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On DenotingAuthor(s): Bertrand RussellSource: Mind, New Series, Vol. 14, No. 56 (Oct., 1905), pp. 479-493Published by: Oxford University Press on behalf of the Mind Association

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II.-ON DENOTING.

By BERTRAND RUSSELL.

By a " denioting hrase " I mean a phrase such as any oneof the following: a man, some man, any man, every man,all men, the present King of England, the present King ofFrance, the centre of mass of the Solar System at the firstinstant of the twentieth entury, he revolution f the earthround the sun, the revolution of the sun round the earth.Thus a phrase is denoting solely in virtue of its form. Wemay distinguish hree cases: (1) A phrase may be denoting,and yet not denote anything; e.g., "{the present King ofFrance (2) A phrase may denote one definite bject; e.g.,"the present King of England " denotes a certain man. (3)A phrase may denote ambiguously; e.g., " a man " denotesnot many men, but an ambiguous man. The interpretationof such phrases s a matter f considerable ifficultv; ndeed,it is very hard to frame ny theory not susceptible of formalrefutation. All the difficulties ith which I am acquaintedare met, so far as I can discover, by the theory which I amabout to explain.

The subject of denoting s of verv great importance, notonly n logic and matheinatics, but also in theory of know-ledge. For example, we know that the centre of m-ass f theSolar System at a definite nstant s some definite oint, andwe can affirm number of propositions abouit t; but wehave no immediate acquaintance with this point, which isonly known to us by description. The distinction betweenacquaintance and knowledgebout is the distinction betweenthe things we have presentations f, and the things we onlyreach by means of denoting phrases. It often happens thatwe know that a certain phrase denotes unambiguously, l-though we have no acquaintance with what it denotes; thisoccurs in the above case of the centre of mass. In percep-tion we have acquaintance with the objects of perception,and in thought we have acquaintance with objects of a moreabstract logical character; but we do not necessarily haveacquaintance with the objects denoted by phrases composed



480 BERTRAND RUSSELL:

of words with whose meanings we are acquainted. To take

a very mportantnstance: There seems no reason to believe

that we are ever acquainted with other people'sminds, eeingthat these are not directly erceived; hence what we knowabout them is obtained through denoting. All thinking hasto start from cquaintance; but it succeeds n thinking boutmany things with which we have no acquaintance.

The course of my argument will be as follows. I shallbegin by stating the theory intend to advocate; I I shallthen discuss the theories of Frege and Meinong, showingwhy neither of them satisfies me; then I shall give thegrounds n favour of my theory; and finally shall brieflyindicate the philosophical consequences of my theory.

My theory, briefly, s as follows. I take the notion of thevariable s fundamental; I use " C (x)" to mean a proposi-tion2 in which x is a constituent, where x, the variable, isessentially nd wholly undetermined. Then we can considerthe two notions "C (x) is always true" and " C (x) is some-times true .3 Then everything nd nothing nd something(which are the most primitive of denoting phrases) are tobe interpreted s follows:C (everything) means " C (x) is always true";C (nothing) means "'C (x) is false' is always true";C (something) means "It is false that ' C (x) is false' is

always true ".Here the notion " C (x) is always true" is taken as ultimateand indefinable, nd the others are defined by means of it.Everything, othing, nd something,re not assumed to have anymeaning in isolation, but a meaning s assigned to every ro-position in which they occur. This is the principle of thetheory f denoting wish to advocate: that denoting phrasesnever have any meaning in themselves, but that every pro-ppsition n whose verbal expression they occur has a mean-ing. The difficulties oncerning denoting are, I believe, atlthe result o*f wrong analysis of propositions whose verbalexpressions contain denoting phrases. The proper analysis,if am not mistaken, may be further et forth s follows.

I I have discussed his subject n Principles of Mathematics, hapterv., and ? 476. The theory here advocated s very nearly the same asFrege's, and s quite different rom he theory o be advocated n whatfollows.

