The Unity of the Proposition

Linsky, Leonard.

Journal of the History of Philosophy, Volume 30, Number 2, April 1992,

pp. 243-273 (Article)

Published by The Johns Hopkins University Press

DOI: 10.1353/hph.1992.0024

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http://muse.jhu.edu/journals/hph/summary/v030/30.2linsky.html

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 The Unity of the Proposition
LEONARD LINSKY

"Es fr~igt sich hier, wie k o m m t

d e r S a t z v e r b a n d zustande."
Wittgenstein

1~

A PERSISTENT THEME in Russell's Principles of Mathematics ~ concerns the unity of

the proposition. Propositions are c o m b i n a t i o n s ("complexes") o f their constitu-
ents, b u t not all c o m p l e x e s are propositions. For e x a m p l e , classes as many are
combinations o f t h e i r m e m b e r s b u t they are not propositions, for classes as
many are j u s t m a n y q t h e y lack the characteristic unity o f propositions., Russell
says, " I n a class as m a n y , the c o m p o n e n t terms, t h o u g h they have s o m e kind o f
unity, have less t h a n is r e q u i r e d f o r a whole. T h e y have, in fact, j u s t so m u c h
unity as is r e q u i r e d to m a k e t h e m m a n y , a n d not e n o u g h to p r e v e n t t h e m
from remaining many."s
A central c o n c e p t in Russell's m e t a p h y s i c s o f propositions in Principles is
the concept o f a term. For p r e s e n t - d a y readers, the t e r m " t e r m " will be u n d e r -
stood to r e f e r to words. B u t that is not Russell's m e a n i n g in Principles. " W h a t -
ever m a y be an object o f t h o u g h t , o r m a y o c c u r in any t r u e or false proposi-
tion, o r can be c o u n t e d as one, I call a term. T h i s is the widest w o r d in the
philosophical vocabulary. I shall use it as s y n o n y m o u s with the words unit,
individual, a n d entity . . . . A m a n , a m o m e n t , a n u m b e r , a class, a relation, a
c h i m e r a , or a n y t h i n g else that can be m e n t i o n e d , is sure to be a term."4
A p r o p o s i t i o n is a u n i t - - i t is one, while the class as m a n y is a m e r e c o m b i n a -
tion o f terms, which c o m b i n a t i o n is n o t 0 n c - - n o t unitary, not a t r u e unit. O f
such c o m p l e x e s Russell says, " T h e c o m b i n a t i o n s are c o m b i n a t i o n s o f terms,

' Bertrand Russell, The Prindples of Mathematics (Cambridge: Cambridge University Press,
x9o3; second edition, New York: W. W. Norton & Company, Inc., 1938).
, Ibid., 68-69.
Ibid., 69.
4 Ibid., 43.

[~43]
244 JOURNAL OF THE HIS T ORY OF P H I L O S O P H Y 30:2 APRIL 1992
effected w i t h o u t the use o f relations."5 W h a t distinguishes c o m p l e x e s which a r e
true unities f r o m those which a r e m e r e a g g r e g a t e s lacking unity is "the use o f
relations." It is b e c a u s e every p r o p o s i t i o n contains at least o n e v e r b that it has its
characteristic unity. Verbs, f o r Russell, in a c c o r d a n c e with the ontologizing o f
g r a m m a r in Print/pies, are relations. 6 Russell is certain that the unity o f the
proposition is e f f e c t e d by relations. T h a t is why he says that the c o m b i n a t i o n o f
t e r m s which is the class as m a n y a n d which lacks t r u e unity is " e f f e c t e d without
the use o f relations." Verbs, however, have a twofold n a t u r e , "as actual v e r b a n d
as verbal noun."v " T h e r e is t h e v e r b in the f o r m which it has as v e r b . . , a n d
t h e r e is the verbal n o u n indicated by the infinitive o r (in English) the p r e s e n t
participle. T h e distinction is t h a t between 'Felton killed B u c k i n g h a m ' a n d 'Kill-
ing no m u r d e r ' "(sic). s T h e d i f f e r e n c e between the v e r b as "actual v e r b " a n d as
verbal n o u n is the d i f f e r e n c e between a relation as actually relating a n d the
inert relation "in itself." Analysis reveals the difference.
Consider, for example, the proposition 'A differs from B'. The constituents of this
proposition, if we analyze it, appear to be only A, difference, B. Yet these constituents,
thus placed side by side, do not reconstitute the proposition. The difference which
occurs in the proposition actually relates A and B, whereas the difference after analysis
is a notion which has no connection with A and B. It may be said that we ought, in the
analysis, to mention the relations which difference has to A and B, relations which are
expressed by/s andfrora when we say 'A is different from B'. These relations consist in
the fact that A is referent and B relatum with respect to difference. 'A, referent,
difference, relatum, B' is still merely a list of terms, not a proposition. A proposition, in
fact, is essentially a unity, and when analysis has destroyed the unity, no enumeration
of constituents will restore the proposition. The verb when used as a verb, embodies
the unity of the proposition, and is thus distinguishable from the verb considered as a
term, though I do not know how to give a clear account of the precise nature of the
distinction.0

J u s t as the t e r m " t e r m " does not, for Russell, r e f e r only to words but is the
widest w o r d o f the philosophical vocabulary a n d thus refers to a n y t h i n g which
is "one," a unit, so the t e r m "verb" does n o t r e f e r only to words, b u t to what
"verbs," in the linguistic sense, stand for, viz., relations. Words, in general, are
o f no interest in Principles. Russell says, "Words all h a v e m e a n i n g , in the simple
sense that they a r e symbols which stand for s o m e t h i n g o t h e r t h a n themselves.
But a p r o p o s i t i o n , unless it h a p p e n s to be linguistic (i.e., to be a b o u t words),

s Ibid., 58.
See Leonard Linsky, "Terms and Propositions in Russell's Principlesof Mathematics,"Journal
o/the Historyo/Philosophy 26 (October, 1988): 621--642; 625.
Russell, Principles of Mathematics, 49-
s Ibid., 48.
9 Ibid., 49-5o.
 UNITY OF THE PROPOSITION 245

does not contain words: it contains the entides indicated by words. T h u s

meaning, in the sense in which words have meaning, is irrelevant to logic. ''~~
In the "ontological assay" o f Russell's Principles two kinds o f terms, things
and concepts, are distinguished. Things are terms indicated by p r o p e r names,
they are the logical subjects o f propositions. Concepts are what are indicated by
"all other words. ''~ A m o n g concepts again, Russell distinguishes between
those indicated by adjectives a n d those indicated by verbs. These terms are
themselves called adjectives and verbs. T h e f o r m e r kind are also called predi-
cates or class-concepts, "the latter are almost always relations. ''~' Every proposi-
tion contains things and concepts as constituents. Russell finds that some
terms, viz., concepts, can occur in a special way in a proposition in which terms
called "things" cannot occur. He contrasts t h e proposition "Socrates is h u m a n "
with the different, t h o u g h equivalent proposition " H u m a n i t y belongs to Socra-
tes." In the first "the notion expressed by human occurs in a d i f f e r e n t way
from that in which it occurs w h e n it is called humanity, the difference being
that in the latter case, but not in the former, the proposition is about this
notion. This indicates that h u m a n i t y is a concept, not a thing. I shall speak o f
the terms of a proposition as those terms, however n u m e r o u s , which occur in a
proposition a n d m a y be r e g a r d e d as subjects about which the proposition is.",s
So, not all o f the terms which are constituents of the proposition occur as
terms o f the proposition. T h i n g s are those terms which cannot occur in a
proposition in any way other t h a n as terms--logical subjects. "Predicates,
then, are concepts, other t h a n verbs which occur in propositions having only
one term or subject.",4 By extension, verbs (relations) are concepts other than
predicates which occur in propositions having more than one term or logical
subject. "Socrates is a thing, because Socrates can never occur otherwise than
as a term in a proposition: Socrates is not capable o f that curious twofold use
which is involved in human a n d humanily."~5 It is in this twofold m o d e of
occurrence o f terms other t h a n things that Russell finds the source o f the
unity of the proposition. W h e n a concept occurs other than as a term in the
proposition o f which it is a c o m p o n e n t it is the source o f u n i t y - - " t h e verb
when used as a verb." Russell's problem is that he has no explanation whatever
for what it is for the verb to be used as a verb rather than as inert verbal n o u n
and logical subject. How do relations relate?

~o I b i d . , 47.
~ I b i d . , 44-
'" I b i d .
u I b i d . , 45.
~, I b i d .
~5 I b i d .
946 JOURNAL OF THE HISTORY OF PHILOSOPHY 30:2 APRIL 1992
Consider a wall consisting o f bricks held t o g e t h e r with cement. T h e c e m e n t
is the source o f the "unity" o f the wall. But what if one felt compelled to think
o f the c e m e n t as itself a n o t h e r kind o f brick? T h e n what holds the bricks
t o g e t h e r is itself in n e e d o f a c e m e n t to hold it together with the o t h e r bricks.
Russell's p r o b l e m is that verbs d o play the double role in p r o p o s i t i o n s - - a s
actual verb and as verbal n o u n . As actual verb, they are the source o f the unity
o f the proposition. But ontoiogically, verbs are also terms like all o t h e r c o m p o -
nents o f the proposition. So we must think o f this c e m e n t as, at the same time,
a kind o f brick and ask the a b s u r d q u e s t i o n - - " W h a t holds the c e m e n t to the
bricks?" While this question a b o u t the unity o f the wall is absurd, Russell
t h o u g h t the question o f the unity o f the proposition not at all absurd, t h o u g h
he did not know how to deal with it. H e had an a r g u m e n t that any t e r m must
be capable o f playing the role o f logical subject. H e n c e relations must be both
the source o f unity, the cement, and, at the same time, they must be c o m p o -
nents, t o g e t h e r with the bricks, o f the propositions in which they occur. The
Principles of Mathematics has only o n e ontological category. " W h a t e v e r m a y be
an object o f t h o u g h t , o r may occur in any true or false proposition, o r can be
c o u n t e d as one, I call a term. '''6
Russell's p r o b l e m was f o r m u l a t e d earlier by Bradley. In the table o f con-
tents for Appearance and Reality, the content o f Part 3 o f C h a p t e r 3 is given as
follows: "Relations with, or without, qualities are unintelligible."'7 Bradley
formulates the p r o b l e m as a p r o b l e m as to how a relation can relate qualities.
" I f it is n o t h i n g to the qualities, then they are not related at all . . . . But if it is to
be something to them, t h e n clearly we now shall require a new c o n n e c t i n g
relation. For the relation h a r d l y can be the m e r e adjective o f one or both o f its
terms . . . . A n d if being s o m e t h i n g itself, it does not itself b e a r a relation to the
terms, in what intelligible way will it succeed in being a n y t h i n g to t h e m ? But
h e r e again we are h u r r i e d o f f into the e d d y o f a hopeless process, since we are
forced to go on finding new relations without end. T h e links are united by a
link and this b o n d o f u n i o n is a link which also has two ends; and these r e q u i r e
each a fresh link to c o n n e c t t h e m with the old. '''8 Bradley declares that "this
problem is insolvable. I f you take the connection as a solid thing, you have got
to show, a n d you c a n n o t show, how the o t h e r solids are j o i n e d to it. And, if
you take it as a kind o f m e d i u m o r insubstantial a t m o s p h e r e , it is a c o n n e c t i o n
no longer."~9 Bradley concludes that "relations are unintelligible." T h e issue is
o f central i m p o r t a n c e for Russell. The Principles of Mathematics is the first fruit

