Russell's
PRINCIPLES OF MATHEMATICS
Bertrand
Russell began work on the book we now know as The Principles of Mathematics in the beginning of 1898, when he was
27, and still under the spell of the Hegelian school of logic that dominated English
philosophy at the time. Russell's
purpose was to address the contradiction inherent in the nature of number.
A number of things is one thing (since we talk of 'a' number or 'a'
collection of things), but is also many (since it, or they, may be two or more
things). He had proposed to deal with
this in the Hegelian manner, by accepting the notion of quantity as inherently
contradictory, then constructing a 'dialectic' around it. (A discussion of the problem survives in
section 74 of the book).
After reading
Whitehead's Universal Algebra, and
Dedekind's Nature and Meaning of Numbers
(a title which I am unable to find in any reference to Dedekind's works, but
which may be Monk's translation of Was
sind und was sollen die Zahlen, Meyer 1891), he became convinced that this
Hegelian approach was wrong, and began work on a book to be called An Analysis of Mathematical Reasoning. He gave up this in 1900, but used the
material extensively in Principles of
Mathematics, a version of which was nearly complete by mid-1900.
At this point,
while he now accepted Cantor's account of the continuum, he was still unable to
accept Cantor's idea of an infinite whole, which underpins all of set
theory. This changed when he went to
present a paper at the International Congress of Philosophy at Paris in July
1899, at the invitation of Louis Couturat.
Peano dominated the conference, and impressed Russell, who immediately
began work on understanding the work of the Italian mathematicians. In October 1900 he rewrote large sections of
the book (according to Monk, ten pages a day and 200,000 words in less than
three months). He wanted to show that
all mathematics could be reduced to a handful of logical concepts, and that
even the concepts of 'zero', 'number' and 'successor' could be constructed out
of the concept of a 'class'. He said
that this period was 'the highest point of my life', and an 'intellectual
honeymoon such as I have never experienced before or since'.
My sensations resembled those one has after climbing a mountain in a
mist, when, on reaching the summit, the mist suddenly clears, and the country
becomes visible for forty miles in every direction. For years I had been endeavouring to analyse
the fundamental notions of mathematics, such as order and cardinal
numbers. Suddenly, in the space if a few
weeks, I discovered what appeared to be definitive answers to the problems
which had baffled me for years
I went about saying to myself that now at last
I had done something worth doing, and I had the feeling that I must be careful
not to be run over in the street before I had written it down.
He pointedly
finished the manuscript on the very last day of 1900 (which, being a
mathematician, he thought of as the last day of the nineteenth century).
Alas, in the
Spring of 1901, he discovered what is now called 'Russell's Paradox', and which
Russell himself modestly called 'The Contradiction'. Russell's description of it is in the tenth
chapter of the book, here.
Russell had been studying Cantor's proof that there is no greatest
ordinal. Russell did not at first accept
this, since he thought the class of all classes must surely have the greatest
number there is. This led him (around
April or May) to consider the class of all classes that are not members of
themselves. Is this class a member of
itself or not? If it is, it isn't, if it
isn't, it is. Contradiction. The honeymoon was over.
Russell worked
for at least a year on the book (Grattan Guinness (2000, 311 ff) has a detailed
account of the chronology) before sending the manuscript to the publisher in
early June 1902, two weeks before he notified Frege of the paradox in a letter
in German.
This clearly
contradicts the idea that Russell became aware of the importance of the paradox
through Frege. Indeed, Russell
discovered Frege fairly late, reading some of his works in detail as late as
June 1902 (according to Grattan Guinness, he told Couturat of his ignorance in
letters of 25 June and 2 July 1902).
Hence his accounts on Frege's work appear as an appendix in sections
§475 to §496 here.
Russell wrote
the preface in December 1902. It
appeared in May 1903, near to Russell's 31st birthday. The print-run was 1,000 copies at 12/6d
each. By June 1909 the last copies were
at the binders. The book had an
important part in publicising the work of Cantor and Frege to the
English-speaking world, and it is undoubtedly a masterpiece (Monk regards it as
one of the most important philosophical works of the twentieth century). However, it is somewhat of a mess, owing to
the disorderly presentation, and Russell wrote to Gilbert Murray that the book
'disgusted' him. However, the book was
well received. The Spectator wrote
"we should say that Mr. Russell has an inherited place in literature or
statesmanship waiting for him if he will condescend to come down to the common
day" (1903). Hardy thought that
Russell had 'proved his point' about logicism, and confessed (interestingly)
that he had never heard of Frege until Russell's book. Stout wrote to Russell that he was impressed
by the book, but thought it fundamentally wrong. He asked W.E.
Johnson to review it for Mind,
but nothing arrived and he instead asked A.T.
Shearman, who gave it 12 pages, hailing it as the most important work since
Boole's Laws of Thought. Shearman also welcomed the account of Frege's
work without the "extreme cumbrousness" of the Begriffschift notation.
The passages
from the book given below are on the subjects of class and class-membership,
the relevance of grammar to philosophy and mathematics ("Although a
grammatical distinction cannot be uncritically assumed to correspond to a
genuine philosophical difference, yet the one is prima facia evidence of
the other, and may often be most usefully employed as a source of
discovery"), the problem of the unity of the proposition, on enumerations
vs. classes, classes as many vs. classes as one, the definition of an infinite
set, Russell's Paradox, and Frege's contribution to logic.
THE
PRINCIPLES OF MATHEMATICS
BERTRAND
RUSSELL
§ 21. The
insistence on the distinction between e and the relation of whole
and part between classes is due to Peano, and is of very great importance to
the whole technical development and the whole of the applications to
mathematics. In the scholastic doctrine of the syllogism, and in all previous
symbolic logic, the two relations are confounded, except in the work of
Frege*. The distinction is the same as that between the relation of
individual to species and that of species to genus, between the relation of
Socrates to the class of Greeks and the relation of Greeks to men. On the
philosophical nature of this distinction I shall enlarge when I come to deal
critically with the nature of classes; for the present it is enough to observe
that the relation of whole and part is transitive, while e is not so; we
have Socrates is a man, and men are a class, but not Socrates is a class.