2 Moreexactly, propositional unction.3The secondof these can be defined y mieans f the first, f we take

it to mean, "It is not true hat C (x)is false' is alwaystrue .4I shall sometimes se, nstead of this complicated hrase, he phrase

"C (x) snot always alse," r " C (x) s sometimes rue," upposed efinedto mean the same as the complicated hrase.



ON DENOTING. 481

Suppose now we wish to interpret he proposition, I meta man ". If this is true,. met some definite man; but thatis not what I affirm. What I affirm s, according o the theoryI advocate:

" ' I met x, and x is human' is not always false ".Generally, defining he class of men as the class of objectshaving the predicate human, we say that:4'C (a man)" means "' C (x) and x is human' is not always

false ".This leaves "a man," by itself, wholly destitute of meaning,but gives a meaning to every proposition n whose verbalexpression "'a man " occurs.

Consider next the proposition " all men are mortal".This proposition is really hypothetical nd states that f any-thing is a man, it is mortal. That is, it states that if x isa man, x is mortal, whatever x may be. Hence, substituting'x is human' for' x is a man,' we find:-"All men are mortal means "' If x is human, x is mortal'

is always true ".This is what is expressed in symbolic logic by saying that" all men are mortal" means "'x is human' implies 'x ismortal' for all values of x". More generally, we say:4CC(all men) means "' If x is human, then C (x)is true' is

always trueSimilarly"C (no men)" means "' If x is human, then C (x) is false'

is always true"."C (some men) " will mean the same as " C (a man)," 2 and''C (a man) " means " It is false that C (x)and x is human'

is always false ."C (every man)" will mean the same as " C (all men)It remains to interpret hrases containing the. These are

by far the most interesting nd difficult f denoting phrases.;Take as an instance " the father f Charles II. was executed .This asserts that there was an x who was the father ofCharles II. and was executed. Now the, when it is strictlyused, nvolvesuniqueness; we do, it is true, peak of " the on,of So-and-so " even when So-and-so has several sons, but it

would be more correct o say " a son of So-and-so". Thusfor our purposes we take the s involving uniqueness. Thuswhen we say " x was the father f Charles II." we not only.assert that x had a certain relation to Charles II., but also

I Ashas been ably rgued n Mr. Bradley'sLogic,book ., chap. ii.2Psychologically " C (a man)" has a suggestion f only nie, nd " C

(some men) has a suggestion f mnore han one; but we may neglect*these uggestions n a preliminary ketch.




that nothing lse had this relation. The relation n ques-tion,without he ssumption f uniqueness, nd without nydenoting hrases, s expressed y " xbegatCharles .". Toget an equivalent f " x was the father f Charles I.," wemNustdd, If y s other hanx, y did niot eget Charles I)"or, what s equivalent, If y begat Charles I., y s identicalwith x". Hence " x is the father f Charles I." becomnes" begatCharles L; and it y begat Charles L, y s identicalwith x' is always true of y"

Thus " the father f Charles Mwas executed becomuesIt is not always alse f x that begatCharles I. and that

x was executed nd that if y begat Charles L, y isidenticalwith x' is always rue ofy".This mnay eema somewhat ncredible nterpretation; utI am not at present iving easons, am merely tating hetheory.

To interpret C (the father f Charles I.)," where Cstands for any statement bout him, we have only o sub-stitute C (x) for x was executed" in the above. Observethat, ccordina o the above nterpretation, hatever tate-ment C may be, " C (the father f Charles I.)" implies:"It is not always falseofx that if y begat Charles I., y is

identicalwith x' is always rueof y,"which swhat sexpressed ncommon anguage y " CharlesII. had one father nd no more". Consequently f this con-(ition fails, very roposition f the formn C (the father fCharles I.)" is false. Thus e.g. every roposition f theformn C (the present Kinig f France) is false. This is agreat advantage n the present heory. I shall show laterthat t is not contrary o the awof contradiction, s lightbe at first upposed.

