WRITERS AND CRITICS - BERTRAND RUSSELL.
by John Watling.
NOTE: The pagination of this text corresponds as closely as possible to the original
1970 edition of Oliver and Boyd, Edinburgh.
Bertrand Russell was born in 1872. His mother died only two
years afterwards, and his father a year and a half after that. His
parents held advanced opinions on matters of social reform. They
supported universal suffrage and rejected the view that birth
control was immoral, and they were actively engaged in
discussing and furthering these views.1 They were acquainted
with some of the leading progressive thinkers of the time,
including John Stuart Mill. His father’s will left both Russell and
his brother in the care of two men – one of them had been tutor
to his brother – who were both atheists. Russell’s paternal
grandparents did not share his parents’ progressive views. They
had little difficulty in obtaining custody of the children, and
Russell was brought up by his grandmother, for his grandfather
died a few years later. This grandfather, Lord John Russell, was
eighty-three when Russell’s father died. He had been Prime
Minister during two Liberal administrations, and before he
became Prime Minister had introduced the First Reform Bill. He
and his wife lived in Pembroke Lodge, a grace-and-favour house
in Richmond Park. There are still such houses in Richmond Park,
but Pembroke Lodge and its gardens are now open to the public.
Russell spent his childhood in this house where his grandparents
were visited by many eminent people, chiefly from political life.
He was not sent to school until he was sixteen, when he went to
an army crammer to prepare for a Cambridge scholarship. In his
later childhood he was intensely interested in mathematics, as
well as in history and poetry. When he was eighteen, he sat the
scholarship examination for Trinity College, Cambridge, gained a
scholarship, and went up to Trinity a year later. Before going up
The Autobiography of Bertrand Russell 1872-1914, pp. 15-17.
he read J. S. Mill’s Autobiography, Political Economy and A
System of Logic, Herbert Spencer’s The Man Versus the State, and
some works of Henry George.
At Cambridge Russell obtained a First in Mathematics
after three years and a First in Moral Sciences after another one.
In his second year, he was elected to a semi-secret discussion
society – known to outsiders as “The Apostles” – which included
both undergraduates and dons and whose members were chosen
as undergraduates, one or two a year, for their intellectual
brilliance. Through this society Russell quickly became
acquainted with people of ability and like interests, for A. N.
Whitehead, who had examined for the scholarships, had been
impressed by Russell’s papers and had suggested to other
members of the society that they should call on him. Besides
Whitehead, he came to know Ellis McTaggart and, later, G. E.
Moore. Between the summers of 1894 and 1895, Russell spent
three months as an attaché at the British Embassy in Paris, got
married, spent three months studying at a marxist socialist
movement in Berlin, and wrote a dissertation on the foundations
of geometry, on which he was awarded a Fellowship of Trinity.
The Fellowship did not entail teaching duties and Russell and his
wife spent only part of their time at Cambridge; however he
lectured on Leibniz in 1899. In the summer of 1900, he attended
an International Congress of Philosophy in Paris with Whitehead,
and so became acquainted with the Italian mathematician G.
Peano. Russell had been thinking about the foundations of
mathematics since he was an undergraduate and Peano’s ideas
inspired him to start work on the book which became The
Principles of Mathematics. He had completed the first draft by the
end of the year, and felt some satisfaction at having brought the
problem of the nature of mathematics so close to a solution
before the end of the nineteenth century. It was also about this
time that he became a friend of the French logician and
philosopher Louis Couturat. Until 1910, Russell was concerned
with the writing of The Principles of Mathematics and the
Principia Mathematica. He undertook the immense labour of
writing out the manuscript of the Principia single-handed, but
still found time to write a few articles on other philosophical
topics and to engage in some political activity. He made public
speeches on behalf of Free Trade and contested a by-election as a
women’s suffrage candidate in 1907. He was elected a member of
the Royal Society in 1908. After the publication of the Principia,
Russell sought again to enter politics. He was a member of the
Fabian Society2 but his views cannot have squared very well with
those of its other members, for Russell has throughout his life
been an opponent of state control. In 1910, he sought to become a
Liberal candidate but was rejected because of his atheism. After
this he turned to academic life and accepted a Lectureship at
Trinity. Russell has said that the writing of the Principia damaged
his capacity for abstract thought, but the period from 1910 until
1914 was, philosophically, one of his most productive. It was
during this time that Wittgenstein came to Cambridge to study
the foundations of mathematics under Russell.
About this time Russell gave away nearly all the money
he had inherited. Since then he has earned his living either by
academic teaching, by writing books, or by journalism.
Russell says that he had become a pacifist in 1901,
having until then supported the Boer War. During the next
decade he was dismayed by the movement towards war with
Germany, and when war was declared he was horrified both by
the immediate and the more distant consequences he believed it
would bring. He became an active anti-war propagandist,
advocating that individuals should refuse war service. He was an
equally active opponent of conscription and of the treatment
meted out to conscientious objectors. In 1916, he was fined £100
for “statements likely to prejudice the recruiting and discipline of
His Majesty’s Forces”. These were alleged to have been made in a
leaflet issued by the No-Conscription Fellowship concerning a
conscientious objector who had been sentenced to two years’
imprisonment. Immediately after this case he was dismissed from
his lectureship at Trinity College. He was refused a passport to
enable him to travel to the United States to lecture at Harvard
University, and restrictions were placed on his freedom of
movement in the British Isles. Another prosecution arose from an
article Russell wrote in the No-Conscription Fellowship’s weekly
Alan Wood, Bertrand Russell the Passionate Sceptic, 1957, p.
paper. The article was held to be defamatory of the British and
American armed forces, and Russell was sentences to six months’
imprisonment. Because of the influence of his friends and
relations he experienced little of the extreme mental and physical
hardship of other prisoners at the time. His cell was furnished by
his brother’s wife and provided with fresh flowers, and he was
able to devote his whole time to study. He wrote his book
Introduction to Mathematical Philosophy. He published no
philosophical work in 1916 and 1917.
In 1919, Russell accepted an offer from Trinity to re-
instate him, but he applied for a year’s leave of absence in order to
lecture at Peking University and, on his return, resigned his
lectureship. He travelled to Russia as well as to China and wrote
books about both countries. Soon after his return he twice stood
as Labour candidate for Chelsea, but was not elected. In 1927, he
started Beacon Hill School with his second wife, Dora. Russell’s
ideas on education fell significantly short of those of the most
advanced thinkers of the time, such as Homer Lane and A. S.
Neill, but the school allowed and encouraged great freedom of
expression. It does not seem to have been a great success, perhaps
because neither Russell nor his wife gave it their full attention,
and perhaps because they lacked the ability to deal with the
difficult children which a progressive school is likely to attract.
Dora Russell continued to run the school after their marriage
broke up in 1932. Russell became the 3rd Earl Russell upon the
death of his brother in 1931. He seems to have inherited little but
liabilities from his brother and at that time it was impossible to
renounce a title.
In 1938, Russell went to the United States and he lived
there until 1944. He held the post of Visiting Professor at the
University of Chicago in 1938, and at the University of California
in 1939. While in California he accepted a post as Professor of
Philosophy at New York City College, but before he took up the
appointment an action was brought against the College by a New
York tax-payer to annul it. Russell was not a party to the
proceedings and was not allowed to take any part in them. The
action was successful and the appointment was annulled on three
grounds: first, that Russell was not an American; second, that he
had not obtained the post by competitive examination; and third,
that he had put forward “immoral and salacious doctrines” in his
books. Harvard University resisted pressure to cancel an
invitation to give the William James Lectures in 1940. These
lectures were published as An Enquiry into Meaning and Truth.
From 1940 until 1943, Russell lectured on the history of
philosophy at the Barnes Foundation in Pennsylvania; he had a
five-year contract but was dismissed at short notice and
successfully sued for wrongful dismissal. His lectures at the
Barnes Foundation formed the basis of The History of Western
Philosophy, which figured several times in the United States best-
sellers lists. In 1944, Trinity College offered him a Fellowship and
Lectureship and he returned to England. He gave lectures at
Trinity which he published as Human Knowledge, its Scope and
Limits. This was his last large-scale contribution to philosophy. In
contrast to his reputation in America, Russell became a respected
figure in Britain, giving the Reith Lectures for the B.B.C. in 1948,
and receiving the Order of Merit in 1949. In 1950, he received the
Nobel Prize for Literature.
In 1940, Russell renounced his pacifism, although not
without misgivings. In 1914, he had thought the domination of
Europe by the Kaiser’s Germany preferable to war; in 1939, he
thought the domination of Europe by Nazi Germany worse than
war. On similar grounds, he supported British and American
rearmament after the war, going so far as to advocate that Russia
should be compelled by the threat of force to accept the
provisions of the Baruch Plan for the internationalisation of
atomic energy.3 He continued in this opinion throughout the
Korean War, but towards the middle of the ’fifties he began to
speak against the possession of atomic weapons by any state. He
argued that it was possible to produce scientific evidence to prove
that in a future war no nation could be the victor, so that no
nation could achieve its aims by war. He proposed an
international conference of scientists to report on the probable
results of a nuclear war.4 This proposal was sponsored by an
American, Cyrus Eaton, and the conference was held on his estate
at Pugwash, Nova Scotia. This was the first Pugwash Conference
Christopher Driver, The Disarmers, 1964, p. 18.
Portraits from Memory, pp. 221-7.
and, according to I.F. Stone’s Weekly, not a single American
newspaper published its report.5 After this, Russell became more
and more active in the movement to ban the testing of atomic
weapons and, later, the weapons themselves. He was a sponsor of
a number of organisations, including the Campaign for Nuclear
Disarmament, of which he became the first president. When
C.N.D. split on the issue of civil disobedience, he resigned his
presidency and was immediately elected president of the more
militant Committee of 100. He took part in the first of the
Committee’s sit-down demonstrations against the government’s
decision to provide facilities for the nuclear-missile-firing
submarines of the United States. Later in 1961, he was summoned
to court, together with other members of the Committee, and
required to bind himself over to keep the peace for one year.
Together with most of the others summoned he refused, and was
sentenced to two months’ imprisonment. This was reduced to one
week on medical evidence. At this time Russell was eighty-nine
Early in the ’sixties, Russell became convinced, not only
that the United States’ cause in Vietnam was unjust, but that it
was being furthered in a barbarous manner. He became actively
engaged in an attempt to bring the facts about the war to the
notice of the American public and the rest of the world. These
efforts culminated in the first session of Russell’s international
War Crimes Tribunal in Stockholm in 1967. This Tribunal was
based upon a speech made at the Nuremburg war crimes trials by
the chief prosecutor, a judge of the U.S. Supreme Court:
If certain acts and violations of treaties are crimes, they
are crimes whether the United States does them or
whether Germany does them.6
The members of the tribunal were writers, scientists, politicians,
and lawyers, many of them with an international reputation. The
Tribunal heard evidence and examined the people who presented
it; these included observers which it had itself sent to Vietnam.
In the ’sixties, the prodigious stream of publications
which Russell produced during the previous seventy years has
Christopher Driver, The Disarmers, p. 29.
War Crimes in Vietnam, p. 125.
diminished, but still averages at least one a year. In 1967, 1968
and 1969, he published three volumes of autobiography. These
take the form of an anthology of autobiographical pieces, written
at different times and many of them reprinted verbatim from
earlier works, together with a large number of letters written or
received by Russell. They succeed by the direct, simple and witty
fashion in which Russell tells many aspects of his life, his relations
and his distinguished friends.
Of all the various subjects on which Russell has written –
and these include politics, pacifism, marriage and education – the
present book considers only one: philosophy. It deals mainly with
his writings between 1900 and 1920, concerning the philosophy of
logic and mathematics and of our apprehension of reality. It is
these books and articles which contain his truly original work.
Note. This book was in the press when Russell died on 2 February
1970 at the age of 97.
Russell’s first book on philosophy was An Essay on the
Foundations of Geometry, the published version of his Fellowship
dissertation. This essay presents an account of the nature of space
which is a modification of Kant’s. I shall begin, however, with his
second book, A Critical Exposition of the Philosophy of Leibniz.
The book arose out of the course of lectures which Russell gave at
Cambridge in 1899, and it was published a year later. It is
important for the light it throws on Leibniz’s philosophy and for
the doctrines which Russell set against those doctrines of Leibniz
he held to be mistaken. It is important also because of the
influence which the study of Leibniz exerted on the development
of Russell’s own philosophy. This influence reveals itself in a
number of details but also in one fundamental respect. Russell
discovered that Leibniz’s metaphysical doctrines had their
foundation in logic: the philosophy which Russell himself
developed had this same foundation.
Russell tells, in the preface to his book, how Leibniz’s
Monadology seemed to him like a fairy tale: a fantastic, although
coherent, picture of the world which there was no reason to
accept, and which gave no clue as to Leibniz’s reasons for
proposing it. Russell found the reasons he was seeking in the
Discourse on Metaphysics and in the letters to the theologian
Arnauld. These works convinced him that the philosophy of the
Monadology was deduced from a few doctrines concerning the
nature of propositions; doctrines which, apart from the
conclusions which Leibniz drew from them, might seem
unexceptionable. What is more, Russell held that the deductions
were, for the most part, valid, and that Leibniz’s logical doctrines
do have as a consequence a large part of the philosophy of the
Monadology. The logical doctrines themselves, however, Russell
The fairy-tale nature of the Monadology can hardly fail
to strike anyone reading it for the first time. Leibniz held that
nature was composed of simple substances which he called
monads. These monads were without parts, so that they did not
have any extension and shape, and were indivisible. They did,
however, have qualities. Indeed each had an infinite number of
qualities, for no two monads were alike and there were an infinite
number of them. In one way each monad was a world apart, for
no change in a monad could be produced by the influence of any
other; but in another way they formed a unity, for each monad
reflected the state of every other, so that in the qualities on any
one monad the truth about the whole universe could be read.
Each monad developed by an internal principle, but the changes
in all the monads fitted together into a perfect order so that the
monads gave the appearance of influencing one another. This
order was pre-established by one of the monads, God, who
Russell did not believe that the whole of this system
followed from the logical doctrines of the Discourse on
Metaphysics. Two non-logical principles were required, and apart
from these, certain inconsistencies were present. Some arose from
Leibniz’s deference to the prevailing religious opinions. For
example, Russell pointed out that Leibniz’s doctrine that men
pursue what appears to them to be the greatest good is
inconsistent with the existence of sin, which Leibniz accepted.
Russell indeed had a low opinion of Leibniz’s honesty, holding
that he chose his arguments for their persuasiveness rather than
their validity. Other doctrines of the Monadology, for example the
view that there is more than one substance, are equally consistent
with Leibniz’s first principles but were genuinely held.
Nevertheless, Russell considered that the logical doctrines of the
Discourse on Metaphysics formed the principle foundation of
That all sound philosophy should begin with an analysis of
propositions, is a truth too evident, perhaps, to demand a proof.
That Leibniz’s began with such an analysis, is less evident, but
seems to be no less true.7
The logical doctrines which, Russell held, constituted Leibniz’s
analysis of propositions were three:
I. Every proposition has a subject and a predicate.
II. A subject may have predicates which are qualities
existing at various times. (Such a subject is called a
III. True propositions not asserting existence at particular
times are necessary and analytic, but such as assert
existence at particular times are contingent and
synthetic. The latter depend upon final causes.8
Premise I needs further amplification. It is doubtful whether
Premise II expresses Leibniz’s most fundamental definition of
what a substance is. There is considerable doubt whether Premise
III correctly presents Leibniz’s view. What is not in doubt is this:
that each of these premises concerns a matter which is
fundamental in Leibniz’s thought and that his views on these
three matters did, to a very large extent, determine his
philosophy. Russell’s identification of these fundamental
problems was, in itself, an important contribution to the study of
Russell had no difficulty in finding, in the Discourse on
Metaphysics and Leibniz’s letters to Arnauld, passages which
contained Premise I. In his letter to Arnauld dated 14 July 1686
Finally, I have given a decisive reason which, in my opinion,
takes the place of a demonstration; that is, that always in every
affirmative proposition whether veritable, necessary or
contingent, universal or singular, the concept of the predicate is
comprised in some sort in that of the subject. Either the predicate
is in the subject or else I do not know what truth is.9
P.L., p. 8.
P.L., p. 4.
Leibniz, Basic Writings, tr. G. R. Montogomery, 1962, p.132.
Russell held that the importance of Premise I lay in its
consequences for relational and for numerical propositions. Both
of these sorts of proposition played a large part in Leibniz’s
philosophy; he did not deny their possibility but supposed that
both were species of subject-predicate proposition and would
appear as such if properly understood. Here again, Russell drew
attention to a passage of considerable importance.
The ratio of proportion between two lines L and M may be
conceived three several ways; as a ratio of the greater L to the
lesser M; as a ratio of the lesser M to the greater L; and lastly, as
something abstracted from both, that is as the ratio between L
and M, without considering which is the antecedent and which
the consequent; which the subject, and which the object. … In
the first way of considering them, L is greater is the subject; in the
second, M the lesser is the subject of that accident which
philosophers call relation or ratio. But which of them is the
subject in the third way of considering them? It cannot be said
that both of them, L and M together, are the subject of such an
accident; for if so, we should have an accident in two subjects,
with one leg in one, and the other in the other; which is contrary
to the notion of accidents. Therefore we must say that this
relation, in this third way of considering it, is indeed out of the
subjects; but being neither a substance, nor an accident, it must
be a mere ideal thing, the consideration of which is nevertheless
Leibniz insists on regarding the fact that L is longer than M as the
fact that L has the predicate “is longer than M”, or that M has the
predicate “is shorter than L”, but not as the fact that the relation
“is longer than” stands between them. Russell produced no
argument against this attitude until his next book The Principles
of Mathematics. There he offers several arguments of which the
following is the simplest, and perhaps the most cogent. No
predicate of L which does not involve a reference to M can imply
a relation between them, while anything which does involve a
reference to M is not merely a predicate of L.11 That Leibniz
Leibniz, The Leibniz-Clarke Correspondence, ed. H. G.
Alexander, 1956, p.71.
P.O.M., p. 222.
regarded “being longer than M” as a mere predicate of L is clear,
as Russell points out, from the passage quoted above. It is also
implied by Leibniz’s view that in the predicates of any one
substance the whole universe is revealed. Once the predicates of
any substance are given, then its relations to all other substances
are given too. I have not found this particular point in Russell,
although he quotes selections from the passage in the letter to
Arnauld of 14 July 1686, in which it occurs.
Russell was able to cite passages which established that
Premise I led Leibniz to his doctrine of the identity of
indiscernibles, but he could nowhere find the argument set out.
He makes a suggestion as to how the argument ran, and it seems
very likely that he is right.12 Consider two substances, let us cal
them A and B. Between A and B, the argument goes, there will
hold a relation which, in the logician’s phrase, implies diversity.
For example, A may be longer than B, or it may be at a distance
from B. Each of these relations implies diversity, for A cannot be
longer than itself, nor at a distance from itself. Doubts about the
assumption that at least one such relation must hold may be
answered by pointing out that whatever relations between A and
B are lacking, one at least must hold: that of not being the very
same thing. This, in fact, is the relation to which Russell points,
but the assumption that “not being the very same thing” is a
relation may reasonably be regarded with suspicion. Premise I
implies that a relation between A and B must be regarded as a
predicate of one of them, or perhaps as a pair of predicates, one of
each of them. The latter seems more reasonable, since the relation
concerns each. If this is so, Premise I implies, for the relation we
are discussing, that if A and B are two different things, then A
lacks the predicate of being the very same thing as B, and B lacks
the predicate of being the very same thing as A. Now A is the very
same thing as itself, and this relation must be regarded as a
predicate of A, so that A will possess the predicate of being the
very same thing as A, and B the predicate of being the very same
thing as B. Therefore the predicate of being the very same thing as
A is a predicate which A has and B lacks, while the predicate of
being the very same thing as B is a predicate which B has and A
P.L., p. 58.
lacks. From the assumption that they are not the same thing we
have deduced that they are not exactly alike.
Against this argument Russell contends that, whatever
relations can be reduced to difference of predicates, the relation of
numerical diversity cannot:
For the numerical diversity of the substances is logically prior to
their diversity as to predicates: there can be no question of their
differing in respect of predicates, unless they first differ
He concludes that if there exist two things, then there will be at
least one relation between them which is not reducible to
predicates, so that Leibniz’s doctrine that there are no relational
propositions should have led him to the conclusion that there is
only one substance. When Russell argues that numerical diversity
is logically prior to diversity as to predicates, his point must be
that having different predicates and having the same predicates
both imply diversity, and hence that the diversity of two things
cannot be due to their possession of different predicates. This
argument of Russell’s is no more than a denial of Leibniz’s
conclusion. It is true to say that two things have the same
predicates implies that they are two, but that does not establish
that their being two is consistent with their having the same
predicates. It is this consistency which Leibniz denies.
In the light of the logic which Russell’s later work did so
much to establish, the doctrine that all propositions have a subject
and a predicate is refuted by the existence of universal
propositions, such as that all men are mortal. In that logic this
proposition has the import that everything is either not a man or
mortal. This proposition does not, on the face of it, have a subject
and a predicate. However, in the present work Russell is prepared
to regard it having as subject the property of being a man, and as
being the proposition that this property implies, or involves, the
property of being mortal.14 It seems that Leibniz believed that his
general view that all propositions had a subject and a predicate
left room for further enquiry as to the nature of such
propositions. Couturat later published a number of papers by
Leibniz in which various approaches to the logic of universal
P.L., pp. 58-9
P.L., p. 15.
propositions are sketched. On the whole, however, it seems that
the view that Leibniz most often expressed and most wished to
hold was the one which Russell attributed to him.