,e Ibid., 43-
~7 F. H. Bradley, Appearance and Reality, (Oxford: Clarendon Press, 1897).
,a Ibid., 28.
'~ Ibid.
 UNITY OF THE PROPOSITION 247
of the revolt against Idealism by Russell and Moore which initiates analytical
philosophy in England. A main tenet of this philosophy is the legitimacy of
analysis and the reality of relations. Russell writes, "On fundamental questions
of philosophy, my position, in all its chief features, is derived from Mr. G. E.
Moore. I have accepted from h i m . . , the pluralism which regards the world,
both that of existents and that of entities, as composed of an infinite number
of mutually independent entities, with relations that are ultimate, and not
reducible to adjectives o f their terms or of the whole which these compose. '',o
Russell's concern with the probiem o f unity is a response to Bradley."
T h e problem had earlier been confronted by Frege, independently of both
Bradley and Russell, although it seems that Russell was not aware of this until
after the main text of Principles was completed. Russell compares his position
with that of Frege in Append/x A, "On the Logical and Arithmetical Doctrines
of Frege," added after the completion of the rest of the book. His differences
with Frege on this point are not mentioned in the body o f Principles itself.
Nevertheless, an account o f Frege's views will help to explain Russell's contrast-
ing position and to illuminate the difficulties in the way of a Russellian solu-
tion. For Frege, there is no concept corresponding to Russell's "terms"--there
is no single widest word in the philosophical vocabulary. Instead there are two
categories which are equally comprehensivemfunction and object. There is
no term or category which includes both functions and objects. There is noth-
ing which both functions and objects are. The essential difference between
them is that functions are incomplete and objects are complete. Concepts are a
special kind of function. They are functions which are singly incomplete and
whose values for all arguments are truth-values. Binary relations are, by con-
trast, functions which are doubly incomplete; and their values for all pairs of
arguments are truth-values. "Complete" and "incomplete" are, of course,
metaphors. Frege says that he cannot explain his concept of functions without
these (or equivalent) metaphors. He acknowledges that the metaphors are not
eliminable. A proposition too is a complete thing--a Russellian unity. Frege
analyzes propositions into two kinds o f constituents, functions and objects.
One of these constituents, the function or concept, is incomplete. It is in terms
of the contrast between complete and incomplete that Frege explains the unity
of the proposition. T h e proposition results from the completion of the incom-
plete function by the complete objects which fill the argument places in the
function. "For not all parts of a thought can be complete; at least one must be
'unsaturated', or predicative; otherwise they would not hold together. For
example, the sense o f the phrase 'the number u' does not hold together with

Russell, PrinciplesofMalhemahcs, p. xviii.

9~ I owe my appreciation of this point to the unpublished work of Mr. Martin Cox.
248 J O U R N A L OF T H E H I S T O R Y OF P H I L O S O P H Y 3 0 : 2 A P R I L 1992
that of the expression 'the concept/n/me number' without a link. We apply such
a link in the sentence 'the number 2 falls under the concept pr/me number'; it is
contained in the words 'falls under', which need to be completed in two
ways--by a subject and an accusative; and only because their sense is thus
'unsaturated' are they capable o f serving as a link."'" Frege then remarks that
words or phrases such as "falls under," in the above example, stand for a
relation. He goes on to develop the theme of infinite regress in this connection
that has already made its appearance in our discussion of Bradley and Russell.
"We now get the same difficulty for the relation that we were trying to avoid
for the concept. For the words 'the relation of an object to the concept it falls
under' designate not a relation but an object; and the three proper names 'the
number 2', 'the concept pr/me number', 'the relation of an object to a concept it
falls under', hold aloof from one another just as much as the first two do by
themselves; however we put them together, we get no sentence."'s
It is essential to Frege's account that functions and concepts are not only
incomplete, but that they are essentially incomplete. Concepts are essentially
predicative. This means, for Frege, that they cannot play the role of logical
subjects. A fallacy arises from the attempt to force this double role upon them.
The proposition 'Bucephalus is a horse' involves the concept horse, but any
attempt to say, as was just done, what concept it involves transforms this
concept into an object. The concept horse is not a concept, precisely because it
is a logical subject in this very proposition. Therefore, the concept horse is an
object. This conclusion is admittedly embarrassing, but Frege embraces it and
says that it exposes a necessary feature of language that it does not provide us
with the means o f saying correcdy what we want to say. "By a kind of necessity
of language, my expressions, taken literally, sometimes miss my thought; I
mention an object, when what I intend is a concept. I fully realize that in such
cases I was relying upon a reader who would be ready to meet me halfway--
who does not begrudge a pinch of salt.",4
By making concepts and functions essentially incomplete, Frege disables
them from playing the role of logical subjects. Consequently they provide the
glue which holds the proposition together. But a further consequence of the
incompleteness of functions is that we cannot talk about them since they cannot
play the role o f logical subjects. Thus the price which Frege pays for his
account of the unity of the proposition is the paradox about the concept horse.
That the connection between Frege's account of unity and the paradox is not a

9" Gottlob Frege, "On Concept and Object," in Translationsfrom the Philosophical Writings of
Gottlob Frege, edited and translated by Max Black and Peter Geach (Oxford: Basil Blackwell,
195~), 54-
,s Ibid., 55.
94 Ibid., 54.
 UNITY OF THE PROPOSITION 949

necessary one he discovered only later as we know from his letters to Russell
discussed in Part 4 of this paper.
Against this background we can appreciate Russell's struggle with the same
problem. Frege embraced a paradox. Russell refused to do that, but he de-
clared himself completely defeated by the problem. The reason was that it
seemed impossible to him that there could be terms which could not play the
role of logical subjects. "Every term," he tells us, "is a logical subject: it is, for
example, the subject of the proposition that itself is one.",5 Russell considers a
Fregean solution to the problem o f unity and argues against it that it produces
the very paradox which Frege embraces. (At this point in Principles, there is no
reference to Frege, and it must be assumed that what he wrote here was
written before his acquaintance with Frege's work.) "It might be thought that a
distinction ought to be made between a concept as such and a concept used as
a term, between e.g., such pairs as/s and being, human and humanity, one in such
a proposition as 'this is one' and 1 in '1 is a number'. But inextricable difficul-
ties will envelop us if we allow such a view.".6 There is, o f course, a grammati-
cal difference, but Russell, in a rare departure from grammar, ignores this
difference. His view is that it is logically the same term which plays the double
role--as predicate, or relation, and as logical subject. O f course, there are, for
Russell, terms which cannot play the role of verbs and adjectives. That is what
distinguishes them from concepts, and Russell calls them "things." "Things"
are essentially logical subjectsmthat is the only role they can play in the propo-
sition. Things are thus like Frege's objects. They cannot play the role of
concepts. For Frege however, nothing can play the double role--the predica-
tive role and the role of logical subject. That enables him to explain the unity
of the proposition. For Russell, whatever plays the predicative role must be
able to play the role of logical subject as well, on pain of contradiction. The
argument is as follows: "For suppose that one as adjective differed from I as
term. In this statement one as adjective has been made into a term; hence
either it has become l, in which case the supposition is self-contradictory; or
there is some other difference between one and t in addition to the fact that
the first denotes a concept not a term while the second denotes a concept
which is a term. But in this latter hypothesis, there must be propositions
concerning one as term, and we shall still have to maintain propositions con-
cerning one as adjective as opposed to one as term; yet all such propositions
must be false, since a proposition about one as adjective makes one the subject,
and is therefore really about one as term.",7 Russell sees the view he is oppos-

,5 Russell, Principles of Mathematics, 44-

,,e I b i d . , 45.
9r I b i d . , 46.
250 JOURNAL OF THE HISTORY OF P H I L O S O P H Y 30:2 APRIL 199 9
ing as contradictory. "s His analysis first leads him to distinguish the double
role of the v e r b - - t h e verb as verb, and the verbal noun. Further analysis leads
him to conclude that it is the same thing that plays both roles. What then is the
glue which holds the proposition together? Russell is defeated by the problem,
and it is important to remember this if we are to appreciate fully Russell's
views about propositions after Principles, and his strenuous efforts to rid him-
self of them altogether.
By insisting on the double role of concepts, Russell avoids Frege's contra-
diction about the concept horse, but he is led to the very brink of the cele-
brated antinomy which bears his name. If concepts can play the double role,
there is nothing to prevent them from playing both roles in the same proposi-
tion. In o r d e r to fall into the antinomy, we have only to ask whether the
concept o f being non-self-applicable which a concept has, if and only if, it does
not apply to itself, does or does not apply to itself. The entire development, up
to this point, requires Russell to treat the question as legitimate. He writes, "It
is a characteristic of the terms o f a proposition that any one of them may be
replaced by any entity without our ceasing to have a proposition."'9 T h e
answer to the question is that the concept o f being non-self-applicable is non-
self-applicable, if and only if, it is not non-self-applicable--Russell's celebrated
antinomy in its intensional (predicative) form. Russell could have discovered it
at this point without the comparatively arduous detour through the diagonal
argument used in Cantor's theorem on the cardinality of the power set that in
fact led him there.so Frege's insistence on the single role of concepts both leads
him into the paradox about the concept horse and enables him to resolve
Russell's antinomy in its intensional form. Russell initiated his correspondence
with Frege in a letter dated J u n e 16, 19o2. In this celebrated letter he commu-
nicates his discovery of his antinomy. He presents the antinomy in both its
extensional form about the class of all classes which are not members of
themselves, and in its intensional form about predicates which cannot be
predicated o f themselves. The difference in the two forms was at this time, of
little significance for Russell; he treated it as a single antinomy which mani-
fests itself in these two versions. When he presents it in Chapter l o o f P r i n c i -
ples, and in subsequent discussions, it is referred to simply as "the contradic-
tion." In an equally famous letter dated J u n e 22, 19o2, Frege responded to
Russell's communication. In this, and in their subsequent correspondence he
treats the two versions of the antinomy, unlike Russell, in radically different
ways. Regarding the extensional form he says this: "Your discovery of the