It is to be observed that the class must be distinguished from the
class-concept or predicate by which it is to be defined: thus men are a class,
while man is a class-concept. The relation e must be regarded as
holding between Socrates and men considered collectively, not between Socrates
and man. I shall return to this point in Chapter VI
[§66-68]. Peano holds that all propositional functions containing only a
single variable are capable of expression in the form "x is an a,"
where a is a constant class; but this view we shall find reason to
doubt.
* See his Begriffschift,
Halle, 1879, and Grundgesetze der Arithmetik, Jena, 1894
§ 38.
We need, in fact, the notion of therefore, which is quite different from
the notion of implies, and holds between different entities. In
grammar, the distinction is that between a verb and a verbal noun, between, say
"A is greater than B" and "A's being greater than B."
In the first of these, a proposition is actually asserted, whereas in the
second it is merely considered. But these are psychological terms,
whereas the difference which I desire to express is genuinely logical. It
is plain that, if I may be allowed to use the word assertion in a
non-psychological sense, the proposition "p implies q" asserts
an implication, though it does not assert p or q.
The p and the q which enter into this proposition are not
strictly the same as the p or the q which are separate propositions,
at least, if they are true. The question is: How does a proposition
differ by being actually true from what it would be as an entity if it were not
true? It is plain that true and false propositions alike are entities of
a kind, but that true propositions have a quality not belonging to false ones,
a quality which, in a non-psychological sense, may be called being asserted.
Yet there are grave difficulties in forming a consistent theory on this point,
for if assertion in any way changed a proposition, no proposition which can
possibly in any context be unasserted could be true, since when asserted it
would become a different proposition. But this is plainly false; for in
"p implies q," p and q are not asserted,
and yet they may be true. Leaving this puzzle to logic, however, we must
insist that there is a difference of some kind between an asserted and an
unasserted proposition.
§ 46. In
the present chapter, certain questions are to be discussed belonging to what
may be called philosophical grammar. The study of grammar, in my opinion,
is capable of throwing far more light on philosophical questions than is
commonly supposed by philosophers. Although a grammatical distinction
cannot be uncritically assumed to correspond to a genuine philosophical
difference, yet the one is prima facia evidence of the other, and may
often be most usefully employed as a source of discovery. Moreover, it
must be admitted, I think, that every word occurring in a sentence must have some
meaning: a perfectly meaningless sound could not be employed in the more or
less fixed way in which language employs words. The correctness of our
philosophical analysis of a proposition may therefore be usefully checked by
the exercise of assigning the meaning of each word in the sentence expressing
the proposition. On the whole, grammar seems to me to bring us much
nearer to a correct logic than the current opinions of philosophers: and in
what follows, grammar, though not our master, will yet be taken as our guide.
§ 52 It
remains to discuss the verb, and to find marks by which it is distinguished
from the adjective
By transforming the verb, as it occurs in a proposition,
into a verbal noun, the whole proposition can be turned into a logical subject,
no longer asserted, and no longer containing in itself truth and
falsehood. But here, too, there seems to be no possibility of maintaining
that the logical subject which results is a different entity from the
proposition. "Caesar died" and "the death of Caesar"
will illustrate this point. If we ask: What is asserted in the
proposition "Caesar died"? the answer must be "the death of
Caesar is asserted". In that case, it would seem, it is the death of
Caesar which is true or false; and yet neither truth nor falsity belongs to a
mere logical subject. The answer here seems to be that the death of
Caesar has an external relation to truth or falsehood
whereas "Caesar
died" in some way or other contains its own truth or falsehood as an
element. But if this is the correct analysis, it is difficult to see how
"Caesar died" differs from "the truth of Caesar's death" in
the case where it is true, or "the falsehood of Caesar's death" in
the other case. Yet it is quite plain that the latter, at any rate, is
never equivalent to "Caesar died". There appears to be an
ultimate notion of assertion, given by the verb, which is lost as soon as we
substitute a verbal noun, and is lost when the proposition in question is made
the subject of some other proposition.
Thus the contradiction which was to
have been avoided, of an entity which cannot be made a logical subject, appears
to have here become inevitable. This difficulty, which seems to be
inherent in the very nature of truth and falsehood, is one with which I do not
know how to deal with satisfactorily. The obvious course would be to say
that the difference between an asserted and an unasserted proposition is not
logical but psychological. In the sense in which false propositions may
be asserted, this is doubtless true. But there is another sense of
assertion, very difficult to bring clearly before the mind, and yet quite
undeniable, in which only true propositions are asserted. True and false
propositions alike are in some sense entities, and are in some sense capable of
being logical subjects; but when a proposition happens to be true, it has a
further quality, over and above that which it shares with false propositions,
and it is this further quality which is what I mean by assertion in a logical
as opposed to psychological sense. The
nature of truth, however, belongs no more to the principles of mathematics than
to the principles of everything else. I therefore
leave this question to the logicians with the above brief indication of a
difficulty.
§ 54. The
twofold nature of the verb, as actual verb and as verbal noun, may be
expressed, if all verbs are held to be relations, as the difference between a
relation in itself and a relation actually relating. Consider, for
example, the proposition "A differs from B." The
constituents of this proposition, if we analyze it, appear to be only A,
difference, B. Yet these constituents, thus placed side by side,
do not reconstitute the proposition. The difference which occurs in the
proposition actually relates A and B, whereas the difference after
analysis is a notion which has no connection with A and B.
It may be said that we ought, in the analysis, to mention the relations which
difference has to A and B, relations which are expressed by is
and from when we say " A is different from B."
These relations consist in the fact that A is referent and B
relatum with respect to difference. But "A, referent,
difference, relatum, B" is still merely a list of terms, not a
proposition. A proposition, in fact, is essentially a unity, and when
analysis has destroyed the unity, no enumeration of constituents will restore
the proposition. The verb, when used as a verb, embodies the unity of the
proposition, and is thus distinguishable from the verb considered as a term,
though I do not know how to give a clear account of the distinction.
