Kenny Anthony. An Illustrated Brief History of Western Philosophy
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shown to be a branch of logic in the sense that it could be formalized without the
use of any non-logical notions or axioms. It was in the Grundlagen der Arithmetik
(1884) that Frege first set out to establish this thesis, which is known by the
name of ‘logicism’.
The Grundlagen begins with an attack on the ideas of Frege’s predecessors and
contemporaries (including Kant and Mill) on the nature of numbers and of math-
ematical truth. Kant had maintained that the truths of mathematics were synthetic
a priori, and that our knowledge of them depended on intuition. Mill, on the
contrary, saw mathematical truths as a posteriori, empirical generalizations widely
applicable and widely confirmed. Frege maintained that the truths of arithmetic
were not synthetic at all, neither a priori nor a posteriori. Unlike geometry –
which, he agreed with Kant, rested on a priori intuition – arithmetic was analytic,
that is to say, it could be defined in purely logical terms and proved from purely
logical principles.
The arithmetical notion of number in Frege’s system is replaced by the logical
notion of ‘class’: the cardinal numbers can be defined as classes of classes with the
same number of members; thus the number two is the class of pairs, and the
number three the class of trios. Despite appearances, this definition is not circular,
because we can say what is meant by two classes having the same number of
members without making use of the notion of number: thus, for instance, a
waiter may know that there are as many knives as there are plates on a table
without knowing how many of each there are, if he observes that there is just one
knife to the right of each plate. Two classes have the same number of members if
they can be mapped one-to-one on to each other; such classes are known as
equivalent classes. A number, then, will be a class of equivalent classes.
Thus, we could define four as the class of all classes equivalent to the class of
gospel-makers. But such a definition would be useless for purposes of reducing
arithmetic to logic, since the fact that there were four gospel-makers is no part
of logic. If Frege’s programme is to succeed, he has to find, for each number, not
only a class of the right size, but a class whose size is guaranteed by logic.
What he did was to begin with zero. Zero can be defined in purely logical
terms as the class of all classes equivalent to the class of objects which are not
identical with themselves. Since there are no objects which are not identical with
themselves, that class has no members; and since classes which have the same
members are the same classes, there is only one class which has no members, the
null-class, as it is called. The fact that there is only one null-class is used in
proceeding to the definition of the number one, which is defined as the class of
classes equivalent to the class of null-classes. Two can then be defined as the class
of classes equivalent to the class whose members are zero and one, three as the class
of classes equivalent to the class whose members are zero and one and two, and
so on ad infinitum. Thus the series of natural numbers is to be built up out of the
purely logical notions of identity, class, class-membership, and class-equivalence.
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Figure 43 A page of Frege’s derivation of arithmetic from logic.
(Conceptual Notation, ed. T. Bynum, 1972; by permission of Oxford University Press)
In the Grundlagen there are two theses to which Frege attaches great import-
ance. One is that each individual number is a self-subsistent object; the other is
that the content of a statement assigning a number is an assertion about a con-
cept. At first sight these theses may seem to conflict, but if we understand what
Frege meant by ‘concept’ and ‘object’ we see that they are complementary. In
saying that a number is an object, Frege is not suggesting that a number is some-
thing tangible like a tree or a table; rather, he is denying that number is a property
belonging to anything, whether an individual or a collection. In saying that a
number is a self-subsistent object he is denying that it is anything subjective, any
mental item or any property of a mental item. Concepts are, for Frege, Platonic,
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mind-independent entities, and so there is no contradiction between the thesis
that numbers are objective, and the thesis that number-statements are statements
about concepts. Frege illustrates this latter thesis with two examples.
If I say ‘Venus has 0 moons’, there simply does not exist any moon or agglomera-
tion of moons for anything to be asserted of; but what happens is that a property is
assigned to the concept ‘moon of Venus’, namely that of including nothing under it.
If I say ‘the king’s carriage is drawn by four horses’, then I assign the number four
to the concept ‘horse that draws the King’s carriage’.
Statements of existence, Frege says, are a particular case of number statements.
