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% The Project Gutenberg EBook of Introduction to Mathematical Philosophy, by 
% Bertrand Russell                                                        %
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% This eBook is for the use of anyone anywhere at no cost and with        %
% almost no restrictions whatsoever.  You may copy it, give it away or    %
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% Title: Introduction to Mathematical Philosophy                          %
%                                                                         %
% Author: Bertrand Russell                                                %
%                                                                         %
% Release Date: February 17, 2013 [EBook #41654]                          %
%                                                                         %
% Language: English                                                       %
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% Character set encoding: ISO-8859-1                                      %
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% *** START OF THIS PROJECT GUTENBERG EBOOK MATHEMATICAL PHILOSOPHY ***   %
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The Project Gutenberg EBook of Introduction to Mathematical Philosophy, by 
Bertrand Russell

This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever.  You may copy it, give it away or
re-use it under the terms of the Project Gutenberg License included
with this eBook or online at www.gutenberg.org


Title: Introduction to Mathematical Philosophy

Author: Bertrand Russell

Release Date: February 17, 2013 [EBook #41654]

Language: English

Character set encoding: ISO-8859-1

*** START OF THIS PROJECT GUTENBERG EBOOK MATHEMATICAL PHILOSOPHY ***
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Produced by Andrew D. Hwang. (This ebook was produced using
OCR text generously provided by the University of Toronto
through the Internet Archive.)
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%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%
\PageSep{i}%
\FrontMatter
%[** TN: Publisher's front matter]
\iffalse
Library of Philosophy

EDITED BY J. H. MUIRHEAD, LL.D.

INTRODUCTION TO MATHEMATICAL
PHILOSOPHY
\PageSep{ii}%
\clearpage

By the same Author.

PRINCIPLES OF SOCIAL RECONSTRUCTION.
\textit{3rd Impression.} Demy 8vo. 7s. 6d.
net.

``Mr Russell has written a big and living book.''---\Title{The
Nation}.

ROADS TO FREEDOM: SOCIALISM,
ANARCHISM, AND SYNDICALISM. Demy 8vo.
7s. 6d. net.

An attempt to extract the essence of these three doctrines,
first historically, then as guidance for the coming reconstruction.

\textit{London: George Allen \&amp; Unwin, Ltd.}
\fi
\PageSep{iii}%
\clearpage
\begin{center}
\bfseries\LARGE INTRODUCTION TO \\
\Huge MATHEMATICAL \\
PHILOSOPHY

\normalfont
\vfill
\footnotesize BY \\
\normalsize BERTRAND RUSSELL
\vfill
%[** Publisher's device
\vfill

LONDON: GEORGE ALLEN \&amp; UNWIN, LTD. \\
NEW YORK: THE MACMILLAN CO.
\end{center}
\PageSep{iv}
\clearpage
\begin{center}
\null\vfill
\footnotesize \textit{First published May} 1919 \\
\textit{Second Edition April} 1920
\vfill

[\textit{All rights reserved}]
\end{center}
\PageSep{v}


\Section{Preface}

\First{This} book is intended essentially as an ``Introduction,'' and
\index{Philosophy, mathematical}%
does not aim at giving an exhaustive discussion of the problems
with which it deals. It seemed desirable to set forth certain
results, hitherto only available to those who have mastered
logical symbolism, in a form offering the minimum of difficulty
to the beginner. The utmost endeavour has been made to
avoid dogmatism on such questions as are still open to serious
doubt, and this endeavour has to some extent dominated the
choice of topics considered. The beginnings of mathematical
logic are less definitely known than its later portions, but are of
\index{Logic!mathematical}%
at least equal philosophical interest. Much of what is set forth
in the following chapters is not properly to be called ``philosophy,''
though the matters concerned were included in philosophy so
long as no satisfactory science of them existed. The nature of
infinity and continuity, for example, belonged in former days
to philosophy, but belongs now to mathematics. Mathematical
\emph{philosophy}, in the strict sense, cannot, perhaps, be held to include
such definite scientific results as have been obtained in this
region; the philosophy of mathematics will naturally be expected
to deal with questions on the frontier of knowledge, as
to which comparative certainty is not yet attained. But
speculation on such questions is hardly likely to be fruitful
unless the more scientific parts of the principles of mathematics
are known. A book dealing with those parts may, therefore,
claim to be an \emph{introduction} to mathematical philosophy, though
it can hardly claim, except where it steps outside its province,
to be actually dealing with a part of philosophy. It does deal,
\PageSep{vi}
however, with a body of knowledge which, to those who accept
it, appears to invalidate much traditional philosophy, and even
a good deal of what is current in the present day. In this way,
as well as by its bearing on still unsolved problems, mathematical
logic is relevant to philosophy. For this reason, as well as on
account of the intrinsic importance of the subject, some purpose
may be served by a succinct account of the main results of
mathematical logic in a form requiring neither a knowledge of
mathematics nor an aptitude for mathematical symbolism.
Here, however, as elsewhere, the method is more important than
\index{Method}%
the results, from the point of view of further research; and the
method cannot well be explained within the framework of such
a book as the following. It is to be hoped that some readers
may be sufficiently interested to advance to a study of the
method by which mathematical logic can be made helpful in
investigating the traditional problems of philosophy. But that
is a topic with which the following pages have not attempted
to deal.

\Signature{Bertrand Russell.}
\PageSep{vii}


\Section{Editor's Note}

\First{Those} who, relying on the distinction between Mathematical
Philosophy and the Philosophy of Mathematics, think that this
book is out of place in the present Library, may be referred to
what the author himself says on this head in the Preface. It is
not necessary to agree with what he there suggests as to the
readjustment of the field of philosophy by the transference from
it to mathematics of such problems as those of class, continuity,
infinity, in order to perceive the bearing of the definitions and
discussions that follow on the work of ``traditional philosophy.''
If philosophers cannot consent to relegate the criticism of these
categories to any of the special sciences, it is essential, at any
rate, that they should know the precise meaning that the science
of mathematics, in which these concepts play so large a part,
assigns to them. If, on the other hand, there be mathematicians
to whom these definitions and discussions seem to be an elaboration
and complication of the simple, it may be well to remind
them from the side of philosophy that here, as elsewhere, apparent
simplicity may conceal a complexity which it is the business of
somebody, whether philosopher or mathematician, or, like the
author of this volume, both in one, to unravel.
\PageSep{viii}


\Section{Contents}

% CHAP. PAGE

\ToCLine{}{Preface}{v}

\ToCLine{}{Editor's Note}{vii}

\ToCLine{1.}{The Series of Natural Numbers}{1}

\ToCLine{2.}{Definition of Number}{11}

\ToCLine{3.}{Finitude and Mathematical Induction}{20}

\ToCLine{4.}{The Definition of Order}{29}

\ToCLine{5.}{Kinds of Relations}{42}

\ToCLine{6.}{Similarity of Relations}{52}

\ToCLine{7.}{Rational, Real, and Complex Numbers}{63}

\ToCLine{8.}{Infinite Cardinal Numbers}{77}

\ToCLine{9.}{Infinite Series and Ordinals}{89}

\ToCLine{10.}{Limits and Continuity}{97}

\ToCLine{11.}{Limits and Continuity of Functions}{107}

\ToCLine{12.}{Selections and the Multiplicative Axiom}{117}

\ToCLine{13.}{The Axiom of Infinity and Logical Types}{131}

\ToCLine{14.}{Incompatibility and the Theory of Deduction}{144}

\ToCLine{15.}{Propositional Functions}{155}

\ToCLine{16.}{Descriptions}{167}

\ToCLine{17.}{Classes}{181}

\ToCLine{18.}{Mathematics and Logic}{194}

\ToCLine{}{Index}{207}
\PageSep{1}
\MainMatter

%[** TN: Text printed by \Chapter macro]
% Introduction to
% Mathematical Philosophy

\Chapter{I}{The Series of Natural Numbers}

\First{Mathematics} is a study which, when we start from its most
\index{Philosophy, mathematical}%
familiar portions, may be pursued in either of two opposite
directions. The more familiar direction is constructive, towards
gradually increasing complexity: from integers to fractions,
real numbers, complex numbers; from addition and multiplication
to differentiation and integration, and on to higher
mathematics. The other direction, which is less familiar,
proceeds, by analysing, to greater and greater abstractness
and logical simplicity; instead of asking what can be defined
and deduced from what is assumed to begin with, we ask instead
what more general ideas and principles can be found, in terms
of which what was our starting-point can be defined or deduced.
It is the fact of pursuing this opposite direction that characterises
mathematical philosophy as opposed to ordinary mathematics.
But it should be understood that the distinction is one, not in
the subject matter, but in the state of mind of the investigator.
Early Greek geometers, passing from the empirical rules of
Egyptian land-surveying to the general propositions by which
those rules were found to be justifiable, and thence to Euclid's
axioms and postulates, were engaged in mathematical philosophy,
\index{Axioms}%
according to the above definition; but when once the
axioms and postulates had been reached, their deductive employment,
as we find it in Euclid, belonged to mathematics in the
\PageSep{2}%
ordinary sense. The distinction between mathematics and
mathematical philosophy is one which depends upon the interest
inspiring the research, and upon the stage which the research
has reached; not upon the propositions with which the research
is concerned.

We may state the same distinction in another way. The
most obvious and easy things in mathematics are not those that
come logically at the beginning; they are things that, from
the point of view of logical deduction, come somewhere in the
middle. Just as the easiest bodies to see are those that are
neither very near nor very far, neither very small nor very
great, so the easiest conceptions to grasp are those that are
neither very complex nor very simple (using ``simple'' in a
\emph{logical} sense). And as we need two sorts of instruments, the
telescope and the microscope, for the enlargement of our visual
powers, so we need two sorts of instruments for the enlargement
of our logical powers, one to take us forward to the higher
mathematics, the other to take us backward to the logical
foundations of the things that we are inclined to take for granted
in mathematics. We shall find that by analysing our ordinary
mathematical notions we acquire fresh insight, new powers,
and the means of reaching whole new mathematical subjects
by adopting fresh lines of advance after our backward journey.
It is the purpose of this book to explain mathematical philosophy
simply and untechnically, without enlarging upon those
portions which are so doubtful or difficult that an elementary
treatment is scarcely possible. A full treatment will be found
in \Title{Principia Mathematica};\footnote
  {Cambridge University Press, vol.~i., 1910; vol.~ii., 1911; vol.~iii., 1913.
  By Whitehead and Russell.}
the treatment in the present volume
is intended merely as an introduction.

To the average educated person of the present day, the
\index{Number!natural|EtSeq}%
obvious starting-point of mathematics would be the series of
whole numbers,
\[
1,\ 2,\ 3,\ 4,\ \dots\ \text{etc.}
\]
\PageSep{3}%
Probably only a person with some mathematical knowledge
would think of beginning with $0$ instead of with~$1$, but we will
presume this degree of knowledge; we will take as our starting-point
the series:
\[
0,\ 1,\ 2,\ 3,\ \dots\ n,\ n + 1,\dots
\]
and it is this series that we shall mean when we speak of the
``series of natural numbers.''

It is only at a high stage of civilisation that we could take
this series as our starting-point. It must have required many
ages to discover that a brace of pheasants and a couple of days
were both instances of the number~$2$: the degree of abstraction
involved is far from easy. And the discovery that $1$~is a number
must have been difficult. As for~$0$, it is a very recent addition;
the Greeks and Romans had no such digit. If we had been
embarking upon mathematical philosophy in earlier days, we
should have had to start with something less abstract than the
series of natural numbers, which we should reach as a stage on
our backward journey. When the logical foundations of mathematics
have grown more familiar, we shall be able to start further
back, at what is now a late stage in our analysis. But for the
moment the natural numbers seem to represent what is easiest
and most familiar in mathematics.

But though familiar, they are not understood. Very few
people are prepared with a definition of what is meant by
\index{Definition}%
``number,'' or~``$0$,'' or~``$1$.'' It is not very difficult to see that,
starting from~$0$, any other of the natural numbers can be reached
by repeated additions of~$1$, but we shall have to define what
we mean by ``adding~$1$,'' and what we mean by ``repeated.''
These questions are by no means easy. It was believed until
recently that some, at least, of these first notions of arithmetic
must be accepted as too simple and primitive to be defined.
Since all terms that are defined are defined by means of other
terms, it is clear that human knowledge must always be content
to accept some terms as intelligible without definition, in order
\PageSep{4}%
to have a starting-point for its definitions. It is not clear that
there must be terms which are \emph{incapable} of definition: it is
possible that, however far back we go in defining, we always
\emph{might} go further still. On the other hand, it is also possible
that, when analysis has been pushed far enough, we can reach
\index{Analysis}%
terms that really are simple, and therefore logically incapable
of the sort of definition that consists in analysing. This is a
question which it is not necessary for us to decide; for our
purposes it is sufficient to observe that, since human powers
are finite, the definitions known to us must always begin somewhere,
with terms undefined for the moment, though perhaps
not permanently.

All traditional pure mathematics, including analytical geometry,
\index{Arithmetising of mathematics}%
\index{Geometry!analytical}%
may be regarded as consisting wholly of propositions
about the natural numbers. That is to say, the terms which
occur can be defined by means of the natural numbers, and
the propositions can be deduced from the properties of the
natural numbers---with the addition, in each case, of the ideas
and propositions of pure logic.

That all traditional pure mathematics can be derived from
the natural numbers is a fairly recent discovery, though it had
long been suspected. Pythagoras, who believed that not only
\index{Pythagoras}%
mathematics, but everything else could be deduced from
numbers, was the discoverer of the most serious obstacle in
the way of what is called the ``arithmetising'' of mathematics.
It was Pythagoras who discovered the existence of incommensurables,
\index{Incommensurables}%
and, in particular, the incommensurability of the
side of a square and the diagonal. If the length of the side is
$1$~inch, the number of inches in the diagonal is the square root
of~$2$, which appeared not to be a number at all. The problem
thus raised was solved only in our own day, and was only solved
\emph{completely} by the help of the reduction of arithmetic to logic,
which will be explained in following chapters. For the present,
we shall take for granted the arithmetisation of mathematics,
though this was a feat of the very greatest importance.
\PageSep{5}%

Having reduced all traditional pure mathematics to the
\index{Peano|EtSeq}%
\index{Premisses of arithmetic}%
\index{Primitive ideas and propositions}%
theory of the natural numbers, the next step in logical analysis
was to reduce this theory itself to the smallest set of premisses
and undefined terms from which it could be derived. This work
was accomplished by Peano. He showed that the entire theory
of the natural numbers could be derived from three primitive
ideas and five primitive propositions in addition to those of
pure logic. These three ideas and five propositions thus became,
as it were, hostages for the whole of traditional pure mathematics.
If they could be defined and proved in terms of others,
so could all pure mathematics. Their logical ``weight,'' if one
may use such an expression, is equal to that of the whole series
of sciences that have been deduced from the theory of the natural
numbers; the truth of this whole series is assured if the truth
of the five primitive propositions is guaranteed, provided, of
course, that there is nothing erroneous in the purely logical
apparatus which is also involved. The work of analysing mathematics
is extraordinarily facilitated by this work of Peano's.

The three primitive ideas in Peano's arithmetic are:
\[
\text{$0$, number, successor.}
\]
By ``successor'' he means the next number in the natural
order. That is to say, the successor of $0$ is~$1$, the successor of
$1$ is~$2$, and so on. By ``number'' he means, in this connection,
the class of the natural numbers.\footnote
  {We shall use ``number'' in this sense in the present chapter. Afterwards
  the word will be used in a more general sense.}
He is not assuming that
we know all the members of this class, but only that we know
what we mean when we say that this or that is a number, just
as we know what we mean when we say ``Jones is a man,''
though we do not know all men individually.

The five primitive propositions which Peano assumes are:
\begin{List}
\item[(1)] $0$ is a number.

\item[(2)] The successor of any number is a number.

\item[(3)] No two numbers have the same successor.
\PageSep{6}%

\item[(4)] $0$ is not the successor of any number.

\item[(5)] Any property which belongs to~$0$, and also to the successor
of every number which has the property, belongs to all
numbers.
\end{List}
The last of these is the principle of mathematical induction.
We shall have much to say concerning mathematical induction
in the sequel; for the present, we are concerned with it only
as it occurs in Peano's analysis of arithmetic.

Let us consider briefly the kind of way in which the theory
of the natural numbers results from these three ideas and five
propositions. To begin with, we define $1$~as ``the successor of~$0$,''
$2$~as ``the successor of~$1$,'' and so on. We can obviously go
on as long as we like with these definitions, since, in virtue of~(2),
every number that we reach will have a successor, and, in
virtue of~(3), this cannot be any of the numbers already defined,
because, if it were, two different numbers would have the same
successor; and in virtue of~(4) none of the numbers we reach
in the series of successors can be~$0$. Thus the series of successors
gives us an endless series of continually new numbers. In virtue
of~(5) all numbers come in this series, which begins with~$0$ and
travels on through successive successors: for (\ia)~$0$~belongs to
this series, and (\ib)~if a number~$n$ belongs to it, so does its successor,
whence, by mathematical induction, every number belongs to
the series.

Suppose we wish to define the sum of two numbers. Taking
any number~$m$, we define $m + 0$ as~$m$, and $m + (n + 1)$ as the
successor of~$m + n$. In virtue of~(5) this gives a definition of
the sum of $m$ and~$n$, whatever number $n$~may be. Similarly
we can define the product of any two numbers. The reader can
easily convince himself that any ordinary elementary proposition
of arithmetic can be proved by means of our five premisses,
and if he has any difficulty he can find the proof in Peano.

It is time now to turn to the considerations which make it
necessary to advance beyond the standpoint of Peano, who
\PageSep{7}%
represents the last perfection of the ``arithmetisation'' of
mathematics, to that of Frege, who first succeeded in ``logicising''
\index{Frege}%
\index{Logicising of mathematics}%
mathematics, \ie\ in reducing to logic the arithmetical notions
which his predecessors had shown to be sufficient for mathematics.
We shall not, in this chapter, actually give Frege's definition of
number and of particular numbers, but we shall give some of the
reasons why Peano's treatment is less final than it appears to be.

In the first place, Peano's three primitive ideas---namely, ``$0$,''
``number,'' and ``successor''---are capable of an infinite number
of different interpretations, all of which will satisfy the five
primitive propositions. We will give some examples.

(1) Let ``$0$'' be taken to mean~$100$, and let ``number'' be
taken to mean the numbers from $100$~onward in the series of
natural numbers. Then all our primitive propositions are
satisfied, even the fourth, for, though $100$~is the successor of~$99$,
$99$~is not a ``number'' in the sense which we are now giving
to the word ``number.'' It is obvious that any number may be
substituted for~$100$ in this example.

(2) Let ``$0$'' have its usual meaning, but let ``number''
mean what we usually call ``even numbers,'' and let the
``successor'' of a number be what results from adding two to
it. Then ``$1$''~will stand for the number two, ``$2$''~will stand
for the number four, and so on; the series of ``numbers'' now
will be
\[
\text{$0$, two, four, six, eight \dots\Add{.}}
\]
All Peano's five premisses are satisfied still.

(3) Let ``$0$'' mean the number one, let ``number'' mean
the set
\[
1,\ \tfrac{1}{2},\ \tfrac{1}{4},\ \tfrac{1}{8},\ \tfrac{1}{16},\ \dots
\]
and let ``successor'' mean ``half.'' Then all Peano's five
axioms will be true of this set.

It is clear that such examples might be multiplied indefinitely.
In fact, given any series
\[
x_{0},\ x_{1},\ x_{2},\ x_{3},\ \dots\ x_{n},\ \dots
\]
\PageSep{8}%
which is endless, contains no repetitions, has a beginning, and
has no terms that cannot be reached from the beginning in a
finite number of steps, we have a set of terms verifying Peano's
axioms. This is easily seen, though the formal proof is somewhat
long. Let ``$0$'' mean~$x_{0}$, let ``number'' mean the whole
set of terms, and let the ``successor'' of~$x_{n}$ mean~$x_{n+1}$. Then

(1) ``$0$~is a number,'' \ie\ $x_{0}$~is a member of the set.

(2) ``The successor of any number is a number,'' \ie\ taking
any term~$x_{n}$ in the set, $x_{n+1}$~is also in the set.

(3) ``No two numbers have the same successor,'' \ie\ if $x_{m}$
and $x_{n}$ are two different members of the set, $x_{m+1}$ and $x_{n+1}$ are
different; this results from the fact that (by hypothesis) there
are no repetitions in the set.

(4) ``$0$ is not the successor of any number,'' \ie\ no term in
the set comes before~$x_{0}$.

(5) This becomes: Any property which belongs to~$x_{0}$, and
belongs to~$x_{n+1}$ provided it belongs to~$x_{n}$, belongs to all the~$x$'s.

This follows from the corresponding property for numbers.

A series of the form
\[
x_{0},\ x_{1},\ x_{2},\ \dots\ x_{n},\ \dots
\]
in which there is a first term, a successor to each term (so that
there is no last term), no repetitions, and every term can be
reached from the start in a finite number of steps, is called a
\emph{progression}. Progressions are of great importance in the principles
\index{Progressions}%
of mathematics. As we have just seen, every progression
verifies Peano's five axioms. It can be proved, conversely,
that every series which verifies Peano's five axioms is a progression.
Hence these five axioms may be used to define the
class of progressions: ``progressions'' are ``those series which
verify these five axioms.'' Any progression may be taken as
the basis of pure mathematics: we may give the name~``$0$''
to its first term, the name ``number'' to the whole set of its
terms, and the name ``successor'' to the next in the progression.
The progression need not be composed of numbers: it may be
\PageSep{9}%
composed of points in space, or moments of time, or any other
terms of which there is an infinite supply. Each different
progression will give rise to a different interpretation of all the
propositions of traditional pure mathematics; all these possible
interpretations will be equally true.

In Peano's system there is nothing to enable us to distinguish
between these different interpretations of his primitive ideas.
It is assumed that we know what is meant by~``$0$,'' and that
we shall not suppose that this symbol means $100$ or Cleopatra's
Needle or any of the other things that it might mean.

This point, that ``$0$'' and ``number'' and ``successor''
cannot be defined by means of Peano's five axioms, but must
be independently understood, is important. We want our
numbers not merely to verify mathematical formulæ, but to
apply in the right way to common objects. We want to have
ten fingers and two eyes and one nose. A system in which ``$1$''
meant~$100$, and ``$2$''~meant~$101$, and so on, might be all right
for pure mathematics, but would not suit daily life. We want
``$0$'' and ``number'' and ``successor'' to have meanings which
will give us the right allowance of fingers and eyes and noses.
We have already some knowledge (though not sufficiently
articulate or analytic) of what we mean by ``$1$'' and ``$2$'' and
so on, and our use of numbers in arithmetic must conform to
this knowledge. We cannot secure that this shall be the case
by Peano's method; all that we can do, if we adopt his method,
is to say ``we know what we mean by `$0$' and `number' and
`successor,' though we cannot explain what we mean in terms
of other simpler concepts.'' It is quite legitimate to say this
when we must, and at \emph{some} point we all must; but it is the
object of mathematical philosophy to put off saying it as long
as possible. By the logical theory of arithmetic we are able to
put it off for a very long time.

It might be suggested that, instead of setting up ``$0$'' and
``number'' and ``successor'' as terms of which we know the
meaning although we cannot define them, we might let them
\PageSep{10}%
stand for \emph{any} three terms that verify Peano's five axioms. They
will then no longer be terms which have a meaning that is definite
though undefined: they will be ``variables,'' terms concerning
\index{Variables}%
which we make certain hypotheses, namely, those stated in the
five axioms, but which are otherwise undetermined. If we adopt
this plan, our theorems will not be proved concerning an ascertained
set of terms called ``the natural numbers,'' but concerning
all sets of terms having certain properties. Such a procedure
is not fallacious; indeed for certain purposes it represents a
valuable generalisation. But from two points of view it fails
to give an adequate basis for arithmetic. In the first place, it
does not enable us to know whether there are any sets of terms
verifying Peano's axioms; it does not even give the faintest
suggestion of any way of discovering whether there are such sets.
In the second place, as already observed, we want our numbers
to be such as can be used for counting common objects, and this
requires that our numbers should have a \emph{definite} meaning, not
merely that they should have certain formal properties. This
definite meaning is defined by the logical theory of arithmetic.
\PageSep{11}%


\Chapter{II}{Definition of Number}

\First{The} question ``What is a number?'' is one which has been
%[** TN: Original entries point to "page 10"]
\index{Frege}%
\index{Number!cardinal|EtSeq}%
\index{Plurality}%
often asked, but has only been correctly answered in our own
time. The answer was given by Frege in~1884, in his \Title{Grundlagen
der Arithmetik}.\footnote
  {The same answer is given more fully and with more development in
  his \Title{Grundgesetze der Arithmetik}, vol.~i., 1893.}
Although this book is quite short, not difficult,
and of the very highest importance, it attracted almost no
attention, and the definition of number which it contains remained
practically unknown until it was rediscovered by the
present author in~1901.

In seeking a definition of number, the first thing to be clear
about is what we may call the grammar of our inquiry. Many
philosophers, when attempting to define number, are really
setting to work to define plurality, which is quite a different
thing. \emph{Number} is what is characteristic of numbers, as \emph{man}
is what is characteristic of men. A plurality is not an instance
of number, but of some particular number. A trio of men,
for example, is an instance of the number~$3$, and the number~$3$
is an instance of number; but the trio is not an instance of
number. This point may seem elementary and scarcely worth
mentioning; yet it has proved too subtle for the philosophers,
with few exceptions.

A particular number is not identical with any collection of
terms having that number: the number~$3$ is not identical with
\PageSep{12}%
the trio consisting of Brown, Jones, and Robinson. The number~$3$
is something which all trios have in common, and which distinguishes
them from other collections. A number is something
that characterises certain collections, namely, those that have
that number.

Instead of speaking of a ``collection,'' we shall as a rule speak
\index{Aggregates}%
of a ``class,'' or sometimes a ``set.'' Other words used in
mathematics for the same thing are ``aggregate'' and ``manifold.''
We shall have much to say later on about classes. For
\index{Classes}%
the present, we will say as little as possible. But there are
some remarks that must be made immediately.

A class or collection may be defined in two ways that at first
sight seem quite distinct. We may enumerate its members, as
when we say, ``The collection I mean is Brown, Jones, and
Robinson.'' Or we may mention a defining property, as when
we speak of ``mankind'' or ``the inhabitants of London.'' The
definition which enumerates is called a definition by ``extension,''
\index{Definition!extensional and intensional}%
and the one which mentions a defining property is called
a definition by ``intension.'' Of these two kinds of definition,
the one by intension is logically more fundamental. This is
shown by two considerations: (1)~that the extensional definition
can always be reduced to an intensional one; (2)~that the
intensional one often cannot even theoretically be reduced to
the extensional one. Each of these points needs a word of
explanation.