§ 66. It
has been customary, in works on logic, to distinguish two standpoints, that of
extension and that of intension. Philosophers have usually regarded the
latter as more fundamental, while Mathematics has been held to deal specially
with the former. .. But as a matter of fact, there are positions intermediate
between pure intension and pure extension, and it is in these intermediate
regions that Symbolic Logic has its lair. It is essential that the classes
with which we are concerned should be composed of terms, and should not be
predicates or concepts, for a class must be definite when its terms are given,
but in general there will be many predicates which attach to the given terms
and no others. We cannot of course attempt an intensional definition of a
class as the class of all predicates attaching to the terms in question and to
no others, for this would involve a vicious circle; hence the point of view of
extension is to some extent unavoidable. On the other hand, if we take
extension pure, our class is defined by enumeration of its terms, and this
method will not allow us to deal, as Symbolic Logic does, with infinite
classes. Thus our classes must in general be regarded as objects denoted
by concepts, and to this extent the point of view of intension is
essential. It is owing to this consideration that the theory of denoting
is of such great importance.
§ 67.
When an object is unambiguously denoted by a concept, I shall speak of the
concept as a concept (or, loosely, as the concept) of the object in
question. Thus it will be necessary to distinguish the concept of a class
from a class-concept. We agreed to call man a class-concept, but man
does not, in its usual employment, denote anything. On the other hand men,
and all men (which I shall regard as synonyms) do denote, and I shall
contend that what they denote is the class composed of all men. Thus man
is the class-concept, men (the concept) is the concept of the class, and
men (the object denoted by the concept men) are the class. It is
no doubt confusing, at first, to use class-concept and concept of a
class in different senses; but so many distinctions are required that some
straining of language seems unavoidable, In the phraseology of the preceding
chapter, we may say that a class is a numerical conjunction of terms
§ 68.
We may, then, imagine a kind of genesis of classes, through the successive
stages indicated by the typical propositions "Socrates is human",
"Socrates has humanity", "Socrates is a man",
"Socrates is one among men". Of these propositions, the last
only, we should say, explicitly contains the class as constituent; but every
subject-predicate proposition gives rise to the other three equivalent
propositions, and thus every predicate gives rise to a class. This is the
genesis of classes from the intensional standpoint.
On the other
hand, when mathematicians deal with what they call a manifold, aggregate, Menge,
ensemble, or some equivalent name, it is common, especially where the
number of terms is finite, to regard the object in question (which is in fact a
class) as defined by the enumeration of its terms, and as consisting possibly
of a single term, which in that case is the class. Here it is not
predicates and denoting that are relevant, but terms connected by the word and,
in the sense in which this word stands for a numerical
conjunction. Thus Brown and Jones are a class, and Brown singly is a
class. This is the extensional genesis of classes.
§ 69. The
best formal treatment of classes in existence is that of Peano. But in
this treatment a number of distinctions of great philosophical importance are
overlooked. Peano, not I think quite consciously, identifies the class
with the class-concept; thus the relation of an individual to its class
is, for him, expressed by is a. For him, "2 is a number"
is a proposition in which a term is said to belong to the class number.
Nevertheless he identifies the equality of classes, which consists in their
having the same terms, with identity a proceeding which is quite illegitimate
when the class is regarded as the class-concept. In order to perceive
that man and featherless biped are not identical, it is quite
unnecessary to take a hen and deprive the poor bird of its feathers. Or,
to take a less complex instance, it is plain that even prime is not
identical with integer next after 1. Thus when we identify the
class with the class-concept, we must admit that two classes may be equal
without being identical. Nevertheless, it is plain that when two
class-concepts are equal, some identity is involved, for we say that they have
the same terms. Thus there is some object which is positively
identical when the two class-concepts are equal; and this object, it would
seem, is more properly called the class. Neglecting the plucked hen, the
class of featherless bipeds, every one would say, is the same as the
class of men; the class of even primes is the same as the class of
integers next after 1. Thus we must not identify the class with the
class-concepts, or regard "Socrates is a man" as expressing the
relation of an individual to a class of which it is a member. This has
two consequences
which prevent the philosophical acceptance of certain points
in Peano's formalism. The first consequence is that there is no such
thing as the null-class, though there are null class-concepts. The second
is, that a class having only one term is to be identified, contrary to Peano's
usage, with that one term. I should not propose, however, to alter his
practice or his notation in consequence of either of these points; rather I
should regard them as proofs that Symbolic Logic should concern itself, as far
as notation goes, with class-concepts rather than with classes.
§ 70. A
class, we have seen, is neither a predicate nor a class-concept, for different
predicates and different class-concepts may correspond to the same class.
A class also, in one sense at least, is distinct from the whole composed of its
terms, for the latter is only and essentially one, while the former, where it
has many terms, is, as we shall see later, the very kind of object of which many
is to be asserted. The distinction of a class as many from a class as a
whole is often made by language: space and points, time and instants, the army
and the soldiers, the Cabinet and the Cabinet Ministers, all illustrate the
distinction. The notion of a whole, in the sense of a pure aggregate
which is here relevant, is, we shall find, not always applicable where the
notion of the class as many applies
In such cases, though terms may be said
to belong to the class, the class must not be treated as itself a single
logical subject [As footnote: a plurality of terms is not the logical subject
when a number is asserted of it: such propositions have not one subject, but
many subjects. See end of § 74]. But this never arises where a
class can be generated by a predicate. Thus we may for the present
dismiss this complication from our minds. In a class as many, the
component terms, though they have some kind of unity, have less than is
required for a whole. They have, in fact, just so much unity as is
required to make them many, and not enough to prevent them from remaining
many. A further reason for distinguishing wholes from classes as many is
that a class as one may be one of the terms of itself as many, as in
"classes are one among classes" (the extensional equivalent of
"class is a class-concept"), whereas a complex whole can never be one
of its own constituents.