‘Affirmation of existence,’ he says, ‘is in fact nothing but denial of the number
nought.’ What he means is that a sentence such as ‘Angels exist’ is an assertion
that the concept angel has something falling under it. And to say that a concept
has something falling under it is to say that the number which belongs to that
concept is something other than zero.
It is because existence is a property of concepts, Frege says, that the ontological
argument for the existence of God breaks down. That-there-is-a-God cannot be a
property of God; if there is in fact a God, that is a property of the concept God.
If number statements are statements about concepts, what kind of object is a
number itself? Frege’s answer is that a number is the extension of a concept. The
number which belongs to the concept F, he says, is the extension of the concept
‘like-numbered to the concept F’. This is equivalent to saying that it is the class
of all classes which have the same number of members as the class of Fs, as was
explained above. So Frege’s theory that numbers are objects depends on the
possibility of taking classes as objects.
Frege’s Philosophy of Logic
It will be seen that Frege’s philosophy of mathematics is closely linked to his
understanding of several key terms of logic and of philosophy; and indeed in the
Begriffschrift and the Grundlagen Frege not only founded modern logic, but also
founded the modern philosophical discipline of philosophy of logic. He did so
by making a clear distinction between the philosophical treatment of logic and,
on the one hand, psychology (with which it had often been confused by philo-
sophers in the empiricist tradition) and, on the other hand, epistemology (with
which it was sometimes conflated by philosophers in the Cartesian tradition).
There is not, however, the same sharp distinction in his work between logic and
metaphysics: indeed the two are closely related.
Corresponding to the distinction in language between functions and arguments,
Frege maintained, a systematic distinction must be made between concepts and
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objects, which are their ontological counterparts. Objects are what proper names
stand for: there are objects of many kinds, ranging from human beings to numbers.
Concepts are items which have a fundamental incompleteness, corresponding to
the gappiness in a function which is marked by its variable. Where other philo-
sophers talk ambiguously of the meaning of an expression, Frege introduced a
distinction between the reference of an expression (the object to which it refers, as
the planet Venus is the reference of ‘The Morning Star’) and the sense of an
expression. (‘The Evening Star’ differs in sense from ‘The Morning Star’ though
both expressions, as astronomers discovered, refer to Venus.) Frege maintained
that the reference of a sentence was its truth-value (i.e. the True, or the False), and
held that in a scientifically respectable language every term must have a reference
and every sentence must be either true or false. Many philosophers since have
adopted his distinction between sense and reference, but most have rejected the
notion that complete sentences have a reference of any kind.
The climax of Frege’s career as a philosopher should have been the publication
of the two volumes of Die Grundgesetze der Arithmetik (1893–1903), in which
he set out to present in a rigorous formal manner the logicist construction of
arithmetic on the basis of pure logic and set theory. This work was to execute the
task which had been sketched in the earlier books on the philosophy of mathem-
atics: it was to enunciate a set of axioms which would be recognizably truths of
logic, propound a set of undoubtedly sound rules of inference, and then present,
one by one, derivations by these rules from these axioms of the standard truths of
arithmetic.
The magnificent project aborted before it was ever completed. The first volume
was published in 1893. By the time that the second volume appeared, in 1903, it
had been discovered that Frege’s ingenious method of building up the series of
natural numbers out of merely logical notions contains a fatal flaw. The discovery
was due to the English philosopher Bertrand Russell.
Russell’s Paradox
Russell was born in 1872, the grandson of the Prime Minister Lord John Russell,
and godson of John Stuart Mill. At Trinity College, Cambridge, he accepted for
a while a British version of Hegelian idealism. Later, in conjunction with his friend
G. E. Moore, he abandoned idealism for an extreme realist philosophy which
included a Platonist view of mathematics. It was in the course of writing a book
to expound this philosophy that Russell encountered Frege’s ideas, and when
the book was published in 1903 as The Principles of Mathematics it included an
account of them. Much as Russell admired Frege’s writings, he detected a radical
defect in his system, which he pointed out to him just as the second volume of
the Grundgesetze was in press.