,s Ibid.
,9 Ibid., 45.
so Ibid., 362 n.
 UNITY OF T H E P R O P O S I T I O N 251

contradiction has surprised me b e y o n d words and I should almost like to say,

left me t h u n d e r s t r u c k , because it has r o c k e d the g r o u n d on which I meant to
build arithmetic . . . . It is all the m o r e serious as the collapse o f my law V seems
to u n d e r m i n e not only the f o u n d a t i o n s o f my arithmetic but the only possible
foundations o f arithmetic as such."3~ It is a c o m m o n p l a c e o f the history o f
logic that Frege never r e c o v e r e d f r o m the shock o f Russell's letter and that he
gave u p his life's work o f f o u n d i n g mathematics on logic as a response to the
discovery o f the antinomy.
Frege's response to the intensional f o r m o f Russell's antinomy was very
different. It is entirely dismissive and, in this same letter, he offers an immedi-
ate resolution o f it. "Incidentally, the expression 'A predicate is predicated o f
itself' does not seem exact to me. A predicate is as a rule a first-level function
which requires an object as a r g u m e n t and which c a n n o t t h e r e f o r e have itself
as a r g u m e n t (subject)."5, Frege is telling Russell that concepts cannot play the
double role o f both predicate and logical subject; they c a n n o t be predicated o f
themselves. Russell r e s p o n d s just two days later in a letter o f J u n e 24, t9o2, "I
know very well what good reasons t h e r e are to be f o u n d for this view; yet it is
self-contradictory . . . . I f t h e r e can be something which is not an object, then
this fact cannot be stated without contradiction; for in the statement, the
something in question becomes an object."ss
T h e problem about the unity o f the proposition and the contrasting treat-
ments o f it by Russell and Frege, the f o u n d e r s o f m o d e r n logic, has thus led to
contrasting treatments o f the intensional f o r m o f Russell's antinomy. Frege's
doctrine o f the essentially predicative role o f concepts is an a r g u m e n t for his
simple theory o f types for functions, which he first p r e s e n t e d in his address on
"Function and Concept" in 1891, m o r e than a decade b e f o r e he became ac-
quainted with Russell's antinorny. H e t h e r e sketches a simple type hierarchy o f
objects, first-level concepts and relations which take objects as arguments, and
second-level concepts and relations whose a r g u m e n t s are first-level concepts
and relations. H e t h e n writes, " O n e might think that this would go on. But
probably this last step is already not so rich in consequences as the earlier
ones . . . . But this does not banish f r o m t h e world the d i f f e r e n c e between first-
level and second-level functions; for it is not made arbitrarily, but f o u n d e d
d e e p in the n a t u r e o f things."s4
In a letter o f August 8, 19o2, Russell presents a relational form o f the
intensional antinomy. H e first defines

sl Gottloh Frege, Philosophical and Mathematical Correspondenceof Gottlob Frege, edited by Brian
Mc'Guinness, translated by Hans Kaal (Chicago: University of Chicago Press, t98o), 132.
3. Ibid.
s3 Ibid., 134.
Frege, "Function and Concept," in Translations of the Philosophical Writings, 41.
252 JOURNAL OF T H E H I S T O R Y OF P H I L O S O P H Y 30"2 APRIL X992

T(R, S) :--: R = S 9 - R ( R , S).

He first substitutes T for S to obtain,

T(R, T) :---: R = T . - - R ( R , T),

and then substitutes T for R to obtain a contradiction.

T(T, T) : - : - - T ( T , T).

T is, by this definition, the relation which holds between the relations R and S,
if any only if R = S and R does not hold between R and S. Frege's response in
his next letter o f September 2 3, 19o~, is to point out that Russell's argument
involves a violation of simple type constraints by allowing T to play both the
role of logical subject and the predicative role. "Evidently, I do not quite
understand your meaning here. A relation between relations is of a different
logical type from one between objects. For the former is a second-level func-
tion, the latter a first-level one."ss Here then, we have Frege, who had ac-
cepted the simple theory o f types, on grounds quite independent of the antino-
mies, trying to persuade Russell of the correctness o f the theory. The com-
monly held view that it was Russell who first both discovered the theory of
types and employed it in the resolution of logical antinomies is quite mistaken
on both of these points.
Frege's view o f the matter did not change. His last published remarks on
Russell's antinomy were written in 19 lO and inserted as a note by Philip E. B.
Jourdain to his article "Gottiob Frege: The Development of the Theories of
Mathematical Logic and the Principles of Mathematics," published in 1912.s6
Frege expresses the view there that Russell's antinomy is an antinomy about
classes and that it does not therefore constitute a threat to logic. Logic for both
Frege and Russell is higher-order logic. For Frege this is based on the simple
type hierarchy of functions which lies "deep in the nature of things." Frege's
mistake, as he later saw it, was to think that the theory of classes was also a part
of logic and that it could be made to be so through his fifth axiom. Russell's
antinomy destroyed that effort by proving the inconsistency of Frege's theory
of extensions, and with that he destroyed Frege's program of founding arith-
metic on logic. These remarks seem to be little known. "And now we know
that, when classes are introduced, a difficulty [Russell's contradiction] arises.
In my fashion of regarding concepts as functions, we can treat the principal

s5 Frege,Philosophical and Mathematical Correspondence, 146.

r Philip E. B. Jourdain, "Gottlob Frege: The Developmentof the Theories of Mathematical
Logicand the Principles of Mathematics,"The QuarterlyJournal of Pure and Applied Mathematica 43
(1912): 237-69; reprinted in Frege, Philosophical and Mathematical Correspondence; my references
are to the reprinted version.
 U N I T Y OF T H E P R O P O S I T I O N 253

parts of Logic without speaking o f classes, as I have done in my Begriffsschrift,

and that difficulty does not then come into consideration . . . . T h e difficulties
which are bound up with the use of classes vanish if we only deal with objects,
concepts, and relations, and this is possible in the fundamental part of
Logic."s7 Frege would have rejected as confused any demand for a proof of
the consistency of his Befffiffsschift. In these remarks he is claiming that his
system of logic of x879 is immune to both the extensional and the intensional
forms of Russell's antinomy. It is therefore of some interest that Susan
Russinoff has shown that a simple and natural explication of the Begriffsschrift,
in which the major results of Frege's work are retained using essentially
Frege's proofs, is demonstrably consistent.ss It was indeed Frege's manner of
introducing extensions that was his greatest mistakel

2.

A central role for propositions is to be bearers of truth-values. A long tradi-

tion, going back to Aristotle, accounts for the truth and falsity of propositions
by their correspondence or failure to correspond to something other than
themselves, viz., facts. Russell and the early Moore both reject the correspon-
dence theory of truth, for neither o f them have an ontological category of
facts with which propositions can be said to correspond. On the contrary,
propositions as structured entities or complexes, absorb into themselves both
the roles of traditional propositions and the facts corresponding to them. By
insisting on the objectivity of propositions, or on the objective character of
propositions, Russell and the early Moore deprive their correlative facts of any
role to play. T h e r e is no space separating facts from propositions. How then
are they to account for the distinction between true propositions and false
ones? It seems that propositions are themselves facts, and hence all true.
Russell faced the problem, at first, with equanimity. Indeed, one might say
that he brazens it out. "It might be said--and this is, I believe, the correct
view--that there is no problem at all in truth and falsehood; that some propo-
sitions are true and some false,just as some roses are red and some white; that
belief is a certain attitude towards propositions, which is called knowledge
when they are true, error when they are false."s9 The position which Russell
here adopts really amounts to a declaration of inability to account for the truth
of propositions at all. Some propositions are true and some are false, just as

sT Frege, Philosophical and Mathematical Correspondence, 191.

ss Susan Russinoff, "On the Brink of a Paradox?" Notre Dame Journal of Formal Logic 98
(January 1987): 1~5-31.
s9 Bertrand Russell, "Meinong's Theory of Complexes and Assumptions," Mind n.s. 13: 204-
19, 336-54, 5o9-24. My references are to the reprint in Essays in Analysis, ed. Douglas Lackey
(New York: George Braziller, 1975), 75.
~54 J O U R N A L OF T H E H I S T O R Y OF P H I L O S O P H Y 3 0 : 2 APRIL 1992

some roses are red a n d some white, a n d that is all he has to say about the
matter. Russell sees that this leaves him unable to account for o u r preference
for truth over falsehood. It is just like o u r preference for red over white roses.
T h e r e is no accounting for matters o f taste. Russell decides that o u r prefer-
ence for truth is based u p o n an ultimate "ethical" proposition: "It is good to
believe true propositions, and bad to believe false ones.'4o
It is not surprising that Russell did not remain content with this for long.
Just two years after declaring that "there is no problem at all in t r u t h a n d
falsehood," he published a p a p e r in the Proceedings of the Aristotelian Society on
" T h e Nature o f Truth."4~ T h e problem is this: "What do we believe w h e n o u r
belief is mistaken? For w h e n we believe truly, o u r belief is to have an object
which is a fact, but w h e n we believe falsely, it can have no object, unless there
are objective non-facts. T h e people who believe that the sun goes r o u n d the
earth seem to be believing something, and this something cannot be a fact.
Thus, if beliefs always have objects, it follows that there are objective non-
facts."4, Now these objective facts and objective non-facts are just the proposi-
tions of Principles, a n d if we recur to Russell's unsolved problem o f the unity o f
the proposition we can see the nature o f the difficulty. T h e unity of the
proposition is provided by the verb as verb, by relations actually relating. T h e
problem of objective non-facts arises when we attempt to reconcile the unity o f
the proposition with its falsehood. Consider the false proposition " B e r t r a n d
Russell = Alfred N o r t h Whitehead." It is a complex unity, which unity is
provided by the identity-relation. For the proposition to have the characteris-
tic unity which enables propositions to be either true or false, the relation o f
identity must actually relate B e r t r a n d Russell and Alfred North Whitehead.
But then the unity o f the proposition excludes its falsehood, for if Bertrand
Russell is actually related by the identity-relation to Alfred North Whitehead,
it must be true that B e r t r a n d Russell = Alfred North Whitehead. It seems that
in order to have objective falsehoods, we must destroy the unity o f the proposi-
tion. In o r d e r for it to be false that B e r t r a n d Russell = Alfred North White-
head, they must not be related by identity. In a false proposition, the relation
cannot actually relate. But then it loses its source of unity; it cannot be a
proposition at all.
In the Aristotelian Society paper of 19o6 Russell faces the problem squarely
and considers as one way out of his d i l e m m a the a b a n d o n m e n t of the idea of
propositions as terms. He is unable, in any case, to account for the unity o f