§ 71 Class
may be defined either extensionally or intensionally. That is to say, we
may define the kind of object which is a class, or the kind of concept which
denotes a class: this is the precise meaning of the opposition of extension and
intension in this connection. But although the general notion can be
defined in this two-fold manner, particular classes, except when they happen to
be finite, can only be defined intensionally, i.e. as the objects
denoted by such concepts. I believe this distinction to be purely
psychological: logically, the extensional definition appears to be equally
applicable to infinite classes, but practically, if we were to attempt it,
Death would cut short our laudable endeavour before it had attained its
goal. Logically, therefore, extension and intension seem to be on a
par. I will begin with the extensional view.
When a class is
regarded as defined by the enumeration of its terms, it is more naturally
called a collection. I shall for the moment adopt this name, as it
will not prejudge the question whether the objects denoted by it are truly
classes or not. Buy a collection I mean what is conveyed by "A and
B", or "A and B and C", or any other enumeration of definite terms.
The collection is defined by the actual mention of the terms, and the terms are
connected by and. It would seem that and represents a
fundamental way of combining terms, and that just this way of combination is
essential if anything is to result of which a number other than 1 can be
asserted. Collections do not presuppose numbers, since they result simply
from the terms together with and: they could only presuppose numbers in
the particular case where the terms of the collection themselves presupposed
numbers. There is a grammatical difficulty which, since no method exists
of avoiding it, must be pointed out and allowed for. A collection,
grammatically, is singular, whereas A and B, A and B and C, etc. are
essentially plural. This grammatical difficulty arises from the logical
fact (to be discussed presently) that whatever is many in general forms a whole
which is one; it is, therefore, not removable by a better choice of technical
terms.
The notion of and
was brought into prominence by Bolzano*. In order to understand what
infinity is, he says,
we
must go back to one of the simplest conceptions of our understanding, in order
to reach an agreement concerning the word we are to use to denote it.
This is the conception which underlies the conjunction and, which,
however, if it is to stand out as clearly as is required, in many cases, both
by the purposes of mathematics and by those of philosophy, I believe to be best
expressed by the words: 'A system (Inbegriff) of certain things,' or ' a
whole consisting of certain parts.' But we must add that every arbitrary
object A can be combined in a system with any others B, C, D,
, or (still
speaking more correctly) already forms a system by itself i.e. the combination
of A with B, C, D
already forms a system], of which some more or less
important truth can be enunciated, provided only that each of the presentations
A, B, C, D,
in fact represents a different object, or in so far as
none of the propositions 'A is the same as B,' 'A is the same as C,' 'A is the
same as D,' etc., is true. For if e.g. A is the same as B, then it is
certainly unreasonable to speak of a system of the things A and B.
The above
passage, good as it is, neglects several distinctions which we have found
necessary. First and foremost, it does not distinguish the many from the
whole which they form. Secondly, it does not appear to observe that the
method of enumeration is not practically [EDB: my emphasis] applicable to
infinite systems. Thirdly, and this is connected with the second point,
it does not make any mention of intensional definition nor of the notion of a
class. What we have to consider is the difference, if any, of a class
from a collection on the one hand, and from the whole formed of the
collection on the other. But let us examine further the notion of and.
Anything of
which a finite number other than 0 or 1 can be asserted would commonly be said
to be many, and many, it might be said, are always of the form "A and B
and C and
" Here A, B, C,
are each one and are all
different. To say that A is one seems to amount to much the same as to
say that A is not of the form "A1 and A2 and A3
and
" To say that A, B, C,
are all different seems to
amount only to a condition as regards the symbols: it should be held that
"A and A" is meaningless, so that diversity is implied by and,
and need not be specially stated.
A term A which
is one may be regarded as a particular case of a collection, namely as a
collection of one term. Thus every collection which is many presupposes
many collections which are each one: A and B presupposes A and
presupposes B. Conversely some collections of one term presuppose many,
namely those which are complex: thus "A differs from B" is one, but
presupposes A and difference and B. But there is no symmetry in this
respect, for the ultimate presuppositions of anything are always simple terms.
[ A and B may
be fictional but have Being]
The question
may now be asked: What is meant by A and B? Does this mean
anything more than the juxtaposition of A with B? That is, does it
contain any element over and above that of A and that of B? To either
answer there are objections. In the first place, and, we might
suppose, cannot be a new concept, for if it were, it would have to be some kind
of relation between A and B; A and B would then be a proposition, or at
least a propositional concept, and would be one, not two. Moreover, if
there are two concepts, there are two, and no third mediating concept
seems necessary to make them two. Thus and would seem
meaningless. But it is difficult to maintain this theory. To begin
with, it seems rash to hold that any word is meaningless. When we use the
word and, we do not seem to be uttering idle breath, but some idea seems
to correspond to the word. Again some kind of combination seems to be
implied by the fact that A and B are two, which is not true of either
separately. When we say "A and B are yellow," we can replace
the proposition by "A is yellow" and "B is yellow"; but
this cannot be done for "A and B are two"; on the contrary, A is one
and B is one. Thus it seems best to regard and as
expressing a definite unique kind of combination, not a relation, and not
combining A and B into a whole, which would be one. This unique kind of
combination will in future be called addition of individuals. It
is important to observe that it applies to terms, and only applies to numbers
in consequence of their being terms. Thus for the present, 1 and 2 are
two, and 1 and 1 is meaningless.
As regards what
is meant by the combination indicated by and, it is indistinguishable
from what we before called a numerical conjunction. That is, A and
B is what is denoted by the concept of a class of which A and B are the only
members. If u be a class-concept of which the propositions "A
is a u" "B is a u" are true, but of which all
other propositions of the same form are false, then "all u's"
is the concept of a class whose only terms are A and B; this concept denotes
the terms A, B combined in a certain way, and "A and B" are
those terms combined in just that way. Thus "A and B" are the
class, but are distinct from the class-concept and from the concept of the
class.
The notion of and,
however, does not enter into the meaning of a class, for a single term is
a class, although it is not a numerical conjunction. If u be a
class-concept, and only one proposition of the form "x is a u"
be true, then "all u's" is a concept denoting a single
term, and this term is the class of which "all u's" is a
concept. Thus what seems essential to a class is not the notion of and,
but the being denoted by some concept of a class. This brings to the
intensional view of classes.
