutaltering ts cardinalnumber. For example,givenany class , the class la hasthe samecardinalnumber,but is of the next typeabovea. Hence,given anyfinitenumberof classesf different ypes,wecanraiseall of them o the typewhichis what we may call the lowest commonnultiple of all the typesin

question;and it canibe shown hat this can be donein sucha way that theresultingclasseshallhaveno commonmembers.We maythen ormthelogicalsum of all the classeso obtained,and its cardinalnumber will be the arith-meticalsum of the cardinalnumbersof the originalclasses.But where we

34




have an infinite seriesof classes f ascendingypes,this method can not be

applied. For this reason,we can not now provethat there must be infinite

classes. For supposehere wereonly n individuals ltogethern the universe,where n is finite. There would hen be2n classes f individuals, nd 22"classesof classes f individuals, nd so on. Thusthe cardinalnumberof terms n eachtype wouldbe finite; and houghhese umberswouldgrowbeyond nyassignedfinitenumber, here wouldbenoway of addingthem so as to get an infinitenumber. Hencewe needan axiom,so t wouldseem,o the effect hat nofiniteclassof individuals ontains ll individuals;but if any one chooseso assumethat the total numberof individualsn the universes (say)10,367, hereseemsno a prioriway of refutinghisopinion.

Fromthe abovemodeof reasoning,t is plain that the doctrineof typesavoidsall difficultiessto thegreatestcardinal. There is a greatestcardinalneach ype, namely hecardinalnumberof the whole of the type; but this isalways surpassedy the cardinalnumber of the next type,since,f a is thecardinalnumberof one ype, that of thenext type is 2a,which,as Cantor hasshown,s always greaterthan a. Sincethere is no way of addingdifferenttypes,wecan not speakof "the cardinal numberof all objects, f whatevertype," andthus hereis noabsolutely reatestcardinal.

Ifit

is admittedhat no finite classof individuals

ontainsall individuals,it followshat there are classes f individualshavinganyfinitenumber. Henceall finite cardinals xistas individual-cardinals;.e., as the cardinalnumbers fclasses f individuals. It follows hat thereis a classof Nocardinals, amely,the class f finitecardinals. HenceN existsas the cardinalof a class f classesof classes f individuals. By formingall classes f finite cardinals,we find

that 2toexistsas the cardinalof a classof classes f classes f classesf indi-dividuals;andsowecanproceedndefinitely. The existenceof Nn for everyfinite valueof n can alsobe proved; but this requiresthe considerationf

ordinals.If, in addition o assuminghat no finite classcontainsall individuals,we

assumehemultiplicative xiom(i. e., the axiom that, given a set of mutuallyexclusive lasses,oneof which are null, there s at leastoneclassconsisting f

onememnberromeachclassn theset), hen we canprove hat there s a classofindividuals ontainingNomembers,o hat Nowill existasanindividual-cardinal.

This somewhateduceshe type to which we haveto go n orderto prove he




existence-theoremor any givencardinal,but it doesnotgiveusany existence-theoremwhichcan not be gototherwise ooneror later.

Many elementaryheorems oncerningardinalsequirethe multiplicativeaxiom.* It is to be observedhat this axiomis equivalent o Zermelo's,tand

therefore to the assumptionhat every class can be well-ordered.t Theseequivalentassumptionsre, apparently,all incapable f proof, hough he mul-

tiplicativeaxiom,at least,appears ighlyself-evident. In the absence fproof,it seemsbest not to assume he mnultiplicativexiomn,ut to state it as a

hypothesis n everyoccasionniwhich t is used.

X.

Ordinal Numbers.An ordinalnumber s a classof ordinallysimilar well-ordered eries, . e.,

of relationsgeneratingsuchseries. Ordinalsimilarityor likenesss definedasfollows:

Smor_PQW[(ElS).SE1.(1S ='Q.P- S I=IQSF Df.,

where" Smor is short or "similar ordinally."The class of serial relations,which we will call " Ser," is definedas

follows:

Ser P 5xP . ),,. (x = y) xPy. yPz. xPz:XeCCP.) .Pxe'x ;Pcx -C'Pt Df.

That is, readingP as"precedes," relation is serial if (1) no term pre-cedestself,(2) a predecessorf a predecessors a predecessor,3) if x is anyterm in the field of the relation,then the predecessorsf x togetherwithx

togetherwith thesuccessorsf x constitutehe whole ieldof the relation.

* Cf. Part III of apaper by the presentauthor, " On someDifficulties in the Theoryof Transfinite Numbers

and Order Types," Proc. LondonMath. Soc. Ser. II, Vol. IV, Part I.

t Cf. loc.cit. for a statement of Zermelo's axiom, and for the proof that this axiom implies the multipli-cative axiom. The converseimplication results as follows: Putting Prod 'k for the multiplicative class

of k, consider

ZcP= R { (Ex) . xEp D'R = tp.q(1tR = tx } Df.,

and assume

yeProd CZcCl'a.Ra =x (S) . Sy . Sx}.

Then 1 is a Zermelo-correlation. Hence if Prod CZcclda is not null, at least one Zermelo-correlation

for a exists.

I See Zermelo, "Beweis, dass jede Menge wohlgeordnet werden kann." Miath. Annalen, Vol. LIX,

pp. 514-516.




Well-ordered erialrelations,whichwe will call Ql, are definedas follows:

fn= PIPSer:aC O'P.!a.3ai.!(a-P"a)t Df.;

i. e., P generates well-ordered eries f P is serial,and any class containednthe fieldof P and not null has a first term. (Note that PC"aare the terms

comingaftersomeerm of a).