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If we are to proceed from number to number in the way Frege proposes we
must be able to form without restriction classes of classes, and classes of classes of
classes, and so on. Classes must themselves be classifiable; they must be capable of
being members of classes. Now can a class be a member of itself? Most classes are
not (e.g. the class of dogs is not a dog) but some apparently are (e.g. the class of
classes is surely a class). It seems therefore that classes can be divided into two
kinds: there is the class of classes that are members of themselves, and the class of
classes that are not members of themselves.
Consider now this second class: is it a member of itself or not? If it is a member
of itself, then since it is precisely the class of classes that are not members of
themselves, it must be not a member of itself. But, if it is not a member of itself,
then it qualifies for membership of the class of classes that are not members of
themselves, and therefore it is a member of itself. It seems that it must either be
a member of itself or not; but whichever alternative we choose we are forced to
contradict ourselves.
This discovery is called Russell’s paradox; it shows that there is something
vicious in the procedure of forming classes of classes ad lib., and it calls into
question Frege’s whole logicist programme.
Russell himself was committed to logicism no less than Frege was, and he
proceeded, in co-operation with A. N. Whitehead, to develop a logical system,
using a notation different from Frege’s, in which he set out to derive the whole
of arithmetic from a purely logical basis. This work was published in the three
monumental volumes of Principia Mathematica between 1910 and 1913.
In order to avoid the paradox which he had discovered, Russell formulated a
Theory of Types. It was wrong to treat classes as randomly classifiable objects.
Classes and individuals were of different logical types, and what can be true or
false of one cannot be significantly asserted of the other. ‘The class of dogs is a
dog’ should be regarded not as false but as meaningless. Similarly, what can
meaningfully be said of classes cannot meaningfully be said of classes of classes,
and so on through the hierarchy of logical types. If the difference of type between
the different levels of the hierarchy is observed, then the paradox will not arise.
But another difficulty arises in place of the paradox. Once we prohibit the
formation of classes of classes, how can we define the series of natural numbers?
Russell retained the definition of zero as the class whose only member is the null-
class, but he now treated the number one as the class of all classes equivalent to
the class whose members are (a) the members of the null-class, plus (b) any
object not a member of that class. The number two was treated in turn as the
class of all classes equivalent to the class whose members are (a) the members of
the class used to define one, plus (b) any object not a member of that defining
class. In this way the numbers can be defined one after the other, and each
number is a class of classes of individuals. But the natural-number series can be
continued thus ad infinitum only if there is an infinite number of objects in the
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universe; for if there are only n individuals, then there will be no classes with
n + 1 members, and so no cardinal number n + 1. Russell accepted this and
therefore added to his axioms an axiom of infinity, i.e. the hypothesis that the
number of objects in the universe is not finite. This hypothesis may be, as Russell
thought it was, highly probable; but on the face of it it is far from being a logical
truth; and the need to postulate it is therefore a sullying of the purity of the
original programme of deriving arithmetic from logic alone.
When he learned of Russell’s paradox, Frege was utterly downcast. He made
more than one attempt to patch up his system, but these were no more success-
ful in salvaging logicism than was Russell’s theory of types. We now know that
the logicist programme cannot ever be successfully carried out. The path from
the axioms of logic via the axioms of arithmetic to the theorems of arithmetic
is barred at two points. First, as Russell’s paradox showed, the naive set the-
ory which was part of Frege’s logical basis was inconsistent in itself, and the
remedies which Frege proposed for this proved ineffective. Thus, the axioms of
arithmetic cannot be derived from purely logical axioms in the way Frege hoped.
Secondly, the notion of ‘axioms of arithmetic’ was itself later called in question
when the Austrian mathematician Kurt Gödel showed that it was impossible to
give arithmetic a complete and consistent axiomatization in the style of Principia
Mathematica. None the less, the concepts and insights developed by Frege and
Russell in the course of expounding the logicist thesis have a permanent interest
which is unimpaired by the defeat of that programme.