* Paradoxien
des Unendlichen, Leipzig, 1854 (2nd ed., Berlin 1889)
§ 72. We
agreed in the preceding chapter that there are not different ways of denoting,
but only different kinds of denoting concepts and correspondingly different
kinds of denoted objects. We have discussed the kind of denoted object
which forms a class; we have now to consider the kind of denoting concept.
The
consideration of classes which results from denoting concepts is more general
than the extensional consideration, and that in two respects. In the
first place it allows, what the other practically excludes, the admission
of infinite classes; in the second place it introduces the null concept of a
class. But, before discussing these matters, there is a purely logical
point of some importance to be examined.
If u be
a class-concept, is the concept "all u's" analyzable into two
constituents, all and u, or is it a new concept, defined by a
certain relation to u, and no more complex than u itself?
We may observe, to begin with, that "all u's" is synonymous
with "u's" at least according to a very common use of the
plural. Our question is, then, as to the meaning of the plural. The
word all has certainly some definite meaning, but it seems highly
doubtful whether it means more than the indication of a relation.
"All men" and "all numbers" have in common the fact that they
both have a certain relation to a class-concept, namely to man and
number respectively. But it is very difficult to isolate any
further element of all-ness which both share, unless we take as this
element the mere fact that both are concepts of classes. It would seem,
then, that "all u's" is not validly analyzable into all
and u, and that language, in this case as in some others, is a
misleading guide. The same remark will apply to every, any,
some, a, and the.
It might
perhaps be thought that a class ought to be considered, not merely as a
numerical conjunction of terms, but as a numerical conjunction denoted by the
concept of a class. This complication, however, would serve no useful
purpose, except to preserve Peano's distinction between a single term and the
class whose only term it is a distinction which is easy to grasp when the
class is identified with the class-concept, but which is inadmissible in our
view of classes. It is evident that a numerical conjunction considered as
denoted is either the same entity as when not so considered, or else is a
complex of denoting together with the object denoted; and the object denoted is
plainly what we mean by a class.
With regard to
infinite classes, say the class of numbers, it is to be observed that the
concept all numbers, though not itself infinitely complex, yet denotes
an infinitely complex object. This is the inmost secret of our power to
deal with infinity. An infinitely complex concept, though there may be
such, can certainly not be manipulated by the human intelligence; but infinite
collections, owing to the notion of denoting, can be manipulated without
introducing any concepts of infinite complexity. Throughout the
discussion of infinity in later Parts of the present work, this remark should
be borne in mind: if it is forgotten, there is an air of magic which causes the
results obtained to seem doubtful.
§ 74. A
question which is very fundamental in the philosophy of Arithmetic must now be
discussed in a more or less preliminary fashion. Is a class which has
many terms to be regarded as itself one or many? Taking the class as
equivalent simply to the numerical conjunction "A and B and C and
etc.," it seems plain that it is many; yet it is quite necessary that we
should be able to count classes as one each, and we do habitually speak of a
class. Thus classes would seem to be one in one sense and many in
another.
There is a
certain temptation to identify the class as many and the class as one, e.g., all
men and the human race. Nevertheless, wherever a class
consists of more than one term, it can be proved that no such identification is
permissible. A concept of a class, if it denotes a class as one, is not
the same as any concept of the class which it denotes. That is to say, classes
of all rational animals, which denotes the human race as one term, is
different from men, which denotes men, i.e. the human race as
many. But if the human race were identical with men, it would follow that
whatever denotes the one must denote the other, and the above difference would
be impossible. We might be tempted to infer that Peano's distinction,
between a term and a class of which the said term is the only member, must be
maintained, at least when the term in question is a class [as footnote: this
conclusion is actually drawn by Frege from an analogous argument: Archiv fur
Systematische Philosophie I (1895) p. 444]. But it is more correct, I
think, to infer an ultimate distinction between a class as many and a class as
one, to hold that the many are only many, and are not also one. The class
as one may be identified with the whole composed of the terms of the class,
i.e. in the case of men, the class as one will be the human race.
But can we now
avoid the contradiction always to be feared, where there is something that
cannot be made a logical subject? I do not myself see any way of
eliciting a precise contradiction in this case. In the case of concepts,
we were dealing with what was plainly one entity; in the present case, we are
dealing with a complex essentially capable of analysis into units. In
such a proposition as "A and B are two" there is no logical subject:
the assertion is not about A, not about B, nor about the whole composed of
both, but strictly and only about A and B. Thus it would seem that
assertions are not necessarily about single subjects, but may be about
many subjects; and this removes the contradiction which arose, in the case of
concepts, from the impossibility of making assertions about them unless they
were turned into subjects. This impossibility being here absent, the
contradiction which was to be feared does not arise.
§ 75. We
may ask, as suggested by the above discussion, what is to be said of the
objects denoted by a man, every man, some man, and any
man. Are these objects one or many or neither? Grammar treats
them all as one. But to this view, the natural objection is, which
one? Certainly not Socrates, nor Plato, nor any other particular
person. Can we conclude that no one is denoted? As well might we
conclude that every one is denoted, which in fact is true of the concept every
man. I think one is denoted in every case, but in an impartial
distributive manner. Any number is neither 1 nor 2 nor any other
particular number, whence it is easy to conclude that any number is not
any one number, a proposition at first sight contradictory, but really
resulting from an ambiguity in any, and more correctly expressed by
"any number is not some one number." There are,
however, puzzles in this subject which I do not yet know how to solve.
A logical
difficulty remains in regard to the nature of the whole composed of all the
terms of a class. Two propositions appear self-evident: (1) Two wholes
composed of different terms must be different; (2) A whole composed of one term
only is that one term. It follows that the whole composed of a class
considered as one term is that class considered as one term, and is therefore
identical with the whole composed of the terms of the class; but this result
contradicts the first of our supposed self-evident principles. The answer
in this case, however, is not difficult. The first of our principles is
only universally true when all the terms composing our two wholes are
simple. A given whole is capable, if it has more than two parts, of being
analyzed in a plurality of ways; and the resulting constituents, so long as
analysis is not pushed as far as possible, will be different for different ways
of analyzing. This proves that different sets of constituents may
constitute the same whole, and thus disposes of our difficulty.