(1) Brown, Jones, and Robinson all of them possess a certain
property which is possessed by nothing else in the whole universe,
namely, the property of being either Brown or Jones or Robinson.
This property can be used to give a definition by intension of
the class consisting of Brown and Jones and Robinson. Consider
such a formula as ``$x$~is Brown or $x$~is Jones or $x$~is Robinson.''
This formula will be true for just three~$x$'s, namely, Brown and
Jones and Robinson. In this respect it resembles a cubic equation
with its three roots. It may be taken as assigning a property
common to the members of the class consisting of these three
\PageSep{13}%
men, and peculiar to them. A similar treatment can obviously
be applied to any other class given in extension.

(2) It is obvious that in practice we can often know a great
\index{Collections, infinite}%
deal about a class without being able to enumerate its members.
No one man could actually enumerate all men, or even all the
inhabitants of London, yet a great deal is known about each of
these classes. This is enough to show that definition by extension
is not \emph{necessary} to knowledge about a class. But when we come
to consider infinite classes, we find that enumeration is not even
theoretically possible for beings who only live for a finite time.
We cannot enumerate all the natural numbers: they are $0$,~$1$, $2$,
$3$, \emph{and so on}. At some point we must content ourselves with
``and so on.'' We cannot enumerate all fractions or all irrational
numbers, or all of any other infinite collection. Thus our knowledge
in regard to all such collections can only be derived from a
definition by intension.

These remarks are relevant, when we are seeking the definition
of number, in three different ways. In the first place, numbers
themselves form an infinite collection, and cannot therefore
be defined by enumeration. In the second place, the collections
having a given number of terms themselves presumably form an
infinite collection: it is to be presumed, for example, that there
are an infinite collection of trios in the world, for if this were
not the case the total number of things in the world would be
finite, which, though possible, seems unlikely. In the third
place, we wish to define ``number'' in such a way that infinite
numbers may be possible; thus we must be able to speak of
the number of terms in an infinite collection, and such a collection
must be defined by intension, \ie\ by a property common to all
its members and peculiar to them.

For many purposes, a class and a defining characteristic of
it are practically interchangeable. The vital difference between
the two consists in the fact that there is only one class having a
given set of members, whereas there are always many different
characteristics by which a given class may be defined. Men
\PageSep{14}%
may be defined as featherless bipeds, or as rational animals,
or (more correctly) by the traits by which Swift delineates the
Yahoos. It is this fact that a defining characteristic is never
unique which makes classes useful; otherwise we could be
content with the properties common and peculiar to their
members.\footnote
  {As will be explained later, classes may be regarded as logical fictions,
\index{Fictions, logical|FNote}%
  manufactured out of defining characteristics. But for the present it will
  simplify our exposition to treat classes as if they were real.}
Any one of these properties can be used in place
of the class whenever uniqueness is not important.

Returning now to the definition of number, it is clear that
number is a way of bringing together certain collections, namely,
those that have a given number of terms. We can suppose
all couples in one bundle, all trios in another, and so on. In
this way we obtain various bundles of collections, each bundle
consisting of all the collections that have a certain number of
terms. Each bundle is a class whose members are collections,
\ie\ classes; thus each is a class of classes. The bundle consisting
of all couples, for example, is a class of classes: each
couple is a class with two members, and the whole bundle of
couples is a class with an infinite number of members, each of
which is a class of two members.

How shall we decide whether two collections are to belong
to the same bundle? The answer that suggests itself is: ``Find
out how many members each has, and put them in the same
bundle if they have the same number of members.'' But this
presupposes that we have defined numbers, and that we know
how to discover how many terms a collection has. We are so
used to the operation of counting that such a presupposition
\index{Counting}%
might easily pass unnoticed. In fact, however, counting,
though familiar, is logically a very complex operation; moreover
it is only available, as a means of discovering how many
terms a collection has, when the collection is finite. Our definition
of number must not assume in advance that all numbers
are finite; and we cannot in any case, without a vicious circle,
\PageSep{15}%
use counting to define numbers, because numbers are used in
counting. We need, therefore, some other method of deciding
when two collections have the same number of terms.

In actual fact, it is simpler logically to find out whether two
\index{Similarity!of classes|EtSeq}%
collections have the same number of terms than it is to define
what that number is. An illustration will make this clear.
If there were no polygamy or polyandry anywhere in the world,
it is clear that the number of husbands living at any moment
would be exactly the same as the number of wives. We do
not need a census to assure us of this, nor do we need to know
what is the actual number of husbands and of wives. We know
the number must be the same in both collections, because each
husband has one wife and each wife has one husband. The
relation of husband and wife is what is called ``one-one.''

A relation is said to be ``one-one'' when, if $x$~has the relation
in question to~$y$, no other term~$x'$ has the same relation to~$y$,
and $x$~does not have the same relation to any term~$y'$ other
than~$y$. When only the first of these two conditions is fulfilled,
the relation is called ``one-many''; when only the second is
fulfilled, it is called ``many-one.'' It should be observed that
the number~$1$ is not used in these definitions.

In Christian countries, the relation of husband to wife is
one-one; in Mahometan countries it is one-many; in Tibet
it is many-one. The relation of father to son is one-many;
that of son to father is many-one, but that of eldest son to father
is one-one. If $n$~is any number, the relation of $n$ to~$n + 1$ is
one-one; so is the relation of $n$ to~$2n$ or to~$3n$. When we are
considering only positive numbers, the relation of $n$ to~$n^{2}$ is
one-one; but when negative numbers are admitted, it becomes
two-one, since $n$ and~$-n$ have the same square. These instances
should suffice to make clear the notions of one-one, one-many,
and many-one relations, which play a great part in the principles
\index{Relations!many-one}%
\index{Relations!one-many}%
\index{Relations!one-one}%
of mathematics, not only in relation to the definition of
numbers, but in many other connections.

Two classes are said to be ``similar'' when there is a one-one
\index{Classes!similar}%
\PageSep{16}%
relation which correlates the terms of the one class each with
one term of the other class, in the same manner in which the
relation of marriage correlates husbands with wives. A few
preliminary definitions will help us to state this definition more
precisely. The class of those terms that have a given relation
to something or other is called the \emph{domain} of that relation:
\index{Domain}%
thus fathers are the domain of the relation of father to child,
husbands are the domain of the relation of husband to wife,
wives are the domain of the relation of wife to husband, and
husbands and wives together are the domain of the relation of
marriage. The relation of wife to husband is called the \emph{converse}
\index{Converse}%
of the relation of husband to wife. Similarly \emph{less} is the converse
of~\emph{greater}, \emph{later}~is the converse of \emph{earlier}, and so on. Generally,
the converse of a given relation is that relation which holds
between $y$ and~$x$ whenever the given relation holds between
$x$ and~$y$. The \emph{converse domain} of a relation is the domain of
its converse: thus the class of wives is the converse domain
of the relation of husband to wife. We may now state our
definition of similarity as follows:---

\begin{Definition}[\itshape]
One class is said to be ``similar'' to another when there is a
one-one relation of which the one class is the domain, while the
other is the converse domain.
\end{Definition}

It is easy to prove (1)~that every class is similar to itself, (2)~that
if a class~$\alpha$ is similar to a class~$\beta$, then $\beta$~is similar to~$\alpha$, (3)~that
if $\alpha$~is similar to~$\beta$ and $\beta$~to~$\gamma$, then $\alpha$~is similar to~$\gamma$. A
relation is said to be \emph{reflexive} when it possesses the first of these
properties, \emph{symmetrical} when it possesses the second, and \emph{transitive}
when it possesses the third. It is obvious that a relation
which is symmetrical and transitive must be reflexive throughout
its domain. Relations which possess these properties are an
\index{Relations!reflexive}%
\index{Relations!symmetrical}%
\index{Relations!transitive}%
important kind, and it is worth while to note that similarity is
one of this kind of relations.

It is obvious to common sense that two finite classes have
\index{Classes!similar}%
the same number of terms if they are similar, but not otherwise.
The act of counting consists in establishing a one-one correlation
\index{Counting}%
\PageSep{17}%
between the set of objects counted and the natural numbers
(excluding~$0$) that are used up in the process. Accordingly
common sense concludes that there are as many objects in the
set to be counted as there are numbers up to the last number
used in the counting. And we also know that, so long as we
confine ourselves to finite numbers, there are just $n$~numbers
from $1$ up to~$n$. Hence it follows that the last number used in
counting a collection is the number of terms in the collection,
provided the collection is finite. But this result, besides being
only applicable to finite collections, depends upon and assumes
the fact that two classes which are similar have the same number
of terms; for what we do when we count (say) $10$~objects is to
show that the set of these objects is similar to the set of numbers
$1$ to~$10$. The notion of similarity is logically presupposed in
the operation of counting, and is logically simpler though less
familiar. In counting, it is necessary to take the objects counted
in a certain order, as first, second, third,~etc., but order is not
of the essence of number: it is an irrelevant addition, an unnecessary
complication from the logical point of view. The
notion of similarity does not demand an order: for example,
we saw that the number of husbands is the same as the number
of wives, without having to establish an order of precedence
among them. The notion of similarity also does not require
that the classes which are similar should be finite. Take, for
example, the natural numbers (excluding~$0$) on the one hand,
and the fractions which have $1$ for their numerator on the other
hand: it is obvious that we can correlate $2$ with~$\frac{1}{2}$, $3$~with~$\frac{1}{3}$, and
so on, thus proving that the two classes are similar.

We may thus use the notion of ``similarity'' to decide when
two collections are to belong to the same bundle, in the sense
in which we were asking this question earlier in this chapter.
We want to make one bundle containing the class that has no
members: this will be for the number~$0$. Then we want a bundle
of all the classes that have one member: this will be for the
number~$1$. Then, for the number~$2$, we want a bundle consisting
\PageSep{18}%
of all couples; then one of all trios; and so on. Given any collection,
we can define the bundle it is to belong to as being the class
of all those collections that are ``similar'' to it. It is very easy
to see that if (for example) a collection has three members, the
class of all those collections that are similar to it will be the
class of trios. And whatever number of terms a collection may
have, those collections that are ``similar'' to it will have the same
number of terms. We may take this as a \emph{definition} of ``having
the same number of terms.'' It is obvious that it gives results
conformable to usage so long as we confine ourselves to finite
collections.

So far we have not suggested anything in the slightest degree
paradoxical. But when we come to the actual definition of
numbers we cannot avoid what must at first sight seem a paradox,
though this impression will soon wear off. We naturally think
that the class of couples (for example) is something different
from the number~$2$. But there is no doubt about the class of
couples: it is indubitable and not difficult to define, whereas
the number~$2$, in any other sense, is a metaphysical entity about
which we can never feel sure that it exists or that we have tracked
it down. It is therefore more prudent to content ourselves with
the class of couples, which we are sure of, than to hunt for a
problematical number~$2$ which must always remain elusive.
Accordingly we set up the following definition:---

\begin{Definition}[\itshape]
The number of a class is the class of all those classes that are
similar to it.
\end{Definition}

Thus the number of a couple will be the class of all couples.
In fact, the class of all couples will \emph{be} the number~$2$, according
to our definition. At the expense of a little oddity, this definition
secures definiteness and indubitableness; and it is not difficult
to prove that numbers so defined have all the properties that we
expect numbers to have.

We may now go on to define numbers in general as any one of
the bundles into which similarity collects classes. A number
will be a set of classes such as that any two are similar to each
\PageSep{19}%
other, and none outside the set are similar to any inside the set.
In other words, a number (in general) is any collection which is
the number of one of its members; or, more simply still:

\begin{Definition}[\itshape]
A number is anything which is the number of some class.
\end{Definition}

Such a definition has a verbal appearance of being circular,
but in fact it is not. We define ``the number of a given class''
without using the notion of number in general; therefore we may
define number in general in terms of ``the number of a given
class'' without committing any logical error.

Definitions of this sort are in fact very common. The class
of fathers, for example, would have to be defined by first defining
what it is to be the father of somebody; then the class of fathers
will be all those who are somebody's father. Similarly if we want
to define square numbers (say), we must first define what we
mean by saying that one number is the square of another, and
then define square numbers as those that are the squares of
other numbers. This kind of procedure is very common, and
it is important to realise that it is legitimate and even often
necessary.

We have now given a definition of numbers which will serve
for finite collections. It remains to be seen how it will serve
for infinite collections. But first we must decide what we mean
by ``finite'' and ``infinite,'' which cannot be done within the
limits of the present chapter.
\PageSep{20}%


\Chapter{III}{Finitude and Mathematical Induction}

\First{The} series of natural numbers, as we saw in \ChapRef{I}., can all
\index{Induction, mathematical|EtSeq}%
be defined if we know what we mean by the three terms ``$0$,''
``number,'' and ``successor.'' But we may go a step farther:
\index{Number!finite|EtSeq}%
we can define all the natural numbers if we know what we mean
by ``$0$'' and ``successor.'' It will help us to understand the
difference between finite and infinite to see how this can be done,
and why the method by which it is done cannot be extended
beyond the finite. We will not yet consider how ``$0$'' and ``successor''
are to be defined: we will for the moment assume that
we know what these terms mean, and show how thence all other
natural numbers can be obtained.

It is easy to see that we can reach any assigned number, say
$30,000$. We first define ``$1$'' as ``the successor of~$0$,'' then we
define ``$2$'' as ``the successor of~$1$,'' and so on. In the case of
an assigned number, such as $30,000$, the proof that we can reach
it by proceeding step by step in this fashion may be made, if we
have the patience, by actual experiment: we can go on until
we actually arrive at~$30,000$. But although the method of
experiment is available for each particular natural number, it
is not available for proving the general proposition that \emph{all} such
numbers can be reached in this way, \ie\ by proceeding from~$0$
step by step from each number to its successor. Is there any
other way by which this can be proved?

Let us consider the question the other way round. What are
the numbers that can be reached, given the terms ``$0$'' and
\PageSep{21}%
``successor''? Is there any way by which we can define the
whole class of such numbers? We reach~$1$, as the successor of~$0$;
$2$,~as the successor of~$1$; $3$,~as the successor of~$2$; and so on. It
is this ``and so on'' that we wish to replace by something less
vague and indefinite. We might be tempted to say that ``and
so on'' means that the process of proceeding to the successor
may be repeated \emph{any finite number} of times; but the problem
upon which we are engaged is the problem of defining ``finite
number,'' and therefore we must not use this notion in our definition.
Our definition must not assume that we know what a
finite number is.

The key to our problem lies in \emph{mathematical induction}. It will
be remembered that, in \ChapRef{I}., this was the fifth of the five
primitive propositions which we laid down about the natural
numbers. It stated that any property which belongs to~$0$, and
to the successor of any number which has the property, belongs
to all the natural numbers. This was then presented as a principle,
but we shall now adopt it as a definition. It is not difficult
to see that the terms obeying it are the same as the numbers
that can be reached from~$0$ by successive steps from next to
next, but as the point is important we will set forth the matter
in some detail.

We shall do well to begin with some definitions, which will be
useful in other connections also.

A property is said to be ``hereditary'' in the natural-number
\index{Hereditary properties}%
series if, whenever it belongs to a number~$n$, it also belongs to~$n + 1$,
the successor of~$n$. Similarly a class is said to be ``hereditary''
if, whenever $n$~is a member of the class, so is~$n + 1$. It is
easy to \emph{see}, though we are not yet supposed to know, that to say
a property is hereditary is equivalent to saying that it belongs
to all the natural numbers not less than some one of them, \eg\
it must belong to all that are not less than~$100$, or all that are
less than~$1000$, or it may be that it belongs to all that are not
less than~$0$, \ie\ to all without exception.

A property is said to be ``inductive'' when it is a hereditary
\index{Inductive properties}%
\PageSep{22}%
property which belongs to~$0$. Similarly a class is ``inductive''
when it is a hereditary class of which $0$~is a member.

Given a hereditary class of which $0$~is a member, it follows
that $1$~is a member of it, because a hereditary class contains the
successors of its members, and $1$~is the successor of~$0$. Similarly,
given a hereditary class of which $1$~is a member, it follows that
$2$~is a member of it; and so on. Thus we can prove by a step-by-step
procedure that any assigned natural number, say $30,000$,
\index{Number!natural}%
is a member of every inductive class.

We will define the ``posterity'' of a given natural number
\index{Posterity|EtSeq}%
with respect to the relation ``immediate predecessor'' (which
is the converse of ``successor'') as all those terms that belong
to every hereditary class to which the given number belongs. It
is again easy to \emph{see} that the posterity of a natural number consists
of itself and all greater natural numbers; but this also we
do not yet officially know.

By the above definitions, the posterity of~$0$ will consist of those
terms which belong to every inductive class.

It is now not difficult to make it obvious that the posterity of~$0$
is the same set as those terms that can be reached from~$0$ by
successive steps from next to next. For, in the first place, $0$~belongs
to both these sets (in the sense in which we have defined
our terms); in the second place, if $n$~belongs to both sets, so does~$n + 1$.
It is to be observed that we are dealing here with the
kind of matter that does not admit of precise proof, namely, the
comparison of a relatively vague idea with a relatively precise
one. The notion of ``those terms that can be reached from~$0$
by successive steps from next to next'' is vague, though it \emph{seems}
as if it conveyed a definite meaning; on the other hand, ``the
posterity of~$0$'' is precise and explicit just where the other idea
is hazy. It may be taken as giving what we \emph{meant} to mean
when we spoke of the terms that can be reached from~$0$ by
successive steps.

We now lay down the following definition:---

\begin{Definition}[\itshape]
The ``natural numbers'' are the posterity of~$0$ with respect to the
\PageSep{23}%
relation ``immediate predecessor'' \emph{(which is the converse of
``successor'')}.
\end{Definition}

We have thus arrived at a definition of one of Peano's three
\index{Peano}%
primitive ideas in terms of the other two. As a result of this
definition, two of his primitive propositions---namely, the one
asserting that $0$~is a number and the one asserting mathematical
induction---become unnecessary, since they result from the definition.
The one asserting that the successor of a natural number
\index{Successor of a number}%
is a natural number is only needed in the weakened form ``every
natural number has a successor.''

We can, of course, easily define ``$0$'' and ``successor'' by means
of the definition of number in general which we arrived at in
\ChapRef{II}. The number~$0$ is the number of terms in a class
which has no members, \ie\ in the class which is called the ``null-class.''
\index{Null-class}%
By the general definition of number, the number of terms
in the null-class is the set of all classes similar to the null-class,
\ie\ (as is easily proved) the set consisting of the null-class all
alone, \ie\ the class whose only member is the null-class. (This
is not identical with the null-class: it has one member, namely,
the null-class, whereas the null-class itself has no members. A
class which has one member is never identical with that one
member, as we shall explain when we come to the theory of
classes.) Thus we have the following purely logical definition:---

\begin{Definition}[\itshape]
$0$~is the class whose only member is the null-class.
\end{Definition}

It remains to define ``successor.'' Given any number~$n$, let
$\alpha$ be a class which has $n$~members, and let $x$ be a term which
is not a member of~$\alpha$. Then the class consisting of~$\alpha$ with $x$
added on will have $n + 1$~members. Thus we have the following
definition:---

\begin{Definition}[\itshape]
The successor of the number of terms in the class~$\alpha$ is the number
%[** TN: "x"s not italicized in the original]
of terms in the class consisting of a together with~$x$, where $x$~is any
term not belonging to the class.
\end{Definition}

Certain niceties are required to make this definition perfect,
but they need not concern us.\footnote
  {See \Title{Principia Mathematica}, vol.~ii.\ *~110.}
It will be remembered that we
\PageSep{24}%
have already given (in \ChapRef{II}.)\ a logical definition of the
number of terms in a class, namely, we defined it as the set of all
classes that are similar to the given class.

We have thus reduced Peano's three primitive ideas to ideas
\index{Peano}%
of logic: we have given definitions of them which make them
definite, no longer capable of an infinity of different meanings,
as they were when they were only determinate to the extent of
obeying Peano's five axioms. We have removed them from the
fundamental apparatus of terms that must be merely apprehended,
and have thus increased the deductive articulation of
mathematics.

As regards the five primitive propositions, we have already
succeeded in making two of them demonstrable by our definition
of ``natural number.'' How stands it with the remaining three?
It is very easy to prove that $0$~is not the successor of any number,
and that the successor of any number is a number. But there
is a difficulty about the remaining primitive proposition, namely,
``no two numbers have the same successor.'' The difficulty
does not arise unless the total number of individuals in the
universe is finite; for given two numbers $m$ and~$n$, neither of
which is the total number of individuals in the universe, it is
easy to prove that we cannot have $m + 1 = n + 1$ unless we have
$m = n$. But let us suppose that the total number of individuals
in the universe were (say)~$10$; then there would be no class of
$11$~individuals, and the number~$11$ would be the null-class. So
would the number~$12$. Thus we should have $11 = 12$; therefore
the successor of~$10$ would be the same as the successor of~$11$,
although $10$~would not be the same as~$11$. Thus we should have
two different numbers with the same successor. This failure of
the third axiom cannot arise, however, if the number of individuals
in the world is not finite. We shall return to this topic
at a later stage.\footnote
  {See \ChapRef{XIII}.}

Assuming that the number of individuals in the universe is
not finite, we have now succeeded not only in defining Peano's
\PageSep{25}%
three primitive ideas, but in seeing how to prove his five primitive
propositions, by means of primitive ideas and propositions belonging
to logic. It follows that all pure mathematics, in so far
as it is deducible from the theory of the natural numbers, is only
a prolongation of logic. The extension of this result to those
modern branches of mathematics which are not deducible from
the theory of the natural numbers offers no difficulty of principle,
as we have shown elsewhere.\footnote
  {For geometry, in so far as it is not purely analytical, see \Title{Principles of
  Mathematics}, part~vi.; for rational dynamics, \ibid, part~vii.}

The process of mathematical induction, by means of which
we defined the natural numbers, is capable of generalisation.
We defined the natural numbers as the ``posterity'' of~$0$ with
respect to the relation of a number to its immediate successor.
If we call this relation~$N$, any number~$m$ will have this relation
to~$m + 1$. A property is ``hereditary with respect to~$N$,'' or
simply ``$N$-hereditary,'' if, whenever the property belongs to a
number~$m$, it also belongs to~$m + 1$, \ie\ to the number to which
$m$~has the relation~$N$. And a number~$n$ will be said to belong to
the ``posterity'' of~$m$ with respect to the relation~$N$ if $n$~has
every $N$-hereditary property belonging to~$m$. These definitions
can all be applied to any other relation just as well as to~$N$. Thus
if $R$~is any relation whatever, we can lay down the following
definitions:\footnotemark---
\footnotetext{These definitions, and the generalised theory of induction, are due to
\index{Frege|FNote}%
  Frege, and were published so long ago as~1879 in his \Title{Begriffsschrift}. In
  spite of the great value of this work, I was, I believe, the first person who
  ever read it---more than twenty years after its publication.}

A property is called ``$R$-hereditary'' when, if it belongs to
a term~$x$, and $x$~has the relation~$R$ to~$y$, then it belongs to~$y$.

A class is $R$-hereditary when its defining property is $R$-hereditary.

A term~$x$ is said to be an ``$R$-ancestor'' of the term~$y$ if $y$~has
\index{Ancestors}%
every $R$-hereditary property that $x$~has, provided $x$~is a term
which has the relation~$R$ to something or to which something
has the relation~$R$. (This is only to exclude trivial cases.)
\PageSep{26}%

The ``$R$-posterity'' of~$x$ is all the terms of which $x$ is an $R$-ancestor.

We have framed the above definitions so that if a term is the
ancestor of anything it is its own ancestor and belongs to its own
posterity. This is merely for convenience.

It will be observed that if we take for~$R$ the relation ``parent,''
``ancestor'' and ``posterity'' will have the usual meanings,
except that a person will be included among his own ancestors
and posterity. It is, of course, obvious at once that ``ancestor''
must be capable of definition in terms of ``parent,'' but until
Frege developed his generalised theory of induction, no one could
have defined ``ancestor'' precisely in terms of ``parent.'' A
brief consideration of this point will serve to show the importance
of the theory. A person confronted for the first time with the
problem of defining ``ancestor'' in terms of ``parent'' would
naturally say that $A$~is an ancestor of~$Z$ if, between $A$ and~$Z$,
there are a certain number of people, $B$, $C$,~\dots, of whom
$B$~is a child of~$A$, each is a parent of the next, until the last, who
is a parent of~$Z$. But this definition is not adequate unless we
add that the number of intermediate terms is to be finite. Take,
for example, such a series as the following:---
\[
1,\ -\tfrac{1}{2},\ -\tfrac{1}{4},\ -\tfrac{1}{8},\ \dots\
\tfrac{1}{8},\ \tfrac{1}{4},\ \tfrac{1}{2},\ 1.
\]
Here we have first a series of negative fractions with no end,
and then a series of positive fractions with no beginning. Shall
we say that, in this series, $-\frac{1}{8}$~is an ancestor of~$\frac{1}{8}$? It will be
so according to the beginner's definition suggested above, but
it will not be so according to any definition which will give the
kind of idea that we wish to define. For this purpose, it is
essential that the number of intermediaries should be finite.
But, as we saw, ``finite'' is to be defined by means of mathematical
induction, and it is simpler to define the ancestral relation
generally at once than to define it first only for the case of the
relation of~$n$ to~$n + 1$, and then extend it to other cases. Here,
as constantly elsewhere, generality from the first, though it may
\PageSep{27}%
require more thought at the start, will be found in the long run
to economise thought and increase logical power.

The use of mathematical induction in demonstrations was,
in the past, something of a mystery. There seemed no reasonable
doubt that it was a valid method of proof, but no one quite
knew why it was valid. Some believed it to be really a case
of induction, in the sense in which that word is used in logic.
\index{Poincaré}%
Poincaré\footnote
  {\Title{Science and Method}, chap.~iv.}
considered it to be a principle of the utmost importance,
by means of which an infinite number of syllogisms could be
condensed into one argument. We now know that all such views
are mistaken, and that mathematical induction is a definition,
not a principle. There are some numbers to which it can be
applied, and there are others (as we shall see in \ChapRef{VIII}.)\
to which it cannot be applied. We \emph{define} the ``natural numbers''
as those to which proofs by mathematical induction can be
applied, \ie\ as those that possess all inductive properties. It
follows that such proofs can be applied to the natural numbers,
not in virtue of any mysterious intuition or axiom or principle,
but as a purely verbal proposition. If ``quadrupeds'' are
defined as animals having four legs, it will follow that animals
that have four legs are quadrupeds; and the case of numbers
that obey mathematical induction is exactly similar.

We shall use the phrase ``inductive numbers'' to mean the
\index{Number!inductive}%
same set as we have hitherto spoken of as the ``natural numbers.''
The phrase ``inductive numbers'' is preferable as affording a
reminder that the definition of this set of numbers is obtained
from mathematical induction.