Russell’s main failure lies in his interpretation of
Leibniz’s views about propositions concerning individual people
and things. Russell himself wrote in his preface to the second
edition of The Philosophy of Leibniz:
Wherever my interpretation of Leibniz differed from that of
previous commentators, Couturat’s work afforded conclusive
confirmation… But Couturat carried inorthodoxy further than I
had done, and where his interpretation differed from mine, he
was able to cite passages which seemed conclusive.15
The trouble arises over Premise III, which is amplified by Russell
in a number of passages. He takes it as equivalent to the view that
a proposition to the effect that being a triangle implies having
angles adding up to 180 degrees is an analytic and necessary
proposition, while one to the effect that Caesar invaded Britain in
55 B.C. is synthetic and contingent. While he is developing an
objection to Leibniz’s doctrine of substance he says:
Moreover, if this were the case, predications concerning actual
substances would be just as analytic as those concerning essences
contrary, Russell implies, to Leibniz’s actual views. Now Leibniz
often speaks of predicates of individual subjects as being “within
the subject”, and Russell regarded a proposition whose predicate
is within the subject as an analytic proposition.17 Moreover, not
only does Leibniz, in passages quoted by Russell, speak of true
propositions about individuals as being those, and only those,
where the predicate is contained within the subject, but at least
one of the deductions by which, according to Russell, Leibniz
obtained his metaphysical conclusions explicitly involves this
premise. I have in mind the passage where Russell outlines the
deduction of the conclusion that each monad, or simple
substance, is a world apart, in which no change could be
produced by any other. Consider the proposition that Caesar
invaded Britain in 55 B.C. If this proposition is true, this can only
P.L., p. v.
P.L., p. 50.
P.L., p. 17.
be because the notion of invading Britain in 55 B.C. is contained
within the notion of the subject, Caesar. But if that is the
explanation of its truth, then the explanation cannot lie in the
influence of any other substance upon Caesar. As Russell puts it,
closely following a passage in the letter to Arnauld dated 14 July
1686, from which I quoted earlier:
Every predicate, necessary or contingent, past, present or future,
is comprised in the notion of the subject. From this proposition it
follows, says Leibniz, that every soul is a world apart; for every
soul, as a subject, has eternally, as predicates, all the states which
time will bring it; and thus these states follow from its notion
alone, without the need of action from without.18
This passage shows that Russell himself doubted that Premise III
correctly represented Leibniz’s views. Yet in other places he
firmly attributes to Leibniz the view of Premise III.
Russell assumed – and it is an assumption which there is
some evidence for but more against – that Leibniz identified
being necessary with being analytic, and hence being contingent
with being synthetic. He held that Leibniz took the propositions
of logic and mathematics to be analytic, but those of physics to be
synthetic. Russell says:
The discovery which determined his views on this point was, that
the laws of motion, and indeed all causal laws… are synthetic,
and therefore in his system, also contingent.19
However, the passage which Russell cites expresses only the
opinion that causal laws are contingent. It says nothing about
whether they are synthetic:
[Dynamics] is to a great extent the foundation of my system; for
there we learn the different between the truths whose necessity is
brute and geometric, and truth which have their source in fitness
and final causes.20
Couturat’s view that Leibniz did not identify necessity with
analyticity but, believing all true propositions to be analytic, saw
P.L., pp. 10-11.
P.L., p. 16.
Leibniz, quoted in P.L., p. 209.
the distinction between necessity and contingency as a distinction
between two kinds of analytic proposition, fits much better with
Leibniz’s writings. Probably the implausibility of regarding an
analytic proposition as contingent prevented Russell from
attributing this view to Leibniz. Russell, indeed, found it
implausible enough to regard necessary propositions as analytic.
He saw a continuous line of development from Spinoza who
regarded all fundamental truths as analytic, via Leibniz who
insisted on the synthetic nature of the causal laws of science, to
Kant who insisted on the synthetic nature of the propositions of
geometry. Russell, as we shall see, himself carried this
development to its end by insisting on the synthetic character of
the laws of logic.
Perhaps the strongest evidence for Couturat’s view is
that Leibniz himself recognises the implausibility of regarding an
analytic proposition as contingent. In Section XIII of the
Discourse on Metaphysics, there is a passage in which Leibniz
insists that all true propositions are analytic and tries to reconcile
this view with his view that propositions about particular
individuals are contingent:
We have said that the concept of an individual substance includes
once and for all everything which can ever happen to it and that
in considering this concept one will be able to see everything
which can truly be said concerning the individual, just as we are
able to see in the nature of a circle all the properties which can be
derived from it. But does it not seem that in this way the
difference between necessary and contingent truths will be
destroyed, that there will be no place for human liberty, and that
an absolute fatality will rule as well over all our actions as over all
the rest of the events of the world?21
Later, to remove this appearance that the difference between
contingent and necessary truths will be destroyed, he says:
In order to meet the objection completely, I say that the
connection or sequence is of two kinds; the one, absolutely
necessary, whose contrary implies contradiction, occurs in the
eternal verities like the truths of geometry; the other is necessary
Leibniz, Basic Writings, p. 20.
only ex hypothesi, and so to speak by accident, and in itself it is
contingent since the contrary is not implied.22
This explanation seems to make matters worse rather than better,
for how can a truth whose contrary does not imply contradiction
be an analytic truth? Undoubtedly it is passages like this that led
Russell to his view that Leibniz held contingent truths to be
synthetic. Leibniz himself sees that this explanation raises the
same difficulty again, and attempts another explanation, this new
one relying on a distinction between what is necessary ex
hypothesi and what is necessary in itself. This, presumably, is a
distinction between “If anyone were Caesar, then he would cross
the Rubicon”, which Leibniz admits to be necessary, and “Caesar
crossed the Rubicon”, which he denies is necessary. However, the
distinction does not exist if propositions about the individual
Caesar are all analytic. In yet another explanation, Leibniz insists
that the predicate of a true contingent proposition, as of a true
necessary one, is contained in the subject, but points to a
distinction in the manner in which it is contained. Couturat
found passages which explained this difference of manner as
arising from the simplicity of the concept of a triangle and the
infinite complexity of the concept of Caesar.
None of these explanations is satisfactory, but their
failure should not have blinded Russell to the attempt that was
being made to reconcile the analytic character of all propositions
with the contingent character of some of them. Leibniz’s view that
the negation of a true contingent proposition is not self-
contradictory cannot be reconciled, as might perhaps be thought,
with his view that in every true proposition the predicate is
contained within the subject, by ceasing to regard this latter
phrase as implying that all true propositions are analytic. In
Section VIII of the Discourse on Metaphysics Leibniz writes:
Thus the content of the subject must always include that of the
predicate in such a way that if one understands perfectly the
concept of the subject, he will know that the predicate appertains
to it also.23
Leibniz, Basic Writings, p. 20.
Leibniz, Basic Writings, p. 13.
Russell’s failure to grasp Leibniz’s views about necessity
and contingency led him to criticise Leibniz for views about
subject-predicate propositions which he never held. Russell was
right on a fundamental point, that Leibniz held nature to be made
up of substances because all true propositions have subjects.
The ground for assuming substances – and this is a very
important point – is purely and solely logical. What Science deals
with are states of substances, and it is these only that can be given
in experience. They are assumed to be states of substances,
because they are held to be of the logical nature of predicates, and
thus to demand subjects of which they may be predicated.24
How right Russell is in this can easily be proved, for the passage
from the Discourse on Metaphysics which I quoted above is
followed immediately by this passage:
This being so, we are able to say that this is the nature of an
individual substance or of a complete being, namely, to afford a
conception so complete that the concept shall be sufficient for the
understanding of it and for the deduction of all the predicates of
which the substance is or may become the subject.25
However, since Russell could not credit Leibniz with the view that
propositions about individual substances are analytic he did not
do justice to Leibniz’s notion of substance. The passage from The
Philosophy of Leibniz which I quoted above continues:
And this brings us back to the distinction, which we made in
Chapter II, between two kinds of subject-predicate proposition.
The kind which is appropriate to contingent truths, to
predications concerning actual substances, is the kind which says
“This is a man”, not “man is rational”. Here this must be
supposed defined, not primarily by predicates, but simply as that
substance which it is…Thus the substance remains, apart from
its predicates, wholly destitute of meaning. As to the way in
P.L., p. 49.
Leibniz, Basic Writings, p. 13.
which a term wholly destitute of meaning can be logically
employed, or can be valuable in Metaphysics, I confess that I
share Locke’s wonder.26
As the passage from the Discourse on Metaphysics shows,
Leibniz’s substances are not destitute of meaning. Exactly the
opposite is true – the substances afforded a conception which
enabled them to be understood. Russell is right in saying that
Leibniz’s doctrine that nature is made up of substances stems
from his logical doctrine that every proposition has a subject, but,
being wrong about Leibniz’s conception of subject-predicate
proposition, he is inevitably wrong about Leibniz’s conception of
substance. This same mistake led him to give Premise II as
Leibniz’s definition of substance rather than the definition quoted
above from the Discourse on Metaphysics – that an individual
substance has, of its nature, a complete conception from which all
its predicates may be deduced.
There is no space here to discuss more than the
fundamental attitudes of Russell’s work on Leibniz. The insight
that these metaphysical views were founded on logical ones, led
Russell to suspect that the same was true of a metaphysician of his
own time, F. H. Bradley. Indeed, he found that Bradley shared
some of Leibniz’s assumptions but was led by them to very
different conclusions. However, this can be discussed better after
we have considered the views which Russell formed of the nature
P.L., pp. 49-50.
3 Geometry and Logic
Russell’s next book, The Principles of Mathematics, is very
different from his work on Leibniz. In the first place, it is not a
work of criticism but treats directly of an important philosophical
subject, the nature of mathematics. In the second place, it deals
with the subject in a comprehensive manner, so that an enormous
number of different problems are raised and discussed. Since the
main thesis of the book is that mathematics is a part of logic, it is
inevitable that among these problems are many which are central
In order to establish that mathematics and logic are one,
Russell sought to show that all the concepts of mathematics could
be defined in terms of concepts which belong to logic, and that all
the theses of mathematics could be deduced from principles
which belong to logic. In particular, he offered definitions of the
notions one, two, three and so on, as they occur in such
propositions as “There is one chocolate in this box”, and of such
notions as equality in number, as it occurs in such propositions as
“There is the same number of chocolates in this box as in that
one”. He offered definitions of such notions as the sum of… and
the product of…, and he offered proofs of the theses of arithmetic
such as “The sum of two and two is equal to the product of two
Since he held that geometry, no less than arithmetic, is a
part of logic, one might expect to find definitions of such notions
as point, straight line, and triangle, as they occur in such
propositions as “These three points lie in a straight line and do
not form a triangle”, and proofs, from principles of logic, of such
theses as “The sum of the angles of a triangle is 180 degrees”.
However, no such definitions and proofs are offered. Russell did
not claim that spatial concepts could be defined in terms of those
of logic, nor that theses such as Euclid’s theorems followed from
principles of logic:
…we are to remain in the region of pure mathematics: the
mathematical entities discussed will have certain affinities
to the space of the actual world, but they will be discussed
without any logical dependence upon these affinities.27
In fact, Russell did not recognise the subject which is ordinarily
called geometry as part of mathematics; he held that it belonged
to applied mathematics, together with such subjects as mechanics.
Applied mathematics, according to Russell, was not a part of logic
but belonged to natural science. Nevertheless he held that
mathematicians did study a subject which was properly, if not
ordinarily, called geometry and which was a part of logic. How is
this subject related to what is ordinarily thought of as geometry?
When Euclid presented his geometry he divided his
theses into two kinds: those based upon other theses, which are
now called theorems, and those not based upon other theses,
which are now called axioms. (He employed another category of
proposition, the definitions, but these can be regarded as theses
and counted among the axioms.) If chosen correctly, the axioms
would imply the whole of Euclid’s geometry. These axioms, as we
have seen, were not regarded by Russell as theses of geometry:
geometry was not concerned with the question of whether these
axioms were true, nor did any geometrical theory assert them to
be true. He suggested that geometry was concerned, not with the
question “Are Euclid’s axioms true?”, but with questions of the
form “If Euclid’s axioms were true, then would … be true?”
Similarly geometry was not concerned, according to Russell, with
the truth of any of Euclid’s theorems but with the question of
whether a theorem was implied by the axioms.28 In just the same
way someone might turn from considering the question “Are all
P.O.M., p. 372.
P.O.M., pp. 372-4.
men mortal?” to the question “Is it true that if all men are mortal,
then nothing not mortal is a man?”
However, these modified geometrical questions involve
spatial concepts such as point and straight line which do not
figure in principles of logic, and which Russell did not regard as
definable in terms of concepts belonging to logic. Russell
interpreted these modified questions strictly, as questions about
implication according to the principles of logic. The questions
with which geometry is concerned always take the form, “Do
Euclid’s axioms imply, according to the principles of logic,
that…?” In the same way one might ask “Does the proposition
that all men are mortal imply, according to the principles of logic,
that nothing not mortal is a man?” One way of answering such a
question would be to argue that since the concepts of man and
mortality do not occur in the principles of logic, and cannot be
defined in terms of concepts which do, then any principle of logic
according to which the implication holds will be a principle
according to which the implication would hold, no matter what
concepts replace those of man and mortality. Therefore we need
only consider the question “Is it true that, whatever concepts X
and Y may be, if all X are Y, then nothing which is not Y is X?”
Russell held that geometrical theses provided answers to such
questions. Variables have replaced the spatial concepts and the
theses are universal; they have consequences for propositions
obtained by taking any concepts as the values of the variables. By
these two steps, which might be called conditionalising and
generalising, Russell obtained a subject which was indeed a
branch of logic from the subject which is ordinarily thought of as
It is clear that this branch of logic is of great help in the
study of what is ordinarily thought of as geometry, and which
Russell called “applied geometry”. A knowledge of it will enable a
scientist who is considering a body of theses of applied geometry,
each of which he holds to be true, to tell that they follow from a
group of axioms. He will know that the axioms embody
everything which needs to be assumed in order to deduce the
theorems. All assumptions, analytic or synthetic, a priori or a
P.O.M., pp. 7-8, 372-4.
posteriori, trivial or important, obvious or doubtful, will be there.
Russell believed that Euclid intended to produce such an
axiomatisation of his theses but that, through lack of rigour, he
failed in the attempt.30 Again, this branch of logic enables a
scientist who is considering a group of axioms to see what they
imply, and so, perhaps, to decide their truth.
It is equally clear how strange it is to give the name
“geometry” to a subject which is not concerned with spatial
concepts. This strangeness was concealed from Russell by his use
of such words as “point” and “straight line” for concepts which do
occur in his subject but which are not the concepts of point and
straight line. How this came about needs to be explained.
Consider the proposition “All flesh is grass, but there is
grass which is not flesh”. This might be thought of as a
proposition to the effect that the concepts flesh and grass are
related in a particular way, that is, as species and genus. A logician
who recognised the similarity between this proposition and
others such as “All squares are rectangles, but there are rectangles
which are not squares” might be said to have appreciated the
species and genus relationship, and if he investigated what
followed from the assertion that two concepts were related as
species then he might be said to be investigating the species and
genus relationship. In a similar way, Euclid’s axioms may be
thought of as a group of propositions to the effect that the
concepts of point, straight line, and so on, stand in a particular
relation. This same relation might be asserted to hold between
other, non-spatial, groups of concepts. Russell characterised it as
one which established a particular sort of two-dimensional order.
A logician who recognised a similarity between Euclid’s axioms
and another group of axioms involving different concepts,
perhaps concepts applying to complex numbers, may be said to
have appreciated what it is for a group of concepts to establish a
two-dimensional order of this sort, and if he investigated what
followed from the assertion that a group of concepts did establish
such an order, then he may be said to be investigating two-
dimensional orders of that particular sort. In short, just as the
investigation of the species and genus relationship is not an
P.O.M., pp. 404-7.
investigation of what relationship holds between the concepts
flesh and grass, but of one particular relationship which might be
said to hold between them, so Russell’s Euclidean geometry is not
an investigation of what relationships hold between spatial
concepts, but of one relation which might be said to hold between
them. This is the force of Russell’s definition of geometry as the
study of orders of two or more dimensions.31
Now in saying that the concepts flesh and grass are
related by the species and genus relationship, we need to indicate
whether we mean that all flesh is grass but there is grass which is
not flesh, or whether we mean that all grass is flesh but there is
flesh which is not grass. We indicate that the former is our
meaning by specifying that flesh is the species and grass the genus.
If we lacked the words “species” and “genus” we might use those
for well-known examples of concepts which have been held to
exhibit this relationship. For example, if we wished to say that the
concepts square and rectangular were related as species and genus
we might say that they are related “as flesh and grass”, meaning
that they are related as the concepts flesh and grass are said to be
related in the proverb. If this usage were adopted, a logician
investigating the species and genus relationship would be said to
be investigating the flesh and grass relationship, although all he is
doing is to investigate a relation which, in a well-known proverb,
is held to relate these two concepts. In the same way, when
someone says of a group of concepts that they are related in the
way in which Euclid’s axioms assert spatial concepts to be related,
he needs to specify which concept takes the place of point, which
takes the place of straight line, and so on. It is natural for him, in
the absence of other terminology, to use the words “point” and
“straight line” to mean “any two concepts which are related as the
concepts point and straight line are said to be related in Euclid’s
axioms”. It then seems as if he is discussing the spatial concepts of
point and straight line although the truth is that he is discussing a
relation which has been asserted to hold between them in a well-
known group of axioms. This usage explains Russell’s assertion32
that the concepts of geometry are definable in logical terms: the
P.O.M., p. 372.
P.O.M., pp. 435-6.
concepts of Russell’s geometry were not spatial concepts,
although they were given the same names.
Russell’s geometry does contribute to our understanding
of Euclid’s axioms, so that if these present the facts about space it
contributes to our understanding, but not our knowledge, of
those facts. However, it does this without contributing to our
understanding of the spatial concepts which the axioms involve.
When logicians perceived a common from between the
propositions “All flesh is grass, but there is grass which is not
flesh”, “All swans are white, but there are white things which are
not swans”, “All squares are rectangles, but there are rectangles
which are not squares”, each being to the effect that the species
and genus relationship holds between two non-logical concepts,
their investigation of this common form contributed to the
understanding of each such proposition, but not to the
understanding of the non-logical concepts. In the same way,
Russell’s geometers who perceive that Euclid’s axioms share a
common form with other axioms concerning, perhaps, numerical
rather than spatial concepts, contribute to an understanding of
Euclid’s axioms but not to an understanding of the spatial
concepts involved. Certainly an investigation of the species and
genus relationship, together with the knowledge that this
relationship held analytically between the concepts square and
rectangle, would contribute to our understanding of these
concepts. If it were true that the relationship which Euclid’s
axioms assert to hold between the spatial concepts held
analytically between them, then Russell’s geometry, together with
a knowledge of this fact, would contribute to an understanding of
the spatial concepts. However, Russell believed, with justice, that
the truth of Euclid’s axioms was a question for natural science.
Russell’s geometry does not contribute to an understanding of the
spatial concepts, even when it is taken in conjunction with other
facts about them.
Indeed, in Russell’s view, the propositions of Euclid
which are commonly called definitions were asserted and
belonged with the axioms.33 Among asserted propositions Russell
distinguished only between those whose truth could be decided
P.O.M., p. 429.
by principles of logic, and those whose truth was a question for
natural science. None of the axioms of Euclid fell into the former
group. He allowed no third group of analytic truths into which
the proposition “All squares are rectangles”, or some of Euclid’s
axioms such as “The whole is greater than the part”, have been
thought to fall. This view has been adopted by W. V. O. Quine as
a basic tenet of his philosophy.34
It might be thought that Russell’s geometry could be
characterised as an investigation of relations which might hold
between spatial concepts, but this is not the case. Just as the
investigation of the species and genus relationship concerns a
relationship which has been said to hold between the concepts
flesh and grass but which, it might be thought, our understanding
of these concepts shows us to be one which could not possibly
hold between them in the literal sense, so spatial concepts might
be said to establish a certain sort of order and that order might be
investigated, but an understanding of spatial concepts might show
that they could not possibly establish an order of that sort. An
example is provided, perhaps, by orders of more than three
dimensions. Four-dimensional orders have been studied, and
someone could assert that the spatial concepts establish a four-
dimensional rather than a three-dimensional order, yet it is very
doubtful whether they could possibly do so. It is very doubtful
whether space could possibly have four dimensions, and an
understanding of the spatial concepts may be sufficient to show it
to be impossible.
This subject which Russell held geometry to be, the study
of orders of more than one dimension, is important and is a part
of logic. It is the subject which a logician would make of
geometry. Russell’s thesis that geometry is a part of logic entails
regarding this de-spatialised subject, not only as part of geometry,
but as the whole of that part of geometry which is not natural
science. To regard it as a part of geometry is not unreasonable
since it is of assistance in geometrical studies. It is the exclusion
from geometry of any studies which contribute to the
understanding of spatial concepts which is the great fault of
W. V. O. Quine, From a Logical Point of View, 1953, Essay II.
Indeed the very same objection can be brought against
his view of geometry that he himself brought against the view of
arithmetic called formalism. The formalists, as Russell explains in
the introduction to the second edition of The Principles of
Mathematics, treat the numerals 0, 1, 2, and so on as variables and
consider whether any concepts, related as the axioms of
arithmetic declare the integers to be, would be related as the
theorems of arithmetic declare the integers to be. The symbols
“0”, “1”, “2”, undergo at the hands of the formalists the same
change of meaning as the words “point”, “straight line”, and so on
underwent at Russell’s hands: they come to represent positions in
one-dimensional order and cease to represent integers. Russell
Accordingly the symbols 0, 1, 2, … do not represent one definite
series, but any progression whatsoever. The formalists have
forgotten that numbers are needed, not only for doing sums, but
The formalists regard arithmetic as the study of one-dimensional
orders and leave no place in it for the attempt to define the
numerical concepts which are used in counting. This criticism of
the formalist view of arithmetic is the same as that which I have
brought against Russell’s formalist view of geometry. Since he
accepted it he should have admitted that, besides the investigation
of whether Euclid’s axioms are true, there is another part of what
is ordinarily called geometry which does not belong to logic. The
nature of this part of geometry, the attempt to contribute to the
understanding of spatial concepts, is left unexplained by Russell’s
thesis that geometry belongs to logic.
There is, however, one strong argument in favour of
Russell’s view of geometry which I have not yet considered.
Russell points out36 that mathematicians came, during the
nineteenth century, to regard the non-Euclidean geometries as no
less a part of geometry than Euclid’s. The axioms of the non-
Euclidean geometries contradict those of Euclid’s, yet
mathematicians considered that the study of all these geometries
P.O.M., p. vi.