§ 76.
Something must be said as to the relation of a term to a class of which it is a
member, and as to the various allied relations. One of the allied relations
is to be called e, and is to be fundamental in Symbolic Logic. But it is
to some extent optional which of them we take as symbolically fundamental
Logically, the
fundamental relation is that of subject and predicate, expressed in
"Socrates is human" a relation which, as we saw in Chapter IV, is
peculiar in that the relatum cannot be regarded as a term in the
proposition. The first relation that grows out of this is the one
expressed by "Socrates has humanity," which is distinguished by the
fact that here the relation is a term. Next comes "Socrates is a
man." This proposition, considered as a relation between Socrates
and the concept man, is the one which Peano regards as fundamental; and
his e expresses the relation is a between Socrates and man.
So long as we use class-concepts for classes in our symbolism, this practice is
unobjectionable; but if we give e this meaning, we must not assume that two
symbols representing equal class-concepts both represent one and the same
entity. We may go on to the relation between Socrates and the human race,
i.e. between a term and its class considered as a whole: this is expressed by
"Socrates belongs to the human race." This relation might
equally well be represented by e. It is plain that, since a class, except
when it has one term, is essentially many, it cannot be as such
represented by a single letter: hence in any possible Symbolic Logic the
letters which do duty for classes cannot represent the classes as many,
but must represent either class-concepts, or the wholes composed of classes, or
some other allied single entities. And thus e cannot represent the
relation of a term to its class as many; for this would be a relation of one
term to many terms, not a two-term relation such as we want. This
relation might be expressed by "Socrates is one among men"; but this,
in any case, cannot be taken to be the meaning of e.
§ 77. A
relation which, before Peano, was almost universally confounded with e, is the
relation of inclusion between classes, as e.g. between men and mortals.
This is a time-honoured relation, since it occurs in the traditional form of
the syllogism: it has been a battleground between intension and extension, and
has been so much discussed that it is astonishing how much remains to be said
about it. Empiricists hold that such propositions mean an actual
enumeration of the terms of the contained class, with the assertion, in each
case, of membership of the containing class. They must, it is to be
inferred, regard it as doubtful whether all primes are integers, since they
will scarcely have the face to say that they have examined all primes one by
one. Their opponents have usually held, on the contrary, that what is
meant is a relation of whole and part between the defining predicates, but
turned in the opposite sense from the relation between the classes: i.e. the
defining predicate of the larger class is part of that of the smaller.
This view seems far more defensible than the other; and wherever such a
relation does hold between the defining predicates, the relation of inclusion
follows. But two objections may be made, first, that in some cases of
inclusion here is no such relation between the defining predicates, and
secondly, that in any case what is meant is a relation between the
classes, not a relation of their defining predicates. The first point may
be easily established by instances.
The concept even
prime does not contain as a constituent the concept integer between 1
and 10; the concept "English King whose head was cut off" does
not contain the concept "people who died in 1649"; and so on through
innumerable obvious cases. This might be met by saying that, though the
relation of the defining predicates is not one of whole and part, it is one more
or less analogous to implication, and is always what is really meant by
propositions of inclusion. Such a view represents, I think, what is said
by the better advocates of intension, and I am not concerned to deny that a
relation of the kind in question does always subsist between defining
predicates of classes one of which is contained in the other. But the
second of the above points remain valid as against any intensional
interpretation. When we say that men are mortals, it is evident that we
are saying something about men, not about the concept man or the
predicate human. The question is, then, what exactly are we
saying?
Peano held, in
earlier editions of his Formulaire, that what is asserted is the formal
implication "x is a man implies x is a mortal." This
is certainly implied, but I cannot persuade myself that it is the same
proposition. For in this proposition, as we saw I n Chapter III, it is
essential that x should take all values, and not only such as are
men. But when we say "all men are mortals, " it seems plain
that we are only speaking of men, and not of all other imaginable terms
§ 80 [the
notion of such that]
Chapter X "The Contradiction"
§ 100. It
is necessary to examine in more detail the singular contradiction, already
mentioned [i.e. Russell's paradox], with regard to predicates not predicable of
themselves. I may mention that I was led to it in the endeavour to
reconcile Cantor's proof that there can be no greatest cardinal number with the
very plausible supposition that the class of all terms
has necessarily the
greatest possible number of members.
§ 101. If
x be a predicate, x may or may not be predicable of itself.
Let us assume that "not predicable of oneself" is a predicate.
Then to suppose either that this predicate is, or that it is not, predicate of
itself, is self-contradictory. The conclusion, in this case, seems
obvious: "not predicable of oneself" is a not predicate.
Let us now
state the same contradiction in terms of class-concepts. A class-concept
may or may not be a term of its own extension. "Class-concept which
is not a term of its own extension" appears to be a class-concept.
But if it is a term of its own extension, it is a class-concept which is not a
term of its own extension, and vice versa. Thus we must conclude,
against appearances, that "class-concept which is not a term of its own
extension" is not a class-concept.
§ 118 Among
finite classes, if one is a proper part of another, the one has a smaller
number of terms than the other. (A proper part is a part not the whole).
Bur among infinite classes, this no longer holds. This distinction is, in
fact, an essential part of the above definitions of the finite and the
infinite
.
we may define
a0 as follows. Let P be a transitive and asymmetrical
relation, and let any two different terms of the field of P have the relation P
or its converse. Further let any class u contained in the field of
P and having successors (i.e. terms to which every term of u has the
relation P) have an immediate successor, i.e. a term whose predecessors either
belong to u or precede some term of u; let there be one term of
the field of P which has no predecessors, but let every term which has
predecessors have successors and also have an immediate predecessor; then the
number of terms in the field pf P is a0. Other definitions may
be suggested, but as all are equivalent it is not necessary to multiply
them. The following characteristic is important: Every class whose number
is a0 can be arranged in a series having consecutive terms, a
beginning but no end, and such that the number of predecessors of any term of
the series is finite; and any series having these characteristics has the
number a0.