Mathematical induction affords, more than anything else,
the essential characteristic by which the finite is distinguished
\index{Finite}%
from the infinite. The principle of mathematical induction
might be stated popularly in some such form as ``what can be
inferred from next to next can be inferred from first to last.''
This is true when the number of intermediate steps between
first and last is finite, not otherwise. Anyone who has ever
\PageSep{28}%
watched a goods train beginning to move will have noticed how
the impulse is communicated with a jerk from each truck to
the next, until at last even the hindmost truck is in motion.
When the train is very long, it is a very long time before the last
truck moves. If the train were infinitely long, there would be
an infinite succession of jerks, and the time would never come
\index{Infinite}%
when the whole train would be in motion. Nevertheless, if
there were a series of trucks no longer than the series of inductive
numbers (which, as we shall see, is an instance of the smallest
%[** TN: [sic] "infinites"]
of infinites), every truck would begin to move sooner or later
if the engine persevered, though there would always be other
trucks further back which had not yet begun to move. This
image will help to elucidate the argument from next to next,
and its connection with finitude. When we come to infinite
numbers, where arguments from mathematical induction will
be no longer valid, the properties of such numbers will help to
make clear, by contrast, the almost unconscious use that is made
of mathematical induction where finite numbers are concerned.
\PageSep{29}%


\Chapter{IV}{The Definition of Order}

\First{We} have now carried our analysis of the series of natural numbers
\index{Order|EtSeq}%
\index{Series|EtSeq}%
to the point where we have obtained logical definitions of the
members of this series, of the whole class of its members, and
of the relation of a number to its immediate successor. We
must now consider the \emph{serial} character of the natural numbers
in the order $0$,~$1$, $2$, $3$,~\dots\Add{.} We ordinarily think of the numbers
as in this \emph{order}, and it is an essential part of the work
of analysing our data to seek a definition of ``order'' or ``series''
in logical terms.

The notion of order is one which has enormous importance
in mathematics. Not only the integers, but also rational fractions
and all real numbers have an order of magnitude, and
this is essential to most of their mathematical properties. The
order of points on a line is essential to geometry; so is the
\index{Geometry}%
slightly more complicated order of lines through a point in a
plane, or of planes through a line. Dimensions, in geometry,
\index{Dimensions}%
are a development of order. The conception of a \emph{limit}, which
\index{Limit}%
underlies all higher mathematics, is a serial conception. There
are parts of mathematics which do not depend upon the notion
of order, but they are very few in comparison with the parts
in which this notion is involved.

In seeking a definition of order, the first thing to realise is
that no set of terms has just \emph{one} order to the exclusion of others.
A set of terms has all the orders of which it is capable. Sometimes
one order is so much more familiar and natural to our
\PageSep{30}%
thoughts that we are inclined to regard it as \emph{the} order of that
set of terms; but this is a mistake. The natural numbers---or
the ``inductive'' numbers, as we shall also call them---occur
to us most readily in order of magnitude; but they are capable
of an infinite number of other arrangements. We might, for
example, consider first all the odd numbers and then all the
even numbers; or first~$1$, then all the even numbers, then all
the odd multiples of~$3$, then all the multiples of~$5$ but not of
$2$~or~$3$, then all the multiples of~$7$ but not of $2$~or $3$ or~$5$, and so
on through the whole series of primes. When we say that we
``arrange'' the numbers in these various orders, that is an
inaccurate expression: what we really do is to turn our attention
to certain relations between the natural numbers, which themselves
generate such-and-such an arrangement. We can no
more ``arrange'' the natural numbers than we can the starry
heavens; but just as we may notice among the fixed stars
either their order of brightness or their distribution in the sky,
so there are various relations among numbers which may be
observed, and which give rise to various different orders among
numbers, all equally legitimate. And what is true of numbers
is equally true of points on a line or of the moments of time:
one order is more familiar, but others are equally valid. We
might, for example, take first, on a line, all the points that have
integral co-ordinates, then all those that have non-integral
rational co-ordinates, then all those that have algebraic non-rational
co-ordinates, and so on, through any set of complications
we please. The resulting order will be one which the
points of the line certainly have, whether we choose to notice
it or not; the only thing that is arbitrary about the various
orders of a set of terms is our attention, for the terms themselves
have always all the orders of which they are capable.

One important result of this consideration is that we must
not look for the definition of order in the nature of the set of
terms to be ordered, since one set of terms has many orders.
The order lies, not in the \emph{class} of terms, but in a relation among
\PageSep{31}%
the members of the class, in respect of which some appear as
earlier and some as later. The fact that a class may have many
orders is due to the fact that there can be many relations holding
among the members of one single class. What properties must
a relation have in order to give rise to an order?

The essential characteristics of a relation which is to give rise
to order may be discovered by considering that in respect of
such a relation we must be able to say, of any two terms in
the class which is to be ordered, that one ``precedes'' and the
other ``follows.'' Now, in order that we may be able to use
these words in the way in which we should naturally understand
them, we require that the ordering relation should have three
properties:---

(1) If $x$ precedes~$y$, $y$~must not also precede~$x$. This is an
\index{Relations!asymmetrical}%
obvious characteristic of the kind of relations that lead to series.
If $x$~is less than~$y$, $y$~is not also less than~$x$. If $x$~is earlier in
time than~$y$, $y$~is not also earlier than~$x$. If $x$~is to the left of~$y$,
$y$~is not to the left of~$x$. On the other hand, relations which
do not give rise to series often do not have this property. If
$x$~is a brother or sister of~$y$, $y$~is a brother or sister of~$x$. If $x$~is
of the same height as~$y$, $y$~is of the same height as~$x$. If $x$~is of a
different height from~$y$, $y$~is of a different height from~$x$. In
all these cases, when the relation holds between $x$ and~$y$, it also
holds between $y$ and~$x$. But with serial relations such a thing
cannot happen. A relation having this first property is called
\emph{asymmetrical}.

(2) If $x$ precedes~$y$ and $y$ precedes~$z$, $x$~must precede~$z$. This
may be illustrated by the same instances as before: \emph{less}, \emph{earlier},
\emph{left of}. But as instances of relations which do \emph{not} have this
property only two of our previous three instances will serve.
If $x$~is brother or sister of~$y$, and $y$~of~$z$, $x$~may not be brother
or sister of~$z$, since $x$ and~$z$ may be the same person. The same
applies to difference of height, but not to sameness of height,
which has our second property but not our first. The relation
``father,'' on the other hand, has our first property but not
\PageSep{32}%
our second. A relation having our second property is called
\index{Relations!transitive}%
\emph{transitive}.

(3) Given any two terms of the class which is to be ordered,
\index{Relations!connected}%
there must be one which precedes and the other which follows.
For example, of any two integers, or fractions, or real numbers,
one is smaller and the other greater; but of any two complex
numbers this is not true. Of any two moments in time, one
must be earlier than the other; but of events, which may be
simultaneous, this cannot be said. Of two points on a line,
one must be to the left of the other. A relation having this
third property is called \emph{connected}.

When a relation possesses these three properties, it is of the
sort to give rise to an order among the terms between which it
holds; and wherever an order exists, some relation having these
three properties can be found generating it.

Before illustrating this thesis, we will introduce a few
definitions.

(1) A relation is said to be an aliorelative,\footnote
  {This term is due to C.~S. Peirce.}
\index{Peirce|FNote}%
or to \emph{be contained
in} or \emph{imply diversity}, if no term has this relation to itself.
%[** TN: Original entry points to page 87]
\index{Diversity}%
Thus, for example, ``greater,'' ``different in size,'' ``brother,''
``husband,'' ``father'' are aliorelatives; but ``equal,'' ``born
\index{Aliorelatives}%
of the same parents,'' ``dear friend'' are not.

(2) The \emph{square} of a relation is that relation which holds between
\index{Relations!squares of}%
two terms $x$ and~$z$ when there is an intermediate term~$y$ such
that the given relation holds between $x$ and~$y$ and between
$y$ and~$z$. Thus ``paternal grandfather'' is the square of ``father,''
``greater by~$2$'' is the square of ``greater by~$1$,'' and so on.

(3) The \emph{domain} of a relation consists of all those terms that
\index{Domain}%
have the relation to something or other, and the \emph{converse domain}
\index{Converse}%
consists of all those terms to which something or other has the
relation. These words have been already defined, but are
recalled here for the sake of the following definition:---

(4) The \emph{field} of a relation consists of its domain and converse
\index{Field of a relation}%
domain together.
\PageSep{33}%

(5) One relation is said to \emph{contain} or \emph{be implied by} another if
it holds whenever the other holds.

It will be seen that an \emph{asymmetrical} relation is the same thing
as a relation whose square is an aliorelative. It often happens
that a relation is an aliorelative without being asymmetrical,
though an asymmetrical relation is always an aliorelative. For
example, ``spouse'' is an aliorelative, but is symmetrical,
since if $x$~is the spouse of~$y$, $y$~is the spouse of~$x$. But among
\emph{transitive} relations, all aliorelatives are asymmetrical as well
as \Foreign{vice versa}.

From the definitions it will be seen that a \emph{transitive} relation
\index{Ancestors}%
is one which is implied by its square, or, as we also say, ``contains''
its square. Thus ``ancestor'' is transitive, because
an ancestor's ancestor is an ancestor; but ``father'' is not
transitive, because a father's father is not a father. A transitive
aliorelative is one which contains its square and is contained
in diversity; or, what comes to the same thing, one whose
square implies both it and diversity---because, when a relation
is transitive, asymmetry is equivalent to being an aliorelative.

A relation is \emph{connected} when, given any two different terms
of its field, the relation holds between the first and the second
or between the second and the first (not excluding the possibility
that both may happen, though both cannot happen if the relation
is asymmetrical).

It will be seen that the relation ``ancestor,'' for example,
is an aliorelative and transitive, but not connected; it is because
it is not connected that it does not suffice to arrange the human
race in a series.

The relation ``less than or equal to,'' among numbers, is
transitive and connected, but not asymmetrical or an aliorelative.

The relation ``greater or less'' among numbers is an aliorelative
and is connected, but is not transitive, for if $x$~is greater
or less than~$y$, and $y$~is greater or less than~$z$, it may happen
that $x$ and~$z$ are the same number.

Thus the three properties of being (1)~an aliorelative, (2)~transitive,
\PageSep{34}%
and (3)~connected, are mutually independent, since
a relation may have any two without having the third.

We now lay down the following definition:---

A relation is \emph{serial} when it is an aliorelative, transitive, and
\index{Relations!serial}%
connected; or, what is equivalent, when it is asymmetrical,
transitive, and connected.

A \emph{series} is the same thing as a serial relation.

It might have been thought that a series should be the \emph{field}
of a serial relation, not the serial relation itself. But this would
be an error. For example,
\[
1,\ 2,\ 3;\quad
1,\ 3,\ 2;\quad
2,\ 3,\ 1;\quad
2,\ 1,\ 3;\quad
3,\ 1,\ 2;\quad
3,\ 2,\ 1
\]
are six different series which all have the same field. If the
field \emph{were} the series, there could only be one series with a given
field. What distinguishes the above six series is simply the
different ordering relations in the six cases. Given the ordering
relation, the field and the order are both determinate. Thus
the ordering relation may be taken to \emph{be} the series, but the field
cannot be so taken.

Given any serial relation, say~$P$, we shall say that, in respect
of this relation, $x$ ``precedes'' $y$ if $x$~has the relation~$P$ to~$y$,
which we shall write ``$xPy$'' for short. The three characteristics
which $P$~must have in order to be serial are:
\begin{List}
\item[(1)] We must never have $xPx$, \ie\ no term must precede
itself.

\item[(2)] $P^{2}$~must imply~$P$, \ie\ if $x$ precedes~$y$ and $y$ precedes~$z$, $x$~must
precede~$z$.

\item[(3)] If $x$ and $y$ are two different terms in the field of~$P$, we shall
have $xPy$ or~$yPx$, \ie\ one of the two must precede the
other.
\end{List}
The reader can easily convince himself that, where these three
properties are found in an ordering relation, the characteristics
we expect of series will also be found, and \Foreign{vice versa}. We are
therefore justified in taking the above as a definition of order
\PageSep{35}%
or series. And it will be observed that the definition is effected
in purely logical terms.

Although a transitive asymmetrical connected relation always
exists wherever there is a series, it is not always the relation
which would most naturally be regarded as generating the series.
The natural-number series may serve as an illustration. The
relation we assumed in considering the natural numbers was
the relation of immediate succession, \ie\ the relation between
consecutive integers. This relation is asymmetrical, but not
transitive or connected. We can, however, derive from it,
by the method of mathematical induction, the ``ancestral''
relation which we considered in the preceding chapter. This
relation will be the same as ``less than or equal to'' among
\index{Successor of a number}%
inductive integers. For purposes of generating the series of
natural numbers, we want the relation ``less than,'' excluding
``equal to.'' This is the relation of $m$~to~$n$ when $m$~is an ancestor
of~$n$ but not identical with~$n$, or (what comes to the same thing)
when the successor of~$m$ is an ancestor of~$n$ in the sense in which
a number is its own ancestor. That is to say, we shall lay down
the following definition:---

An inductive number~$m$ is said to be \emph{less than} another number~$n$
when $n$~possesses every hereditary property possessed by the
successor of~$m$.

It is easy to see, and not difficult to prove, that the relation
``less than,'' so defined, is asymmetrical, transitive, and connected,
and has the inductive numbers for its field. Thus by
means of this relation the inductive numbers acquire an order
in the sense in which we defined the term ``order,'' and this order
is the so-called ``natural'' order, or order of magnitude.

The generation of series by means of relations more or less
resembling that of $n$ to~$n + 1$ is very common. The series of the
Kings of England, for example, is generated by relations of each
to his successor. This is probably the easiest way, where it is
applicable, of conceiving the generation of a series. In this
method we pass on from each term to the next, as long as there
\PageSep{36}%
is a next, or back to the one before, as long as there is one before.
This method always requires the generalised form of mathematical
induction in order to enable us to define ``earlier'' and
``later'' in a series so generated. On the analogy of ``proper
fractions,'' let us give the name ``proper posterity of~$x$ with respect
\index{Posterity!proper}%
to~$R$'' to the class of those terms that belong to the $R$-posterity
of some term to which $x$~has the relation~$R$, in the sense which
we gave before to ``posterity,'' which includes a term in its own
posterity. Reverting to the fundamental definitions, we find that
the ``proper posterity'' may be defined as follows:---

The ``proper posterity'' of~$x$ with respect to~$R$ consists of
all terms that possess every $R$-hereditary property possessed by
every term to which $x$~has the relation~$R$.

It is to be observed that this definition has to be so framed
as to be applicable not only when there is only one term to which
$x$~has the relation~$R$, but also in cases (as \eg\ that of father and
child) where there may be many terms to which $x$~has the relation~$R$.
We define further:

A term~$x$ is a ``proper ancestor'' of~$y$ with respect to~$R$ if $y$~belongs
to the proper posterity of~$x$ with respect to~$R$.

We shall speak for short of ``$R$-posterity'' and ``$R$-ancestors''
when these terms seem more convenient.

Reverting now to the generation of series by the relation~$R$
between consecutive terms, we see that, if this method is to be
possible, the relation ``proper $R$-ancestor'' must be an aliorelative,
transitive, and connected. Under what circumstances will
this occur? It will always be transitive: no matter what sort
of relation $R$ may be, ``$R$-ancestor'' and ``proper $R$-ancestor''
are always both transitive. But it is only under certain circumstances
that it will be an aliorelative or connected. Consider,
for example, the relation to one's left-hand neighbour at a round
dinner-table at which there are twelve people. If we call this
relation~$R$, the proper $R$-posterity of a person consists of all who
can be reached by going round the table from right to left. This
includes everybody at the table, including the person himself, since
\PageSep{37}%
twelve steps bring us back to our starting-point. Thus in such
a case, though the relation ``proper $R$-ancestor'' is connected,
and though $R$~itself is an aliorelative, we do not get a series
because ``proper $R$-ancestor'' is not an aliorelative. It is for
this reason that we cannot say that one person comes before
another with respect to the relation ``right of'' or to its ancestral
derivative.

The above was an instance in which the ancestral relation was
connected but not contained in diversity. An instance where
it is contained in diversity but not connected is derived from the
ordinary sense of the word ``ancestor.'' If $x$~is a proper ancestor
of~$y$, $x$~and~$y$ cannot be the same person; but it is not true that
of any two persons one must be an ancestor of the other.

The question of the circumstances under which series can be
generated by ancestral relations derived from relations of consecutiveness
\index{Consecutiveness}%
is often important. Some of the most important
cases are the following: Let $R$ be a many-one relation, and let
us confine our attention to the posterity of some term~$x$. When
so confined, the relation ``proper $R$-ancestor'' must be connected;
therefore all that remains to ensure its being serial is that it shall
be contained in diversity. This is a generalisation of the instance
of the dinner-table. Another generalisation consists in taking
$R$ to be a one-one relation, and including the ancestry of~$x$ as
well as the posterity. Here again, the one condition required
to secure the generation of a series is that the relation ``proper
$R$-ancestor'' shall be contained in diversity.

The generation of order by means of relations of consecutiveness,
though important in its own sphere, is less general than the
method which uses a transitive relation to define the order. It
often happens in a series that there are an infinite number of intermediate
terms between any two that may be selected, however
near together these may be. Take, for instance, fractions in order
\index{Fractions}%
of magnitude. Between any two fractions there are others---for
example, the arithmetic mean of the two. Consequently there is
no such thing as a pair of consecutive fractions. If we depended
\PageSep{38}%
upon consecutiveness for defining order, we should not be able
\index{Consecutiveness}%
to define the order of magnitude among fractions. But in fact
the relations of greater and less among fractions do not demand
generation from relations of consecutiveness, and the relations
of greater and less among fractions have the three characteristics
which we need for defining serial relations. In all such cases
the order must be defined by means of a \emph{transitive} relation, since
only such a relation is able to leap over an infinite number of
intermediate terms. The method of consecutiveness, like that
of counting for discovering the number of a collection, is appropriate
to the finite; it may even be extended to certain infinite
series, namely, those in which, though the total number of terms is
infinite, the number of terms between any two is always finite;
but it must not be regarded as general. Not only so, but care
must be taken to eradicate from the imagination all habits of
thought resulting from supposing it general. If this is not done,
series in which there are no consecutive terms will remain difficult
and puzzling. And such series are of vital importance for the
understanding of continuity, space, time, and motion.

There are many ways in which series may be generated, but
all depend upon the finding or construction of an asymmetrical
transitive connected relation. Some of these ways have considerable
importance. We may take as illustrative the generation
of series by means of a three-term relation which we may
call ``between.'' This method is very useful in geometry, and
may serve as an introduction to relations having more than two
terms; it is best introduced in connection with elementary
geometry.

Given any three points on a straight line in ordinary space,
\index{Between|EtSeq}%
there must be one of them which is \emph{between} the other two. This
will not be the case with the points on a circle or any other closed
curve, because, given any three points on a circle, we can travel
from any one to any other without passing through the third.
In fact, the notion ``between'' is characteristic of open series---or
series in the strict sense---as opposed to what may be called
\PageSep{39}%
``cyclic'' series, where, as with people at the dinner-table, a
sufficient journey brings us back to our starting-point. This
notion of ``between'' may be chosen as the fundamental notion
of ordinary geometry; but for the present we will only consider
its application to a single straight line and to the ordering of the
points on a straight line.\footnote
  {Cf.\ \Title{Rivista di Matematica}, iv.\ pp.~55 ff.; \Title{Principles of Mathematics}, p.~394
  (§\;375).}
Taking any two points $a$,~$b$, the line~$(ab)$
consists of three parts (besides $a$~and $b$ themselves):

(1) Points between $a$~and~$b$.

(2) Points $x$ such that $a$~is between $x$ and~$b$.

(3) Points $y$ such that $b$~is between $y$ and~$a$.

Thus the line~$(ab)$ can be defined in terms of the relation
``between.''

In order that this relation ``between'' may arrange the points
of the line in an order from left to right, we need certain assumptions,
namely, the following:---

(1) If anything is between $a$ and~$b$, $a$~and~$b$ are not identical.

(2) Anything between $a$ and~$b$ is also between $b$ and~$a$.

(3) Anything between $a$ and $b$ is not identical with~$a$ (nor,
consequently, with~$b$, in virtue of~(2)).

(4) If $x$~is between $a$ and~$b$, anything between $a$ and~$x$ is also
between $a$ and~$b$.

(5) If $x$~is between $a$ and~$b$, and $b$~is between $x$ and~$y$, then $b$~is
between $a$ and~$y$.

(6) If $x$ and $y$ are between $a$ and~$b$, then either $x$ and $y$ are
identical, or $x$~is between $a$ and~$y$, or $x$~is between $y$ and~$b$.

(7) If $b$~is between $a$ and~$x$ and also between $a$ and~$y$, then either
$x$ and~$y$ are identical, or $x$~is between $b$ and~$y$, or $y$~is between
$b$ and~$x$.

These seven properties are obviously verified in the case of points
on a straight line in ordinary space. Any three-term relation
which verifies them gives rise to series, as may be seen from the
following definitions. For the sake of definiteness, let us assume
\PageSep{40}%
that $a$~is to the left of~$b$. Then the points of the line~$(ab)$ are (1)~those
between which and~$b$, $a$~lies---these we will call to the left
of~$a$; (2)~$a$~itself; (3)~those between $a$ and~$b$; (4)~$b$~itself; (5)~those
between which and~$a$ lies~$b$---these we will call to the right
of~$b$. We may now define generally that of two points $x$,~$y$, on
the line~$(ab)$, we shall say that $x$~is ``to the left of''~$y$ in any
of the following cases:---

\begin{List}
\item[(1)] When $x$ and $y$ are both to the left of~$a$, and $y$~is between
$x$ and~$a$;

\item[(2)] When $x$ is to the left of~$a$, and $y$~is $a$ or $b$ or between $a$ and
$b$ or to the right of~$b$;

\item[(3)] When $x$ is~$a$, and $y$~is between $a$ and $b$ or is~$b$ or is to the
right of~$b$;

\item[(4)] When $x$ and $y$ are both between $a$ and~$b$, and $y$~is between
$x$ and~$b$;

\item[(5)] When $x$ is between $a$ and~$b$, and $y$~is $b$ or to the right of~$b$;

\item[(6)] When $x$ is~$b$ and $y$~is to the right of~$b$;

\item[(7)] When $x$ and $y$ are both to the right of~$b$ and $x$~is between
$b$ and~$y$.
\end{List}

It will be found that, from the seven properties which we have
assigned to the relation ``between,'' it can be deduced that the
relation ``to the left of,'' as above defined, is a \emph{serial} relation as
we defined that term. It is important to notice that nothing
in the definitions or the argument depends upon our meaning
by ``between'' the actual relation of that name which occurs in
empirical space: any three-term relation having the above seven
purely formal properties will serve the purpose of the argument
equally well.

Cyclic order, such as that of the points on a circle, cannot be
\index{Order!cyclic}%
generated by means of three-term relations of ``between.'' We
need a relation of four terms, which may be called ``separation
of couples.'' The point may be illustrated by considering a
journey round the world. One may go from England to New
Zealand by way of Suez or by way of San Francisco; we cannot
\PageSep{41}%
say definitely that either of these two places is ``between''
England and New Zealand. But if a man chooses that route
to go round the world, whichever way round he goes, his times in
England and New Zealand are separated from each other by his
times in Suez and San Francisco, and conversely. Generalising,
if we take any four points on a circle, we can separate them into
two couples, say $a$ and~$b$ and $x$ and~$y$, such that, in order to get
from $a$ to~$b$ one must pass through either $x$ or~$y$, and in order to
get from $x$ to~$y$ one must pass through either $a$ or~$b$. Under these
circumstances we say that the couple $(a, b)$ are ``separated'' by
the couple $(x, y)$. Out of this relation a cyclic order can be generated,
in a way resembling that in which we generated an open
order from ``between,'' but somewhat more complicated.\footnote
  {Cf.\ \Title{Principles of Mathematics}, p.~205 (§\;194), and references there given.}

The purpose of the latter half of this chapter has been to suggest
\index{Series!generation of}%
the subject which one may call ``generation of serial relations.''
When such relations have been defined, the generation of them
from other relations possessing only some of the properties
required for series becomes very important, especially in the
philosophy of geometry and physics. But we cannot, within
the limits of the present volume, do more than make the reader
aware that such a subject exists.
\PageSep{42}%


\Chapter{V}{Kinds of Relations}

\First{A great} part of the philosophy of mathematics is concerned with
\emph{relations}, and many different kinds of relations have different
kinds of uses. It often happens that a property which belongs
to \emph{all} relations is only important as regards relations of certain
sorts; in these cases the reader will not see the bearing of the
proposition asserting such a property unless he has in mind the
sorts of relations for which it is useful. For reasons of this
description, as well as from the intrinsic interest of the subject,
it is well to have in our minds a rough list of the more
mathematically serviceable varieties of relations.

We dealt in the preceding chapter with a supremely important
class, namely, \emph{serial} relations. Each of the three properties which
we combined in defining series---namely, \emph{asymmetry}, \emph{transitiveness},
and \emph{connexity}---has its own importance. We will begin by saying
something on each of these three.

\emph{Asymmetry}, \ie\ the property of being incompatible with the
\index{Relations!asymmetrical}%
converse, is a characteristic of the very greatest interest and
importance. In order to develop its functions, we will consider
various examples. The relation \emph{husband} is asymmetrical, and
so is the relation \emph{wife}; \ie\ if $a$~is husband of~$b$, $b$~cannot be husband
of~$a$, and similarly in the case of \emph{wife}. On the other hand, the
relation ``spouse'' is symmetrical: if $a$~is spouse of~$b$, then $b$~is
spouse of~$a$. Suppose now we are given the relation \emph{spouse}, and
we wish to derive the relation \emph{husband}. \emph{Husband} is the same as
\emph{male spouse} or \emph{spouse of a female}; thus the relation \emph{husband} can
\PageSep{43}%
be derived from \emph{spouse} either by limiting the domain to males
or by limiting the converse to females. We see from this instance
that, when a symmetrical relation is given, it is sometimes possible,
without the help of any further relation, to separate it into two
asymmetrical relations. But the cases where this is possible are
rare and exceptional: they are cases where there are two mutually
exclusive classes, say $\alpha$~and~$\beta$, such that whenever the relation
holds between two terms, one of the terms is a member of~$\alpha$ and
the other is a member of~$\beta$---as, in the case of \emph{spouse}, one term
of the relation belongs to the class of males and one to the class
of females. In such a case, the relation with its domain confined
to~$\alpha$ will be asymmetrical, and so will the relation with its domain
confined to~$\beta$. But such cases are not of the sort that occur
when we are dealing with series of more than two terms; for in
a series, all terms, except the first and last (if these exist), belong
both to the domain and to the converse domain of the generating
relation, so that a relation like \emph{husband}, where the domain and
converse domain do not overlap, is excluded.