P.O.M., p. 373.
yielded truths. Russell’s view of geometry explains easily enough
how this is possible: if these various geometries are concerned
only with the question of whether their axioms imply their
theorems, there is no contradiction between them. But this
explanation raises a problem. On the ordinary view, according to
which geometers are interested in whether Euclid’s axioms are
true, there was a conflict between Euclidean and, for instance,
Riemannian geometry, so that it was clear that both were theories
about the same subject: one asserted what the other denied. But
on Russell’s view the two geometries make independent
assertions. Why should they be regarded as parts of the same
subject? Russell’s answer was that just as Euclid’s axioms assert of
a group of spatial concepts that they establish a two-dimensional
order of a particular sort, so the axioms of one of the non-
Euclidean geometries assert, of the same group of spatial
concepts, that they establish a two-dimensional order of a
different sort. A person who pursues the subject which Russell
calls Euclidean geometry investigates one sort of two-dimensional
order, a person who pursues one of the non-Euclidean geometries
Russell is right that geometers came to consider that
both Euclid’s geometry and the non-Euclidean geometries yielded
truths, but there is another explanation of how it is that they were
able to do so. The fact that two assertions cannot both be true
does not imply that they cannot both be possibilities. It may be
that Euclid’s geometry and the non-Euclidean geometries present
various possibilities for space. Each of the geometries may be
possibly true, not merely in the sense that its axioms are not
contradictory according to the principles of logic, but in the sense
that the relations which its axioms assert to hold between the
spatial concepts are relations in which the spatial concepts could
stand. The invention of the non-Euclidean geometries would then
be the realisation of spatial possibilities which had not been
realised before: that lines which point in the same direction may
yet meet, or that a straight line may be of finite length and yet
have no ends. The truths of both the Euclidean and the non-
Euclidean geometries may have concerned neither logical
possibilities nor spatial actualities but spatial possibilities. This
explanation fits, just as Russell’s does, without the fact that
scientists have suggested that one of the other geometries, and not
Euclid’s, presents the truth about space. It fits too, as Russell’s
does not, with something which many people feel: that the non-
Euclidean geometries are spatial possibilities whereas the five-
dimensional geometries are not. Russell’s explanation can make
no distinction between those non-Euclidean geometries which are
not as a matter of fact true of space and the geometries of more
than three dimensions which could not be true of space, for none
of them contradicts the principles of logic.
In The Principles of Mathematics Russell not only argued
for the thesis that mathematics is a part of logic, but presented a
view of the nature of logic itself. The view is implied by the
distinction he made between applied mathematics, which did not
belong to logic, and pure mathematics, which did. In pure
mathematics, the process of generalising has replaced every
concept which does not belong to logic by a variable: this absence
of any non-logical concepts constituted the defining characteristic
of a proposition of logic. Russell called this characteristic
“formality”, so that his definition of logic can be put by saying
that it is a purely formal discipline. Since this definition relies on
a distinction between those concepts which belong to logic and
those which do not, it raises the question of how this distinction is
to be made. Russell sometimes suggests that it can be done by
reference to the process of generalisation. After describing an
example of an application of the process, he writes:
Here at last we have a proposition of pure mathematics…So long
as any term in our proposition can be turned into a variable, our
proposition can be generalised; and so long as this is possible, it is
the business of mathematics to do it.37
This passage implies that the logical concepts cannot be replaced
by variables, but this is not true. Perhaps the most outstanding
example of the use of variables is provided by algebra, where they
replace particular numbers which, Russell held, can be defined in
terms of logical concepts. The logical concepts, or “logical
constants” as Russell often calls them, cannot be defined as the
residue of the process of generalisation.
P.O.M., p. 7.
Elsewhere38 Russell asserts that the logical concepts
cannot be defined but only enumerated. He did not mean that
they have no character in common which distinguishes them
from other concepts. If that were true there would be no way of
recognising a new concept as one belonging to logic, unless it
could be defined in terms of those already admitted. That
consequence fits the traditional conception of logic as a
completed doctrine, but not Russell’s conception of it as a
developing branch of enquiry. Russell meant, not that the
concepts of logic have no character in common, but that it is
impossible to define what that character is. He thought that the
concepts of logic were so fundamental that they must enter into
the definition of anything whatever and he thought it
objectionable that they should enter into the definition of their
own common character. It would be as if the concept of triangle
were employed in the definition of shape.
Russell held that with his definition of logic he had
accounted for the differences between logic and the natural
sciences. He held that the propositions of logic and mathematics
concerned the fundamental logical concepts and other concepts
of logic and mathematics which could be defined in terms of
them. He held that they concerned such concepts and nothing
else. He treated the process of generalisation as if it yielded
propositions which have no concern with the subject matter of
the original proposition, but are concerned with a new subject
matter, the concepts of logic. Many of these are concepts of the
different types of propositions, or of the different forms
propositions may have:
The process of transforming constants in a proposition into
variables leads to what is called generalisation, and gives us, as it
were, the formal essence of a proposition. Mathematics is
interested exclusively in types of propositions….39
This view makes a convincing distinction between logic and
natural science but it is not consistent with Russell’s definition of
logic. The subject matter of a generalised proposition must
P.O.M., pp. 8-9.
P.O.M., p. 7.
include the subject matter of the original. For example, if the
theorem “No straight lines meet at more than one point” is
generalised so that it yields “No matter what X, Y and Z may be,
no two things which are X will Z more than one thing which is
Y”, then the result implies the original theorem, and it also
implies that no two sign-posts point to more than one town.
Although generalisation will yield a proposition of logic, it is
unlikely to yield a true proposition of logic. The process of
conditionalisation will be needed before a logical truth is
obtained, and conditionalisation does yield a proposition having
different consequences from the original. It does not, however,
yield one with a more restricted subject matter. Whatever
conditional clauses are added to the above generalised theorem
the result, provided that it is not tautological, will still have
consequences concerning signposts and towns. J. S. Mill held, like
Russell, that mathematics was the most universal discipline, but
he did not make the mistake of combining this with the view that
it had a narrower subject matter than the other sciences.40
This difficulty leads to another. Russell held that because
logic has as its subject matter the different forms which
propositions may have, whereas the natural sciences have
particular things and non-logical properties, the principles of
logic could be known a priori – that is, without experience – while
those of the natural sciences could not. However, Russell’s
definition of logic as a completely general subject means that
logical principles imply some propositions concerning particular
things and non-logical properties. If these cannot be known a
priori, as his distinction between logic and the natural sciences
requires, then neither can the principles of logic which imply
In different works Russell gave different solutions to
these problems. A discussion in The Problems of Philosophy41
hesitates between two solutions. In one of these he argues that a
general proposition does not imply the existence of particulars of
any kind, so that it does not imply any propositions concerning
particular things. This inference, as we shall see later, is not in
J. S. Mill, A System of Logic, 1965, Book III, Chapters V and
P.P., pp. 103-6.
accordance with Russell’s views on propositions, but even if the
inference is valid the argument does not show that general
propositions can be known a priori, for there are many
propositions which cannot be known a priori but which deny,
rather than imply, the existence of particulars of a certain kind.
The proposition that there are no immortal men is an example. In
the other solution, he argues that whether or not completely
general propositions imply propositions concerning particulars or
non-logical properties, they do not themselves concern those
particulars or properties. They cannot do so, because we can
understand the completely general propositions without any
acquaintance with those particulars or properties, or any
knowledge of what kinds of them exist. Therefore one fact which
makes it impossible to have a priori knowledge of propositions
concerning particulars or non-logical properties does not make it
impossible to have a priori knowledge of completely general
propositions implying them: the fact that we cannot even
understand a proposition concerning particulars or non-logical
properties without acquaintance with them, or knowledge of what
kinds of them exist. However, this fact is not the only one which
makes it impossible to know such propositions a priori. Even if a
person is acquainted with certain particulars and so understands
a proposition concerning them, there remains the question of
whether he can know that proposition to be true without
observing it to be true. Any reasons for holding that he cannot,
apply equally to any general proposition which implies it.
In another discussion,42 Russell asserts that the
propositions of logic apply to all things and all properties, and
that the proposition that if Socrates is a man, and all men are
mortal, then Socrates is mortal is true “in virtue of its form”. He
asserts that because the truth of logic of which this proposition is
an instance does not mention any particular thing or particular
quality it is wholly independent of the accidental facts of the
existent world. This last assertion seems inconsistent with the
first, and suggests the arguments he employed in Problems of
Philosophy. However, it seems likely that by the qualification
“accidental” Russell intended no more than “not true in virtue of
O.K.E.W., pp. 65-7.
its form”, so that the latter assertion implies only that the truths of
logic are independent of those truths concerning particulars
which are not true in virtue of truths of logic. This assertion is not
a complete tautology; it means that the only facts concerning
particulars and non-logical concepts which follow from truths of
logic are those which arise from, and are explained by, truths of
logic. Russell here admits that some facts concerning particulars
do follow from truths of logic. This is the solution which is most
consistent with his view of logic as having a subject matter
peculiar to itself, but it is not consistent with his view that the
truths of logic can be characterised as completely general truths.
This is because a completely general proposition does not explain
its instances, but merely summarises them. Russell’s views on the
nature of logic are, in fact, a strange combination of rationalism
and empiricism; the empiricist element – that the truths of logic
can be characterised as completely general truths – best fits a
philosophy in which no distinction is made between logic and
mathematics on the one hand and the natural sciences on the
other, except that the truths of the former are more general. The
truths of logic and mathematics have no necessity which those of
the natural sciences lack, their subject matter is no different, nor
can they be known in any different way. It is a consequence of
such a philosophy that there is no distinction, such as Russell
makes, between those propositions concerning particulars and
non-logical properties which are true by virtue of their form and
those which have some other foundation. For example, there
would be no distinction between the proposition that if Socrates
is a man and all men are mortal, then Socrates is mortal, on the
one hand, and the proposition that all men are mortal on the
other. Such a view was held about mathematics by J. S. Mill before
Russell and about logic and mathematics by W. V. O. Quine after
There is an argument against this view which Russell was
unable to advance, because it would have led him to the
conclusion that the truths of logic were trivialities. It might be
argued that the process of generalisation will never yield a true
proposition unless the proposition about particular things to
which the process is applied is a triviality. For example, if the
proposition that Socrates is mortal is generalised it yields the
proposition that all things have all properties, which is false. Only,
this contention goes, if the process of conditionalisation has first
yielded a triviality, such as that if Socrates is a man and all men
are mortal, then Socrates is mortal, will generalisation yield a
truth. This argument is not obviously correct. For one thing, it
might well be denied that such conditionals are trivialities. Again,
if the process of generalisation is applied to an existential
proposition it may yield a completely general proposition which
is not obviously false. For example, the proposition that there is
something mortal becomes the proposition that every property is
exemplified by at least one thing. This contention cannot be
decided without a more precise specification of the process of
generalisation than Russell gives, but whether or not it is true it
would offer no solution of his difficulties. Generalisations of
trivialities are themselves trivialities, so that this contention
would have the consequence that logic and mathematics were
composed of trivialities and embodied no knowledge whatever,
no knowledge, therefore, of the different forms which
propositions may have.
In the same way, Russell rejected the view that the truths
of logic are analytic for, as we saw in Chapter 2, he accepted
Kant’s view that analytic propositions are trivialities. He makes
the same point in Problems of Philosophy.43 However, in a later
work – “The Philosophy of Logical Atomism”44 – Russell seems
inclined to revise his opinion. He reiterates the view that logic is
concerned with the forms that propositions may take, but he also
says that the propositions of logic must in some sense be
tautologies.45 He was presumably influenced here by a doctrine of
Wittgenstein’s which was a principal element of the Tractatus
Logico-Philosophicus. This doctrine was intended to reconcile the
view that the truths of logic are empty with the view that they are
not trivial. Wittgenstein held that the truths of logic are
tautologies, or analytic propositions,46 and he regarded
tautologies as empty – to utter a tautology is to assert nothing.
However, the utterance of a tautology may show what it does not
P.P., pp. 82-3.
L.K., pp. 177-281.
L.K., pp. 239-40.
L. Wittgenstein, Tractatus Logico-Philosophicus, 1922, p. 155.
assert. Tautologies are not trivialities, for what they show is
important. What Wittgenstein believed tautologies to show
corresponded very closely with what Russell held the facts of logic
to be: “the formal – logical – properties of language and the
world”.47 It appears, therefore, that this doctrine of Wittgenstein’s
– that there are truths which can be shown but not stated –
would, if it were true, present an escape from the dilemma which
confronted Russell, so long as he maintained his view that logic
can be characterised as a completely general subject: either to
accept that logic and mathematics belong with the natural
sciences or to accept that they are wholly composed of trivialities.
However, in his introduction48 to the Tractatus Russell says that
he is not convinced of the truth of Wittgenstein’s doctrine
although he did not believe that there was any respect in which it
was obviously mistaken. It was at this point that Russell left the
problem of the nature of logic.
L. Wittgenstein, Tractatus Logico-Philosophicus, p. 157.
L. Wittgenstein, Tractatus Logico-Philosophicus, p. 21.
4 Bradley’s Idealism and the Doctrine of Internal Relations
Russell and Moore became united in opposition to a philosophy
which both had at one time accepted, the idealist philosophy of F.
H. Bradley. The central theses of this philosophy are two: first,
that what common sense and science take to be facts about nature
are no more than misleading appearances; second, that this is
inevitable, since it arises from the character of thought. Idealism
shares with scepticism the thesis that knowledge of nature is
impossible, but whereas the sceptic admits that we might believe
what happens to be the truth – denying only that we can ever
know it to be so – the idealist holds that this too is impossible.
Indeed he derives this possibility from a more fundamental one,
that of conceiving what happens to be the truth. No proposition
whose truth we might consider could present nature as it really is.
Bradley begins his book Appearance and Reality, which
was first published in 1893, by arguing this thesis for propositions
of subject-predicate form. In Chapters II and III he argues that a
proposition such as “This lump of sugar is white” can only be
explained as expressing a relation between the various qualities of
the lump of sugar. Then he argues that it cannot be understood in
that way, since propositional relationships cannot be understood
…a relational way of thought – any one that moves by the
machinery of terms and relations – must give appearance,
and not truth.49
F. H. Bradley, Appearance and Reality, 1930, p. 28.
In Chapter III, Bradley considers a number of ways in which
relational propositions might be understood and rejects each of
them. He must have been dissatisfied with his discussion, for in
an appendix to the second edition he takes up the subject again.
There he appears to have modified his conclusion and to be
arguing, not against the relational way of thought in general, but
only against one form of it: that in which relations are treated as
external to their terms. Russell and other commentators have
taken Bradley in this way and supposed him to conclude in the
appendix that there are relations but that they are internal, not
external, to their terms. This conclusion contradicts the
discussion in Chapter III. It implies, as Russell pointed out, that
there are relational propositions which are reducible to subject-
predicate propositions, yet in Chapter III, Bradley explicitly
The relation is not the adjective of one term, for, if so, it does not
relate. Nor for the same reason is the adjective of each term taken
apart, for then again there is no relation between them. Nor is the
relation their common property, for what then keeps them
What Bradley in fact contends in the appendix is not that there
are relations which are internal to their terms, but that relations
could not be real unless they were internal to their terms. This, he
believes, they cannot be, for if there are terms, those terms must
be related, and must be related by external relations. Therefore
relational propositions cannot adequately represent reality. The
upshot is the same as in Chapter III, that a relational way of
thought, “any which proceeds by the machinery of terms and
relations”, must give appearance and not truth. However, in
Chapter III this conclusion is based upon the contention that
relational propositions cannot be understood; in the appendix, on
the contention that relational propositions cannot adequately
In one place in the appendix where Bradley is developing
this argument he takes the example of spatial relations. These
appear to be external to physical things. The very same chair
F. H. Bradley, Appearance and Reality, p. 27n.
which stands near the table might, it seems, have stood elsewhere.
What is more it might have stood elsewhere and retained the
properties it now possesses.
Why this thing is here and not there, what the connexion is in the
end between spatial position and the quality that holds it and is
determined by it, remains unknown… But any such irrationality
and externality cannot be the last truth about things. Somewhere
there must be a reason why this and that appear together. And
this reason and reality must reside in the whole from which
terms and relations are abstractions, a whole in which their
internal connexion must lie… The merely external is, in short,
our ignorance set up as reality, and to find it anywhere, except as
an inconsistent aspect of fact, we have seen is impossible.51
Our attempt to present reality as a spatial arrangement of physical
things is an attempt to present it as constituted by terms in
external relations. We realise that this picture could not
adequately present reality unless the relations were internal to
their terms: that is, unless the nature of the things in space
determined their spatial position. Reality has a character which
could only be presented by terms in relation, if the relations were
internal to their terms. Yet the externality of relations is essential
to the relational way of thought. Therefore, reality has a character
which cannot be presented by the machinery of terms and
relations. Bradley did argue, in the appendix, that our feeling that
relations must be internal to their terms is a recognition of a
feature which belongs to reality but which we cannot properly
express. He did not claim to be able to express this feature of
reality, so he was not able to show directly that it could not be
presented by the relational way of thought. Instead, he falls back
on the arguments which, like those used in Chapter III, are
directed against the intelligibility of that way of thought.
The doctrine that relations must be internal to their
terms would imply, as Russell saw, that if there are relational
propositions they are reducible to subject-predicate propositions.
However, in the appendix, Bradley’s words often suggest a further
implication, and this is confirmed by what he writes elsewhere in
F. H. Bradley, Appearance and Reality, p. 517.
the book. For example, he says in one place, in an argument
against external relations:
But if the terms from their inner nature do not enter into the
relation, then, so far as they are concerned, they seem related for
no reason at all….52
When he speaks here of the “inner nature” of the terms he might
mean their predicates, or adjectives, but he might mean more
than this. A passage in Chapter XXIV indicates that he does mean
more. In the passage he is concerned to prove that nothing which
fails to give the whole truth can be true at all:
For that which is not all-inclusive must by virtue of its essence
His argument is this:
That which exists in a whole has external relations. Whatever it
fails to include within its own nature, must be related to it by the
whole, and related externally. Now these extrinsic relations, on the
one hand, fall outside of itself, but, upon the other hand, cannot
do so. For a relation must at both ends affect, and pass into, the
being of its terms. And hence the inner essence of what is finite
itself both is, and is not, the relations which limit it. Its nature is
hence incurably relative, passing, that is, beyond itself, and
importing, again, into its own core a mass of foreign connexions.
This to be defined from without is, in principle, to be distracted
Here Bradley asserts that the relations which a thing has to other
things must be included in what he calls, variously, its nature,
essence, or definition, and yet cannot be included in this. This is
the further implication of the doctrine that if there are relations
they are internal to their terms: in the definition of a term its
relations must be contained.
F. H. Bradley, Appearance and Reality, p. 514.
F. H. Bradley, Appearance and Reality, p. 322.
F. H. Bradley, Appearance and Reality, p. 322.
Bradley, no doubt, held that the second of these two
parts of the doctrine would imply the first. For the fact that two
things were related in a certain way would have to be explained,
he held, by the nature, essence, or definition of the things and if it
was explained by the definitions of each of the two things
separately then the same fact would have two explanations. Or he
might argue that it is essential to the fact of two things being two
that one of them can exist without the other existing. Hence the
relation between them cannot follow from the nature, essence, or
definition of either. But with certain categories of individual it is
not true, when two of them are related, that one can exist without
the other existing. The natural numbers are an example: the
existence of any number implies the existence of another number
related to it. Bradley may have been wrong, therefore, to suppose
that the first part of this doctrine would follow from the second.
Russell discussed Bradley’s doctrines in The Principles of
Mathematics55 and in a paper which he read to the Aristotelian
Society in 1907, called “The Nature of Truth”.56 In both of these
discussions he puts great emphasis on the impossibility of
reducing relational propositions to subject-predicate ones. His
intention was to disprove the view that the subject-predicate form
was the only form which propositions could take.
It is a common opinion – often held unconsciously, and
employed in argument, even by those who do not
explicitly advocate it – that all propositions, ultimately,
consist of a subject and a predicate.57
In The Principles of Mathematics, he distinguishes two ways in
which the reduction of relational propositions to subject-
predicate ones might be thought to hold. One he called the
monadistic view of relations. It is the view that a relation between
terms is reducible to predicates of the separate terms. As we have
seen he attributed this view, with justification, to Leibniz. The
other he called the monistic view. It is the view that a relation is a
predicate of the whole made up by its terms taken together. This
P.O.M., Chapter XXVI.
M.P.D., pp. 55-61.
view he attributed, with less justification, to Bradley. We have
already considered one of Russell’s arguments against the
monadistic view. It is an amplification of an argument which
Bradley himself employed in the passage quoted above58. Against
the monistic view, Russell argued that there was one sort of
relation for which the monistic reduction failed. These are
relations, such as “greater than”, which if they hold between A
and B do not hold between B and A. These relations are called
asymmetrical. Now a proposition which has as its subject the
whole composed of A and B together makes no distinction
between “A is greater than B” and “B is greater than A”: it can
express no more than that A and B differ in size. Hence
asymmetrical relations cannot be reduced to subject-predicate
ones in the monistic way. The inadequacy of both the monadistic
and the monistic ways proved, Russell held, that relational
propositions cannot always be reduced to subject-predicate ones.
This conclusion is the contradictory of one of the two things
which Bradley would have held to follow from the thesis that
there are relations which are internal to their terms. Russell
ignored the other, and called this one the “axiom of internal
relations”. His argument convincingly refutes it.
The impossibility of reducing relational propositions to
subject-predicate ones established that if there are relational
propositions they stand in no need of reduction. In The Principles
of Mathematics, Russell showed that many of the propositions
with which mathematics is concerned, for example, propositions
about number, about space, and about infinite series, involve
asymmetrical relations. Therefore, if these propositions are
intelligible, the relational form of proposition must be admitted as
legitimate in its own right, alongside the subject-predicate form.