If wedenoteby NocP the ordinalnumber of a well-ordered elationP,

and by NO the classof ordinal numbers,we shallhave

No 04PjPe2.a=SmorcPt Df.

NO= No"cf.

From thedefinitionof No we have

PF ?iL) .NoCP= Smor cP

- (P e .. - E! No'P.

If we nowexamineourdefinitionswith a view to their connectionith the

theory of types,we see, o begin with, that the definitionsof "Ser" and Ql

involvethefieldsof serial relationis. Now the field s only significantwhenthe

relation is homogeneous;encerelationswhichare not homogeneouso not

generateseries.For example, he relation might bethought o generateseries

of ordinal number c, suchas

andwemightattempt o prove n thisway the existenceof X and No. But x

and Atxareof differentypes,and thereforehere snosuchseriesaccordingo

the definition.

The ordinalnumberof a seriesof individualss, by the abovedefinitionof

No,a classof relationsof individuals. It is thereforeof a differenttype from

any individual,and can not formpart of any seriesn which ndividuals ccur.

Again, suppose ll the finite ordinalsexist as individual-ordinals;. e.,astheordinalsof seriesof individuals. Then the finite ordinalsthemselvesorm a

serieswhoseordinalnumber s ca;thus X existsas an ordinal-ordinal,. e.,as

the ordinalof a seriesof ordinals. But the typeof anordinal-ordinals thatof

classesf relationsof classesf relationsof individuals. Thus heexistence f

cahasbeenprovedn a higher ypethanthat of the finite ordinals. Again,the

cardinalnumberof ordinialnumbersof well-orderederieshat canbemadeout

of finiteordinalss j; henceK, exists n the typeof classesf classesf classes



RUSSELL: Mathematical Logic as based on the Theory of Types. 261

of relations of classes f relations of individuals. Also the ordinal numbers of

well-ordered series comiposedof finite ordinals can be arranged in order of

magnitude, and the result is a well-ordered series whose ordinal niumber isw,.Hence ca,exists as an ordinal-ordinal-ordinal. This processcanbe repeated any

finite number of times, and thus we can establish the existence,in appropriate

types, of MN and c, for any finite value of n.

But the above processof generationno longer leads to any totality of all

ordinals, because, f we take all the ordinals of any given type, there are always

greater ordinals in higher types; and we can not add together a set of ordinals

of which the type rises above any finite limit. Thus all the ordinals in any

type can be arranged by order of magnitude in a well-orderedseries,which has

an ordinal number of higher type than that of the ordinalscomposing he series.In the new type, this new ordinal is not the greatest. In fact, there is no

greatest ordinal in any type, but in every type all ordinals are less than some

ordinals of higher type. It is impossible to complete the series of ordinals,

since t rises to types above every assignablefinite limit; thus although every

segmentof the seriesof ordinalsis well-ordered, we cannot say that the whole

series s well-ordered, because he "whole series" is a fiction. Hence Burali-

Forti's contradiction disappears.

From the last two sections t appears that, if it is allowed that the number

of individuals is not finite, the existence of all Cantor's cardinal and ordinal

numbers can be proved, short of N. and c.. (It is quite possiblethat the

existenceof thesemay also be demonstrable.) The existenceof all finite car-

dinals and ordinals can be proved without assuminghe existenceof anything.

For if the cardinal nunmber f terms in any type is n, that of terms in the next

type is 2n. Thus if there are no individuals, there will be one class namely,

the null-class),two classesof classesnamely, that containingno class and that

containing the null-class), our classesof classesof classes, nd generally 2n-1

classesof the nth order. But we can not add togetherterms of different types,and thus we can not in this way prove the existence of any infinite class.

We can now sumup our whole discussion. After stating someof the para-

doxesof logic, we found that all of them arise from the fact that an expression

referring to all of soinecollectionmay itself appear to denote one of the col-

lection; as, for example, " all propositionsare either true or false" appearsto

be itself a proposition. We decided that, where this appears o occur,we are

dealing with a falsetotality, and that in fact nothing whatever can significantly




be said about all of the supposedollection. In order to give effect to this

decision,we explaineda doctrineof typesof variables,proceeding pon the

principle hat any expression hichrefers o all of somieypemust, f it denotes

anything,denotesomethingf a higher ype thanthat to all of which it refers.

Where all of sometype is referred to, there is an apparentvariablebelongingto

that type. Thus any expressionontainingan apparentvariable is of higher type

than thatvariable. This is the fundamentalprincipleof the doctrineof types.

A changen the mannerin which the typesare constructed,hould t prove

necessary, ould eavethe solutionof contradictionsntouchedo long as this

fundamental rinciples observed. The methodof constructingypesexplained

abovewasshowno enableusto stateall the fundamentaldefinitions f mathe-

matics,and at the samiieime to avoid all known contradictions.And itappearedhat in practice he doctrineof types s never relevant exceptwhere

existence-theoremsre concerned,r where applicationsre to be made o some

particularcase.

Thetheoryof typesraisesa numberofdifficultphilosophicaluestionson-

cerningts interpretation. Suchquestions re,however,essentiallyseparable

from the mathematicaldevelopmentf the theory,and,like all philosophical

questions,ntroduce lements f uncertaintywhichdonot belongo the theory

itself. It seemedbetter, therefore, o state the theory without referenceo

philosophicaluestions,eaving hese o be dealtwith independently.

Documents

Theory of Types