The above gives a reduction f all propositions n whichlenoting hrases occur to forms n which no such phrasesoccur. Why t is imperative oeffect uch a reduction, hesubsequent iscussion willendeavour o show.

The evidence for the above theory s derived from hedifficulties hich seem unavoidable f we regard denotingphrases s standing or enuine onstituents f the proposi-

tions n whose verbal expressions hey ccur. Of the pos-sible theories which dmit uch constituents he implest sthat of Meinong.1 This theory egards ny grammaticallycorrect denoting phrase as standing for an object. Thus" the present ing of France," the round quare," tc., re

I See L'ntersw.Atuinqe.nur (egenstandatheorie tnd sychologie, eip-zig, 1904,the first hree rticles by Mliuiong, maeseder nd Mally re-spectively).



ON DENOTING. 483

supposed to be genuine objects. It is admitted that suchobjects do not su6bsist, ut nevertheless they are supposed tobe objects. This is in itself a difficult iew; but the chiefobjection s that such objects, admittedly, re apt to infriogethe law of contradiction. It is contended, for example, thatthe existent present King of France exists, and also does notexist; that the round square is round, and also not round;etc. But this s intolerable; and if any theory an be foundto avoid this result, t is surely to be preferred.

The above breach of the law of contradiction s avoided byFrege's theory. He distinguishes, n a denoting phrase, twoelements, which we may call the meaninzg nd the denotation.'Thus " the centre of mass of the Solar System at the begin-ning of the twentieth century is highly omplex n meaning,but its denotation s a certain point, which is simple. TheSolar System, the twentieth entury, tc., are constituents fthe meaning; but the denotation as no constituents t all.2One advantage of this distinction s that it shows why t isoften worth while to assert identity. If we say " Scott isthe author of Waverley," e assert an identity f denotationwith a difference f meaning. I shall, however, not repeatthe grounds in favour of this theory, as I have urged itsclaims elsewhere loc.cit.), and am now concerned to disputethose claims.

One of the first ifficulties hat confront s, when we adoptthe view that denoting phrases express meaning and denotea denotation,3 oncerns the cases in which the denotationappears to be absent. If we say " the King of England isbald," that is, it would seem, not a statement about thecomplex meaning the King of England," but about theactual man denoted by the meaning. But now consider" the King of France is bald ". By parity of form, his alsoouight o be about the denotation of the phrase " the King ofFrance ". But this phrase, though t has a neaning rovided

' Seehis " Ueber Sinn und Bedeutung," eitschrift ilr Phil. untd hil.Kritik, ol. 100.

2Frege distinguishes he two elements of meaning and denotationeverywhere, nd not only n complex denoting hrases. Thus it is themeanings f the constituents f a denoting omplex that enter nto tsmeaning, ot their denotation. In the proposition Mont Blanc is over1,000metres igh," t s, according ohim, hemeaning f "cMont Blanc,"not the actual mountain, hat s a constituent f the nmeaning f the pro-position.

3In this theory, e shall say that the denoting phrase expresses ,meaning; and we shall say both of the phrase nd of the meaning hatthey denote denotation. In the other heory, hich advocate, hereis no meaning, nd only ometimes denotation.




" the King of England " has a meaning, has certainly no de-notation, at least in any obvious sense. Hence one wouldsuppose that "the King of France is bald" ought to benonsense; but it is not nonsense, since it is plainly false.Or again consider such a proposition s the following: " If uis a class which has only one member, hen that one memberis a member of u," or, as we may state it, " If u is a unitclass, the u is a u ". This proposition ught to be alwaystrue, since the conclusion s true whenever the hypothesis strue. But " the u " is a denoting phrase, and it is the de-notation, not the meaning, that is said to be a u. Now if uis not a unit class, "the u " seems to denote nothing; henceour proposition would seem to become nonsense as soo-n smus not a unit class.