Russell, very reasonably, assumed the intelligibility of these
propositions about space, number and infinite series, and claimed
to have established that relational propositions are legitimate in
their own right. In this way he could claim to have established
that unless the view that all propositions are of subject-predicate
See note 2, above.
form abandoned, there could be no adequate philosophy of
Since, as we have seen, the doctrine that if there are
relational propositions then they must be reducible to subject-
predicate ones was a central part of Bradley’s idealism, Russell’s
argument, by establishing that relational propositions are
legitimate in their own right, removed one of the main supports
of that philosophy. However, Russell is mistaken in suggesting
that it was the discovery of asymmetrical relations which refuted
idealism. Bradley himself argued that no reduction of relational
propositions to subject-predicate ones was possible. Russell’s
argument concerning asymmetrical relations does no more than
reinforce arguments which Bradley himself put forward. The
difference between Russell and Bradley lay, rather, in the fact that
Russell accepted the relational form of thought as intelligible
whilst Bradley denied the intelligibility of any proposition which
was not of subject-predicate form. The impossibility of reducing
relational propositions to subject-predicate ones, in conjunction
with their respective premises, led Russell to reject Bradley’s
conclusion that only the subject-predicate form of thought was
intelligible and led Bradley to reject Russell’s conclusion that the
relational form of thought was intelligible. What refuted idealism
was Russell’s perception of the absurdity of rejecting the
intelligibility of all propositions about space and number, which
compose a large part of science and common sense, on the
grounds of the doctrine that all propositions are of subject-
predicate form. Russell often suggests that he has removed the
reasons for accepting idealism, but the only argument he had
against those reasons lay in the rejection of the conclusions which
Bradley himself drew from them. This is indeed a good argument,
and Russell sometimes brings it out clearly.
Asymmetrical relations are involved in all series – in space and time,
greater and less, whole and part, and many others of the most
important characteristics of the actual world. All these aspects,
therefore, the logic which reduces everything to subjects and
predicates is compelled to condemn as error and mere appearance.
P.O.M., p. 226.
To those whose logic is not malicious, such a wholesale
condemnation appears impossible.
But later in the same paragraph he suggests that he has other
reasons, besides the absurdity of the wholesale condemnation, for
rejecting the bases of idealism.
It is impossible to argue against what professes to be an insight, so
long as it does not argue in its own favour. As logicians, therefore, we
may admit the possibility of the mystic’s world…. But when he
contends that our world is impossible, then our logic is ready to repel
his attack. And the first step in creating the logic which is to perform
this service is the recognition of the reality of relations.60
Of course it is not impossible to argue against what professes to
be an insight. The fact that a man offers no arguments in favour
of a contention does not make it impossible to offer arguments
against it. What Russell intended was, presumably, that he had no
arguments against the idealist rejection of the reality of relations,
but did have arguments against any reasons the idealists might
produce in favour of that rejection. He is mistaken, however, in
implying that he has any argument other than that which consists
in asserting that relations are indeed real, and his last sentence
Russell’s recognition of asymmetrical relations appears
to be a refutation of Bradley, because he presents Bradley as
asserting that there are relations which are internal to their terms:
that is, he presents him as a philosopher who accepts the
intelligibility of relational propositions, but reconciles himself to
them by the belief that they can be reduced to subject-predicate
ones. Such a philosopher might be convinced by the example of
asymmetrical relations that relational propositions are intelligible
in their own right. This was not Bradley’s position. He held both
that if there are relations they are internal to their terms, and that
if there are relations they are external to their terms. The example
of asymmetrical relations reveals nothing to him that he was not
already aware of. Bradley had in fact already done all the work of
drawing out the consequences of the doctrine that all
O.K.E.W., p. 59
propositions are of subject-predicate form. It only remained to
Russell to point out the absurdity of those consequences. By
doing this and rejecting the doctrine he freed logic and
philosophy from the dilemma either of engaging in the hopeless
struggle to reduce relations to predicates, or of rejecting all
apparently relational facts as unreal. This was Russell’s
G. E. Moore wrote in an autobiographical essay61 that he
devoted a long time to reading and trying to understand The
Principles of Mathematics. It seems possible that he came to grasp
the character of this argument of Russell’s more clearly than
Russell himself did, for such arguments are very prominent in his
work. He saw clearly that the conclusion which a philosopher
draws from certain doctrines may constitute a reason for rejecting
those doctrines, even if no other reason against them exists. For
example, when considering the principles of Hume’s philosophy
Hume does not, therefore, bring forward any arguments at all
sufficient to prove either that he cannot know any one object to be
causally connected with any other or that he cannot know any
external fact. And, indeed, I think it is plain that no conclusive
argument could possibly be advanced in favour of these positions. It
would always be at least as easy to deny the argument as to deny that
we do know external facts….62
We must now turn to Russell’s discussion of the second part of
Bradley’s doctrine concerning relations: that if a thing stands in
relations to other things, then those relations must be included in
its nature, essence, or definition. Russell considers two arguments
for this doctrine, the second of which is the more important. This
argument rests on the alleged fact that if two things have a certain
relation, they cannot but have it, or, as Russell also puts it, if two
things are related in a certain way, then if they were not so related
they would be other than they are. The phrase “would be other
than they are” must be taken to mean “would be different things”,
P. A. Schilpp (ed.), The Philosophy of G. E. Moore, 1952, pp. 3-
G. E. Moore, Philosophical Studies, 1922, p. 163.
and not “would have different qualities”, if the second of these
expressions of the alleged fact is to square with the first. That they
would be different things if not so related has the consequence
that their not being so related implies a contradiction, hence that
they cannot but be so related. That they would have different
qualities if not so related does not imply that they could not but
be so related. This premise implies, the argument goes, that if two
things are related in a certain way, then it follows from their
nature, essence or definition that they are related in that way.
Russell does not cite any use of this argument by Bradley but it is
not of much importance whether he ever employed it, for not
only does the premise imply the conclusion but the conclusion
implies the premise. Consequently whether or not any
considerations which tell against the premise tell against a reason
which Bradley relied upon, they tell against his conclusion. The
premise indeed implies, and is implied by, the conclusion, for the
premise is equivalent to “If two things are related in a certain way,
then if they were those two things they would be related in that
way”. Now the consequence that if they were those two things
they would be related in that way will be true if, and only if, their
being those two things implies that they stand in that relation.
That is, it will be true if, and only if, their standing in that relation
follows from their nature, essence, or definition.
Is the premise true? Ambiguities which arise over such
words as “must”, “cannot” and certain “if…, then…”
constructions can make it seem plausible. The sentence “If two
things are related in a certain way, then they must be related in
that way” may be taken to express the triviality that if two things
are related in a certain way then they are related in that way, but it
may be taken to express the premise of the argument Russell is
examining, that if two things are related in a certain way then they
have to be related in that way. In the first interpretation the word
“must” is taken to express no more than the necessity of the trivial
implication. In the second interpretation it is taken to express the
necessity of the consequent of that implication. Now to assert that
this consequent is necessary is to assert that the two things have
to be related in that way, that being those two things implies
being related in that way. Therefore, in the second interpretation
the word “must” is taken to express the necessity, not of the trivial
implication that if two things are related in a certain way then
they are related in that way, but of the quite different implication,
asserted to be a consequence of the circumstance that two things
are related in a certain way, that being those two things implies
being related in that way.
This ambiguity is especially confusing in an argument
from an assertion expressed by the sentence we are considering,
for when the word “must” becomes separated from the “if…,
then…” construction it can more easily be taken to apply to the
consequent of the more trivial implication. For example, “If St
Paul’s Cathedral and the Eiffel Tower are at least a hundred miles
apart, then they must be at least a hundred miles apart. St Paul’s
Cathedral and the Eiffel Tower are at least a hundred miles apart.
Therefore, St Paul’s Cathedral and the Eiffel Tower must be at
least a hundred miles apart.” Of course, it is natural to employ the
word “must” in expressing the conclusion of an argument, just in
order to indicate that it is the conclusion of an argument, but if
the word remains in the sentence when it is used to assert the
proposition which, once established, is needed as the premise of a
new argument, then it may mislead someone into thinking that
what was established was the impossibility of St Paul’s and the
Eiffel Tower being less than a hundred miles apart.
A similar ambiguity over which of two implications is
being expressed can arise over certain “if..., then…”
constructions. This ambiguity is directly relevant to the second of
the two ways in which Russell puts the premise of the argument
he wishes to examine. The sentence “If A and B are related in a
certain way, then if any two things are not related in that way they
are not the two things A and B” can be taken to mean that if two
things, A and B, are related in a certain way, then any two things
are either related in that way or are not the two things A and B.
This seems the most natural way to take the sentence and it is an
obvious truth, but there is another interpretation in which it
expresses the premise of the argument Russell wishes to examine.
In this interpretation it expresses the validity of the inference
from the premise that two things, A and B, are related in a certain
way to the conclusion, which is itself an implication, that if any
two things are not related in that way then they are not the two
things A and B. Once these interpretations have been separated it
is possible to accept the first and reject the second. This is what
Russell does or, rather, seems about to do. He says that the force
of the argument depends upon “a fallacious form of statement”
and suggests that the premise “If A and B are related in a certain
way, then if they were not so related they would be other than
they are”, which is in the fallacious form, might be replaced by “If
A and B are related in a certain way, then anything not so related
must be other than A and B.”63 Russell seems to recognise the
distinctions we have been considering, reject the premise, and
suggest that it gains its plausibility from its similarity to the
acceptable, and evidently true, proposition by which he replaces
it. The sentence he uses for the replacement does express an
acceptable and true proposition if we take the word “must” to
express the necessity of the implication and not that of what is
implied. It seems that the word should be taken in that way, for
otherwise there would be no contrast with the premise which
Russell declares to be fallacious. However, once he has made this
point he restricts himself to the remark:
But this only proves that what is not related as A and B are must be
numerically diverse from A or B; it will not prove difference of
adjectives, unless we assume the axiom of internal relations.64
Remember that to say of two things that they are numerically
diverse is to say that they are different things, while to speak of
adjectives of a thing is Bradley’s way of speaking of those of its
properties which do not concern its relation with other things.
The word “quality” is perhaps a more natural one to use for such
non-relational properties. Russell’s comment seems irrelevant, for
the conclusion of the argument did not concern difference of
adjectives. The conclusion was that not being related as A and B
are, implies having a different nature, essence or definition from
A and B. To reach the further conclusion that not being related as
A and B are, implies having different qualities from A and B we
need to assume, as we have already seen, that only the qualities of
a thing, and not its relations, can be part of its nature, or essence,
or can follow from its definition. Therefore even the fallacious
M.P.D., pp. 58-9.
M.P.D., p. 59.
premise did not prove difference of adjectives, and the fault to
which Russell points is additional to, not consequent upon, the
fallacious form of the premise. The implication which the
fallacious premise has, and Russell’s revised premise does not
have, is that things not related as A and B are could not but be
numerically distinct from A and B. Russell’s comment leaves it
doubtful whether he saw that this was so. Of course, it may well
be that he took it for granted that only the qualities of a thing, and
not its relations to other things, can be part of its nature. Together
with that assumption the original premise does imply difference
of adjectives while, even with the assumption, the revised premise
does not, so that if the assumption is justified Russell’s comment
draws a correct, if rather indirect, distinction between the force of
the two premises.
The suspicion that Russell did not grasp the distinction
properly is strengthened by the explanation he gives of why the
original premise is fallacious in form. After he has cited the
premise “If A and B are related in a certain way, then if they were
not so related they would be other than they are”, he goes on:
Now if two terms are related in a certain way, it follows that, if they
were not so related, every imaginable consequence would ensue. For,
if they are so related, the hypothesis that they are not so related is
false, and from a false hypothesis anything can be deduced. Thus the
above form of statement must be altered.65
Far from showing why the premise is of fallacious form this
explanation implies that it is true and of an acceptable form.
Moreover, Russell was committed to the view that anything
follows from a false hypothesis by the account of implication
which he gave in The Principles of Mathematics.66 According to
that account, whether one proposition implies another is
determined solely by whether each of them is true or false. This is
not ordinarily held to be the case. Of course, the fact that a
premise is true and a conclusion false does mean that the premise
does not imply the conclusion, but it is ordinarily held that the
fact that both premise and conclusion are true, the fact that the
M.P.D., p. 59.
P.O.M., pp. 33-4, 36.
premise is false and the conclusion true, and the fact that both
premise and conclusion are false, all leave it open whether the
premise implies the conclusion. According to Russell’s account,
each of these three facts about the truth of the premise and of the
conclusion means that the premise implies the conclusion and if
the premise implies the conclusion, one of these three facts must
hold. The relation in which a proposition stands to another when
either it is false or the other is true, Russell called material
implication. According to his account, implication and material
implication are the very same thing. Since implication is the
relation which justifies inference, Russell’s contention that a false
proposition implies every proposition means that to infer from a
false proposition is to infer validly. Of course, it does not mean
that an argument with a false premise is a proof of its conclusion,
for an argument fails if one of its premises is false, just as it does if
the inference it involves is invalid. There is no similar mitigation
of Russell’s contention that a true proposition is implied by every
There is another consideration which seems to lessen the
divergence between Russell’s view of implication and that which
is ordinarily held. Russell puts it forward rather tentatively, after a
complicated discussion, in The Principles of Mathematics,67 but he
became more convinced of it later. For example, he makes use of
it in the passage from Our Knowledge of the External World which
we discussed in Chapter 3. He held that there is only one kind of
implication: the proposition that no green things are red implies
that no red things are green in no stronger, or different, a manner
from that in which it implies that Paris is the capital of France. He
held, however, that there are differences between propositions
which express implications, for some have a greater degree of
generality than others. The most general have no constants except
logical constants and are completely formal propositions of logic
and mathematics. Therefore, Russell can distinguish those
principles concerning implications which belong to logic from
those which do not. He assumes that this enables him to
distinguish, among all the implications which hold, those which
do so for logical reasons, that is, by virtue of their form. For
P.O.M., pp. 38-9, 41.
example, he would say that the proposition that no red things are
green implies that Paris is the capital of France for no better
reason than that Paris is the capital of France, while it implies that
no green things are red because of the logical law that whatever
properties F and G may be, any premise to the effect that no
things which have F have G materially implies a conclusion to the
effect that no things which have G have F. This latter implication,
he would say, is grounded in a purely formal fact. This comes out
in his discussion of the correct expression of the traditional
syllogism in the final chapter of Introduction to Mathematical
Philosophy.68 Unfortunately the view that material implication is
the only relation of implication between propositions plays such
havoc with the notion of reasons, or grounds, that once it has
been adopted, there is no possibility of distinguishing one true
proposition rather than another as the reason, or ground, of a
truth. Since it is true that the proposition that no red things are
green implies that Paris is the capital of France, and true that it
implies that no green things are red, the logical law is equally the
reason, or ground, of both implications. Russell can distinguish
between a logical and a non-logical principle concerning
implication, but not between implications which hold for logical
reasons and those which hold for non-logical ones.
Once implication has been identified with material
implication, the premise of the argument Russell presented for
consideration must be accepted. There is no possibility of
distinguishing between “If A and B are related in a certain way,
then if any two things were not related in that way they would not
be the two things A and B” and “If A and B are related in a certain
way, then any two things are either related in that way or are not
the two things A and B”. Therefore, to be consistent, Russell
should have accepted that every relation which A and B have does
follow from their being A and B and is part of the nature, essence,
or definition of A and B. He should have agreed with Bradley and
accepted the second of the two parts of the doctrine that if there
are relations, then those relations are internal to their terms: the
view that relations would have to be part of the nature, essence, or
definition of the related things. He could have rejected the further
I.M.P., p. 197
consequence that A and B could not but be related in the way that
they are, for, according to his account of implication, the fact that
one proposition implies another means that it is not the case that
it is true and the other false, but does not mean that it could not
be the case that it is true and the other false. However, although
Russell rejected any distinction between the necessity of a
proposition and its truth, he does not seem sure whether to reject
the notion of necessity or to accept it and identify it with truth.
Everything is in a sense a mere fact…. On the other hand, there
seems to be no true proposition of which there is any sense in saying
that it might have been false.69
If, as this suggests, Russell made no distinction between truth and
necessity, it would help to explain how he could accept material
implication as implication, for he would make no distinction
between it not being the case that a certain premise was true and a
conclusion false, and it not being possible that the premise should
be true and the conclusion false. So, perhaps, if he had accepted
that the relations between A and B are part of their natures he
would not have jibbed at the conclusion that A and B could not
but be related as they are. Russell’s quarrel was with the first part
of the doctrine that if there are relations they are internal to their
terms, not the second. He should have considered whether the
second part implies the first, that is, whether it could be part of
the nature of one thing that another thing existed.
Russell’s treatment of the argument he presents for
consideration can be explained in the following way. He saw the
distinction between the idealist premise and the revised form he
proposed and he saw that the idealist premise gained its
plausibility from the obvious truth of the revised form. However,
his commitment to material implication made it impossible for
him to explain the distinction, or, indeed, had he grasped the
distinction clearly, to accept it. Therefore he contented himself
with the remark that the revised premise does not imply the view
that relations are reducible to qualities. Since he fails to explain
why that view follows from the original premise any more than
the revised one, the relevance of this remark to the distinction he
P.O.M., p. 454
made between the premises is obscure. However, there is some
justification for thinking that the premise in its original form does
imply that view, and he should have considered more fully
whether it does so, for, unless he abandons material implication,
he is committed to that premise.
Russell’s interest lay in the first part of the doctrine of
internal relations. However, it seems that the second part too
should be rejected and his account of implication commits him to
accepting it, at least in part. Certainly it prevents him from
distinguishing between relations – and for that matter qualities –
which are part of the nature, essence, or definition of a thing and
those which are not. In an article called “Internal and External
Relations”70, published much later than Russell’s article, G. E.
Moore argued that such a distinction could be made, and that
while many relations were internal to their terms, many were not.
He gave the relation “being father of” as an example of one which
is external to its terms, and the relation “being intermediate in the
shade between” as one which is internal to its terms. It does not
follow, he asserted, from the fact that two individuals were,
respectively, Edward VII and George V that the one was the
father of the other. No one would think this followed unless he
confused “Edward VII was father of George V and A was not
father of B, implies that A and B were not, respectively, Edward
VII and George V” with “Edward VII was father of George V
implies that the proposition that A was not the father of B implies
that A and B were not, respectively, Edward VII and George V”.
The former is an obvious truth, the latter is not true, yet
according to Russell’s account of implication they are equivalent.
On the other hand, from the fact that three colours are
respectively, orange, red, and yellow, it does follow that the
former is intermediate between the two latter: these three colours
could not have been related otherwise. Moore was correct in
thinking that the identification of implication and material
implication was responsible for this confusion but he was not
correct in thinking that Russell was committed to the
consequence that Edward VII could not but have been father of
George V. If implication is material implication, the fact that one
G. E. Moore, Philosophical Studies, pp. 276-309.
proposition implies another does not have the consequence that it
could not be that the latter was false and the former true. At least,
it does not have that consequence unless, as Russell seems to have
been inclined to hold, every truth is a necessary truth.
Moore’s example of an internal relation is of a relation
between colours, which are universals. His example of an external
relation is of a relation between people, that is, between
particulars. Wittgenstein claims that internal relations stand
between such entities as propositions, possibilities and numbers:
The series of numbers is ordered not by an external, but by an
It is distinctions such as that between the way in which the
relation “being twice” is internal to the numbers four and two and
the way in which the relation “being father of” is not internal to
Edward VII and George V, that Russell’s account of implication
confuses. It is perhaps surprising that one such distinction plays
an important part in Russell’s own philosophy. This is the
distinction between those entities which are constituents of a
certain proposition and those which are not.
These, then, were the arguments which led Russell and
Moore to reject the doctrine that relations would have to be
internal to their terms, and, since one of the central theses of
Bradley’s idealism – that no proposition can ever present more
than an illusory appearance – rests on this doctrine, led them to
reject the idealist philosophy. The arguments reduce to two. First,
against the doctrine that relations would have to be qualities of
their own terms, Russell brought the demonstration of the
fundamental importance of relations in mathematics and the
absurdity of rejecting mathematics and much of common-sense
belief in order to preserve the logical principle that all
propositions are of the subject-predicate form. Second, against
the doctrine that all relations would have to be part of the nature,
essence, or definition of their terms, Russell in a confused way,
and later Moore with greater clarity, pointed to the distinction
between the triviality that it must be the case that if two things are
related in a certain way then they are related in that way, and
L. Wittgenstein, Tractatus Logico-Philosophicus, p. 83.
what is not a triviality, that two things which are related in a
certain way could not but have been related in that way. Besides
the confusion between these two, Moore argued, there is no
reason to believe the latter proposition. This leaves us free to
believe what seems evidently true – that relations are not
inevitably internal to their terms.
The rejection of idealism left Russell with the task of
developing a philosophy according to which it is possible to
formulate propositions which present reality, not appearance, and
so to conceive, and believe, the truth.
5 The Theory of Descriptions
Bradley undertook his criticism of relations in order to
demonstrate that there could be no facts of subject-predicate
form. He says towards the beginning of Appearance and Reality:
We find the world’s contents grouped into things and their qualities.
The substantive and adjective is a time-honoured distinction and
arrangement of facts, with a view to understand them and to arrive at
reality. I must briefly point out the failure of this method, if regarded
as a serious attempt at theory.72
A remark which he makes a few sentences later, concerning a
lump of sugar which we might say is white, hard and sweet,
indicates that he found the subject-predicate form of proposition
as suspect as the subject-predicate form of fact.
The sugar, we say, is all that; but what the is can really mean seems
Nevertheless Russell was not mistaken in attributing Bradley’s
rejection of relations to his preoccupation with the subject-
predicate form: Bradley thought it of fundamental importance
but believed it had not been properly understood. The distinction
between subject and predicate, he held, was not a distinction
within the reality which we think about – facts do not divide into
subject and predicate – nor was it a distinction within ideas.
Rather it was a distinction between the reality we think about and
the ideas we apply to it, between what we think about and what
we think it to be, between reality on the one hand and thought on
the other. The distinction is essential to Bradley’s account of the
nature of thought. He held that all thought is either complete or
incomplete judgement. Judgement is an activity, half-way
between believing and asserting, in which people engage. A
F. H. Bradley, Appearance and Reality, p. 16.
person makes a judgement when he applies an idea to a reality or,
in other words, when he predicates an idea of a reality.73 It is
because this is the only way in which a judgement can be made
that the subject-predicate distinction is fundamental to
judgement. All other thinking, such as questioning or supposing,
is incomplete judgement. Perhaps Bradley would have regarded
questioning as hesitation over whether to apply an idea to reality.
On Bradley’s view, what is ordinarily expressed by saying that
someone thinks that the lump of sugar is white would be more
correctly expressed by saying he thinks whiteness of the lump of
sugar. However, as we shall see in a moment, there is a further
aspect of Bradley’s view which means that this expression, too, is
not correct. It is, therefore, neither facts nor propositions which
are of subject-predicate form, but thoughts and judgements.
Indeed his view of thought dispensed with such entities as views,
or hypotheses, which have been held to be the objects of
questioning, supposition, belief, or assertion: in short, it
dispensed with propositions.