40 Ibid., 76 .
4~ Bertrand Russell, " T h e Nature of Truth," Proceedings of the Aristotelian Society, New Series
(London: Williams and Norgate), Vol. 7 (a9~ ~8-49.
4, Ibid., 45-46 .
 UNITY OF THE PROPOSITION 255
propositions, and their unity seems to be incompatible with falsehood, h e n c e
he considers a b a n d o n i n g the claims o f unity, and propositions as unitary
entities. " T h u s a belief, if this view is a d o p t e d , will not consist o f o n e idea with
a c o m p l e x object, but will consist o f several related ideas. T h a t is, if we believe
(say) that A is B, we shall have the ideas o f A and B, and these ideas will be
related in a certain m a n n e r ; b u t we shall not have a single c o m p l e x idea which
can be described as the idea o f 'A is B'."4s This f r a g m e n t i n g o f propositions,
breaking t h e m up, is h e r e tentatively suggested as a way out o f the difficulties.
W h e n in 191o Russell n e x t writes about the problem o f truth, the tentative
suggestion o f 19o6 is fully e m b r a c e d . In the essay "On the N a t u r e o f T r u t h
and Falsehood"44 a n d in the i n t r o d u c t i o n to the first edition ofPrincipia Mathe-
matica, both published in x9 l o, Russell presents his celebrated "Multiple Rela-
tion T h e o r y o f Judgement."4s T h e r e is a n o t h e r presentation o f the view in The
Problems of Philosophy o f 19 xz. H e r e he says, " T h e impossibility o f allowing f o r
falsehood makes it impossible to r e g a r d belief as a relation o f the mind to a
single object, which could be said to be what is believed."46 Russell's Multiple
Relation T h e o r y o f J u d g m e n t was suggested to him by his t r e a t m e n t o f defi-
nite descriptions. Definite descriptions are incomplete symbols, lacking inde-
p e n d e n t meaning. T h e y c o n t r i b u t e to the m e a n i n g o f the whole proposition
containing them but d i s a p p e a r o n analysis and are n o longer p r e s e n t as single
constituents in the propositions in whose verbal expressions they occur. Simi-
larly propositions are said to be incomplete symbols in the Multiple Relation
T h e o r y . T h e larger wholes o f which they are parts are j u d g m e n t s . Proposi-
tions c a n n o t survive as logical or semantical units if they are withdrawn f r o m
the j u d g m e n t s containing them. W h e n Othello j u d g e s that D e s d e m o n a loves
Cassio, t h e r e does not exist a binary relation o f j u d g i n g between Othello and
the proposition that D e s d e m o n a loves Cassio; r a t h e r t h e r e exists a multiple
relation o f j u d g i n g which unites Othello, D e s d e m o n a , loves, a n d Cassio. T h e
multiple relation o f j u d g i n g is a mental act which occurs, at a definite time, in

4s Ibid., 46. Kenneth Olson also notes a tension between Russell's account of unity in Prmd-
ples and the possibilityof false propositions. "In a Russellian proposition the reladng of the terms
by the relation is what makes the proposition a unity. Since even a false proposition is a unity, a
further explanation has to be given for the truth or falsity of the proposition." Kenneth Russell
Ohon, An Essay on Facts, CSLI Lecture Notes, Number 6, Center for the Study of Language and
Information (Stanford University, Stanford, California, 1987), 46-47. My point is not the same as
Oison's, for I find Russell's account of the unity of propositions incompatible with their falsity.
Bertrand Russell, "On the Nature of Truth and Falsehood," in PhilosophicalEssays (Lon-
don: George Allen & Unwin; revised edition, 1966).
4s A. N. Whitehead and Bertrand Russell, Pr/vw/p/aMathematica, Vol. i (Cambridge, En-
gland: Cambridge University Press, 191o; second edition, 19z5).
46 Bertrand Russell, The Problemsof Philosophy (Oxford: Oxford University Press, 1974; first
edition 191Z), 193.
256 JOURNAL OF T H E H I S T O R Y OF P H I L O S O P H Y 30:2 APRIL 1992
the mind o f Othello. T h e only complex is the unity effected by the relation o f
judging. O f course there is a n o t h e r relation involved, viz., the relation loves.
But this relation is not present in the j u d g m e n t as a relation actually relating.
It is present in that other way in which verbs can occur, as verbal noun. T h e
constituents o f the j u d g m e n t are: Othello, judging, Desdemona, loves, and
Cassio. It is j u d g i n g that is the relation that actually relates. T h e other relation,
loves, is an inert constituent o f the j u d g m e n t . " T h e relation 'loving', as it
occurs in the act o f believing, is one o f the objects--it is a brick in the struc-
ture, not the cement. T h e c e m e n t is the relation 'believing'."47 (Russell's ac-
count is the same w h e t h e r in terms o f believing or j u d g i n g or a n o t h e r proposi-
tional attitude.) Now Russell can present his account of truth. " W h e n the
belief is true, there is a n o t h e r complex unity, in which the relation which was
one o f the objects o f the belief relates the other objects. Thus, e.g., if Othello
believes truly that D e s d e m o n a loves Cassio, then there is a complex unity
'Desdemona's love for Cassio', which is composed exclusively o f the objects o f
the belief, in the same o r d e r as they had in the belief, with the relation which
was one o f the objects occurring now as the cement that binds together the
other objects o f the belief. On the other hand, when a belief is false, there is no
such complex unity composed only o f the objects of the belief. I f Othello
believes falsely that D e s d e m o n a loves Cassio, then there is no such complex
unity as 'Desdemona's love for Cassio'."48
This example o f " D e s d e m o n a loves Cassio" is, o f course, a very special case,
for it does not involve generality. (We assume for the sake o f the example that
"Desdemona" and "Cassio" are logically p r o p e r names.) Russell calls such j u d g -
ments "elementary." In Principia, he gives the following account of truth for
such j u d g m e n t s . "We will call a j u d g m e n t e/ementary when it merely asserts such
things as 'a has the relation R to b', 'a has the quality q' or 'a and b and c stand in
the relation S'. T h e n an elementary j u d g m e n t is true when there is a corre-
sponding complex, and false when there is no corresponding complex."49 Con-
sider an elementary j u d g m e n t by the person A that R.(a,, a, . . . . a,). Following
Cocchiarella this j u d g m e n t m a y be represented as having this form:5o

J.+.(A, R . ( ~ , :f. . . . . . :~,,), a~, a,, a,,).

T h e j u d g m e n t J.+. is an n + z-place relation, which relates A, the n place
relation R,,(x,, x . . . . . . x.), and the n objects a~, a . . . . . . a..5, But, o f course, not

4~ Ibid., 9oo.
48 Ibid., 2 o o - - ~ o l .
49 Whitehead and Russell, Principia Mathema~ica, 44 (first edition).
so Nino Cocchiarella, "The Development of the Theory of LogicalTypes and the Notion of a
LogicalSubject in Russell'sEarly Philosophy,"Synthese 45 (September s98o): 71--1 15.
s, Ibid., lo 3.
 U N I T Y OF T H E P R O P O S I T I O N ~57

all judgments are elementary, for they may involve generality or truth-
functional complexity. Consider the judgment "All men are mortal."
Cocchiarella analyzes this as having the form:
J4(A, (x)[~l(x) D ~!(x)], ~ is human, ~ is mortal).~'
The constituents of the judgment are A, the binary relation o f formal implica-
tion, (x)[~[ (x) D ~!(x)] and the two singularly predicative propositional func-
tions, s is a man, and s mortal. Russell does not tell us how negation and other
truth-functions are to be treated hut in the spirit of this account of generality,
it would seem that the j u d g m e n t by A that -(p) or that (p v q) would contain
the truth functions negat/on and disjunction themselves as constituents o f mo-
lecular propositions. This is the kind of view against which Wittgenstein is
protesting when in the Tractatus he writes that his "fundamental thought" is
that the logical constants are not representatives of anything (4.0 3 1~). In all of
his writing on the subject, however, Russell confines his account of the multi-
ple relation theory of j u d g m e n t exclusively to the case where what is j u d g e d is
a proposition of atomic (elementary) form.
Russell's search for the constituents of propositions is governed by the
principle: "Every proposition which we can understand, must be composed
wholly of constituents with which we are acquainted."55 As these analyses
show, we are required to be acquainted with such abstract objects as ~(p), and
(x)[#1(x) D r Here we have another expression of Russell's realism, and
Russell states explicitly that among objects of acquaintance are universals.s4
Thus, the objective proposition has been repudiated. T h e r e is no complex
consisting of Bertrand Russell and Alfred North Whitehead actually related
by identity. Still, someone can j u d g e or believe that Bertrand Russell is Alfred
North Whitehead. What unites Russell, Whitehead, and identity is a mental
act having the logical structure of a quaternary relation. We have a psychologi-
cal solution to the problem of the unity of the proposition. T h e relating

s0 Ibid., lo 4.
s3 Russell, The Problemsof Philosophy, 91 .
"54 Ibid., 81. What are the domains and ranges of the suggested truth functions? T h e question
is raised by an anonymous referee for this j o u r n a l who observes that these cannot be ftmctions
from propositions to propositions, as usually understood, for on the theory u n d e r discussion,
there are no propositions. Russell never deals with this question, as he abandoned work on the
theory before reaching the part of the Theo~ of Knowledge manuscript concerned with it. But
among the papers in the Russell Archives at McMaster University are a n u m b e r of oudines
projecting the contents of this part of the work. T h e sheet published in the Theory of Knowledge as
Appendix A.5 ends with the following remark, "Observe. A j u d g m e n t requires acquaintance with
one form of atomic complex. An inferential consciousness requires acquaintance with such terms
as or and not i.e., with a form of comptex in which propositions are constituent." This supports the
suggestion made in the body of this paper but it leaves us without an answer to the referee's
question. Perhaps there lurks here another reason for abandoning the project.
258 JOURNAL OF T H E H I S T O R Y OF P H I L O S O P H Y 30"9 APRIL 1992
relation is a mental act--a propositional attitude of judgment, or belief, or
supposition, or o f some other kind. T h e relating relation can, like all other
relations, play the double role. Suppose that George IV judges that George V
judges that Bertrand Russell is Alfred North Whitehead. In this iterated judg-
ment, the relation o f judging plays the double role, once as object of a judg-
ment and once as the "cement" which unites that object together with the
other objects of the j u d g m e n t into a complex unity. Suppose now that we raise
Russell's problem of unity again. What accounts for the unity of the judg-
ment? What is the difference between the relation of judging when it plays the
role of the relating relation and when it plays the inert role of object, with
other objects, in the multiple relation? Here progress has been made. We
understand the difference between the state of affairs in which Desdemona
loves Cassio, and the state o f affairs in which she is only believed to love
Cassio. In the latter case, Desdemona and Cassio are united by love only in the
jealous mind of Othello. The difference between j u d g i n g as inert object, and
judging as relating relation in our iterated judgment, is as clear as the differ-
ence between its actually being the case that Desdemona loves Cassio and
Othello merely believing that she does. We must suppose that Russell adopted
the Multiple Relation Theory for two reasons: (1) it enables him to give an
account o f truth and falsehood which avoids the objective falsehoods o f his
earlier theory; (9) it enables him to rid himself of his intractable problem of
the unity of the proposition. But Russell never succeeded in elaborating this
theory in a satisfactory way. For example, he never gave an account of judg-
ments of propositions which were not of atomic f o r m - - o f judgments of mo-
lecular and general propositions. It should be noted that Russell himself never
mentions the problem of the unity o f the proposition as of concern to him in
citing the advantages o f his Multiple Relation Theory or as part of his motiva-
tion for developing this theory.
Russell's efforts to rid himself of propositions are curiously out of har-
mony with the actual logic ofPrincipia Mathematica. We have noticed above the
passage in the introduction to the first edition of that work in which Russell
gives one of his earliest expositions of the Multiple Relation Theory, but
propositions are as real as can be in the body of Principia. Bound variables
ranging over propositions appear in theorems, e.g., * 14. 3. Russell shows some
reluctance about this: "In this proposition, however, the use of propositions as
apparent variables involves an apparatus not required elsewhere, and we have
therefore not used this proposition in subsequent proofs."55 It is true that
elsewhere propositional variables always appear as free ("real") variables, and

55 Whiteheadand Russell,Principia Mathematica, 185.

 U N I T Y OF T H E P R O P O S I T I O N 259

they always appear as free variables in the first five chapters of Principia
devoted to "The Theory of Deduction." Principia Mathematica, however, con-
tains axiom *9.13, which provides that for any theorem o f Principia containing
a free (real) variable, there is also a theorem obtained by universal closure over
that variable. *9. x3 is as follows: "In any assertion containing a real variable,
this real variable may be turned into an apparent 7r of which all possible
values are asserted to satisfy the function in question." (See also axiom *xo. l l
in Principia's alternative development of the theory of quantification.) Thus
there are infinitely many theorems of Principia containing bound proposi-
tional variables. This gives them a very solid place in Russell's logical universe
despite his efforts to free himself from them. "To be is to be a value of a
bound variable."