Now it is plain that such propositions do not becomenonsense merely because their hypotheses are false. TheKing in " The Tempest " might say, " If Ferdinand is notdrowned, Ferdinand is my only son ". Now " my only son "is a denoting phrase, which, on the face of t, has a denota-tion when, and only when, I have exactly one son. But theabove statement would nevertheless have remained true ifFerdinand had been in fact drowned. Thus we must eitherprovide a denotation in cases in which it is at first sightabsent, or we must abandon the view that the denotation swhat is concerned in propositions which contain denotingphrases. The latter is the course that I advocate. Theformer ourse may be taken, as by Meinong, by admittingobjects which do not subsist, and denying that they obeythe law of contradiction; this, however, s to be avoided ifpossible. Another way of taking the same course (so far asour present alternative s concerned) is adopted by Frege,who provides by definition ome purely conventional denota-tion for the cases in which otherwise there would be none.Thus "the King of France," is to denote the null-class;" the only son of Mr. So-and-so " (who has a fine family often), is to.denote the class of all his sons; and so on. Butthis procedure, hough t may not lead to actual ogical error,is plainly artificial, nd does not give an exact analysis of

the matter. Thus if we allow that denoting phrases, ingeneral, have the two sides of meaning and denotation, hecases where there eems to be no denotation ause difficultiesboth on the assumption that there really s a denotation andon the assumption that there really s none.

A logical theory may be tested by its capacity for dealingwith puzzles, and it is a wholesome plan, in thinking boutlogic, to stock the mind with as many puzzles as possible,



ON DENOTING. 485

since these serve much the same purpose as is served byexperiments n physical cience. I shall therefore tate threepuzzles which a theory as to denoting ought to be able tosolve; and I shall show later that my theory solves them.

(1) If a is identical with b, whatever is true of the one istrue of the other, nd either may be substituted for he otherin any proposition without altering the truth or falsehood ofthat proposition. Now George IV. wished to know whetherScott was the author of Wcaverley; nd in fact Scott wasthe author of Waverley. Hence we may substitute Scott forthe uthor f " TWaverley,"nd thereby prove that George IV..wished to know whether Scott was Scott. Yet an interestin the law of identity can hardly be attributed o the first.gentleman of Europe.

(2) By the law of excluded middle, either "A is B " or"A is not B" must be true. Hence either "the present.King of France is bald " or " the present King of France isnot bald" must be true. Yet if we enumerated the thingsthat are bald, and then the things that are not bald, weshould not find the present King of France in either list.Hegelians, who love a synthesis, will probably conclude thathe wears a wig.

(3) Consider the proposition A differs romB ". If thisis true, there is a difference etween A and B, which factmay be expressed n the form the difference etween A andB subsists ". But if it is false that A differs rom B, thenthere s no difference etween A and B, which fact may beexpressed in the form the difference etween A and B does.not subsist ". But how can a non-entity e the subject ofa proposition "I think, herefore am " is no more evidentthan "I am the subject of a proposition, therefore am,"provided "I am" is taken to assert subsistence or being,1-not existence. Hence, it would appear, it must always beself-contradictory o deny the being of anything; but wehave seen, in connexion with Meinong, that to admit beingalso sometimes leads to contradictions. Thus if A and Bdo not differ, o suppose either that there s, or that there s.not, such an object as "the difference etween A and B"

seems equally impossible.The relation of the meaning to the denotation involves,certain rather curious difficulties, hich seem in themselvessufficient o prove that the theory which leads to such diffi--culties must be wrong.

When we wish to speak about the meaning f a denoting

I use these as synonyms.




phrase, as opposed to its denotation, the natural mode ofdoing so is by inverted ommas. Thus we say:The centre of mass of the Solar System is a point, not a

denoting complex;"The centre of mass of the Solar System" is a denoting

complex, not a point.Or again,The first ine of Gray's Elegy states a proposition." The first ine of Gray's Elegy " does not state a proposi-tion. Thus taking any denoting phrase, say C, we wish toconsider the relation between C and " C," where the differ-ence of the two is of the kind exemplified n the above twoinstances.