Bradley took this view of thought to have the
consequence that a person can think of a subject only by applying
ideas to it. Because thought is essentially the application of ideas
to a subject it cannot be that a person thinks of, or has in mind, a
subject to which he goes on to apply ideas. Rather it must be that
he applies ideas to a subject which he does not otherwise think of
at all. Russell endorses this inference of Bradley’s when he says in
The Principles of Mathematics, that the doctrine that every
…an immediate this, and a general concept attached to it by way of
description… develops by an internal logical necessity into the
theory of Mr Bradley’s Logic, that all words stand for ideas having
what he calls meaning, and that in every judgement there is a
something, the true subject of the judgement, which is not an idea
and does not have meaning.74
This consequence of Bradley’s view leads to another. Because the
subject of a person’s thought cannot enter his thought otherwise
F. H. Bradley, Appearance and Reality, p. 144
than through the ideas he predicates of it, a man cannot make one
judgement by having in mind St Paul’s Cathedral and predicating
whiteness of it and another by having in mind the Eiffel Tower
and predicating whiteness of that. These two buildings cannot
enter his thought, and any ideas by which he might be supposed
to single them out will be part of the predicates of his judgements.
So Bradley came to hold that all judgements have the same
subject: the whole of reality. St Paul’s is not the subject of the
judgement expressed by the sentence “St Paul’s is white”. The
subject is the whole of reality and of this subject a complex idea is
predicated, involving not only the idea of whiteness but also that
of a cathedral dedicated to the first apostle. Any difference
between two judgements which the same person might make at
the same time must reside, on this view of Bradley’s, in the ideas
which would be predicated in making it. He admits no
propositions as the objects of thought and belief, but he cannot
ignore the difference which he is unable to express in saying that
in one judgement one proposition is judged true and in another,
another. Instead, he expresses this difference by saying that the
two judgements have different contents. It is the content of a
person’s judgement which plays an analogous role to that played,
in other views of thought, by the proposition he would be said to
believe. The content of a judgement is made up of the ideas which
are predicated in making the judgement: it is a complex idea. The
proposition is made up of the things, qualities and relations which
the person is considering: it is a complex of realities. Bradley’s
view that the ideas a person predicates determine, at least in part,
the content of his judgement, stands opposed to the view that the
things, qualities and relations a person has in mind determine, in
part, the proposition which he holds to be true.
Russell rejected Bradley’s view of the nature of thought.
He believed that when a person thinks, he is directly related to
things in the world, to the qualities which they might have, and to
the relations in which they might stand. This constitutes the
realism which is the main characteristic of Russell’s mature
philosophy, although in The Analysis of Mind, nearly twenty years
later, he modified it so greatly that he could scarcely be said to
hold it any longer. In The Principles of Mathematics, Russell
accepts that when a person considers a question, makes a
supposition, holds a belief, or makes an assertion he is related to a
proposition. He often speaks of propositions as containing or not
containing entities of various kinds. For example, he says:
Whatever may be an object of thought, or may occur in any true or
false proposition, or can be counted as one, I call a term.75
And it seems undeniable that every constituent of every proposition
can be counted as one, and that no proposition contains less than two
This notion of the constituents – Russell also speaks of the terms –
of a proposition is one which he made considerable use of to
express views both about the nature of propositions in general
and about the interpretation of propositions of some particular
kind. The notion of constituents is a correlative of the notion of
logical form. We saw in Chapter 3 how Russell held that different
propositions may share the same logical form. Euclid’s axioms,
for example, share their logical form with any group of
propositions which differ from them only in that other concepts
stand in place of Euclid’s spatial concepts. The definition of
logical form to which such examples point – that it is what a
proposition shares with every other which differs from it only in
respect of non-logical particulars of concepts – closely resembles
Russell’s definition of a proposition of logic and suffers from the
same circularity. The constituents of a proposition are the non-
logical particulars or concepts whose replacement yields a
different proposition, but one of the same logical form.77 The view
that propositions have a logical form and constituents implies
that every proposition can be defined. Each definition must
specify a logical form, the constituents, and the manner in which
the form relates the constituents. For example, a proposition
might be defined as having Socrates as subject and wisdom as
P.O.M., p. 43.
predicate: its form is subject-predicate and its constituents
Socrates and wisdom.
If each proposition can be defined, then a distinction can
be drawn between those of its characteristics which follow from
its definition and those which do not. The former will be internal
to it, the latter external. It is internal to a proposition that it has a
certain logical form and certain constituents, but that it is true, if
it is, is external. The proposition just defined could not have had a
different subject but it could have been false. The definition of a
proposition someone has singled out must not be confused with
the characteristics by which he singles it out. It is no part of the
definition of the hypothesis called after Avogadro that it is called
after Avogadro. It could have been called after someone else and
probably would have been if someone else had discovered it. It is
part of the definition of Avogadro’s hypothesis that it is of
universal form and concerns gases. It could not have been
singular and concerned rocks. As we have seen, Russell’s account
of implication has the consequence that there can be no
distinction between what is internal and what is external. This did
not prevent him from employing these notions in other guises.
Russell may, of course, be wrong about the
characteristics of a proposition which must be included in its
definition. Different views may be held about what are the
internal characteristics of a proposition. Among these internal
characteristics certain relationships to other things might be
included. In that case, the proposition would be internally related
to these things but would not have them as terms. A word is
needed for this more general relationship and it is natural to
speak of such a thing being a part of, or occurring in, a
proposition. Russell does speak in this way but sometimes,
confusingly, he uses the word “constituent” in this more general
Russell’s realism is faced with three difficulties. First, it
seems that a person can think of things that do not exist. Second,
the proposition he considers sometimes seems determined, not by
the object he is thinking about, but by the characteristics he
singles it out by. Third, a person can think of things which he has
never perceived, sometimes could never perceive, such as events
which took place in the distant past or the experiences of other
people. In his first presentation of this view in Chapters IV and V
of The Principles of Mathematics, he ignored the last problem and
the solutions he gave to the first to are inconsistent with his
realism. In “On Denoting”78, he repudiated these solutions,
attacking views of Meinong and Frege which he took to be
versions of them, and put forward a new solution. This was
provided by the theory of descriptions. In “On Denoting” he is
concerned mainly with the first two problems, and certain other
difficulties. The solution which the theory of descriptions
provides to the third problem was developed in Problems of
Philosophy, and will be considered in the next chapter.
In The Principles of Mathematics79 Russell includes
chimaeras in his list of things which may be terms (that is,
constituents) of propositions and again, later in the same chapter,
he speaks of terms which do not exist, such as the points in a non-
Euclidean space or the characters of a novel.80 The need for such
non-existent terms is evident. On Russell’s theory a person
cannot wonder whether Socrates was wise unless the proposition
that Socrates was wise exists, and that proposition cannot exist
unless it has Socrates as a term. Now it is at least plausible to
assume that the non-existence of Socrates is no bar to wondering
whether Socrates was wise. Certainly it is no bar to wondering
whether Socrates existed. If Socrates could be a term of the
proposition without existing, the problem would be solved.
However, that Socrates is a term of the proposition implies that if
he did not exist the proposition would not exist, so the suggested
solution is a contradictory one. It is impossible that a proposition
should have non-existent terms.
When Russell argues, in “On Denoting”81, against his
former theory, he points out that it is possible to consider
propositions whose subjects, if they existed, would have
contradictory properties – for example, the proposition that the
round square is round. If this proposition is to exist, it must have
the round square as a term but that is an entity which not only
does not exist, but could not exist. Russell finds the notion of a
L.K., pp. 41-56.
P.O.M., p. 43.
P.O.M., p. 45.
L.K., p. 45
term which could not exist more objectionable than that of a term
which does not exist, although the notions are equally
contradictory. He was here considering a view he attributed to
Meinong, according to which some things exist and others do
not. The latter were granted a lower degree of existence and were
said to subsist. Russell had earlier regarded subsistence as
sufficient to allow a thing to be a term of a proposition but in fact
nothing short of existence is consistent with his notion of a term.
However, the objection he presses is the absurdity of allowing any
degree of existence to such things as round squares.
The second inconsistent feature of the view of thought
put forward in The Principles of Mathematics, arises over
propositions expressed by such sentences as “I met a man”,
“Every man is mortal”, and others involving such words as “all”,
“any”, “some” and “the”. In part, Russell’s view of these
propositions is in accordance with his realism. He says that the
proposition that I met a man is not about the concept of being
…this is a concept which does not walk the streets, but lives in the
shadowy limbo of the logic-books. What I met was a thing, not a
concept, an actual man with a tailor and a bank account, or a public-
house and a drunken wife.82
On the other hand he says a little later that the actual man he met
is no more a constituent of the proposition than any other man:
What is asserted is merely that some one of a class of concrete events
took place. The whole human race is involved in my assertion: if any
man who ever existed or will exist had not existed or been going to
exist, the purport of my proposition would have been different….
What is denoted is essentially not each separate man, but a kind of
combination of all men.83
A combination – more specifically, a disjunction – of all men is a
strange entity. He seems to have postulated it in order to avoid
accepting that the single sentence “I met a man” might express a
P.O.M., p. 53
P.O.M., p. 62
disjunction of propositions: either I met Jones, or I met Smith,
or… He certainly held that the proposition expressed by that
sentence implies and is implied by the disjunction, so the
strangeness of the combination of all men does not make his view
unacceptable. Equally strange entities are postulated as
constituents of propositions expressed with the help of the words
“all” etc., but “the” is treated differently. He held that a
proposition expressed with the help of the phrase “the Prime
Minister” had the one man who was Prime Minister as
constituent, not a combination of men.
So far this view is consistent with Russell’s realism, but it
is open to an objection which led him to add other features which
were not. Of course it also raises the first difficulty in another
form, for it seems that the sentence “A round square is round”
expresses a proposition, even though no round squares exist to be
constituents of the disjunction of propositions this sentence is
held to express. The view as I have so far expounded it does
distinguish the proposition that I met a man from the proposition
that I met a mathematician, even when the man I met happens to
have been a mathematician, because it is all men, not merely the
man I met, who are constituents of the former proposition and
not every man is a mathematician. It cannot distinguish the
proposition that I met a man from the proposition that I met a
featherless biped, because every man is a featherless biped and
every featherless biped a man. Nor can it distinguish the
proposition that I met the Prime Minister from the proposition
that I met the leader of the party in power, since both will have
the same constituents. Russell believed that each of these
propositions should be distinguished. To achieve this he allowed
that the concept of being a man occurred in the proposition that I
met a man, and the concept of being Prime Minister occurred in
the proposition that I met the Prime Minister. He allowed a
special class of concepts which he called denoting concepts. Such a
concept could occur in a subject-predicate proposition while
being neither its subject nor its predicate. For example, Russell
regarded the concept the Prime Minister as occurring in the
proposition expressed by the sentence “The Prime Minister is
speaking” although he did not regard that concept as subject or
predicate of this proposition. Again, he regarded the denoting
concept a man as occurring in the relational proposition that I
met a man although he did not regard it as one of the terms of the
But such concepts as a man have meaning in another sense: they are,
so to speak, symbolic in their own logical nature, because they have
the property which I call denoting.84
Besides providing him with this distinction, the view that there
are denoting concepts enabled him to argue that the implications
of a subject-predicate proposition involving a denoting concept
depend upon that concept and not upon what subject it happens
to denote. The proposition that the Prime Minister is speaking
implies that a minister is speaking, but it does not imply that
Jones is speaking even when Jones is the Prime Minister. He also
held that denoting explained why it is worth while to assert
identity. It is a triviality that Jones is Jones, but, even when Jones
is Prime Minister, it is not a triviality that Jones is the Prime
Minister. His explanation depends on the occurrence of a
denoting concept in the non-trivial identity, but it is not clear
what the explanation is.
This view that there can be denoting concepts which
occur in propositions is inconsistent with Russell’s realist view of
thought, for that view has the consequence that a person is
concerned in thought with a proposition which is determined
once its terms, its logical form, and the manner in which the form
relates the terms are given. There can be no difference between
propositions which have these three characteristics in common.
Presumably it is this inconsistency which prevented Russell from
explaining how denoting concepts enter into propositions. He
seems to have held that a proposition had two parts – a symbolic
part, or meaning, to which the denoting concepts belong, and an
objective part to which the terms, or constituents, belong. This
raises a number of problems. For example, if the proposition is
composed of both the symbolic and the objective parts, why is it
only the symbolic part which determines the implications of the
P.O.M., p. 47
When he returned to these problems in “On Denoting”,
Russell confirmed two of his earlier assumptions and rejected
two. He remained convinced that a person could, in some sense,
consider questions and hold beliefs about non-existent things. On
his realist view, thought is concerned with propositions, so this
meant that the fact that there are no unicorns does not imply that
there is no proposition to the effect that I met a unicorn, nor does
the fact that France has no king imply that there is no proposition
to the effect that the King of France is bald. Second, he remained
convinced that the proposition that the Prime Minister is
speaking is distinct from the proposition that the leader of the
party in power is speaking, although the Prime Minister is the
leader of the party in power. He reinforces this view with the new
argument that the proposition that Scott was the author of
Waverley must be distinct from the proposition that Scott was
Scott, since George IV wished to know whether the former was
true but, not being a logician, showed no interest in the latter.85
However, as we have already seen, he rejected the assumption that
the King of France could be a term of a proposition without
existing, and he rejected the view that concepts can occur in
propositions in the form of denoting concepts, that is, as
meanings rather than terms. We will consider his reasons for this
presently. These four views left him with no alternative but to
seek an account according to which propositions expressed with
the help of such phrases as “a unicorn” and “the King of France”,
do not have unicorns, or disjunctions of unicorns, or the man
who is King of France, as terms, but do have as terms the
properties of being a unicorn, and being a King of France. Since
the question a person raises when he wonders whether he has
ever met a wise man cannot depend on whether any wise men
exist – although the answer to his question may depend upon that
– the account of the propositions expressed by the sentences “I
met a man” and “I met a unicorn” must be parallel. Russell
discovered an account of all propositions expressed with the help
of the words “a”, “all”, “the”, etc., which, he claimed, fulfilled all of
his conditions. This account came to be called the theory of
descriptions. It enabled him to solve all three of the problems
L.K., p. 50.
which faced his realism. Before discussing this theory we must
consider some criticism of the arguments in “On Denoting”.
Russell begins by referring to the theory concerning the
words “a”, “all”, “the”, etc. which he put forward in The Principles
of Mathematics. He says that this earlier theory is nearly the same
as the theory which Frege suggested in an article called “Uber
Sinn und Bedeutung”86 and he goes on to criticise a theory which
he says is Frege’s. The title of Frege’s article has been translated in
various ways; Russell would have given it as “On Meaning and
Denotation”. Russell’s earlier theory is not the same as Frege’s
and the theory he criticises in “On Denoting” is not Frege’s but
his own. Russell takes Frege to hold, as he himself had done, that
propositions have two parts – a meaning and a denotation. The
concept of being the Prime Minister belongs to the meaning of
the proposition that the Prime Minister is speaking, the man who
is Prime Minister belongs to the denotation.
Frege distinguishes the two elements of meaning and denotation
everywhere, and not only in complex denoting phrases… In the
proposition “Mont Blanc is over 1,000 metres high”, it is, according
to him, the meaning of “Mont Blanc”, not the actual mountain, that is
a constituent of the meaning of the proposition.87
Frege did not hold that a proposition consisted of two parts, a
meaning and a denotation. He held that a sentence had a
meaning, and sometimes a denotation, and he identified the
meaning of the sentence with the proposition. For this reason he
did not hold that the meaning of the phrase “Mont Blanc”
belonged to the meaning of the proposition. He held that it
belonged to the meaning of the sentence, that is, to the
proposition itself. The denotation of the sentence, to which the
actual mountain belonged, was no part of the proposition. In
Frege’s theory, as in Bradley’s, the real things we think about do
not enter into our thought.
This misunderstanding, which is by no means without
excuse, explains why Russell criticises Frege’s theory for
G. Frege, “On Sense and Reference”, in Philosophical Writings
of Gottlob Frege, tr. Max Black & P. Geach, 1952, pp. 56-78.
L.K., p. 46n.
consequences which that theory was designed to avoid. The
passage in which he does this is perhaps the best known in his
If we say “the King of England is bald”, that is, it would seem, not a
statement about the complex meaning “the King of England”, but
about the actual man denoted by the meaning. But now consider “the
King of France is bald”. By parity of form, this also ought to be about
the denotation of the phrase “the King of France”. But this phrase,
though it has a meaning provided “the King of England” has a
meaning, certainly has no denotation, at least in any obvious sense.
Hence one would suppose that “the King of France is bald” ought to
be nonsense; but it is not nonsense, since it is plainly false.88
Russell is considering the theory that a meaning and a denotation
are needed to make up a proposition and argues that the meaning
being King of England is not what the proposition that the King of
England is bald is about, so that it is not part of the denotation.
The King himself is part of the denotation – in fact, he is the
subject of the proposition – so that if England had no king a
constituent of the denotation, and hence of the proposition,
would be lacking. But if a constituent of the proposition is
lacking, then the proposition itself is lacking. Russell actually
writes that, on this theory, since there is no King of France, “ ‘The
King of France is bald’ ought to be nonsense”, but this is his way
of saying that the proposition that the King of France is bald
ought to be nonsense, that is, that the proposition that the King of
France is bald ought not to exist. He has been taken, particularly
by P. F. Strawson89, to be saying that the sentence “the King of
France is bald” ought to be meaningless but, although he is not
consistent in his use of quotation marks, he cannot here be
speaking of the sentence for since, earlier in the passage, he
admits that the phrase “the King of France” would have a
meaning, he cannot have thought that he had given any reason for
concluding that the sentence containing the phrase would have
none. It is not unnatural to say that a proposition is nonsense and
L.K., p. 46
P. F. Strawson, “On Referring”, in Classics of Analytic
Philosophy, ed. Robert R. Ammerman, 1965, pp. 315-34.
to mean that there is no such proposition. In fact, Russell asserts
in this passage exactly what Strawson himself maintains: that on a
theory according to which denoting phrases, such as “the King of
France”, have a meaning and denote a denotation, the sentences
in which the phrases occur will have a meaning but will express
no proposition if the denotation is lacking. Strawson differs from
Russell only in his attitude towards this consequence. Russell held
that, since it is plainly false that the King of France is bald, there
plainly is a proposition to the effect that the King of France is
bald, and the theory must be rejected. Strawson held that it is
neither true nor false that the King of France is bald, so that the
theory can stand. Strawson is mistaken in attributing Russell’s
rejection of the theory to a mistake about meaning. He claims that
Russell confused having meaning with expressing a proposition
which is true or false and that he argued from the premise that the
sentence “the King of France is bald” has meaning to the
conclusion that it expresses a proposition which is false. However,
in this passage Russell rejects his former theory on the grounds
that while it allows meaning to the phrase “the King of France” it
cannot explain how the sentence “the King of France is bald” can
express a false proposition. The closing words of the passage are
“since it is plainly false”. Whether the “it” stands for the sentence
“the King of France is bald” or the proposition that the King of
France is bald, these words express the premise, not an
intermediate step, of his argument. What is interesting in
Strawson’s criticism is the suggestion that Russell should not have
been so ready to reject the conclusion that when there is no King
of France there is no proposition to the effect that the King of
France is bald. Probably Russell argued that if he accepted it, then
he would have to accept the conclusion that when there is no
King of France there is no proposition to the effect that the King
of France does not exist. Strawson’s own theory is a revised form
of Russell’s earlier theory. Whether it avoids the other objections
Russell brought against that theory cannot be considered here.
The view that denoting phrases have meanings which, in
favourable circumstances, single out an entity or group of entities
which may be called their denotation might be thought to leave
three possibilities open. Either the meaning alone, or the
denotation alone, or both meaning and denotation, might be
regarded as internal to a proposition expressed with the help of
the phrase. If he had believed that this view of denoting phrases
did leave open all three possibilities, Russell would have accepted
the first and rejected the second and third. He rejects the third in
the passage we have been considering, because it has the
consequence that if there is no denotation there is no proposition.
He rejects the second for the same reason and, independently,
because it has the consequence that, although the sentences “Scott
is Scott” and “Scott is the author of Waverley” have different
meanings, they express the same proposition. He would accept
the first because it avoids both these difficulties. That, too, was
Frege’s reason for accepting it. However, Russell did not believe
that the view did leave either the first or the third possibility open.
He argues that if the meaning of a denoting phrase were part of a
proposition then that proposition would have as a term, not the
meaning, but the entity singled out by the meaning. Therefore the
meaning cannot be part of a proposition. He says that whenever a
denoting phrase occurs in a sentence
…what is said is not true of the meaning, but only of the denotation,
as when we say: The centre of mass of the solar system is a point.
A little later he concludes:
…so long as we adhere to this point of view, we are compelled to
hold that only the denotation can be relevant.90
He is arguing from his realist position that a proposition is
composed only of the particulars, qualities and relations which
are its terms. If the phrase “the centre of mass of the solar system”
has a meaning and a denotation, then one or the other of them
would be the subject of the proposition. Clearly that one would be
the denotation, not the meaning, so that it would be the
denotation which was internal to the proposition. Frege, holding
a very different view of thought, was able to accept that the
meaning and not the denotation of the denoting phrase could be
internal to the proposition expressed with its help, and so hold
that the view that denoting phrases have meaning and denotation
L.K., p. 50
had just the consequences which Russell was seeking to embody
in a theory.
Here again there is an unrecognised agreement between
Strawson’s views and Russell’s. Like Russell, Strawson seems to
find only the second possibility consistent with this view of
denoting phrases. Unlike Russell he is prepared to accept it.
According to Strawson it is the fact that denoting phrases have
meaning that enables people to use them to single out entities –
he calls this activity “referring” – and so to express propositions
about those entities. However, he does not seem willing to admit
that this view has the consequences which led Russell to reject it,
for example, that by the sentences “Scott is Scott” and “Scott is the
author of Waverley”, a person expresses the very same
proposition, although by different means.