Having acknowledged his inability to solve the problem of the unity o f the
proposition when it is first presented in Principles, Russell does not return in
later publications to make further attempts at a resolution. Russell did this
with other problems of central importance for him in the philosophy o f
logic--denoting and especially the antinomies of logic and the theory of
classes are prominent examples. He came back to these problems over and
over again in both published and unpublished work until he arrived at a
solution that satisfied him. But concerning the problem of the unity of the
propositions, the case is quite different. After admitting his inability to solve it
in 19o 3, he never subsequently attempts to resolve it. In subsequent work,
Russell alludes briefly and en passant to the problem without ever going into it
so as to advance from the thoroughly defeatist conclusion of Principles. In his
discussion of Meinong, published in 19o 4, he accepts Bradley's argument for
the infinite regress, though here, as in Principles, he finds the regress not
vicious: "Hence we shall have to conclude that a relational proposition aRb
does not include in its meaning any relation of a or b to R, and that the endless
regress, though undeniable, is logically quite harmless."# Concerning the
problem of the unity of the proposition, Russell apparently accepted Bradley's
conclusion that "this problem is unsolvable."
Russell, we have seen, did condnue to seek an adequate account of proposi-
tions. From 19o6 until "Lectures on Logical Atomism" in 1918, these efforts
were driven by his desire to get rid o f propositions. Russell's main expressed
reason for dispensing with propositions is, as we have seen, the difficulty o f

Russell, Principles of Mathematics, 1oo. See also Russell, "Meinong's Theory of Complexes
and Assumptions," in Essays in Analysis, 28, and Russell, The Theoryof Knowledge: The ~9 x3 Manu-
script, The CollectedPapers of Bertrand Russell (London: George Allen and Unwin, i983), 7:183.
260 JOURNAL OF T H E H I S T O R Y OF P H I L O S O P H Y 30:2 APRIL 1992

giving an a d e q u a t e account o f false propositions. H e never, after Principles,

refers to the p r o b l e m o f the unity o f the proposition as motivating this effort.
I would suggest that o n e reason for this silence a b o u t the p r o b l e m o f the
unity o f the proposition is that even before he finished writing Principles, Rus-
sell discovered the inconsistency o f the most f u n d a m e n t a l concept d e p l o y e d in
his f o r m u l a t i o n o f the p r o b l e m - - t h e concept o f a term, for t h e r e is a p o r t e n -
tous attack o n the d o c t r i n e o f terms in Principles itself.57 This has to d o with
Cantor's t h e o r e m , which Russell states as follows: "Every class has m o r e
subclasses than t e r m s " (title o f section 347). Section 34 8 is entitled "But this is
impossible in certain cases," a n d section 349 is e n t i t l e d "Resulting contradic-
tions." After p r e s e n t i n g Cantor's p r o o f and a c k n o w l e d g i n g its correctness, Rus-
sell says, "Yet t h e r e are certain cases in which the conclusion seems plainly false.
T o begin with the class o f all terms."ss T h e difficulty is that classes, like all
constituents o f propositions, are terms. H e n c e the p o w e r set o f the class o f all
terms can be m a p p e d into the class o f terms by simply m a p p i n g each class in the
power set o n t o itself. Conversely the class o f terms can be m a p p e d into its p o w e r
set by m a p p i n g each t e r m o n t o its unit class. Russell now cites the S c h r o d e r -
Bernstein t h e o r e m a n d concludes that, c o n t r a r y to Cantor's t h e o r e m , the class
o f all terms has the same cardinality as its p o w e r set.59 W h a t Russell has shown is
that his doctrine o f t e r m s is inconsistent with Cantor's t h e o r e m . Russell d e v o t e d
a great deal o f t h o u g h t to Cantor's t h e o r e m . It was consideration o f it which,
Russell tells us, led him to the discovery o f the a n t i n o m y about the class o f all
classes which are not m e m b e r s o f themselves. 6~ Russell at first reacted to the
discovery by a t t e m p t i n g to find an e r r o r in Cantor's p r o o f o f his t h e o r e m , and
on D e c e m b e r 8, 19oo h e wrote to Louis Couturat, "I have f o u n d a mistake in
Cantor, who holds that t h e r e is no greatest cardinal n u m b e r . TM By the time o f
Principles, Russell had given u p the a t t e m p t to d i s p r o v e C a n t o r and he now
admits that "the above a r g u m e n t [Cantor's], it must be confessed, appears to
contain no dubitable assumption. ''6'
T h e obvious r e s p o n s e would be the rejection o f the doctrine o f terms.
Russell n e v e r explicitly takes that step until " O n Denoting," and then he takes
it only partially, w h e n he has n o f u r t h e r n e e d o f the full doctrine. In Principles
he decides that "this case is adequately met by the doctrine o f types," as he

5~ The remainder of this paragraph and most of the two paragraphs following are taken
from my paper "Terms and Propositions in Russell's Principlesof Mathematics," 639-34.
58 Russell, Principlesof Mathematics, 366.
Ibid., 367.
6o Ibid., 369 n.
61 The passage from Russell's letter to Couturat is taken from J. A. Coffa, "The Humble
Origins of Russell's Paradox," in Russell: The Journal of the Bertrand Russell Archives 0979): 33.
6, Russell, Principlesof Mathematics, 366.
 UNITY OF THE PROPOSITION 261
presents it in Append/x B to Principles.6s (It is to be noted that the a r g u m e n t
which Russell gives also shows that Frege's concept o f an object is inconsistent
with Cantor's theorem, for Frege's value-ranges of functions are objects. In
particular the extensions of concepts [classes] are objects, and hence the class
of all objects is one-to-one mappable onto its power set. Apparently Frege was
ignorant of this part o f Cantor's work, for had he known about it he would
have known that his system was inconsistent years before the publication in
~9o3 o f the second volume o f his Grundgesetze der Arithmetik.64 Cantor's most
mature presentation o f his Mengenlehre is to be f o u n d in his Beitr'age zur
Begriindung der transfiniten Mengenlehre, which was published in 1895.ss)
" T h e case," says Russell, "is adequately met by the doctrine o f types." In
Append/x A to Principles, devoted to the logical doctrines of Frege, Russell gives
this account o f the relevant aspects o f his earliest version o f the theory o f
types. This version is now called " T h e Simple T h e o r y o f Types," and Russell
was in full possession of its central features by 19o 3 . "It will now be necessary
to distinguish (x) terms, (2) classes, (3) classes o f classes, a n d so on ad infinitum;
we shall have to hold that no m e m b e r of one set [type] is a m e m b e r of any
other set [type], a n d that xeg requires that x should be o f a set [type] o f a
degree lower by one than the set [type] to which g belongs. T h u s x~jf will
become a meaningless proposition; and in this way the contradiction is
avoided. ''66 By "the contradiction" Russell means his antinomy about classes.
Russell does not tell us how this doctrine of simple types deals with the para-
dox about terms. But his t h o u g h t must have been as follows. According to the
doctrine of types we distinguish terms from classes. Hence the power set o f
the class of all terms is not a set of terms, and the p u r p o r t e d p r o o f of the one-
to-one m a p p i n g o f these two sets onto each other is destroyed. This response,
far from adequately dealing with the problem, simply abandons the idea of
terms as the maximally comprehensive ontological category which includes all
there is and excludes n o t h i n g whatever. It acknowledges the existence o f
things that are not terms, viz., classes, and it also implicitly acknowledges the
incoherence o f the very idea of a single all-comprehensive.ontological cate-
gory of any kind whatever. Russell sometimes characterized the revolt against
Idealism which he, together with Moore, initiated in the o p e n i n g years o f the

6~ Ibid., 367.
Gottlob Frege, Grundgesetze der A~thmetik, Voi. I, 1893; Vol. II, 19o3 (Jena: Pohle; re-
printed, Hildesheim: Olms, 1962).
Georg Cantor, "Beitr~ge zur Begriindung der transfiniten Mengenlehre," in Ernst
Zermelo, ed. Gesammelte Abhandlungen mathema~chen und philosophischenInlml.ts (Berlin, 1932;
reprinted, Hildesheim: Olms, 1966).
~6 Russell,Principles of Mathematics, 449.
262 JOURNAL OF T H E H I S T O R Y OF P H I L O S O P H Y ' 3 0 : 2 APRIL 199~
twentieth century, as a "revolt into pluralism."c~ It was a protest, in the first
place, against the monism of Bradley who maintained that only the Absolute is
real. But the revolt in Principles, while maintaining what Russell calls
"pluralism"--the doctrine that many things are real--is yet a timid pluralism,
for the many belong to the one all-inclusive ontological category of terms. It is
clear then why Russell was fascinated by Cantor's theorem and the diagonal
reasoning used in its proof. It shows the most fundamental concept of the
metaphysics o f Principles to be logically inconsistent, because o f an uncritical
use of an absolute and unrestricted notion of all. At the same time, we have
here the beginning of Russell's attachment to the theory of logical types which,
in all of its many varieties, has at its center a rejection of this unrestricted
notion of eveothing.
One of the underlying themes of Principles is that mathematics can be used
in resolving philosophical problems--that mathematics is able to provide sup-
port for or a refutation o f metaphysical doctrines. A great deal of use of
Cantor's work is made by Russell. He says, "In Mathematics, my chief obliga-
tions, as is indeed evident, are to Georg Cantor and to Professor Peano. ''6s
Russell used mathematical analyses of Cantor and Weierstrass to refute Idealis-
tic doctrines about the contradictory character of space and time which con-
demn them as unreal and belonging to the realm o f Appearance. It is ironic
that before he had completed Principles, Russell discovered that Cantor's work
in set-theory could be used to refute fundamental doctrines of his own meta-
physics.