§ 119 If n
be any finite number, the number obtained by adding 1 to n is also
finite, and is different from n. Thus beginning with 0 we can form
a series of numbers by successive additions of 1. We may define finite
numbers, if we choose, as those numbers that can be obtained from 0 by such
steps, and that obey mathematical induction. That is, the class of finite
numbers is the class of numbers which is contained in every class s to
which belongs 0 and the successor of every number belonging to s, where
the successor of a number is the number obtained by adding 1 to the given
number. Now a0 is not such a number, since, in virtue of
propositions already proved, no such number is similar to part of itself.
Hence also no number greater than a0 is finite according to the new
definition. But it is easy to prove that every number less than a0
is finite with the new definition as with the old. Hence the two
definitions are equivalent. Thus we may define finite numbers either as
those that can be reached my mathematical induction, starting from 0 and
increasing by 1 at each step, or as those of classes which are not similar to
the parts of themselves obtained by taking away single terms. These two
definitions are both frequently employed, and it is important to realise that
either is a consequence of the other.
§ 180
at this
point, the philosopher is apt to step in, and to declare that, by all true
philosophic principles, every well-defined series of terms must have a last
term. If he insists on creating this last term, and calling it infinity,
he easily deduces intolerable contradictions, from which he infers the
inadequacy of mathematics to deduce absolute truth.
§ 182 [the
argument of petitio principii]
§ 183 Of all
the philosophers who have inveighed against infinite number, I doubt whether
there is one who has known the difference between finite and infinite
numbers. The difference is simply this. Finite numbers obey the law
of mathematical induction; infinite numbers do not
It is in this alone,
and in its consequences, that finite and infinite numbers differ.
The principle
may otherwise be stated thus: If every proposition which holds concerning 0,
and also holds concerning the immediate successor of every number of which it
holds, holds concerning the number n, then n is finite; if not,
not. This is the precise sense of what may be popularly expressed by
saying that every finite number can be reached from 0 by successive steps, or
by successive additions of 1. This the principle which the philosopher
must be held to lay down as obviously applicable to all numbers, though he will
have to admit that the more precisely his principle is stated, the less obvious
it becomes.
§ 184 It
may be worth while to show exactly how mathematical induction enters into the
above proofs. Let us take the proof of (a) [that there are consecutive
magnitudes], and suppose there are n magnitudes between A and B.
Then to begin with, we supposed these magnitudes capable of enumeration, i.e.
of an order in which there consecutive terms and a first term, and a term immediately
preceding any tem except the first. This property presupposes
mathematical induction, and in fact was the very property in dispute.
Hence we must not presuppose the possibility of enumeration, which would be a petitio
principii.
Every series
having a finite number of terms can be shown by mathematical induction to have
a first and a last term; but no way exists of proving this concerning other
series, or of proving that all series are finite
The philosopher's finitist
arguments, therefore, rest on a principle of which he is ignorant, which there
is no reason to affirm, and every reason to deny.
§ 292.
we may define the infinite ordinals or cardinals
as those which, starting
from 0 or 1, can be reached by mathematical induction. This principle,
therefore, is not to be taken as an axiom or postulate, but as the definition
of finitude. It is to be observed that, in virtue of the principle
that every number has an immediate successor, we can prove that any
assigned number, say 10,397, is finite provided, of course, that the number
assigned is a finite number. That is to say, every proposition concerning
10,397 can be proved without the use of mathematical induction, which, as most
of us can remember, was not mentioned in the Arithmetic of our childhood.
There is therefore no kind of logical error in using the principle as a
definition of the class of finite numbers, nor is there a shadow of a reason
for supposing that the principle applies to all ordinal or all
cardinal numbers.
We can now
define finite numbers either by the fact that mathematical induction can reach
them
or by the fact that they are the numbers of collections such that no
proper part of them has the same number as the whole. These two
conditions may be easily shown to be equivalent. But they alone precisely
distinguish the finite form the infinite, and any discussion off infinity which
neglects them must be more or less frivolous.
§ 338
every
propositional function defines a class, and the actual enumeration of the
members of a class is not necessary for its definition.
.. it is
capable of formal proof that the number of the finite numbers themselves cannot
be a term in the progression of finite numbers. A number not
belonging to this progression is called infinite.
§ 339.
That there are infinite classes is so evident that it will scarcely be
denied. Since, however, it is capable of formal proof, it may be as well
to prove it. A very simple proof is that suggested in the Parmenides,
which is as follows. Let it be granted that there is a number 1.
Then 1 is, or has Being, and therefore there is Being. But 1 and Being
are two: hence there is a number 2 and so on. Formally, we have proved
that 1 is not the number of numbers; we prove that n is the number of
numbers from 1 to n, and that these numbers together with Being form a
class which has a new finite number, so that n is not the number of
finite numbers.
Hence the finite numbers, by mathematical induction, are
all contained in the class of things which are not the number of finite
numbers. Since the relation of similarity is reflexive for classes, every
class has a number; therefore the class of finite numbers has a number which,
not being finite, is infinite. A better proof, analogous to the above, is
derived from the fact that, if n be any finite number, the number of
numbers from 0 up to and including n is n + 1, whence it follows
that n is not the number of numbers. Again, it may be proved
directly, by the correlation of whole and part, that the number of propositions
or concepts is infinite [footnote: Cf. Bolzano Paradoxien des Unendlichen,
§ 13; Dedekind, Was sind und was sollen die Zahlen? No. 66]
§ 343. It
is very important to realise, as regards w or a0, that neither has a
number immediately preceding it. This characteristic they share with all
limits, for the limit of a series is never immediately preceded by any term of
the series which it limits
When it is forgotten that w has no immediate predecessor,
all sorts of contradictions emerge. For suppose n to be the last
number before w; then n is a finite number, and the number of finite
numbers is n+1. In fact, to say that w has no predecessor is merely to
say that the finite numbers have no last term. Though w is preceded by
all finite numbers, it is not preceded immediately by any of them: there is
none next to w.