We say, to begin with, that when C occurs it is thedenotation hat we are speaking about; but when " C " occurs,it is the meaning. Now the relation of meaning and denota-tion is not merely inguistic through he phrase: there mustbe a logical relation involved, which we express by savingthat the meaning denotes the denotation. But the difficultywhich confronts s is that we cannot succeed in both reserv-ing the connexion of meaning and denotation nd preventingthem from being one and the same; also that the meaningcannot be got at except by means of denoting phrases. Thishappens as follows.

The one phrase C was to have both meaning and denota-tion. But if we speak of " the meaning of C," that gives usthe meaning (if any) of the denotation. "The meaning ofthe first ine of Gray's Elegy " is the same as "The meaningof The curfew olls the knell of parting day,' and is not thesame as " The meaning of the first ine of Gray's Elegy "'.Thus in order to get the meaning we want, we must speaknot of "the meaning of C," but of "the meaning of ' C,' "which is the same as " C " by itself. Similarly " the denota-tion of C " does not mean the denotation we want, but meanssomething which, f it denotes at all, denotes what is denotedby the denotation we want. For example, let " C " be " thedenoting complex occurring n the second of the above in-stances". Then

C =" the first ine of Gray's Elegy," andthe denotation f C= The curfew olls the knell of parting day.But what we meant ta have as the denotation was " the firstline of Gray's Elegy". Thus we have failed to get whatwe wanted.

The difficulty n speaking of the meaning of a denoting,complex may be stated thus; The moment we put ihe coin-plex in a proposition, he proposition s about the denotation;



ON DENOTING. 487

and if we make a proposition n which the subject is " themeaning of C," then the subject is the meaning (if any) ofthe denotation, which was not intended. This leads us tosay that, when we distinguish meaning and denotation, wemust be dealing with the meaning: the meaning has denota-tion and is a complex, and there s not something ther thanthe meaning, which can be called the complex, and be saidto have both meaning and denotation. The right phrase,on the view in question, is that some meanings have de-notations.

But this only makes our difficulty n speaking of meaningsmore evident. For suppose C is our complex; then we areto say that C is the meaning of the complex. Nevertheless,whenever C occurs without inverted commas, what is saidis not true of the meaning, but only of the denotation, aswhen we say: The centre of mass of the Solar System s apoint. Thus to speak of C itself, .e., to make a propositionabout the meaning, ur subject must not be C, but somethingwhich denotes C. Thus " C," which is what we use whenwe want to speak of the meaning, must be not the meaning,but something which denotes the meaning. And C must notbe a constituent f this complex as it is of " the meaning ofC "); for f C occurs n the complex, t will be its denotation,not its meaning, that will occur, and there is no backwardroad from denotations o meanings, because every object canbe denoted by an infinite umber of different enoting hrases.

Thus it would seem that " C " and C are different ntities,such that " C " denotes C; but this cannot be an explanation,because the relation of " C " to C remains wholly mysterious;and where are we to find the denoting complex " C " whichis to denote C ? Moreover, when C occurs in a proposition,it is not only he denotation that occurs (as we shall see inthe next paragraph); yet, on the view in question, C is onlythe denotation, he meaning being wholly relegated to " C '.This is an inextricable tangle, and seems to prove that thewhole distinction f meaning and denotation has been wronglyconceived.

That the meaning is relevant when a denoting phraseoccurs in a proposition is formally proved by the puzzleabout the author of Waverley. The proposition Scott wasthe author of Wtverley" has a property not possessed by" Scott was Scott," namely the property hat George IV.wished to know whether t was true. Thus the two are notidentical propositions; hence the meaning of " the author ofWcaverley"must be relevant as well as the denotation, f weadhere to the point of view to which this distinction elongs.