Russell’s Theory of Descriptions
In his realist period, when he wrote The Principles of Mathematics, Russell had
believed that in order to save the objectivity of concepts and judgements it was
necessary to accept the existence of Platonic ideas and of propositions which
subsisted independently of their expression in sentences. Like Frege, he accepted
that concepts were something independent of our thinking; but he went beyond
Frege because he believed that not only relations and numbers, but also chimaeras
and the Homeric gods all had being of some kind, for if not it would be impossible
to make propositions about them. ‘Thus being is a general attribute of everything,
and to mention anything is to show that it is.’
By the time he wrote Principia Mathematica Russell had changed his mind.
He wrote:
Suppose we say ‘the round square does not exist’. It seems plain that this is a true
proposition, yet we cannot regard it as denying the existence of a certain object
called ‘the round square’. For if there were such an object, it would exist: we cannot
first assume that there is a certain object, and then proceed to deny that there is such
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an object. Whenever the grammatical subject of a proposition can be supposed not
to exist without rendering the proposition meaningless, it is plain that the gram-
matical subject is not a proper name, i.e. not a name directly representing some
object. Thus in all such cases the proposition must be capable of being so analysed
that what was the grammatical subject shall have disappeared. Thus, when we say
‘The round square does not exist’ we may, as a first attempt at such analysis,
substitute ‘It is false that there is an object x which is both round and square’.
So far, this account is similar to Frege’s method of treating statements of exist-
ence; but Russell saw that it was necessary to give an account of the meaning of
vacuous expressions such as ‘the round square’ and ‘the present King of France’
when they occurred in contexts other than statements of existence; for instance,
in the sentence ‘The present King of France is bald’. Russell called expressions
like ‘the present King of France’ and ‘the man who discovered oxygen’ by the
name ‘definite descriptions’. In his article On Denoting of 1905 he worked out a
general theory of the meaning of definite descriptions, which would take care
both of the cases where there was some object answering to the description (as in
‘the man who discovered oxygen’) and of the cases where the description was
vacuous (as in ‘the present King of France’).
Frege had treated definite descriptions simply as complex names, so that ‘The
author of Hamlet was a genius’ had the same logical structure as ‘Shakespeare
was a genius’. This meant that he had to provide for arbitrary rules to be laid
down in order to ensure that a sentence containing an empty name or vacuous
definite description did not lack a truth-value. Russell thought this was unsatis-
factory, and proposed to analyse sentences containing definite descriptions quite
differently from those containing names. It is a mistake, he believed, to look for
the meaning of definite descriptions in themselves; only the propositions in whose
verbal expression they occur have a meaning.
For Russell there is a big difference between a sentence such as ‘James was
deposed’ (containing the name ‘James’) and a sentence such as ‘The brother of
Charles II was deposed’. An expression such as ‘The brother of Charles II’ has no
meaning in isolation; but the sentence ‘The brother of Charles II was deposed’
has a meaning none the less. It asserts three things:
(a) that some individual was brother to Charles II
(b) that only this individual was brother to Charles II
(c) that this individual was deposed.
Or, more formally:
For some x, (a) x was brother to Charles II
and (b) for all y, if y was brother to Charles II, y = x
and (c) x was deposed.
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The first element of this formulation says that at least one individual was a
brother of Charles II, the second that at most one individual was a brother of
Charles II, so that between them they say that exactly one individual was brother
to Charles II. The third element goes on to say that that unique individual was
deposed. In the analysed sentence nothing appears which looks like a name of
James II; instead, we have a combination of predicates and quantifiers.
What is the point of this complicated analysis? To see this we have to consider
a sentence which, unlike ‘The brother of Charles II was deposed’, is not true.
Consider the following two sentences:
Figure 44 Bertrand Russell as a young man.
(National Portrait Gallery, London)
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(1) The sovereign of the United Kingdom is male.
(2) The sovereign of the United States is male.
Neither of these sentences is true, but the reason differs in the two cases. Every-
one would agree that the first sentence is not true, but plain false, because the
sovereign of the United Kingdom is female. The second fails to be true because
the US has no sovereign, and on Russell’s view this second sentence is not just
untrue but positively false; and consequently its negation ‘It is not the case that
the sovereign of the US is male’ is true. Sentences containing empty definite
descriptions differ sharply in Russell’s system from sentences containing empty
names, i.e. apparent names which name no objects. For Russell a would-be
sentence such as ‘Slawkenburgius was a genius’ is not really a sentence at all,
and therefore neither true nor false, since there was never anyone of whom
‘Slawkenburgius’ was the proper name.