The question how to \emph{construct} relations having some useful
property by means of operations upon relations which only have
rudiments of the property is one of considerable importance.
Transitiveness and connexity are easily constructed in many cases
where the originally given relation does not possess them: for
example, if $R$~is any relation whatever, the ancestral relation
derived from~$R$ by generalised induction is transitive; and if $R$~is
a many-one relation, the ancestral relation will be connected
if confined to the posterity of a given term. But asymmetry is
a much more difficult property to secure by construction. The
method by which we derived \emph{husband} from \emph{spouse} is, as we have
seen, not available in the most important cases, such as \emph{greater},
\emph{before}, \emph{to the right of}, where domain and converse domain overlap.
In all these cases, we can of course obtain a symmetrical relation
by adding together the given relation and its converse, but we
cannot pass back from this symmetrical relation to the original
asymmetrical relation except by the help of some asymmetrical
\PageSep{44}%
relation. Take, for example, the relation \emph{greater}: the relation
\emph{greater or less}---\ie\ \emph{unequal}---is symmetrical, but there is nothing
in this relation to show that it is the sum of two asymmetrical
relations. Take such a relation as ``differing in shape.'' This
is not the sum of an asymmetrical relation and its converse, since
shapes do not form a single series; but there is nothing to show
that it differs from ``differing in magnitude'' if we did not already
know that magnitudes have relations of greater and less. This
illustrates the fundamental character of asymmetry as a property
of relations.

From the point of view of the classification of relations, being
asymmetrical is a much more important characteristic than
implying diversity. Asymmetrical relations imply diversity,
but the converse is not the case. ``Unequal,'' for example,
implies diversity, but is symmetrical. Broadly speaking, we
may say that, if we wished as far as possible to dispense with
relational propositions and replace them by such as ascribed
predicates to subjects, we could succeed in this so long as we
confined ourselves to \emph{symmetrical} relations: those that do not
\index{Relations!symmetrical}%
imply diversity, if they are transitive, may be regarded as asserting
a common predicate, while those that do imply diversity
may be regarded as asserting incompatible predicates. For
example, consider the relation of \emph{similarity between classes},
by means of which we defined numbers. This relation is symmetrical
and transitive and does not imply diversity. It would
be possible, though less simple than the procedure we adopted,
to regard the number of a collection as a predicate of the collection:
then two similar classes will be two that have the same
numerical predicate, while two that are not similar will be two
that have different numerical predicates. Such a method of
replacing relations by predicates is formally possible (though
often very inconvenient) so long as the relations concerned are
symmetrical; but it is formally impossible when the relations
are asymmetrical, because both sameness and difference of predicates
are symmetrical. Asymmetrical relations are, we may
\PageSep{45}%
say, the most characteristically relational of relations, and the
most important to the philosopher who wishes to study the
ultimate logical nature of relations.

Another class of relations that is of the greatest use is the
class of one-many relations, \ie\ relations which at most one
term can have to a given term. Such are father, mother,
husband (except in Tibet), square of, sine of, and so on. But
parent, square root, and so on, are not one-many. It is possible,
formally, to replace all relations by one-many relations by means
of a device. Take (say) the relation \emph{less} among the inductive
numbers. Given any number~$n$ greater than~$1$, there will not
be only one number having the relation \emph{less} to~$n$, but we can
form the whole class of numbers that are less than~$n$. This
is one class, and its relation to~$n$ is not shared by any other class.
We may call the class of numbers that are less than~$n$ the ``proper
ancestry'' of~$n$, in the sense in which we spoke of ancestry and
posterity in connection with mathematical induction. Then
``proper ancestry'' is a one-many relation (\emph{one-many} will always
\index{Relations!one-many}%
be used so as to include \emph{one-one}), since each number determines
a single class of numbers as constituting its proper ancestry.
Thus the relation \emph{less than} can be replaced by \emph{being a member of
the proper ancestry of}. In this way a one-many relation in which
the one is a class, together with membership of this class, can
always formally replace a relation which is not one-many. Peano,
who for some reason always instinctively conceives of a relation
as one-many, deals in this way with those that are naturally
not so. Reduction to one-many relations by this method,
however, though possible as a matter of form, does not represent
a technical simplification, and there is every reason to think
that it does not represent a philosophical analysis, if only because
classes must be regarded as ``logical fictions.'' We shall therefore
\index{Fictions, logical}%
continue to regard one-many relations as a special kind of
relations.

One-many relations are involved in all phrases of the form
``the so-and-so of such-and-such.'' ``The King of England,''
\PageSep{46}%
``the wife of Socrates,'' ``the father of John Stuart Mill,'' and
so on, all describe some person by means of a one-many relation
to a given term. A person cannot have more than one father,
therefore ``the father of John Stuart Mill'' described some one
person, even if we did not know whom. There is much to
say on the subject of descriptions, but for the present it is
relations that we are concerned with, and descriptions are only
relevant as exemplifying the uses of one-many relations. It
should be observed that all mathematical functions result from
\index{Functions}%
one-many relations: the logarithm of~$x$, the cosine of~$x$, etc.,
are, like the father of~$x$, terms described by means of a one-many
relation (logarithm, cosine, etc.) to a given term~($x$). The
notion of \emph{function} need not be confined to numbers, or to the
uses to which mathematicians have accustomed us; it can be
extended to all cases of one-many relations, and ``the father of~$x$''
is just as legitimately a function of which $x$~is the argument as
is ``the logarithm of~$x$.'' Functions in this sense are \emph{descriptive}
\index{Functions!descriptive}%
functions. As we shall see later, there are functions of a still
more general and more fundamental sort, namely, \emph{propositional}
\index{Functions!propositional}%
functions; but for the present we shall confine our attention
to descriptive functions, \ie\ ``the term having the relation~$R$
to~$x$,'' or, for short, ``the~$R$ of~$x$,'' where $R$~is any one-many
relation.

It will be observed that if ``the~$R$ of~$x$'' is to describe a definite
term, $x$~must be a term to which something has the relation~$R$,
and there must not be more than one term having the relation~$R$
to~$x$, since ``the,'' correctly used, must imply uniqueness.
Thus we may speak of ``the father of~$x$'' if $x$~is any human being
except Adam and Eve; but we cannot speak of ``the father
of~$x$'' if $x$~is a table or a chair or anything else that does not
have a father. We shall say that the~$R$ of~$x$ ``exists'' when
there is just one term, and no more, having the relation~$R$ to~$x$.
Thus if $R$~is a one-many relation, the~$R$ of~$x$ exists whenever
$x$~belongs to the converse domain of~$R$, and not otherwise.
Regarding ``the~$R$ of~$x$'' as a function in the mathematical
\PageSep{47}%
sense, we say that $x$~is the ``argument'' of the function, and if
\index{Argument of a function}%
$y$~is the term which has the relation~$R$ to~$x$, \ie\ if $y$~is the~$R$ of~$x$,
then $y$~is the ``value'' of the function for the argument~$x$. If
\index{Value of a function}%
$R$~is a one-many relation, the range of possible arguments to
the function is the converse domain of~$R$, and the range of values
is the domain. Thus the range of possible arguments to the
function ``the father of~$x$'' is all who have fathers, \ie\ the converse
domain of the relation \emph{father}, while the range of possible
values for the function is all fathers, \ie\ the domain of the relation.

Many of the most important notions in the logic of relations
are descriptive functions, for example: \emph{converse}, \emph{domain}, \emph{converse
domain}, \emph{field}. Other examples will occur as we proceed.

Among one-many relations, \emph{one-one} relations are a specially
\index{Relations!one-one}%
important class. We have already had occasion to speak of
one-one relations in connection with the definition of number,
but it is necessary to be familiar with them, and not merely
to know their formal definition. Their formal definition may
be derived from that of one-many relations: they may be
defined as one-many relations which are also the converses of
one-many relations, \ie\ as relations which are both one-many
and many-one. One-many relations may be defined as relations
such that, if $x$~has the relation in question to~$y$, there is no other
term~$x'$ which also has the relation to~$y$. Or, again, they may
be defined as follows: Given two terms $x$ and~$x'$, the terms to
which $x$~has the given relation and those to which $x'$~has it have
no member in common. Or, again, they may be defined as
relations such that the relative product of one of them and
its converse implies identity, where the ``relative product''
of two relations $R$ and~$S$ is that relation which holds between
$x$ and~$z$ when there is an intermediate term~$y$, such that $x$~has
the relation~$R$ to~$y$ and $y$~has the relation~$S$ to~$z$. Thus, for
example, if $R$~is the relation of father to son, the relative product
of $R$ and its converse will be the relation which holds between
$x$~and a man~$z$ when there is a person~$y$, such that $x$~is the father
of~$y$ and $y$~is the son of~$z$. It is obvious that $x$ and~$z$ must be
\PageSep{48}%
the same person. If, on the other hand, we take the relation
of parent and child, which is not one-many, we can no longer
argue that, if $x$~is a parent of~$y$ and $y$~is a child of~$z$, $x$~and~$z$ must
be the same person, because one may be the father of~$y$ and the
other the mother. This illustrates that it is characteristic of
one-many relations when the relative product of a relation and
its converse implies identity. In the case of one-one relations
this happens, and also the relative product of the converse and
the relation implies identity. Given a relation~$R$, it is convenient,
if $x$~has the relation~$R$ to~$y$, to think of $y$ as being reached from~$x$
by an ``$R$-step'' or an ``$R$-vector.'' In the same case $x$~will
be reached from~$y$ by a ``backward $R$-step.'' Thus we may
state the characteristic of one-many relations with which we
have been dealing by saying that an $R$-step followed by a backward
$R$-step must bring us back to our starting-point. With
other relations, this is by no means the case; for example, if
$R$~is the relation of child to parent, the relative product of $R$ and
its converse is the relation ``self or brother or sister,'' and if $R$~is
the relation of grandchild to grandparent, the relative product
of $R$ and its converse is ``self or brother or sister or first cousin.''
It will be observed that the relative product of two relations
is not in general commutative, \ie\ the relative product of $R$
and~$S$ is not in general the same relation as the relative product
of $S$ and~$R$. \Eg\ the relative product of parent and brother is
uncle, but the relative product of brother and parent is parent.

One-one relations give a correlation of two classes, term for
term, so that each term in either class has its correlate in the
other. Such correlations are simplest to grasp when the two
classes have no members in common, like the class of husbands
and the class of wives; for in that case we know at once whether
a term is to be considered as one \emph{from} which the correlating
relation~$R$ goes, or as one \emph{to} which it goes. It is convenient
to use the word \emph{referent} for the term \emph{from} which the relation
\index{Referent@\emph{Referent}}%
goes, and the term \emph{relatum} for the term \emph{to} which it goes. Thus
\index{Relatum@\emph{Relatum}}%
if $x$~and~$y$ are husband and wife, then, with respect to the relation
\PageSep{49}%
``husband,'' $x$~is referent and $y$~relatum, but with respect to the
relation ``wife,'' $y$~is referent and $x$~relatum. We say that a
relation and its converse have opposite ``senses''; thus the
\index{Converse}%
``sense'' of a relation that goes from $x$ to~$y$ is the opposite of
that of the corresponding relation from $y$ to~$x$. The fact that a
relation has a ``sense'' is fundamental, and is part of the reason
why order can be generated by suitable relations. It will be
observed that the class of all possible referents to a given relation
is its domain, and the class of all possible relata is its converse
\index{Domain}%
domain.

But it very often happens that the domain and converse
domain of a one-one relation overlap. Take, for example,
the first ten integers (excluding~$0$), and add~$1$ to each; thus
instead of the first ten integers we now have the integers
\[
2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8,\ 9,\ 10,\ 11.
\]
These are the same as those we had before, except that $1$~has
been cut off at the beginning and $11$~has been joined on at the
end. There are still ten integers: they are correlated with
the previous ten by the relation of $n$ to~$n + 1$, which is a one-one
relation. Or, again, instead of adding~$1$ to each of our original
ten integers, we could have doubled each of them, thus obtaining
the integers
\[
2,\ 4,\ 6,\ 8,\ 10,\ 12,\ 14,\ 16,\ 18,\ 20.
\]
Here we still have five of our previous set of integers, namely,
$2$,~$4$, $6$, $8$,~$10$. The correlating relation in this case is the relation
of a number to its double, which is again a one-one relation.
Or we might have replaced each number by its square, thus
obtaining the set
\[
1,\ 4,\ 9,\ 16,\ 25,\ 36,\ 49,\ 64,\ 81,\ 100.
\]
On this occasion only three of our original set are left, namely,
$1$,~$4$,~$9$. Such processes of correlation may be varied endlessly.

The most interesting case of the above kind is the case where
our one-one relation has a converse domain which is part, but
\PageSep{50}%
not the whole, of the domain. If, instead of confining the domain
to the first ten integers, we had considered the whole of the
inductive numbers, the above instances would have illustrated
this case. We may place the numbers concerned in two rows,
putting the correlate directly under the number whose correlate
it is. Thus when the correlator is the relation of $n$ to~$n + 1$, we
have the two rows:
\begin{align*}
&amp;1,\ 2,\ 3,\ 4,\ 5,\ \dots\ n\Add{,}\ \dots \\
&amp;2,\ 3,\ 4,\ 5,\ 6,\ \dots\ n + 1\Add{,}\ \dots\Add{.}
\end{align*}
When the correlator is the relation of a number to its double,
we have the two rows:
\begin{align*}
&amp;1,\ 2,\ 3,\ 4,\ \Z5,\ \dots\ n\Add{,}\ \dots \\
&amp;2,\ 4,\ 6,\ 8,\ 10,\ \dots\ 2n\Add{,}\ \dots\Add{.}
\end{align*}
When the correlator is the relation of a number to its square,
the rows are:
\begin{align*}
&amp;1,\ 2,\ 3,\ \Z4,\ \Z5,\ \dots\ n\Add{,}\ \dots \\
&amp;1,\ 4,\ 9,\  16,\  25,\ \dots\ n^{2}\Add{,}\ \dots\Add{.}
\end{align*}
In all these cases, all inductive numbers occur in the top row,
and only some in the bottom row.

Cases of this sort, where the converse domain is a ``proper
part'' of the domain (\ie\ a part not the whole), will occupy us
again when we come to deal with infinity. For the present, we
wish only to note that they exist and demand consideration.

Another class of correlations which are often important is
the class called ``permutations,'' where the domain and converse
\index{Permutations}%
domain are identical. Consider, for example, the six possible
arrangements of three letters:
\begin{gather*}
a,\ b,\ c\Add{;} \\
a,\ c,\ b\Add{;} \displaybreak[0] \\
b,\ c,\ a\Add{;} \\
b,\ a,\ c\Add{;} \displaybreak[0] \\
c,\ a,\ b\Add{;} \\
c,\ b,\ a\Add{.}
\end{gather*}
\PageSep{51}%
Each of these can be obtained from any one of the others by
means of a correlation. Take, for example, the first and last,
$(a, b, c)$ and $(c, b, a)$. Here $a$~is correlated with~$c$, $b$~with itself,
and $c$ with~$a$. It is obvious that the combination of two permutations
is again a permutation, \ie\ the permutations of a given
class form what is called a ``group.''

These various kinds of correlations have importance in various
connections, some for one purpose, some for another. The
general notion of one-one correlations has boundless importance
in the philosophy of mathematics, as we have partly seen already,
but shall see much more fully as we proceed. One of its uses
will occupy us in our next chapter.
\PageSep{52}%


\Chapter{VI}{Similarity of Relations}

\First{We} saw in \ChapRef{II}.\ that two classes have the same number
\index{Relations!similar|EtSeq}%
\index{Similarity!of relations|EtSeq}%
of terms when they are ``similar,'' \ie\ when there is a one-one
relation whose domain is the one class and whose converse
domain is the other. In such a case we say that there is a
``one-one correlation'' between the two classes.

In the present chapter we have to define a relation between
relations, which will play the same part for them that similarity
of classes plays for classes. We will call this relation ``similarity
of relations,'' or ``likeness'' when it seems desirable to use a
\index{Likeness}%
different word from that which we use for classes. How is
likeness to be defined?

We shall employ still the notion of correlation: we shall
\index{Maps}%
assume that the domain of the one relation can be correlated
with the domain of the other, and the converse domain with the
converse domain; but that is not enough for the sort of resemblance
which we desire to have between our two relations.
What we desire is that, whenever either relation holds between
two terms, the other relation shall hold between the correlates
of these two terms. The easiest example of the sort of thing
we desire is a map. When one place is north of another, the
place on the map corresponding to the one is above the place
on the map corresponding to the other; when one place is west
of another, the place on the map corresponding to the one is
to the left of the place on the map corresponding to the other;
and so on. The structure of the map corresponds with that of
\PageSep{53}%
the country of which it is a map. The space-relations in the
map have ``likeness'' to the space-relations in the country
mapped. It is this kind of connection between relations that
we wish to define.

We may, in the first place, profitably introduce a certain
\index{Field of a relation}%
restriction. We will confine ourselves, in defining likeness, to
such relations as have ``fields,'' \ie\ to such as permit of the
formation of a single class out of the domain and the converse
domain. This is not always the case. Take, for example,
the relation ``domain,'' \ie\ the relation which the domain of a
relation has to the relation. This relation has all classes for its
domain, since every class is the domain of some relation; and
it has all relations for its converse domain, since every relation
has a domain. But classes and relations cannot be added together
to form a new single class, because they are of different
logical ``types.'' We do not need to enter upon the difficult
\index{Types, logical}%
doctrine of types, but it is well to know when we are abstaining
from entering upon it. We may say, without entering upon
the grounds for the assertion, that a relation only has a ``field''
when it is what we call ``homogeneous,'' \ie\ when its domain
and converse domain are of the same logical type; and as a
rough-and-ready indication of what we mean by a ``type,''
we may say that individuals, classes of individuals, relations
between individuals, relations between classes, relations of
classes to individuals, and so on, are different types. Now the
notion of likeness is not very useful as applied to relations that
are not homogeneous; we shall, therefore, in defining likeness,
simplify our problem by speaking of the ``field'' of one of the
relations concerned. This somewhat limits the generality of
our definition, but the limitation is not of any practical importance.
And having been stated, it need no longer be remembered.
We may define two relations $P$ and~$Q$ as ``similar,'' or as
having ``likeness,'' when there is a one-one relation~$S$ whose
domain is the field of~$P$ and whose converse domain is the field
of~$Q$, and which is such that, if one term has the relation~$P$
\PageSep{54}%
to another, the correlate of the one has the relation~$Q$ to the
correlate of the other, and \Foreign{vice versa}. A figure will make this
\Diagram{2in}{1.875in}{diagr54}
clearer. Let $x$~and~$y$ be two
terms having the relation~$P$.
Then there are to be two terms
$z$,~$w$, such that $x$~has the relation~$S$
to~$z$, $y$~has the relation~$S$
to~$w$, and $z$~has the relation~$Q$
to~$w$. If this happens with
every pair of terms such as $x$
and~$y$, and if the converse happens with every pair of terms such
as $z$ and~$w$, it is clear that for every instance in which the relation~$P$
holds there is a corresponding instance in which the relation~$Q$
holds, and \Foreign{vice versa}; and this is what we desire to secure by
our definition. We can eliminate some redundancies in the
above sketch of a definition, by observing that, when the above
conditions are realised, the relation~$P$ is the same as the relative
product of $S$~and~$Q$ and the converse of~$S$, \ie\ the $P$-step from
$x$ to~$y$ may be replaced by the succession of the $S$-step from
$x$ to~$z$, the $Q$-step from $z$ to~$w$, and the backward $S$-step from
$w$ to~$y$. Thus we may set up the following definitions:---

A relation~$S$ is said to be a ``correlator'' or an ``ordinal
\index{Correlators}%
correlator'' of two relations $P$~and~$Q$ if $S$~is one-one, has the
field of~$Q$ for its converse domain, and is such that $P$~is the
relative product of $S$~and~$Q$ and the converse of~$S$.

Two relations $P$ and~$Q$ are said to be ``similar,'' or to have
``likeness,'' when there is at least one correlator of $P$~and~$Q$.

These definitions will be found to yield what we above decided
to be necessary.

It will be found that, when two relations are similar, they
share all properties which do not depend upon the actual terms
in their fields. For instance, if one implies diversity, so does
the other; if one is transitive, so is the other; if one is connected,
so is the other. Hence if one is serial, so is the other.
Again, if one is one-many or one-one, the other is one-many
\PageSep{55}%
or one-one; and so on, through all the general properties of
relations. Even statements involving the actual terms of the
field of a relation, though they may not be true as they stand
when applied to a similar relation, will always be capable of
translation into statements that are analogous. We are led
by such considerations to a problem which has, in mathematical
philosophy, an importance by no means adequately recognised
hitherto. Our problem may be stated as follows:---

Given some statement in a language of which we know the
grammar and the syntax, but not the vocabulary, what are the
possible meanings of such a statement, and what are the meanings
of the unknown words that would make it true?

The reason that this question is important is that it represents,
much more nearly than might be supposed, the state of our
knowledge of nature. We know that certain scientific propositions---which,
in the most advanced sciences, are expressed
in mathematical symbols---are more or less true of the world,
but we are very much at sea as to the interpretation to be put
upon the terms which occur in these propositions. We know
much more (to use, for a moment, an old-fashioned pair of
terms) about the \emph{form} of nature than about the \emph{matter}.
Accordingly, what we really know when we enunciate a law
of nature is only that there is probably \emph{some} interpretation of
our terms which will make the law approximately true. Thus
great importance attaches to the question: What are the
possible meanings of a law expressed in terms of which we do
not know the substantive meaning, but only the grammar and
syntax? And this question is the one suggested above.

For the present we will ignore the general question, which
will occupy us again at a later stage; the subject of likeness
itself must first be further investigated.

Owing to the fact that, when two relations are similar, their
properties are the same except when they depend upon the
fields being composed of just the terms of which they are composed,
it is desirable to have a nomenclature which collects
\PageSep{56}%
together all the relations that are similar to a given relation.
Just as we called the set of those classes that are similar to a
given class the ``number'' of that class, so we may call the set
of all those relations that are similar to a given relation the
``number'' of that relation. But in order to avoid confusion with
the numbers appropriate to classes, we will speak, in this case, of
a ``relation-number.'' Thus we have the following definitions:---

The ``relation-number'' of a given relation is the class of all
\index{Number!relation}%
\index{Relation numbers|EtSeq}%
those relations that are similar to the given relation.

``Relation-numbers'' are the set of all those classes of relations
that are relation-numbers of various relations; or, what comes to
the same thing, a relation number is a class of relations consisting
of all those relations that are similar to one member of the class.

When it is necessary to speak of the numbers of classes in
\index{Number!cardinal}%
a way which makes it impossible to confuse them with relation-numbers,
we shall call them ``cardinal numbers.'' Thus cardinal
numbers are the numbers appropriate to classes. These include
the ordinary integers of daily life, and also certain infinite
numbers, of which we shall speak later. When we speak of
``numbers'' without qualification, we are to be understood as
meaning \emph{cardinal} numbers. The definition of a cardinal number,
it will be remembered, is as follows:---

The ``cardinal number'' of a given class is the set of all
those classes that are similar to the given class.

The most obvious application of relation-numbers is to \emph{series}.
Two series may be regarded as equally long when they have
the same relation-number. Two \emph{finite} series will have the
same relation-number when their fields have the same cardinal
number of terms, and only then---\ie\ a series of (say) $15$~terms
will have the same relation-number as any other series of fifteen
terms, but will not have the same relation-number as a series
of $14$~or $16$~terms, nor, of course, the same relation-number
as a relation which is not serial. Thus, in the quite special case
of finite series, there is parallelism between cardinal and relation-numbers.
The relation-numbers applicable to series may be
\PageSep{57}%
called ``serial numbers'' (what are commonly called ``ordinal
\index{Number!serial}%
numbers'' are a sub-class of these); thus a finite serial number
is determinate when we know the cardinal number of terms
in the field of a series having the serial number in question.
If $n$~is a finite cardinal number, the relation-number of a series
which has $n$~terms is called the ``ordinal'' number~$n$. (There
are also infinite ordinal numbers, but of them we shall speak
in a later chapter.) When the cardinal number of terms in
the field of a series is infinite, the relation-number of the series
is not determined merely by the cardinal number, indeed an
infinite number of relation-numbers exist for one infinite cardinal
number, as we shall see when we come to consider infinite series.
When a series is infinite, what we may call its ``length,'' \ie\
its relation-number, may vary without change in the cardinal
number; but when a series is finite, this cannot happen.

We can define addition and multiplication for relation-numbers
as well as for cardinal numbers, and a whole arithmetic
of relation-numbers can be developed. The manner in which
this is to be done is easily seen by considering the case of series.
Suppose, for example, that we wish to define the sum of two
non-overlapping series in such a way that the relation-number
of the sum shall be capable of being defined as the sum of the
relation-numbers of the two series. In the first place, it is clear
that there is an \emph{order} involved as between the two series: one
of them must be placed before the other. Thus if $P$~and~$Q$
are the generating relations of the two series, in the series which
is their sum with $P$~put before~$Q$, every member of the field of~$P$
will precede every member of the field of~$Q$. Thus the serial
relation which is to be defined as the sum of $P$~and~$Q$ is not
``$P$~or~$Q$'' simply, but ``$P$~or $Q$ or the relation of any member
of the field of~$P$ to any member of the field of~$Q$.'' Assuming
that $P$~and~$Q$ do not overlap, this relation is serial, but ``$P$~or~$Q$''
is not serial, being not connected, since it does not hold between
a member of the field of~$P$ and a member of the field of~$Q$. Thus
the sum of $P$ and~$Q$, as above defined, is what we need in order
\PageSep{58}%
to define the sum of two relation-numbers. Similar modifications
are needed for products and powers. The resulting arithmetic
does not obey the commutative law: the sum or product
\index{Commutative law}%
of two relation-numbers generally depends upon the order in
which they are taken. But it obeys the associative law, one
\index{Associative law}%
form of the distributive law, and two of the formal laws for
\index{Distributive law}%
powers, not only as applied to serial numbers, but as applied to
relation-numbers generally. Relation-arithmetic, in fact, though
recent, is a thoroughly respectable branch of mathematics.