We must now turn to the theory of descriptions,
Russell’s own solution of these difficulties. He found propositions,
whose expression did not involve denoting phrases, which, he
claimed, clearly did not have the entity which he had formerly
regarded as the denotation of the denoting phrase as a term, and
clearly did have as terms the qualities and relations which he had
formerly regarded as the meaning of the denoting phrase. These
propositions he identified with the propositions expressed with
the help of denoting phrases. For example, he identified the
proposition expressed by the sentence “I met a man”, which
involves the denoting phrase “a man”, with that expressed by the
sentence “there was something which was both human and met
by me”. This procedure he calls presenting an analysis of the
proposition. Although he gives different analyses for the different
denoting phrases, they have a common feature. In the unanalysed
expression of the propositions the denoting phrases occur as units
such as “a man”, “all men”, “the men”, and they seem to work as
units, having a meaning by which they single out an entity or
group of entities as a term of the proposition. In the analysed
expression the words “a”, “all”, “the”, are no longer united with
the concept words, nor do the pairs work as units. The word “the”
disappears entirely. For example, the proposition that I met a
man is identified with the proposition that a thing was both
human and met by me. The proposition that I met all men is
identified with the proposition that all things were either not
human or met by me. The analysis of the proposition that I met
the man is more complicated. Russell held that this proposition
implies the proposition that I met a manrom it solely in having
the further implication that only one man exists. If it were true
that there was one and only one thing which was human Russell
would be prepared to speak of that man as the denotation of the
phrase “the man”. That man, however, is not a term of the
proposition expressed by the sentence “I met the man”, nor does
the phrase “the man” have a meaning as a unit because if it had,
its function would be to single out an entity as a term of the
proposition expressed by the sentence.91
These analyses enabled Russell to explain in what sense it
is that we can consider questions about non-existent things. If a
person considers whether he has ever met a wise man he
considers whether there was something which was wise, human,
and met by him. He can do this even if no wise men exist.
Similarly, he can consider whether the author of the Iliad was
blind because he can consider whether there was no more than
one person who wrote the Iliad and whether there was a person
who both wrote the Iliad and was blind. He can do this even if the
Iliad was in fact a collection of folk tales without an author.
However, if the analysis is to be used to explain in what sense we
can think of Homer if he did not exist then the proposition that
Homer was blind must be identified with some proposition
expressed by means of a denoting phrase. It might be identified
with the proposition that the author of the Iliad was blind. Russell
held that we can think about non-existent things or people only in
such ways as this. He says that in the sentence “Homer was
blind” the word “Homer” is a description and not a proper name.
He sometimes says that it is not a logically proper name, meaning
that it is grammatically a proper name but does not introduce an
entity as subject, or other constituent, of the proposition. He held
that a person can understand a proposition only if he can have its
constituents in mind directly. This, he thought, a person can only
do if he has been in some direct cognitive relation to them. A
person cannot understand a proposition having a particular as
constituent unless he has perceived that particular, nor one
L.K., p. 51
having a quality or relation has constituent unless he has acquired
an intellectual grasp of that quality or relation. These were the
two ways in which a person could have acquaintance with an
If the proposition that the author of the Iliad did not
exist is analysed in the way we have been considering, it would be
identified with the contradiction that only one person wrote the
Iliad and did not exist. For such propositions Russell proposed a
different analysis: it is not true that one and only one person
wrote the Iliad. Even existential propositions concerning
particulars which are objects of acquaintance should, perhaps, be
analysed in this way. A particular with which a person is
acquainted might not have existed. If it had not, then it could not
have been a constituent of the proposition that it exists. Some
philosophers take it as a principle that the thoughts a person
thinks, and consequently the propositions he considers, cannot be
restricted by non-logical facts such as the non-existence of a
particular. This principle stems from Leibniz. Philosophers who
adopt it, for example Quine93 and A. J. Ayer94, do not allow that
any particular is a constituent of a proposition and, with
qualifications, regard all proper names as descriptions.
It has been pointed out by Martin Shearn95 that Russell’s
analysis of propositions concerning the existence of particulars
sometimes yields implausible results. An example similar to one
of Shearn’s would be the proposition that the god feared by the
ancient Israelites did not exist. Russell’s analysis would be: it is
not true that there was one and only one god feared by the
Israelites. Yet Russell very likely accepts, surely without self-
contradiction, both that the god feared by the ancient Israelites
did not exist and that the ancient Israelites feared one and only
one god. In fact many propositions concerning human attitudes
and activities pose difficulties for the theory of descriptions. One
which Russell mentions is the proposition that George IV asked
whether Scott was the author of Waverley. Does this imply, as it
L.K., p. 41.
W. V. O. Quine, From a Logical Point of View, p.2
A. J. Ayer, The Concept of a Person, 1963, p. 134.
M. Shearn, “Russell’s Analysis of Existence”, in Analysis vol.
11, 1950-51, p. 124ff.
would on the analysis we have been considering, that someone
wrote Waverley? Russell deals with this difficulty by introducing
another element into his analysis: the distinction between
primary and secondary occurrence of denoting phrases.96 If the
proposition is analysed as: only one person wrote Waverley and of
a person who wrote Waverley George IV asked whether he was
Scott, then the denoting phrase “the author of Waverley” has been
treated as having a primary occurrence. If it is analysed as: George
IV asked whether only one person wrote Waverley and Scott
wrote Waverley, then the denoting phrase has been treated as
having secondary occurrence. Quine97 has suggested a means of
extending this distinction of Russell’s to other propositions which
raise difficulties. An example is the proposition that Hudson tried
to find the North West Passage, which would be true even if no
North West Passage existed. Quine would analyse this
proposition as: Hudson tried to bring it about that he knew where
the North West Passage lay. The phrase “the North West Passage”
can now be treated as having secondary occurrence.
The theory of descriptions provides nearly all that
Russell asked of it. It explains in what way we can think and speak
of things which do not exist. It distinguishes between the
proposition that the Prime Minister is speaking and the
proposition that the leader of the party in power is speaking. It
explains both these without implying that concepts can occur in
propositions other than as terms. The only doubt arises over
whether, according to the theory, propositions expressed with the
help of denoting phrases do not have as constituents those entities
which the denoting phrases denote. This doubt arises in the
following way. It is not internal to the proposition that there was
something which was both human and met by me, that there was
something human, i.e. that there were any men. The proposition
implies, of course, that there were men, but it would have existed
whether or not there had been men, and could have been
considered, believed or disbelieved. This is a point Russell often
stresses. The same argument applies to the proposition that I met
the man. However, the question whether Brown is a constituent
L.K., pp. 52-3.
W. V. O. Quine, Word and Object, 1960, pp. 151-6.
of the proposition that there was something which was both
human and met by me, is not the same as the question whether it
is internal to that proposition that there are men. Brown might
not have been a man, so that his existence might be necessary to
that of the proposition although the existence of men is not.
What, according to the theory of descriptions, are the
constituents of the proposition that I met a man? Russell’s
account98 of them suffers from complete circularity. It is founded
upon the notion of a variable and that notion is inseparable from
another, that of a propositional function. A propositional
function, as Russell explains it, is an entity which is got by
omitting a term from a proposition. It is symbolised by an
expression in which the gap left when the name of the missing
term is removed is filled by a letter such as x. For example, if we
take the proposition that Brown is human and omit Brown we get
the propositional function x is human. Again, if we take the
proposition that Brown is both human and met by me and omit
Brown we get the propositional function x is both human and met
by me. Once he has introduced this notion Russell goes on to
consider three properties which propositional functions may have
– the property of being always true, the property of being
sometimes true and the property of being never true. He says that
the last two can be defined in terms of the first, but that he is
going to treat the first as indefinable. He then explains the
proposition that there is something which is both human and met
by me as the proposition that the propositional function x is both
human and met by me is sometimes true. This completes his
account of the constituents of the proposition that I met a man. It
seems that he intends that the subject of this proposition is a
propositional function and the predicate is the property of being
Although Russell says at the beginning of “On Denoting”
that he intends to take the notion of being always true as
“ultimate and indefinable” he does, of course, expect us to
understand what he means by it. In a later work99 he explains that
to say that a propositional function is always true is to say that,
L.K., pp. 42-3.
I.M.P., p. 158
whatever entity we take to complete the propositional function,
the result is always a true proposition. Similarly, to say that a
propositional function is sometimes true is to say that some one
of the entities which might be taken to complete the propositional
function would yield a true proposition. But these notions were
introduced for the very purpose of giving an analysis of
propositions such as that something is human or that everything
is human which would make it clear what their constituents were.
Since the analysis proposed is itself one of these propositions it
leaves us no clearer about their constituents.
There is, however, another explanation of the notions of
being sometimes true and being always true which is not circular
and which Russell may have intended. It might be held that the
propositional function x is human is always true if, and only if,
the conjunction of all the propositions which can be obtained by
completing the propositional function is true. Since the gap
symbolised by x might be filled by one of the men but might
equally be filled by any other entity this conjunction will be very
long, perhaps infinitely long. Similarly, the propositional function
x is human will sometimes be true if, and only if, the disjunction
of all these propositions is true. This is suggested by the following
It is to be observed that “all S is P” does not apply only to those terms
that actually are S’s; it says something equally about terms which are
not S’s…. In order to understand “all S is P”, it is not necessary to be
able to enumerate what terms are S’s; provided we know what is
meant by being an S and what by being a P, we can understand
completely what is actually affirmed by “all S is P”, however little we
may actually know of actual instances of either. This shows that it is
not merely the actual terms that are S’s that are relevant in the
statement “all S is P”, but all the terms concerning which the
supposition that they are S’s is significant.100
This suggests that whereas on his earlier view the proposition that
I met a man was taken to be equivalent to a disjunction of
propositions involving all men, on his later view it is taken to be
equivalent to a disjunction of rather different propositions
I.M.P., pp. 161-2.
involving all entities. The original disjunction was that either I
met Brown, or I met Jones, or I met Robinson, or… The revised
disjunction is that either Brown is human and I met Brown, or
Fido is human and I met Fido, or Excalibur is human and I met
This difficulty concerns only one of the consequences
Russell drew from the theory of descriptions but it is an
important one, for he relied on it in the explanation of how it was
possible for a person to understand propositions concerning
entities which did exist but with which he had no acquaintance.
The theory cannot explain this unless it has the consequence that,
even when a denoting phrase does denote an entity, that entity is
not a constituent of the proposition expressed with the help of the
phrase. However, in considering Russell’s views concerning a
person’s knowledge of the world beyond his own sensations – in
which this consequence plays a central part – I shall make no
more mention of the difficulty.
6 The External World
The theory of descriptions enabled Russell to avoid the
implausible consequences of his view that when we think, we are
directly related to entities which are independent of us and our
minds. It did this at the price of re-introducing a distinction
between thought and knowledge which Bradley had made much
use of: that between appearance and reality. In the theory of
descriptions this took the particular form of a distinction between
the apparent constituents of a proposition and its actual
constituents. These are confused, Russell held, because of a
confusion between the subject of a sentence and the subject of the
proposition expressed by that sentence. The phrase “the golden
mountain” is the grammatical subject of the sentence “The golden
mountain does not exist”, but the proposition expressed by that
sentence does not have the golden mountain as its subject, for
there is no such mountain. In misleading us about the
constituents of our propositions, grammar misleads us about
their logical form. We are misled by grammar into attributing to
the proposition expressed by the sentence “The golden mountain
does not exist” the subject-predicate rather than the existential
form. The idea that the grammar of a language can mislead us
about the logical form of the propositions which it enables us to
express, suggests that a language might be designed with a perfect
grammar which would not mislead us. Russell regarded the
symbolic language of Principia Mathematica as a first step
towards such a perfect language. He never succeeded in
improving on the unsatisfactory account of logical form which he
gave in The Principles of Mathematics, and this difficulty of
explaining what it is that grammar misleads us about led later
philosophers, inspired by the logical insights of the Principia, to
seek a different account of the inadequacies of natural languages.
Rudolf Carnap101 identified grammatical form with logical form
and held that the fault in the grammar of a natural language lay in
its ambiguities and inconsistencies rather than in its failure to
express the true logical form of propositions. Tarski102 and
Quine103 adopted a similar approach.
To identify the actual constituents of the proposition
expressed by a sentence Russell employs his principle that
…in every proposition that we can apprehend… all the constituents
are really entities with which we have immediate acquaintance.104
For example, if we can understand the proposition expressed by
the sentence “Hadrian visited Britain”, that proposition cannot be
that only one person became Emperor of Rome in A.D. 117 and
someone both became Emperor of Rome in A.D. 117 and visited
Britain, for we no more have immediate acquaintance with
ancient Rome than with Hadrian. The proposition expressed
might perhaps be that only one person built Hadrian’s Wall and
someone both built Hadrian’s Wall and visited Britain, if
Hadrian’s Wall is a thing which we can, at this time, have
immediate acquaintance with. This illustrates one way of
interpreting a sentence which seems to express a proposition
having constituents with which we are not acquainted: the actual
constituents include entities which the apparent constituents are
implied to have brought into being.
There are at least two other ways in which such
sentences may be interpreted. The actual constituents of the
proposition expressed may be taken to be qualities which, the
proposition is held to imply, are possessed by, and only by, the
apparent constituents. For example, the sentence “Pegasus rose
into the air” might be interpreted as expressing the proposition
that only one thing ever was a winged horse and that at least one
thing both was a winged horse and rose into the air. In order to
R. Carnap, The Logical Syntax of Language, 1937.
A. Tarski, Logic, Semantics and Metamathematics, 1956.
W. V. O. Quine, Word and Object.
L.K., p. 56
understand this proposition we need acquaintance with the
qualities of being a horse and having wings, but not with Pegasus.
Again, the actual constituents of the proposition expressed by a
sentence might be taken to include certain universals with which
we have acquaintance and with which the apparent constituents
are implied to be uniquely related. For example, the sentence
“Newton investigated light” might be taken to express the
proposition that only one person first distinguished the properties
of mass and weight and that at least one person both did this and
If we knew one of these propositions to be true we would
know that there existed an entity of a certain kind or an entity
related to the constituents of the proposition in a particular way.
We would know that there once existed someone who had
Hadrian’s Wall built, that there once existed a winged horse, and
that there once existed someone who first distinguished the
properties of mass and weight. Russell says that if we had this
knowledge we would have “knowledge by description” of these
three entities. He contrasts the manner in which our knowledge
that only one person had Hadrian’s Wall built and someone both
had Hadrian’s Wall built and visited Britain is knowledge of the
man who had Hadrian’s Wall built, with the manner in which
Hadrian’s knowledge that he himself had the Wall built is
knowledge of the man who had Hadrian’s Wall built. The former
knowledge is knowledge of the man who had the Wall built
because it includes the knowledge that one and only one man had
the Wall built; the latter knowledge is knowledge of the man who
had the Wall built because that man is a constituent of the
proposition which is known to be true.105
This is the distinction which Russell intended by the
phrases “knowledge by description” and “knowledge by
acquaintance”. It is a distinction between two ways in which our
knowledge can be of an actual entity. The former is made possible
by our knowing that one and only one entity satisfies a certain
description, the latter is made possible by our having
acquaintance with the entity itself. Unfortunately some of the
things Russell says when he is discussing this topic in Problems of
P.P., pp. 54-7.
Philosophy suggest that the contrast he has in mind lies between
acquaintance as one sort of knowledge and description as another
Knowledge of things, when it is of the kind we call knowledge by
acquaintance, is essentially similar than any knowledge of truths….
Knowledge of things by description, on the contrary, always
involves… some knowledge of truths as its source and ground.106
A little later he says, speaking of a particular shade of colour
which he is seeing:
…so far as concerns knowledge of the colour itself, as opposed to any
knowledge of truths about it, I know the colour perfectly and
completely when I see it, and no further knowledge of it itself is even
The first passage means, I think, that that knowledge which is
knowledge of a thing because we have acquaintance with it is not
knowledge of that thing because of anything we know to be true,
while that knowledge of a thing which Russell calls knowledge by
description is knowledge of that thing because of something we
know to be true. However, the second passage does explicitly state
that when a person has acquaintance with a thing, he knows that
thing perfectly and completely. This view is irrelevant to the
distinction between knowledge by description and knowledge by
acquaintance which Russell goes on to make, but criticisms of the
view have sometimes been regarded as criticisms of the
distinction. G. E. Moore criticises Russell’s use of the phrase
“knowledge by acquaintance” on the grounds that acquaintance is
not knowledge at all, but direct perception.108 He is right about
acquaintance but the criticism is beside the point: knowledge by
acquaintance of an entity is knowledge which is of it because we
have acquaintance with it, not knowledge of it which is
acquaintance with it.
P.P., p. 46.
P.P., p. 47.
G. E. Moore, Some Main Problems of Philosophy, 1953, p.
Acquaintance with an entity does not prevent us, Russell
held, from having knowledge by description of it. He illustrated
this by the sentence “The candidate who gets the most votes will
be elected”. Someone who utters this sentence may be acquainted
with all the candidates, yet the proposition he intends to express
is unlikely to be one which has the candidate who will get the
most votes as subject. The proposition which he is most likely to
have intended implies that anyone, if he gets most votes, will be
elected. However, this example throws doubt on Russell’s notion
of knowledge of an entity by description, for it hardly seems that
someone who knows that anyone, if he gets most votes, will be
elected, has knowledge of the candidate who will get the most
votes. On the other hand, if the seat was a safe one, someone
might employ the sentence “The candidate who will get the most
votes is a scoundrel” to express something he knows to be true,
and then, it seems, his knowledge would be knowledge of the
candidate who will get most votes. However, the proposition he
intends is one which does have the candidate who will get the
most votes as its subject, not one which implies that anyone, if it
is true that he will get the most votes, is a scoundrel. This suggests
that the plausibility of Russell’s view that there is such a thing as
knowledge of an entity by description rests on sentences which
are ambiguous. In one interpretation they express propositions of
such a character that anyone who knows them to be true knows
that there is one and only one entity of a certain kind. In the other
interpretation they express propositions of such a character that
anyone who knows them to be true has knowledge which is
knowledge of a certain entity.
The existence of such ambiguities can be shown by
another example. When we speak of a painting as being Turner’s
we mean that it was painted by Turner, so by the sentence
“Turner’s paintings were painted by Turner” we may mean that
Turner painted some paintings and that anything, if it was
painted by Turner, was painted by Turner. In fact we might
mean, besides the fact that Turner painted some paintings,
nothing more than a tautology. However, we might wish to
express the fact that the paintings which were painted by Turner
might have been painted by someone else, and, if we knew of
nothing in common between Turner’s paintings except that they
were all painted by Turner, we might express this fact by the
sentence “It might not have been the case that Turner’s paintings
were painted by Turner”. Since we would, in that case, be using
this sentence to express a fact, and not merely the fact that Turner
might have painted nothing, it cannot be that the proposition we
would express by the words “Turner’s paintings were painted by
Turner” is, besides the implication that Turner painted some
paintings, a tautology. This latter sentence is therefore
ambiguous. In its first meaning it expresses knowledge which,
according to Russell, is knowledge of Turner’s paintings (by
description) and the knowledge it expresses is that they were
painted by Turner. However, it cannot in fact embody this
knowledge, for nothing about any paintings in particular is
entailed by the fact that Turner painted some paintings. In its
second meaning the sentence expresses a proposition which
embodies knowledge of certain paintings – the knowledge that
they were painted by Turner – but it is not a conjunction of an
existential statement and a tautology. It seems that it is a
proposition which has Turner’s paintings as subject. Only the
confusion of these two propositions gives plausibility to Russell’s
view that if our knowledge includes the fact that there is one and
only one entity, or group of entities, of a certain kind, then our
knowledge is knowledge of that entity.
Russell often writes as if, although we can have no
acquaintance with people or things which ceased to exist before
we were born, or will come into existence only after our death, yet
we do have acquaintance with people and things when we see or
touch them, and do have acquaintance with ourselves. He takes
this position only for the sake of simplicity of exposition. In
reality he believed that we can have acquaintance with particular
things only if we can perceive them directly, and he did not
believe that we have direct perception of physical objects, of other
people, or even of ourselves.
…the real table, if there is one, is not the same as what we
immediately experience by sight or touch or hearing. The real table,
if there is one, is not immediately known to us at all, but must be an
inference from what is immediately known.109
P.P., p. 11.
If we see a thing, as opposed to seeing something which that thing
has brought into being, we have immediate experience, or direct
perception, of that thing. If we see that a thing has a certain
property, as opposed to inferring that fact from another which we
see to hold, then we have immediate knowledge of that fact.
Russell held both that we have no direct perception of physical
objects and that we have no immediate knowledge of facts
concerning physical objects. The reasons which led him to this
view have a long history in philosophy. He sets them out in the
first chapter of Problems of Philosophy. Three arguments can be
distinguished. A table appears to have very different colours in
different lights. Therefore there is no colour which it appears to
be, and hence no colour which we see it to be.110 The colour a
table appears to have does not depend only upon the colour it has.
It depends upon many other factors such as the incident light and
the eyesight of the observer. Therefore we do not see what colour
the table has. Russell concludes that we only see what colour it
seems to have in a certain light to an observer of a certain kind.111
The shape a table appears to be changes as we shift our position,
even if the shape it is remains unchanged. Therefore what we see
from one position has one shape, what we see from another, has
another. Hence we do not see the table itself. Russell concludes
that we see various appearances of the table.112
Since Russell wrote, these arguments have received
considerable attention from both philosophers and psychologists.
They cannot be discussed very fully here. The work of the Gestalt
psychologists such as Wolfgang Köhler,113 has shown that the
premise of the first argument needs qualification. There is also
considerable doubt whether the conclusion follows from the
premise. After a person has looked at a table from various angles
there is often a definite colour and shape which it appears to him
to have. The premise of the second argument is undoubtedly
correct, but the different factors upon which the appearance of
the table depends do not all have the same status. For example,
P.P., pp. 8-9.
P.P., pp. 9-10.
P.P., pp. 10-11.
W. Köhler, The Gestalt Psychology, 1947.
the fact that it depends upon the incident light perhaps supports
the conclusion that when we look at an object in different lights,
we see the reaction of the object to light of different kinds, and
not one thing – the colour of the object. On the other hand, the
fact that the appearance of the object depends upon the condition
of the observer shows that some observers cannot see what colour
an object is, but not that no observer can see this. Russell’s third
argument is perhaps the strongest, but certain doubts arise. The
inference is invalid unless the premise that a table appears to have
one shape from one angle and another from another implies that
what we see from one angle does not have the same shape as what
we see from another. Yet the fact that what a person saw on one
occasion did not seem to have the same shape as what he saw on
another is not a very strong argument for the conclusion that
what he saw on one occasion had a different shape from what he
saw on the other. There is, however, some plausibility in Russell’s
contention that when we look at the same thing on two occasions
we may see colours and shapes on one of the occasions, which we
do not see on the other.