Yet, as we have just seen, so long as we adhere to this point,

of view, we are compelled to hold that only the denotationcan be relevant. Thus the point of view in question mustbe abandoned.

It remains to show how all the puzzles we have been con-sidering re solved by the theory xplained at the beginningof this article.

According to the view which I advocate, a denoting phraseis essentially part of a sentence, and does not, like mostsingle words, have any significance n its own account. IfI say " Scott was a man," that is a statement of the formn" x was a man," and it has " Scott " for ts subject. Butif I say " the author of Wcaverleyas a man," that is not a,statement f the form x was a man," and does not have " theauthor of Waverley for ts subject. Abbreviating he state-ment made at the beginning f this article, we may put, inplace of "the author of Waverley as a man," the follow-ing: " One and only one entity wrote Waverley, nd thatone was a man". (This is not so strictly what is meant aswhat was said earlier; but t is easier to follow.) And speak-ing generally, suppose we wish to say that the author ofWaverley ad the property , what we wish to say is equiva-lent to " One and only one entity wrote Waverley, nd thatone had the property ".

The explanation of denotation s now, as follows. Everyproposition in which " the author of Waverley occursbeing explained as above, the proposition "Scott was theauthor of Waverley" i.e. " Scott was identical with theauthor of Waverley") becomes " One and only one entitywrote Waverley, nd Scott was identical with that one"; or,reverting to the wholly explicit form: "It is not alwaysfalse of x that x wrote Waverley, hat it is always true of ythat if y wrote Waverley is identical with x, and that Scottis identical with x ". Thus if " C " is a denoting phrase, tmay happen that there s one entity x (there cannot be morethan one) for which the proposition x is identical with C"is true, this proposition being interpreted as above. Wemay then say that the entity x is the denotation of thephrase "C ". Thus Scott is the denotation of " the authorof Waverley". The "-C " in inverted ommas will be merelythe phrase, not anything-that an be called the meaning. Thephrase per se has no meaning, because in any proposition nwhich it occurs the proposition, fully expressed, does notcontain the phrase, which has been broken up.

The puzzle about George IV.'s curiosity s now seen tohave a very simple solution. The proposition " Scott was



ON DENOTING. 489

the author of Waverley," which was written out in its un-abbreviated form n the preceding paragraph, does not con-tain any constituent "the author of Waverley for whichwe could substitute " Scott ". This does not interfere withthe truth f nferences esulting rommaking what is verballythe substitution f " Scott " for " the author of Waverley," olong as " the author of Waaverley" as what I call a primaryoccurrence n the proposition considered. The difference fprimary and secondary occurrences of denoting phrases isas follows:-

When we say: " George IV. wished to know whether so-and-so," or when we say " So-and-so is surprising" or " So-and-so is true," etc., the " so-and-so must be a proposition.Suppose now that " so-and-so contains a denoting phrase.We may either eliminate this denoting phrase from the;subordinate roposition so-and-so," or from he whole pro-position n which " so-and-so is a mere constituent. Differ-ent propositions result according to which we do. I haveheard of a touchy owner of a yacht to whom a guest, on firstseeing it, remarked, I thought your yacht was larger than

it is"; and the owner replied, " No, my yacht is not largerthan it is ". What the guest meant was, " The size that Ithought your yacht was is greater than the size your yachtis "; the meaning attributed o him is, " I thought the sizeof your yacht was greater han the size of your yacht". Toreturn to George IV. and Waverley, hen we say, "GeorgeIV. wished to know whether Scott was the author ofWoaverley," e normally mean " George IV. wished to knowwhether one-and only one man wrote Waverley nd Scott

was that man"; but we may also mean: "One and onlyone -man wrote Waverley, nd George IV. wished to knowwhether Scott was that man ". In the latter, " the authorof Waverley" has a primary occurrence; in the former,secondary. The latter might be expressed by "George IV.wished to know, concerning the man who in fact wroteWaverley, hether he was Scott ". This would be true, forexample, if George IV. had seen Scott at a distance, andhad asked "Is that Scott?" A secondary ccurrence of a