It must not be supposed, merely because series afford the
most obvious application of the idea of likeness, that there are
no other applications that are important. We have already
mentioned maps, and we might extend our thoughts from this
illustration to geometry generally. If the system of relations
by which a geometry is applied to a certain set of terms can be
brought fully into relations of likeness with a system applying
to another set of terms, then the geometry of the two sets is
indistinguishable from the mathematical point of view, \ie\ all
the propositions are the same, except for the fact that they are
applied in one case to one set of terms and in the other to another.
We may illustrate this by the relations of the sort that may be
called ``between,'' which we considered in \ChapRef{IV}. We
\index{Between}%
there saw that, provided a three-term relation has certain formal
logical properties, it will give rise to series, and may be called
a ``between-relation.'' Given any two points, we can use the
between-relation to define the straight line determined by those
two points; it consists of $a$~and~$b$ together with all points~$x$,
such that the between-relation holds between the three points
$a$,~$b$,~$x$ in some order or other. It has been shown by O.~Veblen
\index{Veblen}%
that we may regard our whole space as the field of a three-term
between-relation, and define our geometry by the properties we
assign to our between-relation.\footnote
  {This does not apply to elliptic space, but only to spaces in which
  the straight line is an open series. \Title{Modern Mathematics}, edited by
  J.~W.~A. Young, pp.~3--51 (monograph by O.~Veblen on ``The Foundations of
  Geometry'').}
Now likeness is just as easily
\PageSep{59}%
definable between three-term relations as between two-term
relations. If $B$~and $B'$ are two between-relations, so that
``$xB(y, z)$'' means ``$x$~is between $y$ and~$z$ with respect to~$B$,''
we shall call~$S$ a correlator of $B$~and $B'$ if it has the field of~$B'$
for its converse domain, and is such that the relation~$B$ holds
between three terms when $B'$~holds between their $S$-correlates,
and only then. And we shall say that $B$~is like~$B'$ when there
is at least one correlator of~$B$ with~$B'$. The reader can easily
convince himself that, if $B$~is like~$B'$ in this sense, there can be
no difference between the geometry generated by~$B$ and that
\index{Geometry}%
generated by~$B'$.

It follows from this that the mathematician need not concern
himself with the particular being or intrinsic nature of his points,
\index{Points}%
lines, and planes, even when he is speculating as an \emph{applied}
mathematician. We may say that there is empirical evidence
of the approximate truth of such parts of geometry as are not
matters of definition. But there is no empirical evidence as to
what a ``point'' is to be. It has to be something that as nearly
as possible satisfies our axioms, but it does not have to be ``very
small'' or ``without parts.'' Whether or not it is those things
is a matter of indifference, so long as it satisfies the axioms. If
we can, out of empirical material, construct a logical structure,
no matter how complicated, which will satisfy our geometrical
axioms, that structure may legitimately be called a ``point.''
We must not say that there is nothing else that could legitimately
be called a ``point''; we must only say: ``This object we have
constructed is sufficient for the geometer; it may be one of
many objects, any of which would be sufficient, but that is no
concern of ours, since this object is enough to vindicate the
empirical truth of geometry, in so far as geometry is not a
matter of definition.'' This is only an illustration of the general
principle that what matters in mathematics, and to a very great
extent in physical science, is not the intrinsic nature of our
terms, but the logical nature of their interrelations.

We may say, of two similar relations, that they have the same
\PageSep{60}%
``structure.'' For mathematical purposes (though not for those
\index{Structure|EtSeq}%
of pure philosophy) the only thing of importance about a relation
is the cases in which it holds, not its intrinsic nature. Just as a
class may be defined by various different but co-extensive concepts---\eg\
``man'' and ``featherless biped,''---so two relations which
are conceptually different may hold in the same set of instances.
An ``instance'' in which a relation holds is to be conceived as a
couple of terms, with an order, so that one of the terms comes
first and the other second; the couple is to be, of course,
such that its first term has the relation in question to its second.
Take (say) the relation ``father'': we can define what we may
call the ``extension'' of this relation as the class of all ordered
couples $(x, y)$ which are such that $x$~is the father of~$y$. From
the mathematical point of view, the only thing of importance
about the relation ``father'' is that it defines this set of ordered
couples. Speaking generally, we say:

The ``extension'' of a relation is the class of those ordered
\index{Extension of a relation}%
couples $(x, y)$ which are such that $x$~has the relation in question
to~$y$.

We can now go a step further in the process of abstraction,
\index{Maps|EtSeq}%
and consider what we mean by ``structure.'' Given any relation,
we can, if it is a sufficiently simple one, construct a map of it.
For the sake of definiteness, let us take a relation of which the
extension is the following couples: $ab$,~$ac$, $ad$, $bc$, $ce$, $dc$,~$de$, where
$a$,~$b$, $c$, $d$,~$e$ are five terms, no matter what. We may make a
``map'' of this relation by taking five points
on a plane and connecting them by arrows,
as in the accompanying figure. What is
revealed by the map is what we call the
``structure'' of the relation.

\Diagram{1.375in}{1.25in}{diagr60}
It is clear that the ``structure'' of the
relation does not depend upon the particular
terms that make up the field of the relation.
The field may be changed without changing the structure, and
the structure may be changed without changing the field---for
\PageSep{61}%
example, if we were to add the couple~$ae$ in the above illustration
we should alter the structure but not the field. Two relations
have the same ``structure,'' we shall say, when the same map
will do for both---or, what comes to the same thing, when either
can be a map for the other (since every relation can be its own
map). And that, as a moment's reflection shows, is the very
same thing as what we have called ``likeness.'' That is to say,
two relations have the same structure when they have likeness,
\ie\ when they have the same relation-number. Thus what we
defined as the ``relation-number'' is the very same thing as is
obscurely intended by the word ``structure''---a word which,
important as it is, is never (so far as we know) defined in precise
terms by those who use it.

There has been a great deal of speculation in traditional
philosophy which might have been avoided if the importance of
structure, and the difficulty of getting behind it, had been realised.
For example, it is often said that space and time are subjective,
\index{Space}%
\index{Time}%
but they have objective counterparts; or that phenomena are
\index{Counterparts, objective}%
subjective, but are caused by things in themselves, which must
have differences \Foreign{inter~se} corresponding with the differences in
the phenomena to which they give rise. Where such hypotheses
are made, it is generally supposed that we can know very little
about the objective counterparts. In actual fact, however, if
the hypotheses as stated were correct, the objective counterparts
would form a world having the same structure as the phenomenal
world, and allowing us to infer from phenomena the truth of all
propositions that can be stated in abstract terms and are known
to be true of phenomena. If the phenomenal world has three
dimensions, so must the world behind phenomena; if the phenomenal
world is Euclidean, so must the other be; and so on.
In short, every proposition having a communicable significance
must be true of both worlds or of neither: the only difference
must lie in just that essence of individuality which always eludes
words and baffles description, but which, for that very reason,
is irrelevant to science. Now the only purpose that philosophers
\PageSep{62}%
have in view in condemning phenomena is in order to persuade
themselves and others that the real world is very different from
the world of appearance. We can all sympathise with their wish
to prove such a very desirable proposition, but we cannot congratulate
them on their success. It is true that many of them
do not assert objective counterparts to phenomena, and these
escape from the above argument. Those who do assert counterparts
are, as a rule, very reticent on the subject, probably because
they feel instinctively that, if pursued, it will bring about too
much of a \Foreign{rapprochement} between the real and the phenomenal
world. If they were to pursue the topic, they could hardly avoid
the conclusions which we have been suggesting. In such ways,
as well as in many others, the notion of structure or relation-number
is important.
\PageSep{63}%


\Chapter{VII}{Rational, Real, and Complex Numbers}

\First{We} have now seen how to define cardinal numbers, and also
relation-numbers, of which what are commonly called ordinal
numbers are a particular species. It will be found that each
of these kinds of number may be infinite just as well as finite.
But neither is capable, as it stands, of the more familiar extensions
of the idea of number, namely, the extensions to negative,
fractional, irrational, and complex numbers. In the present
chapter we shall briefly supply logical definitions of these various
extensions.

One of the mistakes that have delayed the discovery of correct
definitions in this region is the common idea that each extension
of number included the previous sorts as special cases. It was
thought that, in dealing with positive and negative integers, the
positive integers might be identified with the original signless
integers. Again it was thought that a fraction whose denominator
is~$1$ may be identified with the natural number which is its
numerator. And the irrational numbers, such as the square
root of~$2$, were supposed to find their place among rational fractions,
as being greater than some of them and less than the others,
so that rational and irrational numbers could be taken together
as one class, called ``real numbers.'' And when the idea of
number was further extended so as to include ``complex''
numbers, \ie\ numbers involving the square root of~$-1$, it was
thought that real numbers could be regarded as those among
complex numbers in which the imaginary part (\ie\ the part
\PageSep{64}%
which was a multiple of the square root of~$-1$) was zero. All
these suppositions were erroneous, and must be discarded, as we
shall find, if correct definitions are to be given.

Let us begin with \emph{positive and negative integers}. It is obvious
\index{Integers, positive and negative}%
on a moment's consideration that $+1$ and~$-1$ must both be
relations, and in fact must be each other's converses. The
obvious and sufficient definition is that $+1$~is the relation of
$n + 1$ to~$n$, and $-1$ is the relation of $n$ to~$n + 1$. Generally, if $m$~is
any inductive number, $+m$~will be the relation of $n + m$ to~$n$
(for any~$n$), and $-m$~will be the relation of $n$ to~$n + m$. According
to this definition, $+m$~is a relation which is one-one so
long as $n$~is a cardinal number (finite or infinite) and $m$~is an
inductive cardinal number. But $+m$~is under no circumstances
capable of being identified with~$m$, which is not a relation, but
a class of classes. Indeed, $+m$~is every bit as distinct from~$m$
as $-m$~is.

\emph{Fractions} are more interesting than positive or negative integers.
\index{Fractions}%
\index{Ratios}%
We need fractions for many purposes, but perhaps most obviously
for purposes of measurement. My friend and collaborator Dr
A.~N. Whitehead has developed a theory of fractions specially
\index{Whitehead}%
adapted for their application to measurement, which is set forth
in \Title{Principia Mathematica}.\footnote
  {Vol.~iii.\ *~300~ff., especially~303.}
But if all that is needed is to define
objects having the required purely mathematical properties, this
purpose can be achieved by a simpler method, which we shall
here adopt. We shall define the fraction~$m/n$ as being that
relation which holds between two inductive numbers $x$,~$y$ when
$xn = ym$. This definition enables us to prove that $m/n$~is a one-one
relation, provided neither $m$ or~$n$ is zero. And of course $n/m$~is
the converse relation to~$m/n$.

From the above definition it is clear that the fraction~$m/1$ is
that relation between two integers $x$ and~$y$ which consists in the
fact that $x = my$. This relation, like the relation~$+m$, is by no
means capable of being identified with the inductive cardinal
number~$m$, because a relation and a class of classes are objects
\PageSep{65}%
of utterly different kinds.\footnote
  {Of course in practice we shall continue to speak of a fraction as (say)
  greater or less than~$1$, meaning greater or less than the ratio~$1/1$. So
  long as it is understood that the ratio~$1/1$ and the cardinal number~$1$ are
  different, it is not necessary to be always pedantic in emphasising the
  difference.}
It will be seen that $0/n$~is always the
same relation, whatever inductive number $n$~may be; it is, in short,
the relation of~$0$ to any other inductive cardinal. We may call
this the zero of rational numbers; it is not, of course, identical
\index{Zero}%
with the cardinal number~$0$. Conversely, the relation~$m/0$ is
always the same, whatever inductive number $m$~may be. There
is not any inductive cardinal to correspond to~$m/0$. We may call
it ``the infinity of rationals.'' It is an instance of the sort of
\index{Infinity!of rationals}%
infinite that is traditional in mathematics, and that is represented
by~``$\infty$.'' This is a totally different sort from the true Cantorian
infinite, which we shall consider in our next chapter. The infinity
of rationals does not demand, for its definition or use, any
infinite classes or infinite integers. It is not, in actual fact, a
very important notion, and we could dispense with it altogether
if there were any object in doing so. The Cantorian infinite, on
\index{Infinite!Cantorian}%
the other hand, is of the greatest and most fundamental importance;
the understanding of it opens the way to whole new realms
of mathematics and philosophy.

It will be observed that zero and infinity, alone among ratios,
are not one-one. Zero is one-many, and infinity is many-one.

There is not any difficulty in defining \emph{greater} and \emph{less} among
\index{Greater and less}%
ratios (or fractions). Given two ratios $m/n$ and~$p/q$, we shall say
that $m/n$~is \emph{less} than~$p/q$ if $mq$~is less than~$pn$. There is no
difficulty in proving that the relation ``less than,'' so defined, is
serial, so that the ratios form a series in order of magnitude. In
this series, zero is the smallest term and infinity is the largest.
If we omit zero and infinity from our series, there is no longer
any smallest or largest ratio; it is obvious that if $m/n$~is any ratio
other than zero and infinity, $m/2n$~is smaller and $2m/n$~is larger,
though neither is zero or infinity, so that $m/n$~is neither the smallest
\PageSep{66}%
nor the largest ratio, and therefore (when zero and infinity are
omitted) there is no smallest or largest, since $m/n$~was chosen
arbitrarily. In like manner we can prove that however nearly
equal two fractions may be, there are always other fractions
between them. For, let $m/n$ and $p/q$ be two fractions, of which
$p/q$~is the greater. Then it is easy to see (or to prove) that
$(m + p)/(n + q)$ will be greater than~$m/n$ and less than~$p/q$. Thus
the series of ratios is one in which no two terms are consecutive,
but there are always other terms between any two. Since there
are other terms between these others, and so on \Foreign{ad~infinitum}, it
is obvious that there are an infinite number of ratios between
any two, however nearly equal these two may be.\footnote
  {Strictly speaking, this statement, as well as those following to the end
  of the paragraph, involves what is called the ``axiom of infinity,'' which
\index{Infinity, axiom of|FNote}%
  will be discussed in a later chapter.}
A series
\index{Series!compact}%
having the property that there are always other terms between
any two, so that no two are consecutive, is called ``compact.''
Thus the ratios in order of magnitude form a ``compact'' series.
Such series have many important properties, and it is important
to observe that ratios afford an instance of a compact series
generated purely logically, without any appeal to space or time
or any other empirical datum.

Positive and negative ratios can be defined in a way analogous
to that in which we defined positive and negative integers.
Having first defined the sum of two ratios $m/n$ and~$p/q$ as
$(mq + pn)/nq$, we define $+p/q$ as the relation of $m/n + p/q$ to~$m/n$,
where $m/n$~is any ratio; and $-p/q$~is of course the converse of~$+p/q$.
This is not the only possible way of defining positive and
negative ratios, but it is a way which, for our purpose, has the
merit of being an obvious adaptation of the way we adopted in
the case of integers.

We come now to a more interesting extension of the idea of
\index{Number!irrational}%
\index{Number!real}%
number, \ie\ the extension to what are called ``real'' numbers,
which are the kind that embrace irrationals. In \ChapRef{I}.\ we
\index{Irrationals}%
had occasion to mention ``incommensurables'' and their discovery
\index{Incommensurables}%
\PageSep{67}%
by Pythagoras. It was through them, \ie\ through
\index{Pythagoras}%
geometry, that irrational numbers were first thought of. A
\index{Geometry}%
square of which the side is one inch long will have a diagonal of
which the length is the square root of $2$~inches. But, as the
ancients discovered, there is no fraction of which the square is~$2$.
This proposition is proved in the tenth book of Euclid, which is
\index{Euclid}%
one of those books that schoolboys supposed to be fortunately lost
in the days when Euclid was still used as a text-book. The proof
is extraordinarily simple. If possible, let $m/n$~be the square root
of~$2$, so that $m^{2}/n^{2} = 2$, \ie\ $m^{2} = 2n^{2}$. Thus $m^{2}$~is an even number,
and therefore $m$~must be an even number, because the square of
an odd number is odd. Now if $m$~is even, $m^{2}$~must divide by~$4$,
for if $m = 2p$, then $m^{2} = 4p^{2}$. Thus we shall have $4p^{2} = 2n^{2}$, where
$p$~is half of~$m$. Hence $2p^{2} = n^{2}$, and therefore $n/p$~will also be the
square root of~$2$. But then we can repeat the argument: if
$n = 2q$, $p/q$~will also be the square root of~$2$, and so on, through
an unending series of numbers that are each half of its predecessor.
But this is impossible; if we divide a number by~$2$, and then
halve the half, and so on, we must reach an odd number after a
finite number of steps. Or we may put the argument even more
simply by assuming that the $m/n$ we start with is in its lowest
terms; in that case, $m$~and~$n$ cannot both be even; yet we have
seen that, if $m^{2}/n^{2} = 2$, they must be. Thus there cannot be any
fraction~$m/n$ whose square is~$2$.

Thus no fraction will express exactly the length of the diagonal
of a square whose side is one inch long. This seems like a
challenge thrown out by nature to arithmetic. However the
arithmetician may boast (as Pythagoras did) about the power
of numbers, nature seems able to baffle him by exhibiting lengths
which no numbers can estimate in terms of the unit. But the
problem did not remain in this geometrical form. As soon as
algebra was invented, the same problem arose as regards the
solution of equations, though here it took on a wider form,
since it also involved complex numbers.

It is clear that fractions can be found which approach nearer
\PageSep{68}%
and nearer to having their square equal to~$2$. We can form an
ascending series of fractions all of which have their squares
less than~$2$, but differing from~$2$ in their later members by
less than any assigned amount. That is to say, suppose I assign
some small amount in advance, say one-billionth, it will be
found that all the terms of our series after a certain one, say the
tenth, have squares that differ from~$2$ by less than this amount.
And if I had assigned a still smaller amount, it might have been
necessary to go further along the series, but we should have
reached sooner or later a term in the series, say the twentieth,
after which all terms would have had squares differing from~$2$
by less than this still smaller amount. If we set to work to
extract the square root of~$2$ by the usual arithmetical rule, we
shall obtain an unending decimal which, taken to so-and-so
many places, exactly fulfils the above conditions. We can
equally well form a descending series of fractions whose squares
are all greater than~$2$, but greater by continually smaller amounts
as we come to later terms of the series, and differing, sooner or
later, by less than any assigned amount. In this way we seem
to be drawing a cordon round the square root of~$2$, and it may
seem difficult to believe that it can permanently escape us.
Nevertheless, it is not by this method that we shall actually
reach the square root of~$2$.

If we divide \emph{all} ratios into two classes, according as their
squares are less than~$2$ or not, we find that, among those whose
squares are \emph{not} less than~$2$, all have their squares greater than~$2$.
There is no maximum to the ratios whose square is less than~$2$,
and no minimum to those whose square is greater than~$2$. There
is no lower limit short of zero to the difference between the
numbers whose square is a little less than~$2$ and the numbers
whose square is a little greater than~$2$. We can, in short, divide
\emph{all} ratios into two classes such that all the terms in one class
are less than all in the other, there is no maximum to the one
class, and there is no minimum to the other. Between these
two classes, where $\sqrt{2}$~ought to be, there is nothing. Thus our
\PageSep{69}
cordon, though we have drawn it as tight as possible, has been
drawn in the wrong place, and has not caught~$\sqrt{2}$.

The above method of dividing all the terms of a series into
two classes, of which the one wholly precedes the other, was
\index{Dedekind}%
brought into prominence by Dedekind,\footnote
  {\Title{Stetigkeit und irrationale Zahlen}, 2nd edition, Brunswick, 1892.}
and is therefore called
a ``Dedekind cut.'' With respect to what happens at the point
of section, there are four possibilities: (1)~there may be a
\index{Section!Dedekindian|EtSeq}%
maximum to the lower section and a minimum to the upper
section, (2)~there may be a maximum to the one and no minimum
to the other, (3)~there may be no maximum to the one, but a
minimum to the other, (4)~there may be neither a maximum to
the one nor a minimum to the other. Of these four cases, the
first is illustrated by any series in which there are consecutive
terms: in the series of integers, for instance, a lower section
must end with some number~$n$ and the upper section must
then begin with~$n + 1$. The second case will be illustrated
in the series of ratios if we take as our lower section all ratios
up to and including~$1$, and in our upper section all ratios greater
than~$1$. The third case is illustrated if we take for our lower
section all ratios less than~$1$, and for our upper section all ratios
from $1$ upward (including $1$~itself). The fourth case, as we have
seen, is illustrated if we put in our lower section all ratios whose
square is less than~$2$, and in our upper section all ratios whose
square is greater than~$2$.

We may neglect the first of our four cases, since it only arises
\index{Limit|EtSeq}%
in series where there are consecutive terms. In the second of
our four cases, we say that the maximum of the lower section
is the \emph{lower limit} of the upper section, or of any set of terms
chosen out of the upper section in such a way that no term of
the upper section is before all of them. In the third of our
four cases, we say that the minimum of the upper section is the
upper limit of the lower section, or of any set of terms chosen
out of the lower section in such a way that no term of the lower
section is after all of them. In the fourth case, we say that
\PageSep{70}%
there is a ``gap'': neither the upper section nor the lower has
\index{Gap, Dedekindian|EtSeq}%
a limit or a last term. In this case, we may also say that we
have an ``irrational section,'' since sections of the series of ratios
have ``gaps'' when they correspond to irrationals.

What delayed the true theory of irrationals was a mistaken
belief that there must be ``limits'' of series of ratios. The
notion of ``limit'' is of the utmost importance, and before
proceeding further it will be well to define it.

A term~$x$ is said to be an ``upper limit'' of a class~$\alpha$ with
respect to a relation~$P$ if (1)~$\alpha$~has no maximum in~$P$, (2)~every
\index{Maximum}%
member of~$\alpha$ which belongs to the field of~$P$ precedes~$x$, (3)~every
member of the field of~$P$ which precedes~$x$ precedes some member
of~$\alpha$. (By ``precedes'' we mean ``has the relation~$P$ to.'')

{\Loosen This presupposes the following definition of a ``maximum'':---}

A term~$x$ is said to be a ``maximum'' of a class~$\alpha$ with respect
to a relation~$P$ if $x$~is a member of~$\alpha$ and of the field of~$P$ and does
not have the relation~$P$ to any other member of~$\alpha$.

These definitions do not demand that the terms to which
they are applied should be quantitative. For example, given
a series of moments of time arranged by earlier and later, their
``maximum'' (if any) will be the last of the moments; but if
they are arranged by later and earlier, their ``maximum'' (if
any) will be the first of the moments.

The ``minimum'' of a class with respect to~$P$ is its maximum
\index{Minimum}%
with respect to the converse of~$P$; and the ``lower limit'' with
respect to~$P$ is the upper limit with respect to the converse of~$P$.

The notions of limit and maximum do not essentially demand
that the relation in respect to which they are defined should
be serial, but they have few important applications except to
cases when the relation is serial or quasi-serial. A notion which
is often important is the notion ``upper limit or maximum,''
to which we may give the name ``upper boundary.'' Thus the
\index{Boundary}%
``upper boundary'' of a set of terms chosen out of a series is
their last member if they have one, but, if not, it is the first
term after all of them, if there is such a term. If there is neither
\PageSep{71}
a maximum nor a limit, there is no upper boundary. The
``lower boundary'' is the lower limit or minimum.

Reverting to the four kinds of Dedekind section, we see that
in the case of the first three kinds each section has a boundary
(upper or lower as the case may be), while in the fourth kind
neither has a boundary. It is also clear that, whenever the
lower section has an upper boundary, the upper section has
a lower boundary. In the second and third cases, the two
boundaries are identical; in the first, they are consecutive
terms of the series.

A series is called ``Dedekindian'' when every section has a
\index{Series!Dedekindian}%
boundary, upper or lower as the case may be.

We have seen that the series of ratios in order of magnitude
\index{Ratios}%
is not Dedekindian.

From the habit of being influenced by spatial imagination,
\index{Postulates}%
people have supposed that series \emph{must} have limits in cases where
it seems odd if they do not. Thus, perceiving that there was
no \emph{rational} limit to the ratios whose square is less than~$2$, they
allowed themselves to ``postulate'' an \emph{irrational} limit, which
was to fill the Dedekind gap. Dedekind, in the above-mentioned
work, set up the axiom that the gap must always be filled, \ie\
that every section must have a boundary. It is for this reason
that series where his axiom is verified are called ``Dedekindian.''
But there are an infinite number of series for which it is not
verified.

The method of ``postulating'' what we want has many advantages;
they are the same as the advantages of theft over honest
toil. Let us leave them to others and proceed with our honest toil.

It is clear that an irrational Dedekind cut in some way ``represents''
an irrational. In order to make use of this, which to
begin with is no more than a vague feeling, we must find some
way of eliciting from it a precise definition; and in order to do
this, we must disabuse our minds of the notion that an irrational
must be the limit of a set of ratios. Just as ratios whose denominator
is~$1$ are not identical with integers, so those rational
\PageSep{72}%
numbers which can be greater or less than irrationals, or can
\index{Irrationals}%
have irrationals as their limits, must not be identified with ratios.
We have to define a new kind of numbers called ``real numbers,''
of which some will be rational and some irrational. Those that
are rational ``correspond'' to ratios, in the same kind of way
in which the ratio~$n/1$ corresponds to the integer~$n$; but they are
not the same as ratios. In order to decide what they are to be,
let us observe that an irrational is represented by an irrational
cut, and a cut is represented by its lower section. Let us confine
ourselves to cuts in which the lower section has no maximum;
in this case we will call the lower section a ``segment.'' Then
those segments that correspond to ratios are those that consist
\index{Segments}%
of all ratios less than the ratio they correspond to, which is
their boundary; while those that represent irrationals are those
that have no boundary. Segments, both those that have
boundaries and those that do not, are such that, of any two
pertaining to one series, one must be part of the other; hence
they can all be arranged in a series by the relation of whole and
part. A series in which there are Dedekind gaps, \ie\ in which
there are segments that have no boundary, will give rise to more
segments than it has terms, since each term will define a segment
having that term for boundary, and then the segments without
boundaries will be extra.

We are now in a position to define a real number and an
\index{Number!irrational}%
\index{Number!real}%
irrational number.

A ``real number'' is a segment of the series of ratios in order
of magnitude.

An ``irrational number'' is a segment of the series of ratios
which has no boundary.

A ``rational real number'' is a segment of the series of ratios
which has a boundary.

Thus a rational real number consists of all ratios less than a
certain ratio, and it is the rational real number corresponding
to that ratio. The real number~$1$, for instance, is the class of
proper fractions.
\PageSep{73}

In the cases in which we naturally supposed that an irrational
must be the limit of a set of ratios, the truth is that it is the limit
of the corresponding set of rational real numbers in the series
of segments ordered by whole and part. For example, $\sqrt{2}$~is
the upper limit of all those segments of the series of ratios that
correspond to ratios whose square is less than~$2$. More simply
still, $\sqrt{2}$~is the segment \emph{consisting} of all those ratios whose square
is less than~$2$.