Whatever the strength of these arguments, they led
Russell to the conclusion that we have immediate perception of
colours, shapes, sounds, smells and textures, but not of tables and
chairs or the bodies of other people. The colours are coloured
patches – non-physical things which are coloured – rather than
the colours red, blue, etc. Russell held that a colour, such as red, is
a universal. He believed that we were acquainted with that
universal, but acquaintance with the universal was a different
thing from acquaintance with the colours we see when we look at
a physical object. In Problems of Philosophy, Russell argued in
these ways for the view that our knowledge of the expressions on
the faces of other people, of the marks they inscribe on paper and
the vibrations they impress on the air, rests upon inference. He
took it for granted that a further inference was involved in our
knowledge of their thoughts and feelings. He believed that we
have acquaintance with our own thoughts and feelings, and direct
knowledge of facts about them, but he was doubtful whether we
have acquaintance with the self whose thoughts and feelings they
are. He says, echoing Hume, that when we look for ourselves we
find nothing but our thoughts and feelings. He was inclined
nevertheless to accept that we do have acquaintance with
ourselves, on the grounds that we often know that we have a
certain feeling and that we could not even understand a
proposition to that effect without acquaintance with ourselves.114
However, he combined this view with the belief that we do not
know directly that we have a continuing self with a past and a
future. Descartes went beyond what he knew when he asserted
that he himself existed, for all we know at any time is that we have
a momentary self experiencing the feelings of the moment.115
Russell found the explanation of how we might have
knowledge concerning physical objects and the feelings of other
people in the possibility of knowledge by description. He
suggested propositions such that anyone knowing them to be true
would have knowledge concerning physical objects but which
could be understood without acquaintance with physical objects,
so that lack of acquaintance with the objects was no bar to
knowing the propositions to be true. However, other difficulties
arose about our knowledge of these propositions. We know,
concerning a table, that it has a certain property if we know that
there exists only one thing which is causing a certain collection of
colours and sounds with which we are acquainted and that
nothing causing those colours and sounds lacks the property. The
problem remained of how we can know that a collection of
colours and sounds has one cause, or what properties that cause
has, if we cannot perceive it directly and cannot perceive directly
any facts about it. This problem presented difficulties because if
we do not perceive physical objects our knowledge of them must
be obtained by inference from our knowledge of the colours and
sounds we do perceive. Now there is no logically valid inference
from the occurrence of certain sounds and colours to the
conclusion that they all have one cause, or to the conclusion that
they all have a cause of a certain kind. Therefore the proposition
which, according to this view of Russell’s, we must know, if we are
to have knowledge concerning physical objects and their
properties, are propositions which we can never infer with logical
validity from the data of perception.
P.P., pp. 50-1.
P.P., p. 19.
In Problems of Philosophy116, Russell met this difficulty
by arguing that although it is logically possible that our sense-data
are not produced by the presence of single physical objects having
a definite colour and shape, yet the fact that a much simpler view
of the world is obtained by supposing that they are so produced is
an adequate reason for supposing that they are. He found further
support for this view in the fact that we instinctively believe we
see and touch physical objects. He thought that this instinctive
belief could not stand up to the arguments he had brought against
it and he argued that the trust we instinctively have in it must be
transferred to that view which can stand up to criticism and
which is closest to the original. This he held to be the view that
our sense-data are produced by the presence of physical objects.
He defined an instinctive belief as one which “we find in ourselves
as soon as we begin to reflect”. His belief that there are such
things probably derives from Descartes. He argued that
philosophy cannot find reasons against our whole body of
instinctive beliefs, but can only find, in some of our instinctive
beliefs, reasons against others. Therefore a consistent system of
instinctive beliefs is worthy of acceptance. This argument may be
questioned at several points. First, why should not philosophy
discover propositions which we do not find ourselves to believe
when we begin to reflect but to which, when they are pointed out
to us, we at once transfer our allegiance? A man must base his
proofs on what he believes to be true, but, as Stuart Hampshire
has argued117, when he is constructing a proof he is not restricted
to those premises which he believes to be true when he begins the
task, for his beliefs may change as he considers the problem.
Indeed this is what, in his own opinion, happened to Russell
when he was led to abandon the view that we see physical objects.
Second, even if he could find no reason against such a system it
would not follow that there was no reason against it. The
explanation of his being able to find no reason against it lies, for
Russell, in his inability to stray far from his instinctive beliefs, but
that inability does not have the consequence that no beliefs
besides those are worthy of acceptance.
P.P., p. 22.
Stuart Hampshire, Freedom and the Individual, 1965, p. 97.
Russell’s argument here is a less plausible version of
Moore’s argument against Hume’s scepticism which I quoted in
Chapter 4. Moore claimed that it was impossible to prove that we
cannot know any external facts on the grounds that it would
always be as easy to deny the argument as to accept its conclusion.
It is doubtful whether Moore’s claim is justified, for how can he
be sure that it will always be at least as easy without having
considered all the arguments which might be brought? However,
Russell’s argument is worse, for he claims not merely that it is as
easy to take the sceptical as the non-sceptical side, but that it is
impossible to take the sceptical side. Later, in “Four Forms of
Scepticism”,118 Moore used a less general and therefore more
acceptable version of this argument against the sceptical views
which, as we shall see, Russell himself had come to hold.
Russell held that these arguments from simplicity and
from instinct show that we had good reason for the belief that
some collections of colours and sounds which we do directly
perceive are caused by a single thing which we do not. Yet we
shall be able to know little concerning a thing we cannot perceive
directly, such as a table, unless we can tell not only that some
collection of colours and sounds has a cause but that whatever is
its cause has certain properties, such as a definite shape, colour
and texture. Now among the arguments which convinced Russell
that we do not perceive directly that a table has such properties,
was one which convinced him that we have no reason to believe
that it has them at all.119 This was the argument that the colour a
table seems to have depends upon many factors besides the
properties of the table itself, such as the character of the light
falling on it, the colour of its surroundings or the medium lying
between it and our eyes. He was also impressed by the scientific
theories which explained light and sound in terms of quite other
properties. He concluded that science provides convincing
reasons for doubting whether physical objects have any such
properties as colour, pitch and warmth. This conclusion is
undoubtedly true, but he goes on to describe in much too extreme
terms the contemporary scientific picture of nature.
G. E. Moore, “Four Forms of Scepticism”, Philosophical
Papers, 1959, pp. 196-226.
P.P., pp. 34-5.
Physical science, more or less unconsciously, has drifted into the view
that all natural phenomena ought to be reduced to motions. Light
and heat and sound are all due to wave-motions, which travel from
the body emitting them to the person who sees light or feels heat or
He is here assuming the elastic solid theory of the propagation of
light rather than the electromagnetic theory which had by that
time replaced it. Sound is transmitted in air because a to-and-fro
movement of air particles communicates itself to neighbouring
particles, so that a wave of particle movement travels through the
air. Thus sound can fairly be said to be reduced to the motion of
particles. Light is transmitted because a variation in the strength
of the electrical and the magnetic fields at one point in space is
communicated to neighbouring points, so that a wave of electrical
and magnetic field variation passes through space. Light is not
reduced to motions. This mistake, which might be regarded as no
more than a slip if it did not play a central part in the view of our
knowledge of the properties of physical objects which Russell was
putting forward, does not affect the validity of his contention that
science denies physical objects such as colour, pitch and warmth.
To conclude from science that we have no knowledge of
the properties of physical objects would be flagrantly circular.
What Russell did was to admit that we could have knowledge of
the relations of physical objects in physical space and time. This
knowledge he thought sufficient for knowledge of the laws of
physical science – hence the importance of the view that science
reduces all natural phenomena to motion, which can be defined
in terms of space and time. The laws of physical science known in
this way had the consequence that physical objects do not have all
the properties they are commonly held to possess. The fact that
science does not reduce all phenomena to motion means that
there are considerable parts of science which Russell cannot
accept that we know to be true unless he accepts that we can
know more about physical objects than their relations in space
and time. He might have invoked considerations of simplicity to
justify our beliefs in this part of science, but he did not do so. This
P.P., pp. 22-8.
may have been because he had in mind another difficulty which
faces his explanation, a problem which he mentions but does not
discuss at any length. He suggested, as we have seen, that it is
possible for us to know concerning a physical object that it has a
certain property, although we cannot perceive that object directly,
because it is possible for us to know that a collection of colours
and sounds has a single cause and that whatever is their cause
possesses that property. This proposition can be understood
without acquaintance with the physical object. The further
difficulty is that it cannot be understood without acquaintance
with the property in question. Since Russell believed that we were
not acquainted with any properties of, or relations between,
physical objects, he was committed to the view that we cannot
understand this proposition or know it to be true. Although he
did not believe that we were acquainted with the spatial
properties of physical objects, such as shape, or with the spatial
relations between them, such as distance, he thought that the
relative positions of objects in physical space might correspond
with those of colours in visual space.121 He thought that a similar
correspondence would exist between the arrangement of colours
in visual space and the arrangement of touch-sensations in tactile
space which arise from seeing an object and exploring that same
object by touch.
We can know all those things about physical space which a man born
blind might know through other people about the space of sight…122
The correspondence Russell was thinking of might also be
illustrated by that between the variations in pressure of the steam
in the boiler and the movement of the needle of its pressure
gauge. When we watch the movement of the needle we perceive a
variation in its position which will, if the gauge is accurate, be the
very same as the variation in the pressure of the boiler. Although
pressure is a very different property from position, a pressure and
a position may vary in the very same way, and an understanding
of the variation of the one provides understanding of the other. It
seems likely that Russell held that our acquaintance with the
P.P., pp. 30-1.
P.P., p. 32
arrangement of colours in our visual field was acquaintance with
arrangements in which physical objects might stand, and that it
was on this acquaintance that he believed our understanding of
the properties science ascribes to physical objects to be based.
The difference Russell found between our understanding
of the spatial and temporal properties and relations of physical
objects and our understanding of their other properties and
relations would arise from his belief that when we are acquainted
with an arrangement of colours in visual space we are acquainted
with an arrangement in which physical objects might stand,
whereas when we are acquainted with the variation in position of
a gauge needle we are acquainted, not with properties and
relations of physical objects, but with relations which may hold
between those properties and relations. However, this difference
is an illusion, as Russell comes to admit later in his discussion.
Just as a relation between pressures may be the same as a relation
between needle movements, so a relation between physical
distances may be the same as a relation between visual distances.
Russell has not shown that a relation between physical objects
could be the same as a relation between colours. He has explained
how we might come to grasp relationships in which relations
between physical objects, such as distance, might stand, but not
how we might come to grasp those relations themselves.
Thus we come to know much more about the relations of distances
in physical space than about the distances themselves.…123
Russell’s explanation of how we can grasp the properties which
science ascribes to physical objects does not, therefore, depend on
the reducibility of all properties to motions, since it explains how
we can understand relations between any properties whatever.
But this difficulty is replaced by another, for it has become clear
that he has failed to explain how we can have any understanding
of the properties themselves.
The correspondence upon which Russell relies in
explaining our understanding of these properties is the very same
P.P., pp. 31-2.
as that Wittgenstein employs in his explanation of how a sentence
can express a possibility.
To the configuration of the simple signs in the propositional sign
corresponds the configuration of the objects in the state of affairs.124
Wittgenstein discusses such correspondences in much more
detail than Russell, but it seems likely that Russell’s use of such
correspondences to explain the relation between our sense-data
and the physical world suggested to Wittgenstein that they might
be used to explain the relation between language and the world.
Russell makes one assertion125 which suggests another
explanation of how we can understand propositions ascribing
properties to physical objects. This is the assertion that we may
have knowledge by description of universals as well as particulars.
He does not develop this suggestion at all, but it would fit the
general attitude of Problems of Philosophy better than an attempt
to show that we are acquainted, if not with properties and
relations between physical objects, at least with relations between
those properties and relations. That general attitude is that there
are facts which we cannot perceive directly to hold and which
concern things with which we are not acquainted. The suggestion
is that just as it might be held that we have knowledge concerning
the property which steam has of being at a certain pressure by
knowing that there is a single cause of the gauge needle, so we
might have knowledge of the shape of the table by knowing that
there is a single cause of the position of the gauge needle, so we
might have knowledge of the shape of a table by knowing that
there is a single cause of the various shapes of some of our sense-
data. This is an important suggestion, but it faces the difficulty
that to speak either of a physical object or of a property of a
physical object as causing our sense-data is artificial. What causes
our sense-data, if they have causes, must be some fact, perhaps
the fact that a physical object has a certain property. If knowledge
of a proposition to the effect that a collection of our sense-data
has a single cause is knowledge concerning that cause, then it is
knowledge concerning a fact. We know that there is a fact which
L. Wittgenstein, Tractatus Logico-Philosophicus, pp. 47-9.
P.P., p. 101.
is a cause of our sense-data but we do not know which physical
object it concerns, nor understand the property ascribed to it. It
hardly seems that this account shows how we can have knowledge
by description of the properties of physical objects.
The explanation which knowledge by description was
intended to provide of how we can have knowledge concerning
particulars, properties and relations with which we have no
acquaintance, is not consistently developed in Problems of
Philosophy. In later works it is abandoned altogether, and
Russell’s view of scientific knowledge and common sense no
longer invoked knowledge of things by description, and hence no
longer had the theory of descriptions as its basis. The change
arose from his coming to take a sceptical attitude towards beliefs
which do not follow with logical validity from any facts we
perceive to be true. The view he arrived at did not dispense with
such beliefs altogether, but minimised their importance. Whereas
before, as we have seen, he was prepared to accept that we could
know to be true propositions to the effect that a collection of our
sense data had a single physical object as cause; in Our Knowledge
of the External World he says there is no good reason to believe
such propositions and refers to beliefs in them as a priori and
unwarranted.126 We saw that if we are to have knowledge by
description of physical objects with which we cannot be
acquainted, we must be able to know the truth of propositions to
the effect that our sense-data have causes. Scepticism about the
possibility of such knowledge led Russell to abandon the view that
we had any knowledge concerning things with which we cannot
be acquainted. Indeed it led him to abandon the view that we had
any knowledge concerning objects with which we might have
been acquainted but as a matter of fact were not. For example, he
speaks in Our Knowledge of the External World of the various
perspectives which make up the appearance of the world to a
single observer. These he calls “actual” perspectives, but says that
we conceive of “ideal” perspectives which are appearances that
the world would have had to an observer who might have been,
but was not, stationed at a particular place at a particular time. Of
these he writes:
O.K.E.W., pp. 110-12.
It is open to us to believe that the ideal elements exist, and there can
be no reason for disbelieving this; but unless in virtue of some a
priori law we cannot know it, for empirical knowledge is confined to
what we actually observe.127
He still accepted that the theory of descriptions could show us a
proposition we could understand by which we could replace a
proposition we seemed to understand, but the replacement it
offered was one which he no longer thought we had any reason to
believe to be true. There was no way out except to take the
proposition we knew and understood to be concerned only with
objects of our acquaintance and to regard a physical object as a
series of sense-data. Our belief that certain propositions concern
physical objects is a convenient fiction, concealing the fact that
they concern nothing but our sense-data.
This view is a form of phenomenalism. Russell summed
it up in his version of Ockham’s razor:
Wherever possible, logical constructions are to be substituted for
It differs from other forms of phenomenalism in one very
important respect. Russell did not regard sense-data – the
colours, shapes and sounds of immediate acquaintance – as
mind-dependent, or mental. Although he thought we had no
reason to believe that any existed when we were not perceiving
them, he did not believe that it was logically impossible that they
should exist unperceived. It was impossible for Russell to find an
interpretation of scientific knowledge according to which it had
no implications which are not actually perceived by somebody to
be true. He asserts that physics has implications concerning how
things would appear at times when they are not appearing to
anyone.129 He thinks, however, that these implications should be
regarded as hypotheticals concerning the sense-data someone
would have had if he had had certain other sense-data, rather
O.K.E.W., p. 117.
Mysticism and Logic, p. 155.
O.K.E.W., p. 116.
than as existential propositions implying the existence of sense-
data which no one experienced. These propositions, whether
hypothetical or existential, have not been perceived to be true by
anyone. It seems that Russell preferred to regard the implications
as hypotheticals because he thought them verifiable, although not
verified, but believed that an unperceived sense-datum could not
have been perceived.130 In fact, the hypotheticals have no
advantage in verifiability: that a sense-datum was unperceived
does not imply that it could not have been perceived.
Russell regarded minds too as constructions out of
sense-data. He held that no person could perceive directly any
entity which was his own mind and which was logically
independent of his sense-data; still less could he perceive directly
an entity which was the mind of another. Nor did he believe that
anyone had any good reason to assume that there did exist a
single entity which was the subject of his experiences, so that no
one could have any knowledge of any mind by description.
Therefore that knowledge which we think of as knowledge of our
minds must be knowledge of series of sense-data. To speak of
certain kinds of knowledge concerning sense-data as knowledge
of minds is no more than a convenient fiction.
Since the sense-data out of which both minds and
physical objects were constructed were themselves neither mental
or physical Russell called this doctrine neutral monism. It took
different forms in later works. Indeed, in Human Knowledge he
again argued, as in Problems of Philosophy, that science includes
principles which we cannot perceive directly to be true and that in
reaching our scientific beliefs we must assume the validity of
inferences which are not logically valid. He attempted to
formulate the minimum of assumptions which would enable
scientists to arrive at those principles which they believe to be
O.K.E.W., pp. 88-9, 116-17.
Human Knowledge, Its Scope and Limits, Part VI, pp. 439-527.
7 Fact and Belief
Russell’s sceptical attitude towards the claim that there exist
causes of the colours, shapes and sounds which we perceive
directly was accompanied by scepticism concerning entities of a
different category, which had played a fundamental role in his
logic. Two of the most important were propositions and classes.
He had supposed these to be objects of acquaintance, so his
rejection of them did not arise from doubts about the validity of
inferences from directly perceived facts. Rather it sprang from
difficulties and contradictions which their existence entailed. Not
only were his reasons for scepticism different, but the scepticism
itself was of a different kind. Whereas he had accepted that there
might be causes of our sense-data, holding only that we had no
reason to believe that there were any, he did not accept that the
existence of propositions or of classes was even a possibility.
However, just as his scepticism about the external world had left
him the task of re-interpreting propositions which seemed to
concern physical objects, so his rejection of propositions and
classes left him the task of re-interpreting many kinds of
proposition from logic and psychology.
Unfortunately there is no space in this book to discuss,
except very briefly, one of the most important parts of Russell’s
philosophy, his definitions of the concepts of arithmetic132, his
discovery of contradictions in the logic of classes, and in other
branches of logic, to which those definitions led him133, and the
theory of types – developed in the Principia Mathematica134 – by
which he sought to free logic from these contradictions. Georg
Cantor defined a cardinal number as a property of a class, a
property which a class shared with all other classes whose
P.O.M., pp. 111-20.
P.O.M., pp. 101-7.
P.M., pp. 37-65.
numbers could be paired off with its members. A cardinal
number is the number of members in a class, an ordinal number
the position of a member in a sequence. For example, on 1
January 1900, the Christian era was one thousand eight hundred
and ninety-nine years old. This is a cardinal number since it is the
number of years which had elapsed since the birth of Christ. But
the year which began on that day was the one-thousand-nine-
hundredth year of the Christian era. This is an ordinal number,
for it gives the position of that year in the sequence of years. Thus
an age is a cardinal number, being the number of years for which
something has been in existence, but a date is an ordinal number,
giving the position of a year, or a day, in a sequence. This explains
why Russell had time to finish The Principles of Mathematics
before the end of the nineteenth century: the twentieth century
did not begin until 1 January 1901.
Russell showed how Cantor’s definition of a cardinal
number could be used to define the concepts of arithmetic in
terms of the concepts of logic and to prove the theorems of
arithmetic from principles of logic. However, the supposition that
classes must be recognised among the entities which make up the
world led, Russell saw, to some paradoxical conclusions. Cantor
had proved that any class with more than one member has more
sub-classes than it has members. This is not surprising for finite
classes. The class whose members are the first four letters of the
alphabet has six two-member sub-classes – A and B, A and C, A
and D, B and C, B and D, C and D. These are only some of its
sub-classes. But Cantor proved that it also held for classes with an
infinite number of members. This theorem, which is called
Cantor’s Theorem, together with the assumption that classes are
among the things which make up the world, has the consequence
that there is no class which is the class of all existing things, for,
whatever we take that class to be, there must be a larger class with
members our class has not got. The difficulty this poses for
Cantor’s and Russell’s definition of cardinal numbers is obvious.
If numbers are properties of classes, and if there are some true
propositions in which numbers are ascribed, then classes must
exist. If classes exist, then surely all the things there are must
make up a class. But Cantor’s Theorem proves that there is no
such class.135 Russell accepted that there were no such things as
classes but preserved his definition of numbers by analysing
propositions which seem to concern classes into propositions
which do not have classes as constituents. Thus he regarded the
definition of numbers as properties of classes as a stage on the
road to a satisfactory definition. It is convenient to speak as if
classes exist, just as it is convenient to speak as if physical objects
exist, but the supposition that such entities exist is no more than a
fiction and the propositions that seem to concern them can be
identified with propositions into which they do not enter. He calls
both classes and physical objects “logical fictions”136.
Russell thought that the rejection of classes gave him a
freedom he would not otherwise have had. If there were classes it
could hardly be that there was not the class of all existing things,
but if propositions which seem to concern classes are to be
analysed as propositions into which classes do not enter, then the
analysis can be done in such a way that certain propositions
which seem to concern classes are declared not to exist at all. This
was what Russell did. The analysis of class propositions provided
by the theory of types allows no analyses for propositions which
seem to concern what Russell calls “classes of mixed type”. These
are, for example, classes having as members both entities which
are not themselves classes and entities which are classes of such
entities. This has the consequence that no analysis is provided for
propositions which seem to concern classes which are members
of themselves. Of course the success of this resolution of the
paradox posed by Cantor’s Theorem depends upon whether
Russell’s analysis of class propositions is one according to which
it is plausible to contend that propositions concerning classes of
mixed type can be given no interpretation at all. This is an
investigation which we must forego here.
Although there are propositions which seem to exist but
which can be proved to be both true and not true, such as the
proposition expressed by the sentence “The proposition expressed
by this sentence is false”, Russell’s scepticism concerning
propositions did not rest on such contradictions and had very
I.M.P., pp. 135-6.