Why did Russell want to ensure that sentences containing vacuous definite
descriptions should count as false? He was, like Frege, interested in constructing
a precise and scientific language for purposes of logic and mathematics. Both
Frege and Russell regarded it as essential that such a language should contain
only expressions which had a definite sense, by which they meant that all sen-
tences in which the expressions could occur should have a truth-value. For if we
allow into our system sentences lacking a truth-value, then inference and deduc-
tion become impossible. It is easy enough to recognize that ‘the round square’
denotes nothing, because it is obviously self-contradictory. But prior to investigation
it may not be clear whether some complicated mathematical formula contains a
hidden contradiction. And if it does so, we will not be able to discover this by
logical investigation unless sentences containing it are assured of a truth-value.
Logical Analysis
In On Denoting and later papers Russell constantly speaks of the activity of
the philosopher as being one of analysis. By analysis he means a technique of
substituting a logically clear form of words for another form of words which
was in some way logically misleading. His theory of descriptions was for long
a paradigm of such logical analysis. But in Russell’s mind, logical analysis was
far more than a device for the clarification of sentences. He came to believe that
once logic had been cast into a perspicuous form it would reveal the structure
of the world.
Logic contained individual variables and propositional functions: correspond-
ing to these the world contained particulars and universals. In logic complex
propositions were built up out of simple propositions as truth-functions of the
simpler propositions. Similarly, in the world there were independent atomic facts
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corresponding to the simple propositions. Atomic facts consisted either in the
possession by a particular of a characteristic, or else in a relation between two or
more particulars. This theory of Russell’s was called ‘logical atomism’.
The theory of descriptions was the great analytic tool of logical atomism.
Russell began to apply it not only to round squares and to Platonic entities, but
also to many things which common sense would regard as perfectly real, such as
Julius Caesar, tables, and chairs. The reason for this was that Russell came to
believe that every proposition which we can understand must be composed wholly
of items with which we are acquainted. ‘Acquaintance’ was Russell’s word for
immediate presentation: we were acquainted, for instance, with our own sense-
data, which correspond in his system to Hume’s impressions or the deliverances
of Cartesian consciousness. But Russell still retained something of his earlier
Platonism: he believed that we had direct acquaintance with the universals which
were represented by the predicates of the reformed logical language. But the
range of things which we could know by acquaintance was limited: we could not
be acquainted with Queen Victoria or our own past sense-data. Those things
which were not known by acquaintance were known only by description; hence
the importance of the theory of descriptions.
In the sentence ‘Caesar crossed the Rubicon’, uttered in England now, we have
a proposition in which there are apparently no individual constituents with which
we are acquainted. In order to explain how we can understand the sentence
Russell analyses the names ‘Caesar’ and ‘Rubicon’ as definite descriptions. The
descriptions, spelt out in full, no doubt include reference to those names, but not
to the objects they named. The sentence is exhibited as being about general
characteristics and relations, and the names with which we become acquainted as
we pronounce them.
For Russell, then, ordinary proper names were in fact disguised descriptions.
A fully analysed sentence would contain only logically proper names (words
referring to particulars with which we are acquainted) and universals (words
referring to characters and relations). It was never altogether clear what counted
as logically proper names. Sometimes Russell seemed to countenance only de-
monstratives such as ‘this’ and ‘that’. An atomic proposition, therefore, would be
something like ‘(this) red’ or ‘(this) beside (that)’.
Logical atomism was presented in a famous series of lectures in 1918. It was
far from being Russell’s last word on philosophy. In the fifty-two years which
remained to him Russell wrote many books and essays, some of which concern
issues of logic and epistemology as well as the moral and educational topics which
began to take up more and more of his attention. In his later life, and particularly
after he inherited an earldom, he was known to a very wide public as a writer and
campaigner on various social and political issues. But most of the work which
established his reputation among professional philosophers and mathematicians
was completed before 1920. Logical atomism itself, as Russell was the first to
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