It is easy to prove that the series of segments of any series
\index{Series!Dedekindian}%
is Dedekindian. For, given any set of segments, their boundary
will be their logical sum, \ie\ the class of all those terms that
belong to at least one segment of the set.\footnote
  {For a fuller treatment of the subject of segments and Dedekindian
  relations, see \Title{Principia Mathematica}, vol.~ii.\ *~210--214. For a fuller
  treatment of real numbers, see \ibid, vol.~iii.\ *~310~ff., and \Title{Principles of
  Mathematics}, chaps.\ xxxiii.\ and~xxxiv.}

The above definition of real numbers is an example of ``construction''
\index{Construction, method of}%
\index{Postulates}%
as against ``postulation,'' of which we had another
example in the definition of cardinal numbers. The great
advantage of this method is that it requires no new assumptions,
but enables us to proceed deductively from the original apparatus
of logic.

There is no difficulty in defining addition and multiplication
for real numbers as above defined. Given two real numbers
$\mu$ and~$\nu$, each being a class of ratios, take any member of~$\mu$ and
any member of~$\nu$ and add them together according to the rule
for the addition of ratios. Form the class of all such sums
obtainable by varying the selected members of $\mu$ and~$\nu$. This
gives a new class of ratios, and it is easy to prove that this new
class is a segment of the series of ratios. We define it as the
sum of $\mu$ and~$\nu$. We may state the definition more shortly as
follows:---

The \emph{arithmetical sum of two real numbers} is the class of the
arithmetical sums of a member of the one and a member of the
other chosen in all possible ways.
\PageSep{74}%

We can define the arithmetical product of two real numbers
in exactly the same way, by multiplying a member of the one by
a member of the other in all possible ways. The class of ratios
thus generated is defined as the product of the two real numbers.
(In all such definitions, the series of ratios is to be defined as
excluding $0$~and infinity.)

There is no difficulty in extending our definitions to positive
and negative real numbers and their addition and multiplication.

It remains to give the definition of complex numbers.

Complex numbers, though capable of a geometrical interpretation,
are not demanded by geometry in the same imperative way
\index{Geometry}%
in which irrationals are demanded. A ``complex'' number means
\index{Number!complex|EtSeq}%
a number involving the square root of a negative number, whether
integral, fractional, or real. Since the square of a negative
number is positive, a number whose square is to be negative has
to be a new sort of number. Using the letter~$i$ for the square
root of~$-1$, any number involving the square root of a negative
number can be expressed in the form $x + yi$, where $x$~and $y$ are
real. The part~$yi$ is called the ``imaginary'' part of this number,
$x$~being the ``real'' part. (The reason for the phrase ``real
numbers'' is that they are contrasted with such as are ``imaginary.'')
Complex numbers have been for a long time habitually
used by mathematicians, in spite of the absence of any precise
definition. It has been simply assumed that they would obey
the usual arithmetical rules, and on this assumption their employment
has been found profitable. They are required less for
geometry than for algebra and analysis. We desire, for example,
to be able to say that every quadratic equation has two roots,
and every cubic equation has three, and so on. But if we are
confined to real numbers, such an equation as $x^{2} + = 0$ has no
roots, and such an equation as $x^{3} - 1 = 0$ has only one. Every
generalisation of number has first presented itself as needed for
some simple problem: negative numbers were needed in order
that subtraction might be always possible, since otherwise $a - b$
would be meaningless if $a$~were less than~$b$; fractions were needed
\PageSep{75}
in order that division might be always possible; and complex
numbers are needed in order that extraction of roots and solution
of equations may be always possible. But extensions of
number are not \emph{created} by the mere need for them: they are
created by the definition, and it is to the definition of complex
numbers that we must now turn our attention.

A complex number may be regarded and defined as simply an
ordered couple of real numbers. Here, as elsewhere, many
definitions are possible. All that is necessary is that the definitions
adopted shall lead to certain properties. In the case of
complex numbers, if they are defined as ordered couples of real
numbers, we secure at once some of the properties required,
namely, that two real numbers are required to determine a complex
number, and that among these we can distinguish a first
and a second, and that two complex numbers are only identical
when the first real number involved in the one is equal to the
first involved in the other, and the second to the second. What
is needed further can be secured by defining the rules of addition
and multiplication. We are to have
\begin{alignat*}{2}
&amp;(x + yi) + (x' + y'i) &amp;&amp;= (x + x') + (y + y')i\Add{,} \\
&amp;(x + yi) (x' + y'i) &amp;&amp;= (xx' - yy') + (xy' + x'y)i.
\end{alignat*}
{\Loosen Thus we shall define that, given two ordered couples of real
numbers, $(x, y)$ and $(x', y')$, their sum is to be the couple \mbox{$(x + x', y + y')$},
and their product is to be the couple \mbox{$(xx' - yy', xy' + x'y)$}.}
By these definitions we shall secure that our ordered couples
shall have the properties we desire. For example, take the
product of the two couples $(0, y)$ and~$(0, y')$. This will, by the
above rule, be the couple $(-yy', 0)$. Thus the square of the
couple~$(0, 1)$ will be the couple~$(-1, 0)$. Now those couples in
which the second term is~$0$ are those which, according to the usual
nomenclature, have their imaginary part zero; in the notation
$x + yi$, they are $x + 0i$, which it is natural to write simply~$x$. Just
as it is natural (but erroneous) to identify ratios whose denominator
is unity with integers, so it is natural (but erroneous)
\PageSep{76}%
to identify complex numbers whose imaginary part is zero with
real numbers. Although this is an error in theory, it is a convenience
in practice; ``$x + 0i$'' may be replaced simply by~``$x$''
and ``$0 + yi$'' by~``$yi$,'' provided we remember that the ``$x$'' is
not really a real number, but a special case of a complex number.
And when $y$ is~$1$, ``$yi$''~may of course be replaced by~``$i$.'' Thus
the couple $(0, 1)$ is represented by~$i$, and the couple $(-1, 0)$ is
represented by~$-1$. Now our rules of multiplication make the
square of~$(0, 1)$ equal to~$(-1, 0)$, \ie\ the square of~$i$ is~$-1$. This
is what we desired to secure. Thus our definitions serve all
necessary purposes.

It is easy to give a geometrical interpretation of complex
numbers in the geometry of the plane. This subject was agreeably
\index{Clifford, W. K.}%
expounded by W.~K. Clifford in his \Title{Common Sense of the
Exact Sciences}, a book of great merit, but written before the
importance of purely logical definitions had been realised.

Complex numbers of a higher order, though much less useful
and important than those what we have been defining, have
certain uses that are not without importance in geometry, as
may be seen, for example, in Dr~Whitehead's \Title{Universal Algebra}.
\index{Whitehead}%
The definition of complex numbers of order~$n$ is obtained by an
obvious extension of the definition we have given. We define a
complex number of order~$n$ as a one-many relation whose domain
consists of certain real numbers and whose converse domain
consists of the integers from $1$ to~$n$.\footnote
  {Cf.\ \Title{Principles of Mathematics}, §\;360, p.~379.}
This is what would ordinarily
be indicated by the notation $(x_{1}, x_{2}, x_{3}, \dots\Add{,} x_{n})$, where the
suffixes denote correlation with the integers used as suffixes, and
the correlation is one-many, not necessarily one-one, because $x_{r}$
and~$x_{s}$ may be equal when $r$ and~$s$ are not equal. The above
definition, with a suitable rule of multiplication, will serve all
purposes for which complex numbers of higher orders are needed.

We have now completed our review of those extensions of
number which do not involve infinity. The application of number
to infinite collections must be our next topic.
\PageSep{77}


\Chapter{VIII}{Infinite Cardinal Numbers}

\First{The} definition of cardinal numbers which we gave in \ChapRef{II}.\
\index{Infinity!of cardinals|EtSeq}%
\index{Number!cardinal|EtSeq}%
\index{Number!infinite|EtSeq}%
was applied in \ChapRef{III}.\ to finite numbers, \ie\ to the ordinary
natural numbers. To these we gave the name ``inductive
numbers,'' because we found that they are to be defined as
numbers which obey mathematical induction starting from~$0$.
But we have not yet considered collections which do not have an
inductive number of terms, nor have we inquired whether such
collections can be said to have a number at all. This is an
ancient problem, which has been solved in our own day, chiefly
by Georg Cantor. In the present chapter we shall attempt to
\index{Cantor, Georg}%
explain the theory of transfinite or infinite cardinal numbers as
it results from a combination of his discoveries with those of
Frege on the logical theory of numbers.
\index{Frege}%

It cannot be said to be \emph{certain} that there are in fact any infinite
collections in the world. The assumption that there are is what
we call the ``axiom of infinity.'' Although various ways suggest
\index{Infinity, axiom of}%
themselves by which we might hope to prove this axiom, there
is reason to fear that they are all fallacious, and that there is no
conclusive logical reason for believing it to be true. At the same
time, there is certainly no logical reason \emph{against} infinite collections,
and we are therefore justified, in logic, in investigating the hypothesis
that there are such collections. The practical form of this
hypothesis, for our present purposes, is the assumption that, if
$n$~is any inductive number, $n$~is not equal to~$n + 1$. Various
subtleties arise in identifying this form of our assumption with
\PageSep{78}%
the form that asserts the existence of infinite collections; but
we will leave these out of account until, in a later chapter, we
come to consider the axiom of infinity on its own account. For
the present we shall merely assume that, if $n$~is an inductive
number, $n$~is not equal to~$n + 1$. This is involved in Peano's
assumption that no two inductive numbers have the same successor;
for, if $n = n + 1$, then $n - 1$ and~$n$ have the same successor,
namely~$n$. Thus we are assuming nothing that was not involved
in Peano's primitive propositions.
\index{Peano}%

Let us now consider the collection of the inductive numbers
themselves. This is a perfectly well-defined class. In the first
place, a cardinal number is a set of classes which are all similar
to each other and are not similar to anything except each other.
We then define as the ``inductive numbers'' those among
\index{Number!inductive}%
cardinals which belong to the posterity of~$0$ with respect to the
relation of $n$ to~$n + 1$, \ie\ those which possess every property
possessed by~$0$ and by the successors of possessors, meaning by
the ``successor'' of~$n$ the number~$n + 1$. Thus the class of
``inductive numbers'' is perfectly definite. By our general
definition of cardinal numbers, the number of terms in the class
of inductive numbers is to be defined as ``all those classes that
are similar to the class of inductive numbers''---\ie\ this set of
classes \emph{is} the number of the inductive numbers according to our
definitions.

Now it is easy to see that this number is not one of the inductive
numbers. If $n$~is any inductive number, the number of numbers
from $0$ to~$n$ (both included) is~$n + 1$; therefore the total number
of inductive numbers is greater than~$n$, no matter which of the
inductive numbers $n$~may be. If we arrange the inductive
numbers in a series in order of magnitude, this series has no last
term; but if $n$~is an inductive number, every series whose field
has $n$~terms has a last term, as it is easy to prove. Such differences
might be multiplied \Foreign{ad~lib}. Thus the number of inductive
numbers is a new number, different from all of them, not possessing
all inductive properties. It may happen that $0$~has a certain
\PageSep{79}%
property, and that if $n$~has it so has~$n + 1$, and yet that this new
number does not have it. The difficulties that so long delayed
the theory of infinite numbers were largely due to the fact that
some, at least, of the inductive properties were wrongly judged
to be such as \emph{must} belong to all numbers; indeed it was thought
that they could not be denied without contradiction. The first
step in understanding infinite numbers consists in realising the
%[** TN: [sic] "mistakenness"]
mistakenness of this view.

The most noteworthy and astonishing difference between an
inductive number and this new number is that this new number
is unchanged by adding~$1$ or subtracting~$1$ or doubling or halving
or any of a number of other operations which we think of as
necessarily making a number larger or smaller. The fact of being
not altered by the addition of~$1$ is used by Cantor for the definition
\index{Cantor, Georg}%
of what he calls ``transfinite'' cardinal numbers; but for
various reasons, some of which will appear as we proceed, it is
better to define an infinite cardinal number as one which does
not possess all inductive properties, \ie\ simply as one which is
not an inductive number. Nevertheless, the property of being
unchanged by the addition of~$1$ is a very important one, and we
must dwell on it for a time.

To say that a class has a number which is not altered by the
addition of~$1$ is the same thing as to say that, if we take a term~$x$
which does not belong to the class, we can find a one-one relation
whose domain is the class and whose converse domain is obtained
by adding~$x$ to the class. For in that case, the class is similar
to the sum of itself and the term~$x$, \ie\ to a class having one extra
term; so that it has the same number as a class with one extra
term, so that if $n$~is this number, $n = n + 1$. In this case, we shall
also have $n = n - 1$, \ie\ there will be one-one relations whose
\index{Relations!one-one}%
domains consist of the whole class and whose converse domains
consist of just one term short of the whole class. It can be shown
that the cases in which this happens are the same as the apparently
more general cases in which \emph{some} part (short of the whole) can be
put into one-one relation with the whole. When this can be done,
\PageSep{80}%
the correlator by which it is done may be said to ``reflect'' the
whole class into a part of itself; for this reason, such classes will
\index{Classes!reflexive}%
be called ``reflexive.'' Thus:

A ``reflexive'' class is one which is similar to a proper part
of itself. (A ``proper part'' is a part short of the whole.)

A ``reflexive'' cardinal number is the cardinal number of a
\index{Number!reflexive}%
reflexive class.

We have now to consider this property of reflexiveness.

One of the most striking instances of a ``reflexion'' is Royce's
\index{Maps}%
\index{Royce}%
illustration of the map: he imagines it decided to make a map
of England upon a part of the surface of England. A map, if
it is accurate, has a perfect one-one correspondence with its
original; thus our map, which is part, is in one-one relation with
the whole, and must contain the same number of points as the
whole, which must therefore be a reflexive number. Royce is
interested in the fact that the map, if it is correct, must contain
a map of the map, which must in turn contain a map of the map
of the map, and so on \Foreign{ad infinitum}. This point is interesting,
but need not occupy us at this moment. In fact, we shall do
well to pass from picturesque illustrations to such as are more
completely definite, and for this purpose we cannot do better
than consider the number-series itself.

The relation of $n$ to~$n + 1$, confined to inductive numbers, is
one-one, has the whole of the inductive numbers for its domain,
and all except~$0$ for its converse domain. Thus the whole class
of inductive numbers is similar to what the same class becomes
when we omit~$0$. Consequently it is a ``reflexive'' class according
to the definition, and the number of its terms is a ``reflexive''
number. Again, the relation of $n$ to~$2n$, confined to inductive
numbers, is one-one, has the whole of the inductive numbers for
its domain, and the even inductive numbers alone for its converse
domain. Hence the total number of inductive numbers is the
same as the number of even inductive numbers. This property
was used by Leibniz (and many others) as a proof that infinite
\index{Leibniz}%
numbers are impossible; it was thought self-contradictory that
\PageSep{81}%
``the part should be equal to the whole.'' But this is one of those
phrases that depend for their plausibility upon an unperceived
vagueness: the word ``equal'' has many meanings, but if it is
taken to mean what we have called ``similar,'' there is no contradiction,
since an infinite collection can perfectly well have parts
similar to itself. Those who regard this as impossible have,
unconsciously as a rule, attributed to numbers in general properties
which can only be proved by mathematical induction,
and which only their familiarity makes us regard, mistakenly,
as true beyond the region of the finite.

Whenever we can ``reflect'' a class into a part of itself, the
same relation will necessarily reflect that part into a smaller
part, and so on \Foreign{ad~infinitum}. For example, we can reflect,
as we have just seen, all the inductive numbers into the even
numbers; we can, by the same relation (that of $n$ to~$2n$) reflect
the even numbers into the multiples of~$4$, these into the multiples
of~$8$, and so on. This is an abstract analogue to Royce's problem
of the map. The even numbers are a ``map'' of all the inductive
numbers; the multiples of~$4$ are a map of the map; the multiples
of~$8$ are a map of the map of the map; and so on. If we had
applied the same process to the relation of $n$ to~$n + 1$, our ``map''
would have consisted of all the inductive numbers except~$0$;
the map of the map would have consisted of all from $2$ onward,
the map of the map of the map of all from $3$ onward; and so on.
The chief use of such illustrations is in order to become familiar
with the idea of reflexive classes, so that apparently paradoxical
arithmetical propositions can be readily translated into the
language of reflexions and classes, in which the air of paradox
is much less.

It will be useful to give a definition of the number which is
\index{Peano}%
\index{Progressions|EtSeq}%
that of the inductive cardinals. For this purpose we will
first define the kind of series exemplified by the inductive cardinals
in order of magnitude. The kind of series which is called a
``progression'' has already been considered in \ChapRef{I}. It is a
series which can be generated by a relation of consecutiveness:
\index{Consecutiveness}%
\PageSep{82}%
every member of the series is to have a successor, but there is
to be just one which has no predecessor, and every member of
the series is to be in the posterity of this term with respect to
the relation ``immediate predecessor.'' These characteristics
%[** TN: Footnote mark precedes dash elsewhere]
may be summed up in the following definition:\footnotemark---
\footnotetext{Cf.\ \Title{Principia Mathematica}, vol.~ii.\ *~123.}

A ``progession'' is a one-one relation such that there is just
one term belonging to the domain but not to the converse domain,
and the domain is identical with the posterity of this one term.

It is easy to see that a progression, so defined, satisfies Peano's
five axioms. The term belonging to the domain but not to the
converse domain will be what he calls~``$0$''; the term to which
a term has the one-one relation will be the ``successor'' of the
term; and the domain of the one-one relation will be what
he calls ``number.'' Taking his five axioms in turn, we have
the following translations:---

(1) ``$0$~is a number'' becomes: ``The member of the domain
which is not a member of the converse domain is a member of
the domain.'' This is equivalent to the existence of such a
member, which is given in our definition. We will call this
member ``the first term.''

(2) ``The successor of any number is a number'' becomes:
``The term to which a given member of the domain has the relation
in question is again a member of the domain.'' This is
proved as follows: By the definition, every member of the
domain is a member of the posterity of the first term; hence
the successor of a member of the domain must be a member of
the posterity of the first term (because the posterity of a term
always contains its own successors, by the general definition of
posterity), and therefore a member of the domain, because by
the definition the posterity of the first term is the same as the
domain.

(3) ``No two numbers have the same successor.'' This is
only to say that the relation is one-many, which it is by definition
(being one-one).
\PageSep{83}%

(4) ``$0$~is not the successor of any number'' becomes: ``The
first term is not a member of the converse domain,'' which is
again an immediate result of the definition.

(5) This is mathematical induction, and becomes: ``Every
member of the domain belongs to the posterity of the first term,''
which was part of our definition.

Thus progressions as we have defined them have the five
formal properties from which Peano deduces arithmetic. It is
easy to show that two progessions are ``similar'' in the sense
defined for similarity of relations in \ChapRef{VI}. We can, of
\index{Similarity!of relations}%
course, derive a relation which is serial from the one-one relation
by which we define a progression: the method used is that
explained in \ChapRef{IV}., and the relation is that of a term to
a member of its proper posterity with respect to the original
one-one relation.

Two transitive asymmetrical relations which generate progressions
are similar, for the same reasons for which the corresponding
one-one relations are similar. The class of all such
transitive generators of progressions is a ``serial number'' in
the sense of \ChapRef{VI}.; it is in fact the smallest of infinite
serial numbers, the number to which Cantor has given the name~$\omega$,
by which he has made it famous.

But we are concerned, for the moment, with \emph{cardinal} numbers.
\index{Alephs}%
Since two progressions are similar relations, it follows that their
domains (or their fields, which are the same as their domains)
are similar classes. The domains of progressions form a cardinal
number, since every class which is similar to the domain of a
progression is easily shown to be itself the domain of a progression.
This cardinal number is the smallest of the infinite cardinal
numbers; it is the one to which Cantor has appropriated the
Hebrew Aleph with the suffix~$0$, to distinguish it from larger
infinite cardinals, which have other suffixes. Thus the name of
the smallest of infinite cardinals is~$\aleph_{0}$.

To say that a class has $\aleph_{0}$~terms is the same thing as to say
that it is a member of~$\aleph_{0}$, and this is the same thing as to say
\PageSep{84}%
that the members of the class can be arranged in a progression.
It is obvious that any progression remains a progression if we
omit a finite number of terms from it, or every other term, or
all except every tenth term or every hundredth term. These
methods of thinning out a progression do not make it cease to
be a progression, and therefore do not diminish the number of
its terms, which remains~$\aleph_{0}$. In fact, any selection from a progression
is a progression if it has no last term, however sparsely
it may be distributed. Take (say) inductive numbers of the form~$n^{n}$,
or~$n^{n^{n}}$. Such numbers grow very rare in the higher parts
of the number series, and yet there are just as many of them as
there are inductive numbers altogether, namely,~$\aleph_{0}$.

Conversely, we can add terms to the inductive numbers without
increasing their number. Take, for example, ratios. One
\index{Ratios}%
might be inclined to think that there must be many more ratios
than integers, since ratios whose denominator is~$1$ correspond
to the integers, and seem to be only an infinitesimal proportion
of ratios. But in actual fact the number of ratios (or fractions)
is exactly the same as the number of inductive numbers, namely,~$\aleph_{0}$.
This is easily seen by arranging ratios in a series on the
following plan: If the sum of numerator and denominator in
one is less than in the other, put the one before the other; if
the sum is equal in the two, put first the one with the smaller
numerator. This gives us the series
\[
1,\ 1/2,\ 2,\ 1/3,\ 3,\ 1/4,\ 2/3,\ 3/2,\ 4,\ 1/5,\ \dots\Add{.}
\]
This series is a progression, and all ratios occur in it sooner or
later. Hence we can arrange all ratios in a progression, and
their number is therefore~$\aleph_{0}$.

It is not the case, however, that \emph{all} infinite collections have
\index{Number!real}%
$\aleph_{0}$~terms. The number of real numbers, for example, is greater
than~$\aleph_{0}$; it is, in fact,~$2^{\aleph_{0}}$, and it is not hard to prove that $2^{n}$~is
greater than~$n$ even when $n$~is infinite. The easiest way of
proving this is to prove, first, that if a class has $n$~members, it
contains $2^{n}$~sub-classes---in other words, that there are $2^{n}$~ways
\index{Sub-classes|EtSeq}%
\PageSep{85}%
\index{Cantor, Georg|FNote}%
of selecting some of its members (including the extreme cases
where we select all or none); and secondly, that the number of
sub-classes contained in a class is always greater than the number
of members of the class. Of these two propositions, the first
is familiar in the case of finite numbers, and is not hard to extend
to infinite numbers. The proof of the second is so simple and
so instructive that we shall give it:

In the first place, it is clear that the number of sub-classes
of a given class (say~$\alpha$) is at least as great as the number of
members, since each member constitutes a sub-class, and we thus
have a correlation of all the members with some of the sub-classes.
Hence it follows that, if the number of sub-classes is
not \emph{equal} to the number of members, it must be \emph{greater}. Now
it is easy to prove that the number is not equal, by showing that,
given any one-one relation whose domain is the members and
whose converse domain is contained among the set of sub-classes,
there must be at least one sub-class not belonging to
the converse domain. The proof is as follows:\footnote
  {This proof is taken from Cantor, with some simplifications: see
  \Title{Jahresbericht der deutschen Mathematiker-Vereinigung}, i.~(1892), p.~77.}
When a one-one
correlation~$R$ is established between all the members of~$\alpha$
and some of the sub-classes, it may happen that a given member~$x$
is correlated with a sub-class of which it is a member; or,
again, it may happen that $x$~is correlated with a sub-class of
which it is not a member. Let us form the whole class, $\beta$~say,
of those members~$x$ which are correlated with sub-classes of which
they are not members. This is a sub-class of~$\alpha$, and it is not
correlated with any member of~$\alpha$. For, taking first the members
of~$\beta$, each of them is (by the definition of~$\beta$) correlated with
some sub-class of which it is not a member, and is therefore not
correlated with~$\beta$. Taking next the terms which are not members
of~$\beta$, each of them (by the definition of~$\beta$) is correlated with
some sub-class of which it is a member, and therefore again
is not correlated with~$\beta$. Thus no member of~$\alpha$ is correlated
with~$\beta$. Since $R$~was \emph{any} one-one correlation of all members
\PageSep{86}%
with some sub-classes, it follows that there is no correlation
of all members with \emph{all} sub-classes. It does not matter to the
proof if $\beta$~has no members: all that happens in that case is that
the sub-class which is shown to be omitted is the null-class.
Hence in any case the number of sub-classes is not equal to the
number of members, and therefore, by what was said earlier,
it is greater. Combining this with the proposition that, if $n$~is
the number of members, $2^{n}$~is the number of sub-classes, we have
the theorem that $2^{n}$~is always greater than~$n$, even when $n$~is
infinite.

It follows from this proposition that there is no maximum
to the infinite cardinal numbers. However great an infinite
number~$n$ may be, $2^{n}$~will be still greater. The arithmetic of
infinite numbers is somewhat surprising until one becomes
accustomed to it. We have, for example,
\begin{align*}
\aleph_{0} + 1 &amp;= \aleph_{0}, \\
\aleph_{0} + n &amp;= \aleph_{0}, \text{where $n$ is any inductive number,} \\
\aleph_{0}^{2} &amp;= \aleph_{0}.
\end{align*}
(This follows from the case of the ratios, for, since a ratio is
determined by a pair of inductive numbers, it is easy to see that
the number of ratios is the square of the number of inductive
numbers, \ie\ it is~$\aleph_{0}^{2}$; but we saw that it is also~$\aleph_{0}$.)
%[** TN: Reformatted slightly from the original]
\begin{alignat*}{2}
&amp;&amp;\llap{$\aleph_{0}^{n}$} &amp;= \aleph_{0}, \text{where $n$ is any inductive number.} \\
&amp;\text{(This follows from } \aleph_{0}^{2} &amp;&amp;= \aleph_{0} \text{ by induction; for if $\aleph_{0}^{n} = \aleph_{0}$,} \\
&amp;\text{then} &amp;\llap{$\aleph_{0}^{n+1}$} &amp;= \aleph_{0}^{2} = \aleph_{0}.)
\end{alignat*}
But
\[
2^{\aleph_{0}} &gt; \aleph_{0}.
\]
In fact, as we shall see later, $2^{\aleph_{0}}$~is a very important number,
namely, the number of terms in a series which has ``continuity''
\index{Continuity}%
in the sense in which this word is used by Cantor. Assuming
\index{Cantor, Georg}%
space and time to be continuous in this sense (as we commonly
\index{Space}%
\index{Time}%
do in analytical geometry and kinematics), this will be the
\index{Geometry!analytical}%
number of points in space or of instants in time; it will also be
the number of points in any finite portion of space, whether
\PageSep{87}%
%[** TN: Entry moved to page 32]
%\index{Diversity}%
line, area, or volume. After~$\aleph_{0}$, $2^{\aleph_{0}}$~is the most important and
interesting of infinite cardinal numbers.