L.K., pp. 265-6, 271-2.
different grounds from his scepticism concerning classes. It
sprang from his adoption, in opposition to Bradley, of the view
that truth consists in correspondence with fact137. Bradley
believed it to be impossible to arrive at a thought which could be
wholly true, for he held that no thought could be entirely self-
consistent. Concentrating on this insuperable bar to making a
true judgement, which he believed could be overcome to some
degree but never altogether, he slipped from the view that the
more a thought was self-consistent, the more truth it could have,
to the view that the more a thought was self-consistent, the more
truth it had. Russell and Moore regarded truth as a necessary, but
not a sufficient, condition of truth. To obtain a sufficient
condition, correspondence with fact must be added138. Now it is
natural to speak of a judgement, or of a proposition, as
corresponding with fact if there exists a fact to which it
corresponds and as failing to correspond if there does not exist a
fact to which it corresponds. In this way, Russell and Moore were
led to identify the proposition that it is a fact that lions are
carnivorous with the proposition that the fact that lions are
carnivorous exists. They came to treat facts as entities which, if
they were not facts, would not exist. Moore discusses this
problem in Some Main Problems of Philosophy139. He sets out to
refute the view that to hold a belief is to be related to an object of
belief, or proposition. He argues that such an object of belief
would have to be something which, if the belief were true, would
be a fact140 and that there can be nothing of this sort. A true
proposition cannot be a fact since, if the proposition were not
true, the proposition would exist but the fact would not141. This
argument rests on the assumption that if it is not a fact that lions
are carnivorous, then the fact that lions are carnivorous does not
exist. Consider a parallel and evidently invalid argument where
the corresponding assumption is false. If Jones is the only man in
his village with any skill with iron someone might argue: Jones
cannot be the blacksmith, since if Jones had had no skill with iron
P.P., p. 123
P.P., pp. 121-3.
G. E. Moore, Some Main Problems of Philosophy, pp. 252-87.
G. E. Moore, Some Main Problems of Philosophy, p. 261.
G. E. Moore, Some Main Problems of Philosophy, p. 260.
Jones would have existed but the blacksmith would not. This
premise is false: if Jones had had no skill with iron there would
have been no blacksmith but the blacksmith would have existed.
This is possible because the blacksmith could have existed
without being the blacksmith; he might, for example, have been
the shoemaker. Similarly, it might be argued against Moore’s
premise that if the proposition that lions are carnivorous were not
true, then the fact that lions are carnivorous would have existed
but, of course, would not have been a fact. The correspondence
theory of truth, by leading Russell and Moore to identify “It is a
fact…” with “The fact that … exists”, led them to treat fact as a
category, just as someone might treat physical object as a category.
A property is a category if something which possesses it would
not exist if it did not possess it.
Russell says: “Time was when I thought there were
propositions”142. What convinced him that there were no such
things was the seemingly innocent observation that “you cannot
say that you believe facts”143. In one interpretation this is
obviously true. Since some beliefs are false, it cannot be that
believing something implies believing something which is a fact.
However, when Russell had believed in propositions he had not
believed anything as silly as this. He had held that in belief a
person believes something that, if true, is a fact. What he is
denying by his seemingly innocent remark is that belief is ever a
belief in fact. He denied this because, as we have seen, he had
come to hold that something which was a fact would not have
existed had it not been a fact. He had accepted propositions
because he had held that relation in thought to them was at least
relation to something which, if true, was a fact, and he now
concluded that there were no such things. Therefore he took the
next best thing and suggested that belief was a relation to parts
which, if they make a whole, make a fact.
In rejecting propositions Russell was rejecting entities
which facts would be if they were not facts and which would be
facts if they were true. He rejects them because he thinks that
there is nothing facts would be if they were not facts. False
L.K., p. 223.
L.K., p. 222.
propositions would, he now thought, be non-existent facts and
hence non-existent. This explains why his reasons for supposing
there are no such things as propositions are all reasons for
supposing that there are no such things as false propositions.
To suppose that in the actual world of nature there is a whole set of
false propositions going about is to my mind monstrous. I cannot
bring myself to suppose it. I cannot believe that they are there in the
sense in which facts are there.144
It also explains why he treats the view that there are false
propositions as no less absurd than Meinong’s view that there is
such an object as the round square but it is not an actuality. To
Meinong, of course, the view that there is such an object as the
round square but it is not an actuality is no less sensible than the
view that there is a proposition that lions are vegetarians but it is
a false proposition. He made an identification which, in a way,
was the opposite of that made by Moore and Russell. He identifies
the proposition that the round square does not exist with the
proposition that the round square is not an actuality. This leads
him to treat actualities as entities which would exist even if they
were not actual. He treated actualities as it is common to treat
facts, while Russell and Moore treated facts as it is common to
In the absence of propositions, Russell and Moore had to
explain what it is that possesses the properties of being true and
being false. The word “belief” may mean a proposition which
someone believes, or it may mean someone’s belief in a
proposition. When a belief is said to be false it is perhaps more
natural to take the word in the former sense, but Russell had the
idea of taking it in the latter. He suggested that when Othello
believed that Desdemona loved Cassio it was not what he believed
that was false, but rather his belief in it. His belief in the
proposition was false because there was no fact to which it
corresponded. According to this suggestion, it is not propositions
but beliefs which are true and false. All that remained was to give
an account, which did not invoke propositions of what it was for
someone to believe something.
L.K., p. 223.
Russell accepted that facts had constituents and logical
form just as propositions did. What is more he thought that had
there been a fact that Desdemona loved Cassio that fact would
have had just the constituents and logical form which he had
formerly attributed to the proposition that Desdemona loved
Cassio. These constituents would exist, of course, even if the fact
did not, so he suggested that belief was a relation between a
person and certain entities which, if the belief was true, were
constituents of a fact. This suggestion preserved the essential
feature of his realism – that thought is a relation to real entities –
but sacrificed the view that it is a relation to something which, if
true, would be a fact. A belief has real entities as constituents, just
as it would have if it were a relation between a person and a
proposition. There are certain puzzles made up of a number of
pieces of wood which can be fitted together to form a single cube.
If someone is busy trying to fit the pieces together he cannot be
said to be related to the cube he is trying to make, since that does
not exist. However, the person who is trying to do the puzzle is
related to the separate pieces of wood. When Russell and Moore
refused to accept that something which is a fact might have
existed and not been a fact they were treating being a fact like
being a physical object. When a physical object has been broken
up it has ceased to exist, but its parts may still exist. They hoped
to explain trying to do the puzzle as a relationship between a
person and the pieces. Before considering this explanation, we
must consider some other consequences of Russell’s rejection of
Propositions had formerly had a place in Russell’s
understanding of the principles of logic as well as in his views on
belief and truth and falsehood. He had held that many principles
of logic concern the implication of one proposition by another.
Perhaps because it is less plausible to take beliefs (as distinct from
what is believed) as the entities between which implications hold,
Russell, in “The Philosophy of Logical Atomism”, took the
principles of logic to concern implications between sentences.
However, this view is difficult to square with his view that truth
and falsehood belong to beliefs, for those entities which
implications hold between must surely be the entities to which
truth and falsehood belong. Russell admitted this inconsistency:
he says that in logic it is natural to treat sentences as the bearers of
truth and falsehood, but that this is not appropriate in discussions
of belief and knowledge.145 Evidently one of these attitudes must
be considered as no more than a convenience. Presumably it is
the view that implications hold between sentences. Russell found
it convenient to alter his terminology and take the word
“proposition” to mean “sentence in the indicative mood”. In this
way he could retain many of the sentences in which he had
expressed principles of logic, but these would now mean that
some sentence implied another rather than that some object of
thought implied another.146 However, he does not keep
consistently to this new usage. When he declares that there are no
such things as propositions he does not intend to say that there
are no such things as sentences in the indicative mood.
Whether it is beliefs or sentences which are to be true or
false, truth and falsehood consist in correspondence with fact.
Logical atomism is a doctrine which may form part of a
correspondence theory of truth. It holds that much of the
complexity of facts is only apparent: much complexity belongs to
beliefs or sentences and not to facts. Logical atomism has two
tasks. Both Russell’s theory and Wittgenstein’s147 succeed partially
in the first, but neither can deal with the second. The first is that
of showing how a complex belief may correspond, not to a
complex fact, but in a complex way to facts. For example, both
Russell and Wittgenstein held that there are no conjunctive facts.
There is no fact that it is raining and it is snowing to correspond
with someone’s belief that it is raining and it is snowing. Rather
that belief is true if there exist both the fact that it is raining and
the fact that it is snowing. If one or other of these two facts fails to
exist, then the belief is false. Again, there are no disjunctive facts.
The belief that either it is raining or it is snowing is true if the fact
that it is raining exists, true if the fact that it is snowing exists,
false if neither exists. A belief which corresponds in such a
complex way as this was said to be a truth-function of the simpler
beliefs that it is raining and it is snowing. Russell and
Wittgenstein held that facts had one or other of a few simple
L.K., pp. 184-5.
L.K., p. 185.
L. Wittgenstein, Tractatus Logico-Philosophicus, passim.
logical forms: the subject-predicate form and the various
relational forms. Therefore they were forced to deny that there is
a fact to the effect that if the Eiffel Tower were undermined it
would fall down, since that fact has none of these forms. They
recognised a belief that if the Eiffel Tower were undermined it
would fall down but held that its truth depended solely on the
existence or non-existence of the two facts – that the Eiffel Tower
will be undermined and that the Eiffel Tower will fall down. Nor
could they admit facts concerning beliefs, and they were hard put
to it to present the truth of beliefs about beliefs as determined by
facts which they did admit.
The second task is that of giving an account of what it is
for someone to hold a complex belief. Both Russell’s and
Wittgenstein’s accounts of beliefs apply only to simple beliefs. It
seems that complex belief should be explained in terms of simple
belief, but neither Russell nor Wittgenstein attempts to do this.
Instead they both fall back on stating what facts make them true
and what facts make them false. This is either to accept
propositions in Russell’s old sense, and explain complex beliefs by
stating what proposition is believed to be true, or to identify a
certain complex belief by stating that it is a belief which would be
made true by certain facts, which does not tell us what it is to hold
that belief. One might ask how, on Russell’s view of belief, it is
possible to hold beliefs which are made true and false by these
facts. The same problem arises for Wittgenstein’s explanation of
thought as the construction of pictures. Although Russell does
not appreciate this difficulty in “The Philosophy of Logical
Atomism” he does do so in An Inquiry into Meaning and Truth.
There he explains what belief-state of the speaker is expressed by
uttering a sentence of what he calls the primary language, and
goes on to ask what is expressed by a sentence in which two
primary language sentences are joined by the word “or”. He is
inclined to answer that such sentences express a state of
hesitation.148 On this account, to believe either that it will rain or
that it will snow is to hesitate between the two beliefs. The
account, as Russell sees, is not entirely satisfactory. A person who
holds such a belief may have no wish to decide whether it will rain
I.M.T., pp. 84-5, 210.
or to decide whether it will snow. He might be quite satisfied with
his belief that one or other of them is true. Russell’s account is
inadequate, but he did at least recognise the existence of the
The account of belief to which, as we have seen, Russell
was led by his rejection of propositions is confronted by a serious
difficulty. When Othello believed that Desdemona loved Cassio
he was related, not to the proposition that Desdemona loved
Cassio, but to Desdemona, to Cassio and to the relation loves.
Now there are several propositions which have Desdemona,
Cassio and the relation loves as constituents, so that the fact that
Othello held some belief concerning these constituents does not
determine what it was that he believed. He might have believed
that Desdemona loved Cassio, that Cassio loved Desdemona, or
that Cassio and Desdemona loved loving. Russell himself held that
to specify a proposition it was necessary to specify not only the
constituents but also the logical form and the manner in which
the logical form relates the constituents. His account of belief
ignores the latter two completely. He attempts to meet this
difficulty with the suggestion that had Othello believed that
Cassio loved Desdemona, he would have been related to the three
constituents in a different way from that in which he actually was
related to them in believing that Desdemona loved Cassio.149
However, in “The Philosophy of Logical Atomism”, he says that
Wittgenstein has convinced him of the inadequacy of his account
of belief and he seems to admit that his view cannot satisfactorily
meet the difficulty we have been considering. On the analogy with
the wooden puzzle, Russell’s account of belief would correspond
to an account which identified trying to assemble the pieces into a
cube as a relation between the person doing the puzzle and the
pieces. The difficulty in finding such an account would be to find
a relation between the person and the pieces, which would imply
that he was trying to assemble them into a cube. Similarly, the
difficulty of finding an adequate account of belief along the lines
Russell suggests is that of finding a relation between a person and
the constituents of a fact which implies that only a fact in which
those constituents were related in a particular way by a particular
P.P., pp. 126-7.
logical form would constitute the truth of the person’s belief. It is
not easy to see whether this can be done.
Rightly or wrongly Russell concluded that this problem
could not be solved and the attempt to find an account of belief
which did not invoke propositions led him to abandon the realist
view of thought which had characterised his philosophy for so
long. According to the view of belief which he put forward in The
Analysis of Mind and An Inquiry into Meaning and Truth, the fact
that a person holds a certain belief no longer has as constituents
either the proposition which he might be said to believe, the fact
which would make his belief true, or the constituents of that fact.
Holding a particular belief was identified with being in a
particular mental state. In The Analysis of Mind,150 this state was
characterised as one in which an image is present in the mind
along with a belief-feeling, or feeling of assent. In An Inquiry into
Meaning and Truth,151 it was characterised as one, perhaps a state
of tension, upon which certain perceptions, if they occurred,
would have a certain result, perhaps that of removing it. In
coming to the former view, Russell was probably influenced by
Wittgenstein’s theory that a person thinks by constructing
pictures. In arriving at the latter he was certainly influenced by
the behaviourist psychology. On these views of belief the
constituents of the fact which would make the belief true are
never constituents of the fact that a person holds a particular
belief. Therefore, a person can hold a belief, even if he has no
acquaintance with the entities which Russell would formerly have
held to be constituents of the proposition he believed, and even if
those entities do not exist. What the theory of descriptions
explained no longer needs to be explained, so that the theory has
become unnecessary and does not figure in these later works.
The problems which led Russell to abandon his
realist view of thought were in some ways artificial. His
rejection of propositions arose from the identification of
“It is a fact that…” with “The fact that … exists”, which
has little in its favour. There is no more excuse for treating
false propositions as non-existent facts – rather than
A.M., pp. 250-2.
I.M.T., pp. 177-80.
treating facts as true propositions – than for Meinong’s
treatment of imaginary things as beings – rather than as
non-existent. But, however that may be, neither
propositions nor facts have position in space and time, and
someone with that vivid sense of reality which Russell held
that every logician must have, might well reject both.
i. Books and Pamphlets
German Social Democracy. London 1896.
An Essay on the Foundations of Geometry. Cambridge 1897.
A Critical Exposition of the Philosophy of Leibniz. Cambridge
1900; second edition, London 1937.
The Principles of Mathematics. Cambridge 1903; second edition,
Principia Mathematica, vol. I, with A. N. Whitehead. Cambridge
Philosophical Essays. London 1910.
Principia Mathematica, vol. II, with A. N. Whitehead. Cambridge
The Problems of Philosophy. London 1912; second edition,
Principia Mathematica, vol. II, with A. N. Whitehead. Cambridge
Our Knowledge of the External Word. London 1914; third edition,
The Philosophy of Bergson. London 1914.
War, the Offspring of Fear. London 1915.
Principles of Social Reconstruction. London 1916.
Justice in War-Time. London 1916.
The Case of Ernest F. Everett. London 1916.
Rex vs. Bertrand Russell. London 1916.
Political Ideals. New York 1917.
Mysticism and Logic. London 1917.
Roads to Freedom. London 1918.
Introduction to Mathematical Philosophy. London 1919.
The Practice and Theory of Bolshevism. London 1920.
The Analysis of Mind. London 1921.
The Problems of China. London 1922.
The Prospects of Industrial Civilization, with Dora Russell.
The ABC of Atoms. London 1923.
Icarus or the Future of Science. London 1924.
Bolshevism and the West. London 1924.
The ABC of Relativity. London 1925.
What I Believe. London 1925.
Why I am not a Christian. London 1927.
The Analysis of Matter. London 1927.
An Outline of Philosophy. London 1927.
Selected Essays of Bertrand Russell. New York 1927.
Sceptical Essays. London 1928.
Marriage and Morals. Lonson 1929.
The Conquest of Happiness. London 1930.
Has Religion Made Useful Contributions to Civilization? London
The Scientific Outlook. New York 1931.
Education and the Social Order. London 1932.
Freedom and Organization 1814-1914. London 1934.
In Praise of Idleness. New York 1935.
Which Way to Peace? London 1936.
Determinism and Physics. Newcastle-upon-Tyne 1936.
The Amberley Papers, with Patricia Russell. London 1937.
Power: a New Social Analysis. New York 1938
An Inquiry into Meaning and Truth. London 1940.
Let the People Think. London 1941.
A History of Western Philosophy. London 1945.
Human Knowledge: its Scope and Limits. London 1948.
Authority and the Individual. London 1949.
Unpopular Essays. London 1950.
The Impact of Science on Society. Columbia 1951.
New Hopes for a Changing World. London 1951.
Satan in the Suburbs. London 1953.
Nightmares of Eminent Persons. London 1954.
Human Society in Ethics and Politics. London 1954.
Logic and Knowledge. London 1956.
Pertraits from Memory. London 1956.
Common Sense and Nuclear Warfare. London 1959.
My Philosophical Development. London 1959.
The Wisdom of the West. London 1959.
Fact and Fiction. London 1961.
Has Man a Future? London 1961.
Unarmed Victory. London 1963.
Political Ideals. London 1963.
War Crimes in Vietnam. London 1967.
Autobiography 1872-1914. London 1967.
Autobiography 1914-1944. London 1968.
Autibiography 1944-1968. London 1969.
Throughout his life Russell has written a great many aritcles on a wide
variety of subjects. This Bibliography includes only articles on
philosophy and of those, with very few exceptions, does not include ones
re-published in books already listed.
“The A Priori in Geometry”, in Proc. Aristotelian Soc. 1896.
“The Logic of Geometry”, in Mind 1896.
“On the Relations of Number and Quantity”, in Mind 1896.
“Les Axiomes Propres à Euclide sont-ils Empiriques?”, in Rev. de
Métaphysique et de Morale 1898.
“Sur les Axiomes de la Géométrie”, in Rev. de
Métaphysique et de Morale 1899.
“L’Idée d’Ordre et la Position Absolute dans l’Espace et les
Temps”, in Congrès int. de philosophie, logique et
histoires des sciences 1901.
“On the Nature of Order”, in Mind 1901.
“Is Position in Time and Space Absolute or Relative?”, in Mind
“Sur la Logique des Relations avec des Applications à la Théorié
des Séries”, in Rev. de Mathématique 1902.
“On some Difficulties in the Theory of Transfinite Numbers and
Order Types”, in Proc. London Math. Soc. 1906.
“The Theory of Implication”, in Amer. Journal of Mathematics
“Les Paradoxes de la Logique”, in Rev. de Métaphysique et de
“On the Nature of Truth”, in Proc. Aristotelian Soc. 1906-7.
“‘If’ and ‘Imply’”, in Mind 1908.
“The Philosophy of William James”, in Living Age 1910.
“Some Explanations in Reply to Mr Bradley”, in Mind 1910.
“La Théorie de Types Logiques”, in Rev. de Métaphysique et de
“The Basis of Realism”, in J. of Philosophy, Psychology and
Scientific Method 1911.
“The Philosophical Importance of Mathematical Logic”, in
“Definitions and Methodological Principles in Theory of
Knowledge”, in Monist 1914.
“Sensation and Imagination” in Monist 1915.
“On the Experience of Time”, in Monist 1915.
“Philosophy of Logical Atomism”, in Monist 1918; reprinted in
Logic and Knowledge.
“Philosophy of Logical Atomism II”, in Monist 1919; reprinted in
Logic and Knowledge.
“Meaning of Meaning”, in Mind 1920.
“Introduction” to L. Wittgenstein, Tractatus Logico-Philosophicus.
“Physics and Perception”, in Mind 1922.
“Doctor Schiller’s Analysis of the Analysis of Mind” in J. of
“Logical Atomism” in Contemporary British Philosophy: Personal
Statements. London 1924.
“Introduction: Materialism: Past and Present”, in F. A. Lange, A
History of Materialism. London 1925.
“On Non-Euclidean Geometries”; “Theory of Knowledge”;
“Philosophical Consequences of the Theory of
Relativity”, in Encyclopedia Britannica, thirteenth
“On Order in Time”, in Proc. Cambridge Phil. Soc. 1936.
“On Verification”, in Proc. Aristotelian Soc. 1937.
“The Relevance of Psychology to Logic”, in Proc. Aristotelian Soc.
“On the Importance of Logical Form”, in International
Encyclopedia of Unified Science, vol I, Chicago 1938.
“Living Philosophy, Revised”, in I Believe, ed. C. Fadiman. New
“Dewey’s New Logic”, in the The Philosophy of John Dewey, ed. P.
A. Schilpp. Chicago 1939.
“The Philosophy of Santayana”, in The Philosophy of George
Santayana, ed. P. A. Schilpp. Chicago 1940.
“My Mental Development”; “Reply to Criticism”, in The
Philosophy of Bertrand Russell, ed. P. A. Schilpp.
“Whitehead and Principia Mathematica”, in Mind 1948.
“Ludwig Wittgenstein”, in Mind 1951.
“Philosophical Analysis”, in Hibbert Journal 1955-6.
“Logic and Ontology”, in J of Philosophy 1957.
“Mr Strawson on Referring” in Mind 1957.
“What is ‘Mind’?”, in J of Philosophy 1958.
ABBREVIATED TITLES BY WHICH RUSSELL’S WORKS ARE
A.M. = The Analysis of Mind.
I.M.P = Introduction to Mathematical Philosophy.
I.M.T. = An Inquiry into Meaning and Truth.
L.K. = Logic and Knowledge.
M.P.D. = My Philosophical Development.
O.K.E.W = Our Knowledge of the External World.
P.L. = A Critical Exposition of the Philosophy of
P.M. = Principia Mathematica.
P.O.M. = The Principles of Mathematics.
P.P. = The Problems of Philosophy.