Frege's account of the problem of the unity of the proposition is best seen not
as an attempted solution, but as a dissolution of it. He wants us to view it as a
muddle felt as a problem. Russell's idea of the single, all inclusive, ontological
category of terms makes the problem both inevitable and intractable for him.
In contrast, Frege's logical universe is severely dualistic. T h e r e is an unbridge-
able gap between functions (including concepts) and objects. So severe is this
gap that nothing that can be said about a function is allowed even to make
sense if said about an object, and conversely. Nothing can play the double role
of predicate (or relation) and logical subject which Russell ascribes to the verb.
The concept horse is not a concept, it is an object. Frege finds this embarrass-
ing, but nevertheless--given a grain of salt--ultimately satisfactory. The em-
barrassment is forced upon us by the nature of everyday language and cannot

Bertrand Russell, My PhilosophicalDevelopment (London: Allen and Unwin, 1959), see

Chapter 5.
6, Russell,PrinciplesofMatheraatics,xviii.
 U N I T Y OF T H E P R O P O S I T I O N 263
be avoided since there is no alternative to our colloquial language in which to
give an account of these matters. Frege did not, in publications subsequent to
"On Concept and Object" in 1893, further develop his treatment of the prob-
lem in that essay. He does, however, deepen his analysis in unpublished work
in his Nachlass and in his correspondence with Russell, to which we will turn
below. Russell discusses Frege's views on the problem of the unity of the
proposition in the appendix on Frege in Principles, and he rejects them be-
cause they lead to contradiction, but he is unable to offer any alternative, and
he published nothing further about the problem of unity after Principles.
The previous discussion was not, however, without influence. The early
Wittgenstein was deeply influenced by it in the Tractatua Logico-Philosophicus.
In the preface to this work, Wittgenstein says, "I am indebted to Frege's great
works and to the writings o f my friend Mr. Bertrand Russell for much of the
stimulation of my thoughts.'~ We find in the Tractatus an elaboration of
Frege's treatment of o u r problem which underlies some o f the most funda-
mental doctrines of Wittgenstein's book.
The problem has its origin with Frege and Russell. This is not to deny that
there are similar or analogous problems for Kant, Ltibniz, and the British
Idealists; and for Plato and Aristotle. But the form which the problem takes
for Frege and Russell is peculiar to them because of their special views about
the nature of propositions. These views have their origin in the fact that
Russell and Frege approach their problem from the point of view of the new
logic of which they are the founders. Logic requires a n analysis of proposi-
tions into their logically relevant parts: into functions and arguments, quanti-
tiers and truth-functional connectives. The task of formulating the correct
logical laws cannot be carried through without analyzing propositions into
their logically relevant components, and giving an account of the nature of
these components. Frege was the first to analyze propositions into function
and argument. It is this step which enables him to proceed to his most impor-
tant contribution to logic--the introduction of quantifiers, and an account of
the logical laws which govern them. In the course of such an investigation it is
readily observed that the complete proposition has characteristics lacking to
any of its components. Frege characterized the conceptual content of the full
declarative sentence as differing from that of any of the sub-sentential parts
because it alone has a content which is "judgable." That opposite magnetic
poles attract each other is judgable. "Magnetic poles" has a conceptual con-
tent, but it cannot be judged. Another way of saying this is that the complete
proposition has a unity which is lacking to any of its parts. T h e "unity" of the

c~ Ludwig Wittgenstein, Tractatus Logico-Philosophicus, translated by D. F. Pears and B. F.

McGuiness (London: Routledge and Kegan Paul, ~92~; this translation 1961), 3.
264 J O U R N A L OF THE HISTORY OF P H I L O S O P H Y 3 0 : 2 APRIL 1 9 9 2

proposition just is its assertability or judgability. The problem of the unity of

the proposition is to explain how the parts of propositions, none of which is
judgable or assertable can combine into a whole which has this characteristic
possessed by none o f them. In forming his conceptual notation in the Be-
griffsschrift of 1879 Frege introduced two bits of notation to mark this special
feature of the full declarative sentence. He introduced the horizontal stroke to
be followed by the sign for ajudgable content, and he introduced the vertical
stroke to mark the actual assertion o f the expression preceded by the horizon-
tal stroke. In Principia Mathematica the two signs are melded into one to be-
come the "assertion sign" of that work.
According to Frege, the sub-sentential parts have sense and reference and
so does the full declarative sentence. There is a functional relation between
them. The sense of the full declarative sentence is a function of the senses of
its logically relevant parts. The same functional relation applies in the case of
reference. T h e reference of the full declarative sentence is a function of the
references of its constituent names. But only the sense of the full declarative
sentence is judgable or assertable. How does this sense acquire this feature
lacking to all o f the others? Frege answered that an examination o f the con-
stituent senses will provide the needed explanation. These senses are of two
kinds--complete and incomplete. If there is to be an assertable content it can
only arise through a combination of a complete sense with an incomplete
sense.7o What is an incomplete sense? Frege acknowledges that the terms
"complete" and "incomplete" are used metaphorically in this context, and that
they cannot be eliminated. Language, he says, "leaves us in the lurch," unable
to say what we mean. The metaphors are of use because they provide hints
which enable his readers to see what he means but cannot say.
A great deal in Wittgenstein's Tractatus Logico-Philosophicus bears on the
problem of the unity of the proposition. At 4.221 he says, "It is obvious that
the analysis of propositions must bring us to elementary propositions which
consist of names in immediate combination." He then raises our problem, "Es
fr~igt sich hier, wie kommt der Satzverband zustande." ("This raises the ques-
tion how the propositional combination comes about.") T h e Tractatus contains
a resolution of the problem which is Fregean in spirit and which improves on
Frege's solution because it develops and makes explicit elements of this ac-
count which do not appear in Frege's published work but only in an undevel-
oped and inceptive form in his Nachlass and correspondence with Russell.
The Tractatus acknowledges the unique role of propositions. There are, for
Wittgenstein, only two logically relevant proper parts into which propositions
decompose: proper names and logical constants (connectives and quantifiers).

Frege, "On Concept and Object," in Translationsfrom the PhilosophicalWritings, 54.

 UNITY OF THE PaOPOSITION 265
For Frege, the proposition decomposes into function and arguments: the
completion of the incomplete function by complete objects is the source of the
unity of the propositon. Wittgenstein, by contrast, analyzes atomic proposi-
tions as consisting o f names alone. An atomic proposition is a concatenation o f
names. It thus appears that the Tr~tatus adopts the very view which Frege had
rejected because, at least in part it makes intractable the problem of the unity
of the proposition. If a proposition is just a list of names, how can it have that
characteristic unity which makes it judgable and which no mere list of names
which "hold aloof" from one another possesses?
There is, however, a serious difficulty in Frege's manipulation of the meta-
phors used in his dissolution of the problem. It arises in connection with his
account of higher-level concepts and functions, for example, quantifiers.
What we call first-order quantifiers are analyzed by Frege as second-level
concepts which take first-level functions as arguments. An object is said to "fall
under" a first-level concept, and a first-level concept is said to "fall within" a
second-level concept. We have here an example of Frege's doctrine that noth-
ing that can be said about an object can also meaningfully be said about a
function and conversely. In this case, however, Frege's metaphors lose what-
ever light they shed. In the case o f an object falling under a first-level concept
we have the completion (Frege sometimes also uses the metaphor of satura-
tion) of an incomplete (unsaturated) function by a complete (saturated) object.
In the case of second-level concepts, such as quantifiers, the only way to
deploy these metaphors is to characterize the application of function to argu-
ment as the completion of an incomplete thing with another incomplete thing,
or the saturation of an unsaturated thing with another unsaturated thing. It is
as though putting one unsaturated sponge together with another would pro-
duce a saturated pair of sponges. T h e metaphors have gone entirely limp;
they cast no light whatever. Frege says, "For not all parts of a thought can be
complete; at least one must be 'unsaturated', or predicative; otherwise they
would not hold together."71 The unsaturated predicative part holds together
with other parts which are saturated. But in the case of propositions involving
quantification all the relevant parts may be incomplete. One kind of incom-
pleteness must be capable of completing another kind of incompleteness.
Here, however, the metaphors do no explanatory work whatever. We have an
internal tension between Frege's account of quantification and second-level
predication, and his account of the unity of the proposition.
How are these matters resolved by Wittgenstein in the Tractatus? Here it is
important to consider Wittgenstein's use of Frege's context principle. Wittgen-
stein repeats Frege's context principle both in the Tractatus Logico-Philosophieus

7, Ibid.
266 JOURNAL OF T H E H I S T O R Y OF P H I L O S O P H Y 30.'2 APRIL I 9 9 2

a n d in t h e Philosophical Investigations. It m a d e a p o w e r f u l i m p r e s s i o n u p o n
him. In the Tractatus the principle asserts that "only p r o p o s i t i o n s h a v e sense;
only in t h e n e x u s o f a p r o p o s i t i o n d o e s a n a m e h a v e a m e a n i n g . " v ' It occurs
f o u r times in F r e g e ' s Grundlagen der Arithmetik, a n d is n e v e r explicitly r e p e a t e d
by h i m a f t e r the publication o f that w o r k in x884. S o m e h a v e concluded f r o m
this silence a n d f r o m o t h e r considerations that F r e g e a b a n d o n e d the c o n t e x t
principle a f t e r Grundlagen. T h i s is not the case. M a n y aspects o f Frege's later
w o r k are best u n d e r s t o o d in t e r m s o f his c o n t i n u e d a d h e r e n c e to the princi-
ple. I n d e e d , Frege's contrast b e t w e e n the c o m p l e t e n e s s o f n a m e s a n d objects
a n d the i n c o m p l e t e n e s s o f functions is best e x p l a i n e d in t e r m s o f the c o n t e x t
principle, a l t h o u g h F r e g e h i m s e l f n e v e r cites the principle to this end. Wittgen-
stein, in t h e Tractatus, did a p p r e c i a t e that the context p r i n c i p l e could be so
used. I n d e e d , the " p i c t u r e t h e o r y o f l a n g u a g e " can be u n d e r s t o o d as an elabo-
ration o f F r e g e ' s principle. At 9.o 191, Wittgenstein writes, " I f things can o c c u r
in states o f affairs, this possibility m u s t be in t h e m f r o m t h e b e g i n n i n g . " A g a i n
he writes at 9.o122, " T h i n g s a r e i n d e p e n d e n t in so far as they can o c c u r in all
poss/ble situations, b u t this f o r m o f i n d e p e n d e n c e is a f o r m o f c o n n e c t i o n with
states o f affairs . . . . (It is impossible f o r w o r d s to a p p e a r in two d i f f e r e n t roles:
by themselves, a n d in propositions.)" At 2.0123 he writes, " I f I know an object
I also k n o w all its possible o c c u r r e n c e s i n states o f affairs." T h i n g s (objects) a r e
what n a m e s stand for. H e r e Wittgenstein explicitly c o n n e c t s their context
d e p e n d e n c e ( " I f I k n o w an object I also know all its possible o c c u r r e n c e s in
states o f affairs"; " I f things can o c c u r in states o f affairs, this possibility m u s t
be in t h e m f r o m the b e g i n n i n g " ) with that o f words (names). Wittgenstein
says, " I f all objects a r e given, t h e n at the s a m e time all possible states o f affairs
are given" (~,.o194). J u s t so, f o r Wittgenstein, if all n a m e s are given all possible
propositions a r e given. All o f this is an ontological version o f Frege's principle.
T h e c o n n e c t i o n o f the context principle with the p r o b l e m o f unity is this:
we begin with p r o p o s i t i o n s a n d t h e i r u n i t y - - " O n l y p r o p o s i t i o n s have sense"
(3-3). T h e c o n c e p t o f a n a m e is logically derivative. All o f the constituents o f