Although addition and multiplication are always possible
with infinite cardinals, subtraction and division no longer give
\index{Subtraction}%
definite results, and cannot therefore be employed as they are
employed in elementary arithmetic. Take subtraction to begin
with: so long as the number subtracted is finite, all goes well;
if the other number is reflexive, it remains unchanged. Thus
$\aleph_{0} - n = \aleph_{0}$, if $n$~is finite; so far, subtraction gives a perfectly
definite result. But it is otherwise when we subtract~$\aleph_{0}$ from
itself; we may then get any result, from $0$ up to~$\aleph_{0}$. This is
easily seen by examples. From the inductive, numbers, take
away the following collections of $\aleph_{0}$~terms:---

(1) All the inductive numbers---remainder, zero.

(2) All the inductive numbers from $n$~onwards---remainder,
the numbers from $0$ to~$n - 1$, numbering $n$~terms in all.

(3) All the odd numbers---remainder, all the even numbers,
numbering $\aleph_{0}$~terms.

All these are different ways of subtracting $\aleph_{0}$ from~$\aleph_{0}$, and
all give different results.

As regards division, very similar results follow from the fact
that $\aleph_{0}$~is unchanged when multiplied by $2$~or $3$ or any finite
number~$n$ or by~$\aleph_{0}$. It follows that $\aleph_{0}$~divided by~$\aleph_{0}$ may have
any value from $1$ up to~$\aleph_{0}$.

From the ambiguity of subtraction and division it results
that negative numbers and ratios cannot be extended to infinite
numbers. Addition, multiplication, and exponentiation proceed
quite satisfactorily, but the inverse operations---subtraction,
division, and extraction of roots---are ambiguous, and the notions
that depend upon them fail when infinite numbers are concerned.

The characteristic by which we defined finitude was mathematical
\index{Induction, mathematical}%
induction, \ie\ we defined a number as finite when it
obeys mathematical induction starting from~$0$, and a class as
finite when its number is finite. This definition yields the sort
of result that a definition ought to yield, namely, that the finite
\PageSep{88}%
numbers are those that occur in the ordinary number-series
$0$,~$1$, $2$, $3$,~\dots\Add{.} But in the present chapter, the infinite numbers
we have discussed have not merely been non-inductive:
they have also been \emph{reflexive}. Cantor used reflexiveness as the
\emph{definition} of the infinite, and believes that it is equivalent to
non-inductiveness; that is to say, he believes that every class
and every cardinal is either inductive or reflexive. This may be
true, and may very possibly be capable of proof; but the proofs
hitherto offered by Cantor and others (including the present
author in former days) are fallacious, for reasons which will be
explained when we come to consider the ``multiplicative axiom.''
At present, it is not known whether there are classes and cardinals
which are neither reflexive nor inductive. If $n$~were such a
cardinal, we should not have $n = n + 1$, but $n$~would not be one
of the ``natural numbers,'' and would be lacking in some of the
inductive properties. All \emph{known} infinite classes and cardinals
are reflexive; but for the present it is well to preserve an open
mind as to whether there are instances, hitherto unknown, of
classes and cardinals which are neither reflexive nor inductive.
Meanwhile, we adopt the following definitions:---

A \emph{finite} class or cardinal is one which is \emph{inductive}.

An \emph{infinite} class or cardinal is one which is \emph{not inductive}.
\index{Number!non-inductive}%
All \emph{reflexive} classes and cardinals are infinite; but it is not known
at present whether all infinite classes and cardinals are reflexive.
We shall return to this subject in \ChapRef{XII}.
\PageSep{89}%


\Chapter{IX}{Infinite Series and Ordinals}
\index{Series!infinite|EtSeq}%

\First{An} ``infinite series'' may be defined as a series of which the field
\index{Infinity!and series and ordinals|EtSeq}%
is an infinite class. We have already had occasion to consider
one kind of infinite series, namely, progressions. In this chapter
we shall consider the subject more generally.

The most noteworthy characteristic of an infinite series is
that its serial number can be altered by merely re-arranging
its terms. In this respect there is a certain oppositeness between
cardinal and serial numbers. It is possible to keep the cardinal
number of a reflexive class unchanged in spite of adding terms
to it; on the other hand, it is possible to change the serial
number of a series without adding or taking away any terms,
by mere re-arrangement. At the same time, in the case of any
infinite series it is also possible, as with cardinals, to add terms
without altering the serial number: everything depends upon
the way in which they are added.

In order to make matters clear, it will be best to begin with
examples. Let us first consider various different kinds of series
which can be made out of the inductive numbers arranged on
various plans. We start with the series
\[
1,\ 2,\ 3,\ 4,\ \dots\ n,\ \dots,
\]
which, as we have already seen, represents the smallest of infinite
serial numbers, the sort that Cantor calls~$\omega$. Let us
\index{Cantor, Georg}%
proceed to thin out this series by repeatedly performing the
\PageSep{90}%
operation of removing to the end the first even number that
occurs. We thus obtain in succession the various series:
\begin{align*}
&amp;1,\ 3,\ 4,\ 5,\ \dots\ n,\ \dots\ 2, \\
&amp;1,\ 3,\ 5,\ 6,\ \dots\ n + 1,\ \dots\ 2,\ 4, \\
&amp;1,\ 3,\ 5,\ 7,\ \dots\ n + 2,\ \dots\ 2,\ 4,\ 6,
\end{align*}
and so on. If we imagine this process carried on as long as
possible, we finally reach the series
\[
1,\ 3,\ 5,\ 7,\ \dots\ 2n + 1,\ \dots\ 2,\ 4,\ 6,\ 8,\ \dots\ 2n,\ \dots,
\]
in which we have first all the odd numbers and then all the even
numbers.

The serial numbers of these various series are $\omega + 1$, $\omega + 2$,
$\omega + 3$,~\dots\ $2\omega$. Each of these numbers is ``greater'' than any
of its predecessors, in the following sense:---

One serial number is said to be ``greater'' than another if
\index{Greater and less}%
any series having the first number contains a part having the
second number, but no series having the second number contains
a part having the first number.

If we compare the two series
\begin{align*}
&amp;1,\ 2,\ 3,\ 4,\ \dots\ n,\ \dots, \\
&amp;1,\ 3,\ 4,\ 5,\ \dots\ n + 1,\ \dots\ 2,
\end{align*}
we see that the first is similar to the part of the second which
omits the last term, namely, the number~$2$, but the second is
%[** TN: [sic] "...obvious, but is easily demonstrated"]
not similar to any part of the first. (This is obvious, but is
easily demonstrated.) Thus the second series has a greater
serial number than the first, according to the definition---\ie\
$\omega + 1$~is greater than~$\omega$. But if we add a term at the beginning
of a progression instead of the end, we still have a progression.
Thus $1 + \omega = \omega$. Thus $1 + \omega$ is not equal to~$\omega + 1$. This is
characteristic of relation-arithmetic generally: if $\mu$ and~$\nu$ are
two relation-numbers, the general rule is that $\mu + \nu$ is not equal
to~$\nu + \mu$. The case of finite ordinals, in which there is equality,
is quite exceptional.

The series we finally reached just now consisted of first all the
odd numbers and then all the even numbers, and its serial
\PageSep{91}%
number is~$2\omega$. This number is greater than $\omega$ or~$\omega + n$, where
$n$~is finite. It is to be observed that, in accordance with the
general definition of order, each of these arrangements of integers
is to be regarded as resulting from some definite relation. \Eg\
the one which merely removes~$2$ to the end will be defined by
the following relation: ``$x$~and~$y$ are finite integers, and either
$y$~is~$2$ and $x$~is not~$2$, or neither is~$2$ and $x$~is less than~$y$.'' The
one which puts first all the odd numbers and then all the even
ones will be defined by: ``$x$~and~$y$ are finite integers, and either
$x$~is odd and $y$~is even or $x$~is less than~$y$ and both are odd or both
are even.'' We shall not trouble, as a rule, to give these formulæ
in future; but the fact that they \emph{could} be given is essential.

The number which we have called~$2\omega$, namely, the number of
a series consisting of two progressions, is sometimes called $\omega · 2$.
Multiplication, like addition, depends upon the order of the
factors: a progression of couples gives a series such as
\[
x_{1},\ y_{1},\ x_{2},\ y_{2},\ x_{3},\ y_{3},\ \dots\ x_{n},\ y_{n},\ \dots,
\]
which is itself a progression; but a couple of progressions gives
a series which is twice as long as a progression. It is therefore
necessary to distinguish between $2\omega$ and~$\omega · 2$. Usage is variable;
we shall use~$2\omega$ for a couple of progressions and~$\omega · 2$ for a progression
of couples, and this decision of course governs our
general interpretation of ``$\alpha · \beta$'' when $\alpha$~and~$\beta$ are relation-numbers:
``$\alpha · \beta$'' will have to stand for a suitably constructed
sum of a relations each having $\beta$~terms.

We can proceed indefinitely with the process of thinning
out the inductive numbers. For example, we can place first
the odd numbers, then their doubles, then the doubles of these,
and so on. We thus obtain the series
\begin{gather*}
1,\ 3,\ 5,\ 7,\ \dots;\quad
2,\ 6,\ 10,\ 14,\ \dots;\quad
4,\ 12,\ 20,\ 28,\ \dots; \\
8,\ 24,\ 40,\ 56,\ \dots,
\end{gather*}
of which the number is~$\omega^{2}$, since it is a progression of progressions.
Any one of the progressions in this new series can of course be
\PageSep{92}%
thinned out as we thinned out our original progression. We can
proceed to $\omega^{3}$, $\omega^{4}$,~\dots\ $\omega^{\omega}$, and so on; however far we have gone,
we can always go further.

The series of all the ordinals that can be obtained in this way,
\ie\ all that can be obtained by thinning out a progression, is
itself longer than any series that can be obtained by re-arranging
the terms of a progression. (This is not difficult to prove.)
The cardinal number of the class of such ordinals can be shown
to be greater than~$\aleph_{0}$; it is the number which Cantor calls~$\aleph_{1}$.
The ordinal number of the series of all ordinals that can
be made out of an~$\aleph_{0}$, taken in order of magnitude, is called~$\omega_{1}$.
Thus a series whose ordinal number is~$\omega_{1}$ has a field whose
cardinal number is~$\aleph_{1}$.

We can proceed from $\omega_{1}$ and~$\aleph_{1}$ to $\omega_{2}$ and~$\aleph_{2}$ by a process
exactly analogous to that by which we advanced from~$\omega$ and~$\aleph_{0}$
to $\omega_{1}$ and~$\aleph_{1}$. And there is nothing to prevent us from advancing
indefinitely in this way to new cardinals and new ordinals. It
is not known whether $2^{\aleph_{0}}$~is equal to any of the cardinals in the
series of Alephs. It is not even known whether it is comparable
\index{Alephs}%
with them in magnitude; for aught we know, it may be neither
equal to nor greater nor less than any one of the Alephs. This
question is connected with the multiplicative axiom, of which
\index{Multiplicative axiom}%
we shall treat later.

All the series we have been considering so far in this chapter
\index{Series!well-ordered}%
have been what is called ``well-ordered.'' A well-ordered
series is one which has a beginning, and has consecutive terms,
and has a term \emph{next} after any selection of its terms, provided
there are any terms after the selection. This excludes, on the
one hand, compact series, in which there are terms between
any two, and on the other hand series which have no beginning,
or in which there are subordinate parts having no beginning.
The series of negative integers in order of magnitude, having
no beginning, but ending with~$-1$, is not well-ordered; but
taken in the reverse order, beginning with~$-1$, it is well-ordered,
being in fact a progression. The definition is:
\PageSep{93}%

A ``well-ordered'' series is one in which every sub-class
(except, of course, the null-class) has a first term.

An ``ordinal'' number means the relation-number of a well-ordered
series. It is thus a species of serial number.

Among well-ordered series, a generalised form of mathematical
\index{Induction, mathematical}%
induction applies. A property may be said to be ``transfinitely
hereditary'' if, when it belongs to a certain selection of the
terms in a series, it belongs to their immediate successor provided
they have one. In a well-ordered series, a transfinitely
hereditary property belonging to the first term of the series
belongs to the whole series. This makes it possible to prove
many propositions concerning well-ordered series which are not
true of all series.

It is easy to arrange the inductive numbers in series which
\index{Series!compact}%
are not well-ordered, and even to arrange them in compact
series. For example, we can adopt the following plan: consider
the decimals from~$.1$ (inclusive) to~$1$ (exclusive), arranged in order
of magnitude. These form a compact series; between any
two there are always an infinite number of others. Now omit
the dot at the beginning of each, and we have a compact series
consisting of all finite integers except such as divide by~$10$. If
we wish to include those that divide by~$10$, there is no difficulty;
instead of starting with~$.1$, we will include all decimals less than~$1$,
but when we remove the dot, we will transfer to the right any
$0$'s that occur at the beginning of our decimal. Omitting these,
and returning to the ones that have no $0$'s at the beginning,
we can state the rule for the arrangement of our integers as
follows: Of two integers that do not begin with the same digit,
the one that begins with the smaller digit comes first. Of two
that do begin with the same digit, but differ at the second digit,
the one with the smaller second digit comes first, but first of all
the one with no second digit; and so on. Generally, if two
integers agree as regards the first $n$ digits, but not as regards
the $(n + 1)$\Ord{th}, that one comes first which has either no $(n + 1)$\Ord{th}
digit or a smaller one than the other. This rule of arrangement,
\PageSep{94}%
as the reader can easily convince himself, gives rise to a compact
series containing all the integers not divisible by~$10$; and,
as we saw, there is no difficulty about including those
that are divisible by~$10$. It follows from this example that
it is possible to construct compact series having $\aleph_{0}$~terms.
In fact, we have already seen that there are $\aleph_{0}$~ratios, and
ratios in order of magnitude form a compact series; thus
we have here another example. We shall resume this topic
in the next chapter.

Of the usual formal laws of addition, multiplication, and exponentiation,
\index{Exponentiation}%
\index{Number!relation}%
all are obeyed by transfinite cardinals, but only
some are obeyed by transfinite ordinals, and those that are obeyed
by them are obeyed by all relation-numbers. By the ``usual
formal laws'' we mean the following:---
\begin{List}
\item[I.] The commutative law:
\index{Commutative law}%
\[
\alpha + \beta = \beta + \alpha \quad\text{and}\quad
\alpha × \beta = \beta × \alpha.
\]

\item[II.] The associative law:
\index{Associative law}%
\[
(\alpha + \beta) + \gamma = \alpha + (\beta + \gamma) \quad\text{and}\quad
(\alpha × \beta) × \gamma = \alpha × (\beta × \gamma).
\]

\item[III.] The distributive law:
\index{Distributive law}%
\[
\alpha(\beta + \gamma) = \alpha\beta + \alpha\gamma.
\]
\end{List}

When the commutative law does not hold, the above form
of the distributive law must be distinguished from
\[
(\beta + \gamma)\alpha = \beta\alpha + \gamma\alpha.
\]

As we shall see immediately, one form may be true and the
other false.

\begin{List}
\item[IV.] The laws of exponentiation:
\[
\alpha^{\beta} · \alpha^{\gamma} = \alpha^{\beta + \gamma},\quad
\alpha^{\gamma} · \beta^{\gamma} = (\alpha\beta)^{\gamma},\quad
(\alpha^{\beta})^{\gamma} = \alpha^{\beta\gamma}.
\]
\end{List}

All these laws hold for cardinals, whether finite or infinite,
and for \emph{finite} ordinals. But when we come to infinite ordinals,
or indeed to relation-numbers in general, some hold and some
do not. The commutative law does not hold; the associative
law does hold; the distributive law (adopting the convention
\PageSep{95}%
we have adopted above as regards the order of the factors in a
product) holds in the form
\[
(\beta + \gamma)\alpha = \beta\alpha + \gamma\alpha,
\]
but not in the form
\[
\alpha(\beta + \gamma) = \alpha\beta + \alpha\gamma;
\]
the exponential laws
\[
\alpha^{\beta} · \alpha^{\gamma} = \alpha^{\beta + \gamma},
\quad\text{and}\quad
(\alpha^{\beta})^{\gamma} = \alpha^{\beta\gamma}
\]
still hold, but not the law
\[
\alpha^{\gamma} · \beta^{\gamma} = (\alpha\beta)^{\gamma},
\]
which is obviously connected with the commutative law for
multiplication.

The definitions of multiplication and exponentiation that
are assumed in the above propositions are somewhat complicated.
The reader who wishes to know what they are and how the
above laws are proved must consult the second volume of
\Title{Principia Mathematica}, *~172--176.

Ordinal transfinite arithmetic was developed by Cantor at
\index{Cantor, Georg}%
\index{Number!cardinal}%
an earlier stage than cardinal transfinite arithmetic, because it
has various technical mathematical uses which led him to it.
But from the point of view of the philosophy of mathematics
it is less important and less fundamental than the theory of
transfinite cardinals. Cardinals are essentially simpler than
ordinals, and it is a curious historical accident that they first
appeared as an abstraction from the latter, and only gradually
came to be studied on their own account. This does not apply
to Frege's work, in which cardinals, finite and transfinite, were
\index{Frege}%
treated in complete independence of ordinals; but it was
Cantor's work that made the world aware of the subject, while
Frege's remained almost unknown, probably in the main on
account of the difficulty of his symbolism. And mathematicians,
like other people, have more difficulty in understanding and
using notions which are comparatively ``simple'' in the logical
sense than in manipulating more complex notions which are
\PageSep{96}%
more akin to their ordinary practice. For these reasons, it was
only gradually that the true importance of cardinals in mathematical
philosophy was recognised. The importance of ordinals,
though by no means small, is distinctly less than that of cardinals,
and is very largely merged in that of the more general conception
of relation-numbers.
\PageSep{97}%


\Chapter{X}{Limits and Continuity}

\First{The} conception of a ``limit'' is one of which the importance in
\index{Continuity|EtSeq}%
\index{Limit|EtSeq}%
mathematics has been found continually greater than had been
thought. The whole of the differential and integral calculus,
indeed practically everything in higher mathematics, depends
upon limits. Formerly, it was supposed that infinitesimals were
involved in the foundations of these subjects, but Weierstrass
\index{Weierstrass}%
showed that this is an error: wherever infinitesimals were thought
to occur, what really occurs is a set of finite quantities having
zero for their lower limit. It used to be thought that ``limit''
was an essentially quantitative notion, namely, the notion of a
quantity to which others approached nearer and nearer, so that
\index{Quantity}%
among those others there would be some differing by less than any
assigned quantity. But in fact the notion of ``limit'' is a purely
ordinal notion, not involving quantity at all (except by accident
when the series concerned happens to be quantitative). A given
point on a line may be the limit of a set of points on the line,
without its being necessary to bring in co-ordinates or measurement
or anything quantitative. The cardinal number~$\aleph_{0}$ is the
\index{Alephs}%
limit (in the order of magnitude) of the cardinal numbers $1$,~$2$,
$3$,~\dots\ $n$,~\dots, although the numerical difference between $\aleph_{0}$
and a finite cardinal is constant and infinite: from a quantitative
point of view, finite numbers get no nearer to~$\aleph_{0}$ as they grow
larger. What makes $\aleph_{0}$ the limit of the finite numbers is the
fact that, in the series, it comes immediately after them, which
is an \emph{ordinal} fact, not a quantitative fact.
\PageSep{98}%

There are various forms of the notion of ``limit,'' of increasing
complexity. The simplest and most fundamental form,
from which the rest are derived, has been already defined, but
we will here repeat the definitions which lead to it, in a general
form in which they do not demand that the relation concerned
shall be serial. The definitions are as follows:---

The ``minima'' of a class~$\alpha$ with respect to a relation~$P$ are
those members of $\alpha$ and the field of~$P$ (if any) to which no member
of~$\alpha$ has the relation~$P$.

The ``maxima'' with respect to~$P$ are the minima with respect
to the converse of~$P$.

The ``sequents'' of a class~$\alpha$ with respect to a relation~$P$ are
\index{Sequent}%
the minima of the ``successors'' of~$\alpha$, and the ``successors'' of~$\alpha$
are those members of the field of~$P$ to which every member of
the common part of $\alpha$ and the field of~$P$ has the relation~$P$.

The ``precedents'' with respect to~$P$ are the sequents with
\index{Precedent}%
respect to the converse of~$P$.

The ``upper limits'' of~$\alpha$ with respect to~$P$ are the sequents
provided $\alpha$~has no maximum; but if $\alpha$~has a maximum, it has no
\index{Maximum}%
upper limits.

The ``lower limits'' with respect to~$P$ are the upper limits with
respect to the converse of~$P$.

Whenever $P$~has connexity, a class can have at most one
maximum, one minimum, one sequent, etc. Thus, in the cases
\index{Minimum}%
we are concerned with in practice, we can speak of ``\emph{the} limit''
(if any).

When $P$~is a serial relation, we can greatly simplify the above
definition of a limit. We can, in that case, define first the
``boundary'' of a class~$\alpha$, \ie\ its limits or maximum, and then
\index{Boundary}%
proceed to distinguish the case where the boundary is the limit
from the case where it is a maximum. For this purpose it is
best to use the notion of ``segment.''

We will speak of the ``segment of~$P$ defined by a class~$\alpha$'' as
\index{Segments}%
all those terms that have the relation~$P$ to some one or more of
the members of~$\alpha$. This will be a segment in the sense defined
\PageSep{99}%
in \ChapRef{VII}.; indeed, every segment in the sense there defined
is the segment defined by \emph{some} class~$\alpha$. If $P$~is serial, the
segment defined by~$\alpha$ consists of all the terms that precede
some term or other of~$\alpha$. If $\alpha$~has a maximum, the segment will
be all the predecessors of the maximum. But if $\alpha$~has no
maximum, every member of~$\alpha$ precedes some other member of~$\alpha$,
and the whole of~$\alpha$ is therefore included in the segment defined
by~$\alpha$. Take, for example, the class consisting of the fractions
\[
\tfrac{1}{2},\ \tfrac{3}{4},\ \tfrac{7}{8},\ \tfrac{15}{16},\ \dots,
\]
\ie\ of all fractions of the form $1 - \dfrac{1}{2^{n}}$ for different finite values
of~$n$. This series of fractions has no maximum, and it is clear
that the segment which it defines (in the whole series of fractions
in order of magnitude) is the class of all proper fractions. Or,
again, consider the prime numbers, considered as a selection from
the cardinals (finite and infinite) in order of magnitude. In this
case the segment defined consists of all finite integers.

Assuming that $P$~is serial, the ``boundary'' of a class~$\alpha$ will be
\index{Boundary}%
the term~$x$ (if it exists) whose predecessors are the segment
defined by~$\alpha$.

A ``maximum'' of~$\alpha$ is a boundary which is a member of~$\alpha$.

An ``upper limit'' of~$\alpha$ is a boundary which is not a member of~$\alpha$.

If a class has no boundary, it has neither maximum nor limit.
This is the case of an ``irrational'' Dedekind cut, or of what is
\index{Dedekind}%
\index{Gap, Dedekindian}%
called a ``gap.''

Thus the ``upper limit'' of a set of terms~$\alpha$ with respect to a
\index{Limiting points}%
series~$P$ is that term~$x$ (if it exists) which comes after all the~$\alpha$'s,
but is such that every earlier term comes before some of the~$\alpha$'s.

We may define all the ``upper limiting-points'' of a set of
terms~$\beta$ as all those that are the upper limits of sets of terms
chosen out of~$\beta$. We shall, of course, have to distinguish upper
limiting-points from lower limiting-points. If we consider, for
example, the series of ordinal numbers:
\[
1,\ 2,\ 3,\ \dots\
\omega,\ \omega + 1,\ \dots\
2\omega,\ 2\omega + 1,\
3\omega,\ \dots\ \omega^{2},\ \dots\ \omega^{3},\ \dots,
\]
\PageSep{100}%
the upper limiting-points of the field of this series are those that
have no immediate predecessors, \ie\
\[
1,\ \omega,\ 2\omega,\ 3\omega,\ \dots\
\omega^{2},\ \omega^{2} + \omega,\ \dots\ 2\omega^{2},\ \dots\
\omega^{3}\Add{,}\ \dots\Add{.}
\]
The upper limiting-points of the field of this new series will be
\[
1,\ \omega^{2},\ 2\omega^{2},\ \dots\
\omega^{3},\ \omega^{3} + \omega^{2}\Add{,}\ \dots\Add{.}
\]
On the other hand, the series of ordinals---and indeed every well-ordered
series---has no lower limiting-points, because there are
no terms except the last that have no immediate successors. But
if we consider such a series as the series of ratios, every member
of this series is both an upper and a lower limiting-point for
suitably chosen sets. If we consider the series of real numbers,
and select out of it the rational real numbers, this set (the
rationals) will have all the real numbers as upper and lower
limiting-points. The limiting-points of a set are called its ``first
derivative,'' and the limiting-points of the first derivative are
\index{Derivatives}%
called the second derivative, and so on.

With regard to limits, we may distinguish various grades of
\index{Geometry}%
what may be called ``continuity'' in a series. The word ``continuity''
had been used for a long time, but had remained without
any precise definition until the time of Dedekind and Cantor.
Each of these two men gave a precise significance to the term,
but Cantor's definition is narrower than Dedekind's: a series
which has Cantorian continuity must have Dedekindian continuity,
but the converse does not hold.

The first definition that would naturally occur to a man seeking
\index{Series!compact}%
a precise meaning for the continuity of series would be to define
it as consisting in what we have called ``compactness,'' \ie\ in the
fact that between any two terms of the series there are others.
But this would be an inadequate definition, because of the
existence of ``gaps'' in series such as the series of ratios. We
saw in \ChapRef{VII}.\ that there are innumerable ways in which
the series of ratios can be divided into two parts, of which one
wholly precedes the other, and of which the first has no last term,
\PageSep{101}%
while the second has no first term. Such a state of affairs seems
contrary to the vague feeling we have as to what should characterise
``continuity,'' and, what is more, it shows that the series of
ratios is not the sort of series that is needed for many mathematical
purposes. Take geometry, for example: we wish to be able to
say that when two straight lines cross each other they have a
point in common, but if the series of points on a line were similar
to the series of ratios, the two lines might cross in a ``gap'' and
have no point in common. This is a crude example, but many
others might be given to show that compactness is inadequate as
a mathematical definition of continuity.

It was the needs of geometry, as much as anything, that led
\index{Continuity!Dedekindian}%
\index{Series!Dedekindian}%
to the definition of ``Dedekindian'' continuity. It will be remembered
that we defined a series as Dedekindian when every
sub-class of the field has a boundary. (It is sufficient to assume
that there is always an \emph{upper} boundary, or that there is always
a \emph{lower} boundary. If one of these is assumed, the other can be
deduced.) That is to say, a series is Dedekindian when there
are no gaps. The absence of gaps may arise either through
terms having successors, or through the existence of limits in the
absence of maxima. Thus a finite series or a well-ordered series
is Dedekindian, and so is the series of real numbers. The former
sort of Dedekindian series is excluded by assuming that our
series is compact; in that case our series must have a property
which may, for many purposes, be fittingly called continuity.
Thus we are led to the definition:

A series has ``Dedekindian continuity'' when it is Dedekindian
and compact.