denoting phrase may be defined s one inwhich the phrase

occurs in a proposition which is a mere counstituent f theproposition we are considering, nd the substitution for thedenoting phrase is to be effected nmp, ot in the whole pro-position concerned. The ambiguity s between primary ndsecondary occurrences s hard to avoid in language; but itdoes no harm if we are on our guard against it. In symboliclogic it is of course easily avoided.

33



490 BERTRAND -RUSSELL:

The distinction f primary nd secondary occurrences lso

enables us to deal with the question whether the presentKing of France is bald or not bald, and generally with thelogical status of denoting phrases that denote nothing. If"C" is a denoting phrase, say " the term having the pro-perty F," then" C has the property b" means "one and only one term

has the property , and that one has the property ".6If now the property belongs to no terms, or to several, tfollows that " C has the property b" is false for all valuesof . Thus " the present King of France is bald " is certainlyfalse; and " the present King of France is not bald" is falseif t means"There is an entity which is now King of France and is not

bald,but is true if t means"It is false that there is an entity which is now King of

France and is bald ".That is, "the King of France is not bald " is false if theoccurrence of " the King of France " is primary, nd true ifit is secondary. Thus all propositions n which " the King ofFrance" has a primary occurrence are false; the denials ofsuch propositions re true, but n them " the King of France "has a secondary occurrence. Thus we escape the conclusionthat the King of France has a wig.

We can now see also how to deny that there is such anobject as the difference etween A and B in the case when Aand B do not differ. If A and B do differ, here is one andonly one entity such that " x is the difference etween A andB " is a true proposition; if A and B do not differ, here sno such entity . Thus according to the meaning of denota-tion lately explained, " the difference etween A and B " hasa denotation when A and B differ, ut not otherwise. Thisdifference pplies to true and false propositions enerally. If"'a R b" stands for " a has the relation R to b," then whenaR b s true, here s such an entity s the relation R betweena and b; when a R b is false, there s no such entity. Thusout of any7 roposition we can make a denoting phrase, whichdenotes an entity f the proposition s true, but does not de-note an entity f the proposition s false. E.g., it is true atleast we will suppose so) that the earth revolves round thesun, and false that the sun revolves round the earth; hence"the revolution of the earth round the sun" denotes an

I his is the abbreviated, not the stricter, nterpretation.



ON DENOTING. 491

entity, while " the revolution of the sun round the earth"

does not denote an entity.'The whole realm of non-entities, uch as " the roundsquare," "the even prime other than 2," " Apollo," " Ham-let," etc., can now be satisfactorily ealt with. All these aredenoting phrases which do not denote anything. A pro-position about Apollo means what we get by substitutingwhat the classical dictionary tells us is meant by Apollo,say " the sun-god . All propositions n which Apollo occursare to be interpreted y the above rules for denoting phrases.If " Apollo" has a primary ccurrence, he proposition on-taining he occurrence s false; if the occurrence s secondary,the proposition may be true. So again " the round square isround " means " there is one and only one entity x which isround and square, and that entity s round," which is afalse proposition, not, as Meinong maintains, a true one." The most perfect Being has all perfections; existence isa perfection; therefore he most perfect Being exists" be-comes:

" There is one and only one entity x which s most perfect;that one has all perfections; existence s a perfection; there-fore that one exists". As a proof, his fails for want of aproof of the premiss "there is one and only one entity xwhich is most perfect '.2

Mr. MacColl (MIND,N.S., No. 54, and again No. 55, p. 401)regards individuals as of two sorts, real and unreal; hencehe defines he null-class as the class consisting of all unrealindividuals. This assumes that such phrases as " thepresent King of France," which do not denote a real in-dividual, do, nevertheless, denote an individual, but an un-real one. This is essentially Meinong's theory, which wehave seen reason to reject because it conflicts with the lawof contradiction. With our theory of denoting, we are ableto hold that there are no unreal individuals; so that thenull-class s the class containing no members, not the classcontaining as members all unreal individuals.