7, Wittgenstein, Tractatus Logico-Philosophicva, 3.3. The account of Wittgenstein's use of the

context principle which follows owes a great deal to Palmer's discussion. Palmer says, "The
opening sections of the Tractatus should have made it clear that in Wittgenstein's view this prob-
lem is only overcome when we cease to think of propositions as having, in any ordinary sense of
the word, constituents at all." Cf. Anthony Palmer, Conceptand Object:The Unity of the Propositionin
Logi~and Psychology(London and New York: Roufledge, t988), 49- Indeed much of the discussion
of the Tractatus presented here is but a reformulation of Palmer's views contained in the chapter
of his book entitled "Colorless Objects." Palmer does not relate these ideas to the resolution of
Frege's paradox about the concept horse. In the large recent secondary literature on the topics
dealt with in this paper there are only two works which make Russell's problem of the unity of the
proposition their central topic. One of these is Palmer's book and the other is Hylton's paper
referred to in footnote 76 .
 U N I T Y OF THE P R O P O S I T I O N 26 7

the proposition in the Tractatus are incomplete. The only complete thing is the
proposition. A name, according to this view, is a constituent whose logical
form consists in its powers of combination with other names to form proposi-
tions. Wittgenstein uses the metaphor of the links in a chain. In the atomic
fact, he tells us, "objects hang one in another like the links in a chain" (9.03).
In a chain, there is no need for cement to hold one link to the next. Rather,
the links themselves hold together. So in the proposition there is no glue
needed to hold names of different logical types together. Wittgenstein uses
the image of the chain again when he tells us that an elementary proposition is
a "concatenation" of names.73 The difference in the logical types of names is
just that difference in their logical valences which allows certain combinations
to adhere together and makes other combinations "hold aloof" from each
other. From this point o f view, the problem of the unity of the proposition
arises from an inversion of logical priorities. It arises from the misguided
effort to begin with the constituents of the proposition conceived as indepen-
dent building blocks and to seek for the cement that will hold them together in
the proposition.
Frege, the author o f the context principle, was well aware that his adher-
ence to the principle marked a fundamental distinction between his approach
to logic and the earlier work of Boole. In his posthumously published essay on
"Boole's Logical Calculus and the Concept-Script" of 1880/81 he compares his
approach to that of Boole: "As opposed to this, I start out from judgements
and their contents, and not from concepts . . . . I only allow the formation of
concepts to proceed from judgements.'b4 We arrive at concepts by extraction
from judgments. At the end of his career in 1919, Frege was still insisting on
the importance of the priority of j u d g m e n t or thought as a guiding principle
in his conception o f logic. "So I do not begin with concepts and put them
together to form a thought or judgement; I come by the parts of a thought by
analyzing the thought. This marks o f f my concept-script from the similar

Cs Wittgenstein, Tractatus, 4.2~.

~4 Gottlob Frege, "Boole's Logical Calculus and the Concept-script" in Posthumous Writings,
ed. H. Hermes, F. Kambartei, and F. Kaulbach, trans. P. Long and R. White (Chicago: The
University of Chicage Press, 1079), 16. The priority of judgments over concepts and its impor-
tance for distinguishing Frege's view of the nature of logic from that of the earlier Boolean
tradition is thoroughly investigated by Hans Sluga in "Frege against the Booleans," Notre Dame
Journal of Formal Log/c ~8 (January, 1987): 8o-98. Sluga coins the useful expression "priority
principle" and connects it with Frege's context principle. Frege's use of the priority principle plays
a central role in Thomas Pickett's "Objectivity and Objecthood: Frege's Metaphysics of Judg-
ment," in Frege Synthesized, ed. Leila Haaparanta and Jaakko Hintikka (Dordrecht: Reidel, 1086).
Rickett's concern is with the issue of Platonism in Frege's philosophy. Neither he nor Sluga deals
with the problem of unity.
268 JOURNAL OF T H E H I S T O R Y OF P H I L O S O P H Y 30"2 APRIL 1992
inventions o f Leibniz and his successors, despite what the n a m e suggests;
perhaps it was n o t a very h a p p y choice o n my part."75
Peter H y l t o n gives d u e weight to the context principle in Frege's t r e a t m e n t
o f unity, a n d at the same time h e seems to miss the real work that it does.
Hylton says, "It is, m o r e o v e r , clear f r o m Frege's writing that he did not hold
that the idea o f u n s a t u r a t e d n e s s has any e x p l a n a t o r y power, f o r he i n t e n d e d
that the n o t i o n o f unsaturatedness, and thus also o f a c o n c e p t (and also, I
believe, o f an object), was to be u n d e r s t o o d in terms o f the p r i o r notion o f a
complete t h o u g h t . " ~ Hylton's thought, it seems, is that the notion o f unsatu-
ratedness (and o f a concept, and also o f an object) c a n n o t be d e p l o y e d in
accounting f o r the unity o f the complete t h o u g h t without circularity, because
these notions themselves p r e s u p p o s e that unity. H e thus sees Frege as unable
to solve the p r o b l e m . But this/s the way in which Frege deals with the p r o b l e m
o f unity. F r e g e gives " p r i d e o f place" to the c o m p l e t e thought. Any way o f
thinking a b o u t functions and objects which makes unity problematic o r impos-
sible is t h e r e f o r e mistaken, f o r functions and objects are only intelligible in
terms o f the p r i o r notion. T h e dissolution o f the p r o b l e m results f r o m exhibit-
ing the d e p e n d e n c y .
It is a basic principle o f Frege's t h e o r y that a symbol for an incomplete
expression o u g h t n e v e r to o c c u r without indication o f its a r g u m e n t places
except w h e n it occurs as a b o u n d variable next to the o p e r a t o r that binds it.
Frege r e p e a t e d l y warns against the fallacy o f isolated function s y m b o l s - -
function symbols o c c u r r i n g without indication o f the n u m b e r and types o f
their a r g u m e n t s . In a letter o f N o v e m b e r 13, 19o 4 he writes to Russell, " T o
use a function sign in isolation is to contradict the n a t u r e o f a function, which
consists in its u n s a t u r a t e d n e s s . . . . T h i s is why function n a m e s must d i f f e r
essentially f r o m p r o p e r names, the d i f f e r e n c e being that they carry with t h e m
at least o n e e m p t y p l a c e - - a n a r g u m e n t place. A n d these a r g u m e n t places
must always be p r e s e r v e d in a function n a m e and be recognizable as such."77
D u m m e t t has observed that in terms o f this notational r e q u i r e m e n t , we can
give an a c c o u n t o f what F r e g e means by the r e p e a t e d claim that functions
themselves contain g a p s - - a r e unselbstiindig. T h e s e characteristics o f function
symbols are, by the context principle, reflected in the " r e a l m o f r e f e r e n c e " by

~5 Frege, "Boole's Logical Calculus," u53. Oison's account of Bradley shows him to be very
close to the view here attributed to Frege. "What he [Bradley] does maintain is that relations as
well as the terms they relate are only an abstraction from the relational fact as a whole." Kenneth
Olson, An Essay on Facts, 63. Oison does not attribute the view to either Frege or Wittgenstein.
"~ Peter Hylton, "The Nature of the Proposition and the Revolt against Idealism," in Philoso-
phy and History,ed. R. Rorty, J. B. Schneewind, and Q. Skinner (Cambridge: Cambridge Univer-
sity Press, ,984), 382 n.
7~ Frege, Philosophical and Mathematical Correspondence, 16a.
 U N I T Y OF THE P R O P O S I T I O N 269

the functions which are the referents. Dummett gives the following justifica-
tion of the requirement. The predicate "~ killed ~" is not literally part of any
English sentence in which it occurs, for it is not a word or string of English
words. It cannot literally be identified with any part of any of the three
sentences "Brutus killed Brutus," "Cassius killed Cassius," "Brutus killed Cae-
sar." Yet the kind of logical analysis initiated by Frege finds the predicate "~
killed ~" as occuring in the first two sentences and not in the third. Such a
predicate then is to be regarded as a feature of the two sentences in which it
occurs rather than as a part of them. The predicate "~ killed ~" must be
distinguished from the predicate "~: killed 7/" for the latter is the predicate
which by double quantification yields "Everyone kills someone." This double
quantification cannot arise from the former predicate. Dummett explains in
this way why isolated function symbols are not to be employed. A function
symbol "cannot literally be removed from a sentence a n d . . , displayed on its
own: we can only indicate the common feature of various sentences which we
have in mind by the use, together with words or symbols belonging to the
language, of the Greek letters which represent argument-places. And it is, in
turn, just because the complex predicate is thus not really an expression--a bit
of language--in its own right, that we are compelled to regard it as formed
from a sentence rather than as built up of its components."Ts Dummett here
both gives a quite unproblematic account of Frege's metaphors of complete-
ness, unsaturatedness, Unselbstandigkeit, as applied to function symbols, and
indicates how we can connect this to Frege's account of functions by the
context principle. By taking the context principle seriously, Wittgenstein, in
the Tractatus, following Frege, reverses the order of explanation. The constitu-
ents of the proposition, names, are only arrived at by extraction from the
unified proposition, just as the various organs of an animal body can be
extracted from it. These organs only perform their function in the whole
healthy animal body, and not in separation from it. Just so, a name has a
meaning only in the context of a proposition. A bit of mechanism is a break
only provided the rest of the mechanism is in place. It is only in the "unity" o f
the whole mechanism that a part functions as a break. The unity of the
mechanism--the animal body, the proposition--is not derivative. Rather the
parts are the parts they are only in the functioning whole containing them.
At Tractatus 4.126, Wittgenstein introduces the notion of formal concepts.
He introduces it, he tells us, "in order to exhibit the source of the confusion
between formal concepts and concepts proper, which pervades the whole of
traditional logic." He explains further, "When something falls under a formal

7e Michael Dummett, Frege:Philosophyof Language(Cambridge, Mass.: Harvard University

Press, 19 8 l), 31.
270 JOURNAL OF T H E H I S T O R Y OF P H I L O S O P H Y 30:2 APRIL 1992

concept as one of its objects, this cannot be expressed by means o f a proposition.