§ 346 [Cantor's
diagonal proof]
§ 348
classes of propositions are only some among objects, yet Cantor's argument
shows that there are more of them than there are propositions. Again, we
can easily prove that there are more propositional functions than
objects. For suppose a correlation of all objects and some propositional
functions to have been affected [sic], and let fx be the correlate
of x. Then "not-fx(x)," i.e. "fx
does not hold of x" is a propositional function not contained in this
correlation; for it is true or false of x according as fx is
false or true of x, and therefore it differs from fx for
every value of x.
§ 475 to § 496
are "The Logical and Arithmetical doctrines of Frege"
§ 475.
Frege's work abounds in subtle distinctions, and avoids all the usual fallacies
which best writers on Logic.
[Disagreements with
Frege] all result from difference on three points: (1) Frege does not think
that there is a contradiction in the notion of concepts which cannot be made
logical subjects (see § 49 supra); (2) he thinks that, if a term a
occurs in a proposition, the proposition can always be analysed into a
and an assertion about a (see Chapter VII); (3) he is not aware of the
contradiction [the Paradox] discussed in Chapter X.
§ 478.
There are some difficulties in the above theory which it will be well to
discuss. In the first place, it seems doubtful whether the introduction
of truth-values marks any real analysis. If we consider, say,
"Caesar died," it would seem that what is asserted is the
propositional concept "the death of Caesar," not "the truth of
the death of Caesar." This latter seems to be merely another
propositional concept, asserted in "the death of Caesar is true,"
which is not, I think, the same proposition as "Caesar died."
There is great difficulty in avoiding psychological elements here, and it would
seem that Frege has allowed them to intrude in describing judgment as the
recognition of truth (Gg p. x). The difficulty is due to the fact that
there is a psychological sense of assertion, which is what is lacking to
Meinong's Annahmen, and that this does not run parallel with the logical
sense. Psychologically, any proposition, whether true or false, may be
merely thought of, or may be actually asserted: but for this possibility, error
would be impossible. But logically, true propositions only are asserted,
though they may occur in an unasserted form as parts of other
propositions. In "p implies q," either or both of
the propositions p, q may be true, yet each, in this proposition,
is unasserted in a logical, and not merely in a psychological sense. Thus
assertion has a definite place among logical notions, though there is a
psychological notion of assertion to which nothing logical corresponds.
But assertion does not seem to be a constituent of an asserted proposition,
although it is, in some sense, contained in an asserted proposition. If p
is a proposition, "p's truth" is a concept which has being
even if p is false, and thus "p's truth" is not the
same as p asserted. Thus no concept can be found which is
equivalent to p asserted, and therefore assertion is not a constituent
in p asserted. Yet assertion is not a term to which p, when
asserted, has an external relation; for any such relation would need to be
itself asserted in order to yield what we want. Also a difficulty arises
owing to the apparent fact, which may however be doubted, that an asserted
proposition can never be part of another proposition: thus, if this be a fact,
where any statement is made about p asserted, it is not really about p
asserted, but only about the assertion of p. This difficulty
becomes serious in the case of Frege's one and only principle of inference (Bs
p.9): "p is true and p implies q; therefore q
is true." Here it is quite essential that there should be three
actual assertions, otherwise the assertion of propositions deduced from
asserted premisses would be impossible; yet the three assertions together form
one proposition, whose unity is shown by the word therefore, without
which q would not have been deduced, but would have been asserted as a fresh
premiss.
It is also
impossible, at least to me, to divorce assertion from truth, as Frege
does. An asserted proposition, it would seem, must be the same as a true
proposition. We may allow that negation belongs to the content of a
proposition (Bs p.4), and regard every assertion as asserting something to be
true. We shall then correlate p and not-p as unasserted
propositions, and regard "p is false" as meaning "not-p
is true." But to divorce assertion from truth seems only possible by
taking assertion in a psychological sense.
§ 478.
(Classes)
the extension of a Begriff, Frege says, is the range (Werthvelauf)
of a function whose value for every argument is a truth-value (FuB.
p.16). Ranges are things, whereas functions are not (p.19). There
would be no null-class, if classes were taken in extension; for the null-class
is only possible if a class is not a collection of terms (KB. pp 436-7).
If x be a term, we cannot identify x, as the extensional view
requires, with the class whose only member is x; for suppose x to
be a class having more than one member, and let y, z be two
different members of x; then if x is identical with the class
whose only member is x, y and z will both be members of
this class, and will therefore be identical with x and with each other,
contrary to the hypothesis (p. 444). The extension of a Begriff
has its being in the Begriff itself, not in the individuals falling
under the Begriff (p. 451). When I say something about all men, I
say nothing about some wretch in the centre of Africa, who is in no way
indicated* (Bedeutet?) and does not belong to the indication (Bedeutung?) of man
(p. 454). Begriffe are prior to their extension, and it is a
mistake to attempt, as Schroeder does, to base extension on individuals; this
leads to the calculus of regions (Gebiete) not to Logic (p. 455).
* Russell
translates Bedeutung as indication.
Bs =
Begriffschift
FT = Uber
formale Theorien der Arithmetik
FuB = Function
und Begriff
BuG Uber
Begriff und Gegenstand
SuB Sinn und
Bedeutung
KB = "A
critical elucidation of some points in E. Schroeder"s Vorlesungen Ueber
Die Algebra der Logik", Archiv fur systematische Philosophie 1895, pp
433-456.
Gl Grundlagen
BP "Ueber
die Begriffschrift des Herrn Peano und meine eigene" (Berichte der math.-physiscen
Classe der Koenigl. Saechse Gesellschaft der Wissenschaften zu Leipzig (1896)
Gg Grundgesetze
References Grattan-Guinness, I. The Search for Mathematical Roots, 1870-1940, Princeton 2000.
Monk, R.., Bertrand Russell: The Spirit of Solitude,
London 1996
Shearman, A.T. Review of Russell's Principle's of Mathematics. Mind ns 16, pp. 254-65. THE
LOGIC MUSEUM Copyright © E.D.Buckner
2005.