It is important to observe the effect f our theory on theinterpretation f definitions hich proceed by means of de-

1 The propositions romh hich such entities re derived re not den-tical either with hese ntities r with hepropositions hat hese ntitieshave being.

2The argument an be made to prove validly hat all members f theclass of most perfect eings exist; it can also be proved formally hatthis class cannot have more han onemember; but, aking hedefinitionof perfection s possessionof all positive predicates, t can be provedalmost equallyformally hat the class does not have even one member.




noting phrases. Most mathematical definitions re of this

sort: for example," n-n means the number which, dded to

n, gives n ". Thus n n is defined as meaning the same asa certain denoting phrase; but we agreed that denotingphrases have no meaning n isolation. Thus what the defini-tion really ought to be is: " Any proposition ontaining rn-nis to mean the proposition which results from substitutingfor rm-n' 'the number which, added to n, gives rn' ". Theresulting proposition s interpreted ccording to the rulesalready given for interpreting ropositions whose verbal ex-pression contains a denoting phrase. In the case where mand n are such that there is one and only one number xwhich, added to n, gives rn, here is a number x which canbe substituted or n - n in any proposition ontaining m nwithout altering the truth or falsehood of the proposition.But in other cases, all propositions n which "in - n" has aprimary ccurrence re false.

The usefulness of dentity s explained by the above theory.No one outside a logic-book ver wishes' to say "x is x," andyet assertions of identity re often made in such forms as" Scott was the author of Waverley or " thou art the man ".The meaning of such propositions cannot be stated withoutthe notion of identity, lthough they are not simply state-ments that Scott is identical with another term, the authorof Waverley, r that thou art identical with another term,the man. The shortest statement of " Scott is the authorof Waverley" seems to be: "Scott, wrote Waverley; and itis always true ofy that if y wrote Waverley, s identical withScott ". It is in this way that identity nters nto " Scott isthe author of Waverley"; and it is owing to such uses thatidentity s worth affirming.

One interesting esult of the above theory of denoting sthis: when there is anything with which we do not haveimmediate acquaintance, but only definition y denotingphrases, then the propositions n which this thing is intro-duced by means of a denoting phrase do not really containthis thing as a constituent, ut contain instead the constitu-ents expressed by the several words of the denoting phrase.

Thus in every proposition that we can apprehend (i.e. notonly n those whose truth or falsehood we can judge of, butin all that we can think bout), all the constituents re reallyentities with which we have immediate acquaintance. Nowsuch things as matter in the sense in which matter occursin physics) and the minds of other people are known to usonly by denoting phrases, i.e., we are not acquaintedwiththem, but we know them as what has such and such proper-



ON DENOTING. 493

ties. Hence, although we can form propositional functionsC (x) which must hold of such and such a material particle,or of So-and-so's mind, yet we are not acquainted with thepropositions which affirm hese things that we know mustbe true, because we cannot apprehend the aetual entitiesconcerned. What we know s " So-and-so has a mind whichhas such and such properties but we do not know " A hassuch and such properties," where A is the mind in question.In such a case, we know the properties of a thing withouthaving acquaintance with the thing tself, nd without, con-sequently, knowing any single proposition f which the thingitself s a constituent.

Of the many other consequences of the view I have beenadvocating, will say nothing. I will only beg the readernot to make up his mind against the view-as he might betempted to do, on account of its apparently xcessive com-plication-until he has attempted to construct a theory ofhis own on the subject of denotation. This attempt, be-lieve, will convince him that, whatever the true theory maybe, it cannot have such a simplicity s one might have ex-pected beforehand.

russel - on denoting

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