Instead it is shown in the very sign for this object" (4.126). Formal concepts are
not genuine concepts. Wittgenstein calls them "pseudo-concepts." He gives, as
an example, "the pseudo-concept object" (4.172). A genuine concept is repre-
sented in a logically correct language (Begriff~chrift), by a predicate, e.g., we can
say that Socrates is wise. But we cannot say, in a such language, that Socrates is
an object, or that wisdom is a concept. When we say that Socrates is an object, we
are trying confusedly to express something which is shown, in a conceptual
notation, by the fact that "Socrates" is allowed to replace an individual variable
and not a predicate variable. That wisdom is a concept is shown by the fact that
"Socrates is wise" is a substitution instance of"q/(Socrates)." But neither "Socra-
tes is an object" nor "Wisdom is a concept" can be written in a language obeying
the rules o f logical syntax. This is what Wittgenstein means when he speaks of
"the pseudo-concept object." Another pseudo-concept is the concept concept.
In a correct logical notation, Frege's embarrassing sentence "The concept
horse is not a concept" cannot be written. That horse is a concept is shown by its
use, but it cannot be said. Thus the "grain of salt" for which Frege asks is
required because of a feature of everyday language which is not reproduced
in any language constructed according to the rules of logical syntax. Frege's
embarrassment can be viewed as due to a defect of ordinary languagemas
Frege said it was. Wittgenstein says, "The same applies to the words 'complex',
'fact', 'function', 'number', etc. They all signify formal concepts, and are repre-
sented in conceptual notation [Begriffsschrift] by variables, not by functions or
classes (as Frege and Russell believed)" (4.1272). The paradox of the concept
horse arises from a defect of ordinary language. Our colloquial language obeys
a very liberal nominalization rule which allows us to nominalize adjectives,
common nouns, and verbs in the freewheeling fashion which produces "the
concept horse" from the predicate of the sentence "Bucephalus is a horse." The
syntax of our language is not a logical syntax.
Much of this is anticipated by Frege in his correspondence with Russell
published posthumously long after the appearance of the Tractatus. In a letter
of June 29, a9o2, he writes, "Here we find ourselves in a situation where the
nature of language forces us to make use of imprecise expressions. The propo-
sition 'A is a function' is such an expression: it is always imprecise; for 'A'
stands for a proper name." Frege then goes on to indicate how in a conceptual
notation we can show correctly what we cannot correctly say. "In a conceptual
notation we can introduce a precise expression for what we mean when we call
something a function . . . . e.g., 'k ~p(e)'. Accordingly, 'k (e 9 3+4) ' would ex-
press precisely what is expressed imprecisely in the proposition '~. 3+4 is a
function'." Wittgenstein makes the distinction between what shows itself in
language and cannot be said, on the one hand, and what can be said on the
 UNITY OF THE PROPOSITION 271

other, central to the Tractatus. It is clearly anticipated in this aspect o f Frege's

philosophy. I n a f o o t n o t e to this passage, the editors o f the c o r r e s p o n d e n c e
explain Frege's notation thus: "By using the spiritus asper instead o f the s/r/titus
/en/s which he customarily uses in this place, Frege a p p a r e n t l y wants to distin-
guish the expressions in question f r o m his n a m e s f o r r a n g e s o f values."79
In a letter o f J u l y 28, 19o2, F r e g e anticipates Wittgenstein's idea o f f o r m a l
concepts: " T h e difficulty in the p r o p o s i t i o n 'A function n e v e r takes the place
o f a subject' is only an a p p a r e n t one, occasioned by the inexactness o f the
linguistic expression; f o r the w o r d s ' f u n c t i o n ' a n d 'concept' should p r o p e r l y
s p e a k i n g be r e j e c t e d . . , they p r e s e n t themselves linguistically as n a m e s o f
first-level functions. It is not s u r p r i s i n g that we r u n into difficulties in using
t h e m . ''s~ F r e g e is saying that the w o r d s ' f u n c t i o n ' a n d 'concept' mislead us, in
colloquial language, by allowing such propositions as 'Horse is a concept.' It is a
short step f r o m this to declare t h e m p s e u d o - c o n c e p t s m f o r m a l c o n c e p t s m a n d
Frege has already indicated how, in c o n c e p t u a l notation, we can show w h a t we
mistakenly try to say with t h e m . F r e g e m a k e s the s a m e points in an u n p u b -
lished review o f 19o6. "So l a n g u a g e b r a n d s a c o n c e p t as an object, since the
only way it can fit the d e s i g n a t i o n f o r a c o n c e p t into its g r a m m a t i c a l s t r u c t u r e
is as a p r o p e r n a m e . B u t in so doing, stricdy s p e a k i n g it falsifies matters. I n the
same way, the w o r d ' c o n c e p t ' itself is, t a k e n strictly, already defective, since no
p r o p e r n a m e can d e s i g n a t e a concept; or b e t t e r still, s o m e t h i n g nonsensical ''8'
We s e e m to be on the h o r n s o f a d i l e m m a . In the sentence " T w o is a p r i m e
n u m b e r " we d e s i g n a t e a relation which the n u m b e r two has to the c o n c e p t
prime number. T h a t relation is the relation o f s u b s u m p t i o n , b u t in these last two
sentences we h a v e r e f e r r e d not to a c o n c e p t a n d a relation but to their object
correlates, so that we h a v e not s u c c e e d e d in saying what was intended. Collo-
quial l a n g u a g e forces us to miss o u r m a r k . But because the concepts relation,
concept, object are all defective p s e u d o - c o n c e p t s they c a n n o t be i n c o r p o r a t e d
into a logically c o r r e c t l a n g u a g e either. T h e r e is no l a n g u a g e in which we can
say what we a t t e m p t to say in o r d e r to give an account o f the w o r k i n g either o f
e v e r y d a y l a n g u a g e o r o f the Begriffsschrifi. Evidendy, what we intend to say
cannot be said.

Frege, Philosophicaland MathematicalCorrespondence, 136 n.

io Ibid., 141.
e, Ibid., 177-78. Thus Frege later came to see that the paradox about the concept horse was
not a necessary consequence of his theory of functions. Dummett has made the same observation.
"The same of course applies to the words 'relation' and 'function'. These words ought therefore
simply to be dropped, as being quite unsuitable for the work they were supposed to do." Michael
Dummett, Frege:Philosophy of Language, 213. Dummett does not connect this with Wittgenstein's
idea of formal concepts and his distinction between saying and showing. Frege's correspondence
with Russell had not been published at the time of the first edition of Dummett's book, so there is
no discussion of the letters quoted here.
:~7 9 J O U R N A L OF T H E H I S T O R Y OF P H I L O S O P H Y 3o:2 APRIL 1992

H e r e we have Frege's diagnosis o f the source o f the infinite regress which

m a d e its a p p e a r a n c e with Bradley and has surfaced again with e v e r y a u t h o r
who has a d d r e s s e d the p r o b l e m o f unity. "This also creates the impression
that the relation o f s u b s u m p t i o n is a third e l e m e n t s u p e r v e n i e n t u p o n the
object and the concept. This isn't the case; the unsaturatedness o f the c o n c e p t
brings it a b o u t that the object, in effecting the saturation, engages immedi-
ately with the concept, without n e e d o f any special cement. Objects and con-
cepts are f u n d a m e n t a l l y m a d e f o r each other, and in s u b s u m p t i o n we have
their f u n d a m e n t a l union. TM
Frege, a n d following him Wittgenstein, both accept the d i l e m m a described
here, but they p r o v i d e a partial escape. I f we give F r e g e the grain o f salt he
requests a n d m e e t him halfway t h e n he is able to use e v e r d a y language, with
all o f its defects, to show us what h e cannot say. Wittgenstein d e v e l o p e d this
distinction, which we find already in Frege's writings, and m a d e it f u n d a m e n -
tal in his Tractatus. Wittgenstein is constrained to convey his doctrines in
colloquial language, which leaves him no alternative but to miss his m a r k in
what he literally says, but he is, nevertheless, able to show what he c a n n o t say.
Wittgenstein generalizes Frege's account to all o f the philosophical re-
marks in his book. "My propositions are elucidations in the following way:
a n y o n e who u n d e r s t a n d s me eventually recognizes them as nonsensical, when
he has used t h e m - - a s s t e p s - - t o climb u p b e y o n d them. (He must, so to speak,
throw away the l a d d e r a f t e r h e has climbed u p it.) H e must transcend these
propositions a n d t h e n h e will see the world aright."ss Wittgenstein's l a d d e r is
Frege's grain o f salt.

University of Chicago

APPENDIX: T H E UNITY OF THE CUBE.

A cube consists o f six sides and twelve edges. It d e c o m p o s e s u n d e r analysis
into these eighteen constituents. W h a t accounts for the unity o f the cube?

s, Ibid., 178.
ss Wittgenstein, Tractatus Logico-Philosophicus, 6.54. I have been much influenced here by
Weiner's account of "elucidations" in the philosophy of Frege. She does not connect this with
Wittgenstein's doctrine of showing. Cf. Joan Weiner, Frege in Perspective(Ithaca and London:
Cornell University Press, x99o), Chap. 6. Other authors have connected Frege's finding that
language leaves him in the lurch and Wittgenstein's doctrine of showing; perhaps the first is
Geach. Cf. Peter Thomas Geach, "Saying and Showing in Frege and Wittgenstein," in Essaysin
Honour of G. H. yon Wright, ed. Jaakko Hintikka, ActaPhilosophtcaFenn/ca 28 (Amsterdam: North
Holland, 1976). Diamond has explored the connection between Frege's salt and Wittgenstein's
ladder in a very subtle essay. Cf. Cora Diamond, "Throwing Away the Ladder," in Philosophy63
(1988): 5-27. All of these accounts differ from the present one in not connecting these issues with
the problem of the unity of the proposition.
 UNITY OF THE PROPOSITION 273
Well, each edge connects two sides. It is the edges that account for the unity.
How does an edge connect two sides? That is the problem of unity. According
to the Bradleyan, each edge attaches to two sides, one on each surface. But in
order for an edge to attach to the sides, there must be two more edges to effect
the attachment to each side. We are thus hurried off into the eddy o f an
infinite process. We were wrong to think that the cube consists of any finite
number of components. Analysis is impossible.
The Russellian distinguishes between the edge as actual edge connecting
the two sides and the edge in itself--as another constituent of the cube along
with the sides. It is the edge as actual edge which accounts for the unity. But it
too is another constituent of the cube. There are parts of the cube, the sides,
which play the single role o f the objects connected and never the role of what
does the connecting. There are other constituents, the edges, which play the
double role. They both do the connecting as actual edges, and they are con-
stituents along with the sides o f the cube. What the difference of the edge as
connecting edge and the edge as constituent of the cube is, the Russellian is
unable to say.
The Fregean says that nothing plays the double role of Russellian edges.
The constituents o f the cube each play a single role, but these roles are radi-
cally different. What are connected are the sides. They are connected by the
edges. Edges are not constituents of the cube in the same sense as the sides.
We can, for example, pick up the sides and move them about so as to form
other objects such as three rectangles, each of which consists of two of the
sides of the cube divided by an edge. But we cannot pick up a side and leave
the edges behind. An edge is really nothing in itself, or we can say that it is an
incomplete thing in contrast to the sides of the cube which are complete
(saturated) objects which can be moved about into different combinations. An
edge is a structural feature of the cube and not a part of it in the way a side is.
It is not as if we could construct a cube by going to one drawer for the six sides
and to another for the twelve edges and putting them together. Rather the
Fregean gives "pride of place" to the whole cube. We start with that and then
discern certain features, the edges, and certain parts, the sides. They are not
all constituents in a univocal sense. A feature is not a part--nothing can play
this double role. An edge is not a selfsubsistent object.