But this definition is still too wide for many purposes. Suppose,
for example, that we desire to be able to assign such properties
to geometrical space as shall make it certain that every point
can be specified by means of co-ordinates which are real numbers:
this is not insured by Dedekindian continuity alone. We want
to be sure that every point which cannot be specified by \emph{rational}
co-ordinates can be specified as the limit of a \emph{progression} of points
\PageSep{102}%
whose co-ordinates are rational, and this is a further property
which our definition does not enable us to deduce.

We are thus led to a closer investigation of series with respect
to limits. This investigation was made by Cantor and formed
\index{Cantor, Georg}%
the basis of his definition of continuity, although, in its simplest
\index{Continuity!Cantorian|EtSeq}%
form, this definition somewhat conceals the considerations which
have given rise to it. We shall, therefore, first travel through
some of Cantor's conceptions in this subject before giving his
definition of continuity.

Cantor defines a series as ``perfect'' when all its points are
\index{Series!perfect}%
limiting-points and all its limiting-points belong to it. But this
definition does not express quite accurately what he means.
There is no correction required so far as concerns the property
that all its points are to be limiting-points; this is a property
belonging to compact series, and to no others if all points are to
be upper limiting- or all lower limiting-points. But if it is only
assumed that they are limiting-points one way, without specifying
which, there will be other series that will have the property
in question---for example, the series of decimals in which a decimal
ending in a recurring~$9$ is distinguished from the corresponding
terminating decimal and placed immediately before it. Such a
series is very nearly compact, but has exceptional terms which
are consecutive, and of which the first has no immediate predecessor,
while the second has no immediate successor. Apart from
such series, the series in which every point is a limiting-point
are compact series; and this holds without qualification if it is
specified that every point is to be an upper limiting-point (or
that every point is to be a lower limiting-point).

Although Cantor does not explicitly consider the matter, we
\index{Series!condensed in itself}%
must distinguish different kinds of limiting-points according to
the nature of the smallest sub-series by which they can be defined.
Cantor assumes that they are to be defined by progressions, or
by regressions (which are the converses of progressions). When
every member of our series is the limit of a progression or regression,
Cantor calls our series ``condensed in itself'' (\Foreign{insichdicht}).
\PageSep{103}%

We come now to the second property by which perfection was
to be defined, namely, the property which Cantor calls that of
being ``closed'' (\Foreign{abgeschlossen}). This, as we saw, was first defined
as consisting in the fact that all the limiting-points of a series
belong to it. But this only has any effective significance if our
series is \emph{given} as contained in some other larger series (as is the
case, \eg\, with a selection of real numbers), and limiting-points
are taken in relation to the larger series. Otherwise, if a series
is considered simply on its own account, it cannot fail to contain
its limiting-points. What Cantor \emph{means} is not exactly what
he says; indeed, on other occasions he says something rather
different, which \emph{is} what he means. What he really means is that
every subordinate series which is of the sort that might be expected
to have a limit does have a limit within the given series;
\ie\ every subordinate series which has no maximum has a limit,
\ie\ every subordinate series has a boundary. But Cantor does
not state this for \emph{every} subordinate series, but only for progressions
and regressions. (It is not clear how far he recognises that
this is a limitation.) Thus, finally, we find that the definition we
want is the following:---

A series is said to be ``closed'' (\Foreign{abgeschlossen}) when every progression
\index{Series!closed}%
or regression contained in the series has a limit in the
series.

We then have the further definition:---

A series is ``perfect'' when it is \emph{condensed in itself} and \emph{closed},
\index{Series!perfect}%
\ie\ when every term is the limit of a progression or regression,
and every progression or regression contained in the series has a
limit in the series.

In seeking a definition of continuity, what Cantor has in mind
is the search for a definition which shall apply to the series of
real numbers and to any series similar to that, but to no others.
For this purpose we have to add a further property. Among
the real numbers some are rational, some are irrational; although
the number of irrationals is greater than the number of rationals,
yet there are rationals between any two real numbers, however
\PageSep{104}%
little the two may differ. The number of rationals, as we saw,
is~$\aleph_{0}$. This gives a further property which suffices to characterise
continuity completely, namely, the property of containing a class
of $\aleph_{0}$~members in such a way that some of this class occur
between any two terms of our series, however near together.
This property, added to perfection, suffices to define a class of
series which are all similar and are in fact a serial number. This
class Cantor defines as that of continuous series.

We may slightly simplify his definition. To begin with,
we say:

A ``median class'' of a series is a sub-class of the field such
\index{Median class}%
that members of it are to be found between any two terms of
the series.

Thus the rationals are a median class in the series of real
numbers. It is obvious that there cannot be median classes
except in compact series.

We then find that Cantor's definition is equivalent to the
following:---

A series is ``continuous'' when (1)~it is Dedekindian, (2)~it
contains a median class having $\aleph_{0}$~terms.

To avoid confusion, we shall speak of this kind as ``Cantorian
continuity.'' It will be seen that it implies Dedekindian continuity,
but the converse is not the case. All series having
Cantorian continuity are similar, but not all series having
Dedekindian continuity.

The notions of \emph{limit} and \emph{continuity} which we have been defining
must not be confounded with the notions of the limit of a function
for approaches to a given argument, or the continuity of a function
in the neighbourhood of a given argument. These are different
notions, very important, but derivative from the above and more
complicated. The continuity of motion (if motion is continuous)
is an instance of the continuity of a function; on the other hand,
the continuity of space and time (if they are continuous) is an
instance of the continuity of series, or (to speak more cautiously)
of a kind of continuity which can, by sufficient mathematical
\PageSep{105}%
manipulation, be reduced to the continuity of series. In view
of the fundamental importance of motion in applied mathematics,
as well as for other reasons, it will be well to deal
briefly with the notions of limits and continuity as applied
to functions; but this subject will be best reserved for a
separate chapter.

The definitions of continuity which we have been considering,
\index{Continuity!in philosophy}%
namely, those of Dedekind and Cantor, do not correspond very
closely to the vague idea which is associated with the word in
the mind of the man in the street or the philosopher. They
conceive continuity rather as absence of separateness, the sort
of general obliteration of distinctions which characterises a thick
fog. A fog gives an impression of vastness without definite
multiplicity or division. It is this sort of thing that a metaphysician
means by ``continuity,'' declaring it, very truly,
to be characteristic of his mental life and of that of children
and animals.

The general idea vaguely indicated by the word ``continuity''
when so employed, or by the word ``flux,'' is one which is certainly
\index{Flux}%
quite different from that which we have been defining. Take,
for example, the series of real numbers. Each is what it is,
quite definitely and uncompromisingly; it does not pass over
by imperceptible degrees into another; it is a hard, separate
unit, and its distance from every other unit is finite, though
it can be made less than any given finite amount assigned in
advance. The question of the relation between the kind of
continuity existing among the real numbers and the kind exhibited,
\eg\ by what we see at a given time, is a difficult and
intricate one. It is not to be maintained that the two kinds
are simply identical, but it may, I think, be very well maintained
that the mathematical conception which we have been
considering in this chapter gives the abstract logical scheme to
which it must be possible to bring empirical material by suitable
manipulation, if that material is to be called ``continuous''
in any precisely definable sense. It would be quite impossible
\PageSep{106}%
to justify this thesis within the limits of the present volume.
The reader who is interested may read an attempt to justify
it as regards \emph{time} in particular by the present author in the
\Title{Monist} for 1914--5, as well as in parts of \Title{Our Knowledge of the
External World}. With these indications, we must leave this
problem, interesting as it is, in order to return to topics more
closely connected with mathematics.
\PageSep{107}%


\Chapter{XI}{Limits and Continuity of Functions}

\First{In} this chapter we shall be concerned with the definition of the
%[** TN: Original entries point to page 106]
\index{Continuity!of functions|EtSeq}%
\index{Limit!of functions|EtSeq}%
limit of a function (if any) as the argument approaches a given
value, and also with the definition of what is meant by a ``continuous
function.'' Both of these ideas are somewhat technical,
and would hardly demand treatment in a mere introduction
to mathematical philosophy but for the fact that, especially
through the so-called infinitesimal calculus, wrong views upon
our present topics have become so firmly embedded in the minds
of professional philosophers that a prolonged and considerable
effort is required for their uprooting. It has been thought
ever since the time of Leibniz that the differential and integral
\index{Leibniz}%
calculus required infinitesimal quantities. Mathematicians
(especially Weierstrass) proved that this is an error; but errors
\index{Weierstrass}%
incorporated, \eg\ in what Hegel has to say about mathematics,
\index{Hegel}%
die hard, and philosophers have tended to ignore the work of
such men as Weierstrass.

Limits and continuity of functions, in works on ordinary
mathematics, are defined in terms involving number. This is
\index{Whitehead}%
not essential, as Dr~Whitehead has shown.\footnote
  {See \Title{Principia Mathematica}, vol.~ii.\ *~230--234.}
We will, however,
begin with the definitions in the text-books, and proceed afterwards
to show how these definitions can be generalised so as to
apply to series in general, and not only to such as are numerical
or numerically measurable.

Let us consider any ordinary mathematical function~$fx$, where
\PageSep{108}%
$x$~and~$fx$ are both real numbers, and $fx$~is one-valued---\ie\ when
$x$~is given, there is only one value that $fx$~can have. We call $x$
the ``argument,'' and $fx$~the ``value for the argument~$x$.'' When
\index{Argument of a function}%
\index{Value of a function}%
a function is what we call ``continuous,'' the rough idea for which
we are seeking a precise definition is that small differences in~$x$
shall correspond to small differences in~$fx$, and if we make the
differences in~$x$ small enough, we can make the differences in~$fx$
fall below any assigned amount. We do not want, if a function
is to be continuous, that there shall be sudden jumps, so that,
for some value of~$x$, any change, however small, will make a
change in~$fx$ which exceeds some assigned finite amount. The
ordinary simple functions of mathematics have this property:
it belongs, for example, to $x^{2}$, $x^{3}$,~\dots\ $\log x$, $\sin x$, and so on.
But it is not at all difficult to define discontinuous functions.
Take, as a non-mathematical example, ``the place of birth of
the youngest person living at time~$t$.'' This is a function of~$t$;
its value is constant from the time of one person's birth to the
time of the next birth, and then the value changes \emph{suddenly}
from one birthplace to the other. An analogous mathematical
example would be ``the integer next below~$x$,'' where $x$~is a real
number. This function remains constant from one integer to
the next, and then gives a sudden jump. The actual fact is
that, though continuous functions are more familiar, they are
the exceptions: there are infinitely more discontinuous functions
than continuous ones.

Many functions are discontinuous for one or several values of
the variable, but continuous for all other values. Take as an
example $\sin 1/x$. The function $\sin \theta$ passes through all values
from $-1$ to~$1$ every time that $\theta$~passes from $-\pi/2$ to~$\pi/2$, or from
$\pi/2$ to~$3\pi/2$, or generally from $(2n - 1)\pi/2$ to $(2n + 1)\pi/2$, where
$n$~is any integer. Now if we consider~$1/x$ when $x$~is very small,
we see that as $x$~diminishes $1/x$~grows faster and faster, so that
it passes more and more quickly through the cycle of values from
one multiple of~$\pi/2$ to another as $x$~becomes smaller and smaller.
Consequently $\sin 1/x$ passes more and more quickly from $-1$
\PageSep{109}%
to~$1$ and back again, as $x$~grows smaller. In fact, if we take
any interval containing~$0$, say the interval from $-\eps$ to~$+\eps$ where
$\eps$~is some very small number, $\sin 1/x$ will go through an infinite
number of oscillations in this interval, and we cannot diminish
the oscillations by making the interval smaller. Thus round
about the argument~$0$ the function is discontinuous. It is easy
to manufacture functions which are discontinuous in several
places, or in $\aleph_{0}$~places, or everywhere. Examples will be found
in any book on the theory of functions of a real variable.

Proceeding now to seek a precise definition of what is meant
by saying that a function is continuous for a given argument,
when argument and value are both real numbers, let us first
define a ``neighbourhood'' of a number~$x$ as all the numbers
\index{Neighbourhood}%
from $x - \eps$ to $x + \eps$, where $\eps$~is some number which, in important
cases, will be very small. It is clear that continuity at a given
point has to do with what happens in \emph{any} neighbourhood of that
point, however small.

What we desire is this: If $a$~is the argument for which we wish
our function to be continuous, let us first define a neighbourhood
($\alpha$~say) containing the value~$fa$ which the function has for the
argument~$a$; we desire that, if we take a sufficiently small
neighbourhood containing~$a$, all values for arguments throughout
this neighbourhood shall be contained in the neighbourhood~$\alpha$,
no matter how small we may have made~$\alpha$. That is to say, if
we decree that our function is not to differ from~$fa$ by more than
some very tiny amount, we can always find a stretch of real
numbers, having $a$ in the middle of it, such that throughout
this stretch $fx$~will not differ from~$fa$ by more than the prescribed
tiny amount. And this is to remain true whatever
tiny amount we may select. Hence we are led to the following
definition:---

The function~$\Chg{f(x)}{fx}$ is said to be ``continuous'' for the argument~$a$
if, for every positive number~$\EPS$, different from~$0$, but as
small as we please, there exists a positive number~$\DELTA$, different
from~$0$, such that, for all values of~$\PARAM$ which are numerically
\PageSep{110}%
less\footnote
  {A number is said to be ``numerically less'' than~$\varepsilon$ when it lies between
  $-\varepsilon$~and~$+\varepsilon$.}
than~$\DELTA$, the difference $f(a + \PARAM) - f(a)$ is numerically less
than~$\EPS$.

In this definition, $\EPS$~first defines a neighbourhood of~$f(a)$,
namely, the neighbourhood from $f(a) - \EPS$ to $f(a) + \EPS$. The definition
then proceeds to say that we can (by means of~$\DELTA$) define a
neighbourhood, namely, that from $a - \DELTA$ to $a + \DELTA$, such that, for
all arguments within this neighbourhood, the value of the function
lies within the neighbourhood horn $f(a) - \EPS$ to $f(a) + \EPS$. If this
can be done, however $\EPS$~may be chosen, the function is ``continuous''
for the argument~$a$.

So far we have not defined the ``limit'' of a function for a
given argument. If we had done so, we could have defined the
continuity of a function differently: a function is continuous
at a point where its value is the same as the limit of its value for
approaches either from above or from below. But it is only
the exceptionally ``tame'' function that has a definite limit as
the argument approaches a given point. The general rule is
that a function oscillates, and that, given any neighbourhood
of a given argument, however small, a whole stretch of values
will occur for arguments within this neighbourhood. As this
is the general rule, let us consider it first.

Let us consider what may happen as the argument approaches
some value~$a$ from below. That is to say, we wish to consider
what happens for arguments contained in the interval from
$a - \DELTA$ to~$a$, where $\DELTA$~is some number which, in important cases,
will be very small.

The values of the function for arguments from $a - \DELTA$ to~$a$ ($a$~excluded)
will be a set of real numbers which will define a certain
section of the set of real numbers, namely, the section consisting
of those numbers that are not greater than \emph{all} the values for
arguments from $a - \DELTA$ to~$a$. Given any number in this section,
there are values at least as great as this number for arguments
between $a - \DELTA$ and~$a$, \ie\ for arguments that fall very little short
\PageSep{111}%
of~$a$ (if $\DELTA$~is very small). Let us take all possible $\DELTA$'s and all
possible corresponding sections. The common part of all these
sections we will call the ``ultimate section'' as the argument
\index{Section!ultimate}%
approaches~$a$. To say that a number~$z$ belongs to the ultimate
section is to say that, however small we may make~$\DELTA$, there are
arguments between $a - \DELTA$ and~$a$ for which the value of the function
is not less than~$z$.

We may apply exactly the same process to upper sections,
\ie\ to sections that go from some point up to the top, instead of
from the bottom up to some point. Here we take those numbers
that are not \emph{less} than all the values for arguments from $a - \DELTA$
to~$a$; this defines an upper section which will vary as $\DELTA$~varies.
Taking the common part of all such sections for all possible $\DELTA$'s,
we obtain the ``ultimate upper section.'' To say that a number~$z$
belongs to the ultimate upper section is to say that, however
small we make~$\DELTA$, there are arguments between $a - \DELTA$ and~$a$ for
which the value of the function is not \emph{greater} than~$z$.

If a term~$z$ belongs both to the ultimate section and to the
\index{Oscillation, ultimate}%
ultimate upper section, we shall say that it belongs to the
``ultimate oscillation.'' We may illustrate the matter by considering
once more the function $\sin 1/x$ as $x$~approaches the
value~$0$. We shall assume, in order to fit in with the above
definitions, that this value is approached from below.

Let us begin with the ``ultimate section.'' Between $-\DELTA$
and~$0$, whatever $\DELTA$~may be, the function will assume the value~$1$
for certain arguments, but will never assume any greater value.
Hence the ultimate section consists of all real numbers, positive
and negative, up to and including~$1$; \ie\ it consists of all negative
numbers together with~$0$, together with the positive numbers
up to and including~$1$.

Similarly the ``ultimate upper section'' consists of all positive
numbers together with~$0$, together with the negative numbers
down to and including~$-1$.

Thus the ``ultimate oscillation'' consists of all real numbers
from $-1$ to~$1$, both included.
\PageSep{112}%

We may say generally that the ``ultimate oscillation'' of
a function as the argument approaches~$a$ from below consists
of all those numbers~$x$ which are such that, however near we
come to~$a$, we shall still find values as great as~$x$ and values as
small as~$x$.

The ultimate oscillation may contain no terms, or one term,
or many terms. In the first two cases the function has a definite
limit for approaches from below. If the ultimate oscillation
has one term, this is fairly obvious. It is equally true if it has
none; for it is not difficult to prove that, if the ultimate oscillation
is null, the boundary of the ultimate section is the same as
that of the ultimate upper section, and may be defined as the
limit of the function for approaches from below. But if the
ultimate oscillation has many terms, there is no definite limit to
the function for approaches from below. In this case we can
take the lower and upper boundaries of the ultimate oscillation
(\ie\ the lower boundary of the ultimate upper section and the
upper boundary of the ultimate section) as the lower and upper
limits of its ``ultimate'' values for approaches from below.
Similarly we obtain lower and upper limits of the ``ultimate''
values for approaches from above. Thus we have, in the general
case, \emph{four} limits to a function for approaches to a given argument.
\emph{The} limit for a given argument~$a$ only exists when all these four
are equal, and is then their common value. If it is also the
\emph{value} for the argument~$a$, the function is continuous for this
argument. This may be taken as defining continuity: it is
equivalent to our former definition.

We can define the limit of a function for a given argument
(if it exists) without passing through the ultimate oscillation
and the four limits of the general case. The definition proceeds,
in that case, just as the earlier definition of continuity proceeded.
Let us define the limit for approaches from below. If there is to
be a definite limit for approaches to~$a$ from below, it is necessary
and sufficient that, given any small number~$\EPS$, two values for
arguments sufficiently near to~$a$ (but both less than~$a$) will differ
\PageSep{113}%
by less than~$\EPS$; \ie\ if $\DELTA$~is sufficiently small, and our arguments
both lie between $a - \DELTA$ and~$a$ ($a$~excluded), then the difference
between the values for these arguments will be less than~$\EPS$.
This is to hold for any~$\EPS$, however small; in that case the
function has a limit for approaches from below. Similarly
we define the case when there is a limit for approaches from
above. These two limits, even when both exist, need not be
identical; and if they are identical, they still need not be identical
with the \emph{value} for the argument~$a$. It is only in this last case
that we call the function \emph{continuous} for the argument~$a$.

A function is called ``continuous'' (without qualification)
when it is continuous for every argument.

Another slightly different method of reaching the definition
of continuity is the following:---

Let us say that a function ``ultimately converges into a
class~$\alpha$'' if there is some real number such that, for this argument
and all arguments greater than this, the value of the function
is a member of the class~$\alpha$. Similarly we shall say that a function
``converges into~$\alpha$ as the argument approaches $x$ from below''
if there is some argument~$y$ less than~$x$ such that throughout
the interval from~$y$ (included) to~$x$ (excluded) the function has
values which are members of~$\alpha$. We may now say that a
function is continuous for the argument~$a$, for which it has the
value~$fa$, if it satisfies four conditions, namely:---

(1) Given any real number less than~$fa$, the function converges
into the successors of this number as the argument
approaches $a$ from below;

(2) Given any real number greater than~$fa$, the function converges
into the predecessors of this number as the argument
approaches $a$ from below;

(3) and (4) Similar conditions for approaches to $a$ from above.

The advantages of this form of definition is that it analyses
the conditions of continuity into four, derived from considering
arguments and values respectively greater or less than the
argument and value for which continuity is to be defined.
\PageSep{114}%

We may now generalise our definitions so as to apply to series
which are not numerical or known to be numerically measurable.
The case of motion is a convenient one to bear in mind. There
is a story by H.~G. Wells which will illustrate, from the case of
\index{Wells, H. G.}%
motion, the difference between the limit of a function for a given
argument and its value for the same argument. The hero of
the story, who possessed, without his knowledge, the power of
realising his wishes, was being attacked by a policeman, but on
ejaculating ``Go to------'' he found that the policeman disappeared.
If $f(t)$~was the policeman's position at time~$t$, and $t_{0}$~the moment
of the ejaculation, the limit of the policeman's positions as $t$~approached
to~$t_{0}$ from below would be in contact with the hero,
whereas the value for the argument~$t_{0}$ was~---. But such occurrences
are supposed to be rare in the real world, and it is assumed,
though without adequate evidence, that all motions are continuous,
\ie\ that, given any body, if $f(t)$~is its position at time~$t$, $f(t)$~is
a continuous function of~$t$. It is the meaning of ``continuity''
involved in such statements which we now wish to define as
simply as possible.

The definitions given for the case of functions where argument
and value are real numbers can readily be adapted for more
general use.

Let $P$ and~$Q$ be two relations, which it is well to imagine
serial, though it is not necessary to our definitions that they
should be so. Let $R$~be a one-many relation whose domain
is contained in the field of~$P$, while its converse domain is contained
in the field of~$Q$. Then $R$~is (in a generalised sense) a
function, whose arguments belong to the field of~$Q$, while its
values belong to the field of~$P$. Suppose, for example, that we
are dealing with a particle moving on a line: let $Q$ be the time-series,
$P$~the series of points on our line from left to right, $R$~the
relation of the position of our particle on the line at time~$a$ to
the time~$a$, so that ``the~$R$ of~$a$'' is its position at time~$a$. This
illustration may be borne in mind throughout our definitions.

We shall say that the function~$R$ is continuous for the argument~$a$
\PageSep{115}%
\index{Intervals}%
if, given any interval~$\alpha$ on the $P$-series containing the value
of the function for the argument~$a$, there is an interval on the
$Q$-series containing $a$ not as an end-point and such that, throughout
this interval, the function has values which are members
of~$\alpha$. (We mean by an ``interval'' all the terms between any
two; \ie\ if $x$~and~$y$ are two members of the field of~$P$, and $x$~has
the relation~$P$ to~$y$, we shall mean by the ``$P$-interval $x$~to~$y$''
all terms~$z$ such that $x$~has the relation~$P$ to~$x$ and $z$~has the relation~$P$
to~$y$---together, when so stated, with $x$~or $y$ themselves.)

We can easily define the ``ultimate section'' and the ``ultimate
oscillation.'' To define the ``ultimate section'' for
approaches to the argument~$a$ from below, take any argument~$y$
which precedes~$a$ (\ie\ has the relation~$Q$ to~$a$), take the values
of the function for all arguments up to and including~$y$, and
form the section of~$P$ defined by these values, \ie\ those members
of the $P$-series which are earlier than or identical with some of
these values. Form all such sections for all~$y$'s that precede~$a$,
and take their common part; this will be the ultimate section.
The ultimate upper section and the ultimate oscillation are then
defined exactly as in the previous case.

The adaptation of the definition of convergence and the
\index{Convergence}%
resulting alternative definition of continuity offers no difficulty
of any kind.

We say that a function~$R$ is ``ultimately $Q$-convergent into~$\alpha$''
if there is a member~$y$ of the converse domain of~$R$ and the
field of~$Q$ such that the value of the function for the argument~$y$
and for any argument to which $y$~has the relation~$Q$ is a member
of~$\alpha$. We say that $R$ ``$Q$-converges into~$\alpha$ as the argument
approaches a given argument~$a$'' if there is a term~$y$ having
the relation~$Q$ to~$a$ and belonging to the converse domain of~$R$
and such that the value of the function for any argument in the
$Q$-interval from~$y$ (inclusive) to~$a$ (exclusive) belongs to~$\alpha$.

Of the four conditions that a function must fulfil in order
to be continuous for the argument~$a$, the first is, putting $b$ for
the value for the argument~$a$:
\PageSep{116}%

Given any term having the relation~$P$ to~$b$, $R$ $Q$-converges
into the successors of~$b$ (with respect to~$P$) as the argument
approaches $a$ from below.

The second condition is obtained by replacing~$P$ by its
converse; the third and fourth are obtained from the first and
second by replacing~$Q$ by its converse.

There is thus nothing, in the notions of the limit of a function
or the continuity of a function, that essentially involves number.
Both can be defined generally, and many propositions about
them can be proved for any two series (one being the argument-series
and the other the value-series). It will be seen that the
definitions do not involve infinitesimals. They involve infinite
classes of intervals, growing smaller without any limit short of
zero, but they do not involve any intervals that are not finite.
This is analogous to the fact that if a line an inch long be halved,
then halved again, and so on indefinitely, we never reach infinitesimals
in this way: after $n$~bisections, the length of our bit is
$\dfrac{1}{2^{n}}$~of an inch; and this is finite whatever finite number $n$~may
be. The process of successive bisection does not lead to
divisions whose ordinal number is infinite, since it is essentially
a one-by-one process. Thus infinitesimals are not to be reached
in this way. Confusions on such topics have had much to do
with the difficulties which have been found in the discussion of
infinity and continuity.
\PageSep{117}%


\Chapter{XII}{Selections and the Multiplicative Axiom}

\First{In} this chapter we have to consider an axiom which can be
\index{Multiplicative axiom|EtSeq}%
\index{Selections|EtSeq}%
enunciated, but not proved, in terms of logic, and which is convenient,
though not indispensable, in certain portions of mathematics.
It is convenient, in the sense that many interesting
propositions, which it seems natural to suppose true, cannot
be proved without its help; but it is not indispensable, because
even without those propositions the subjects in which they
occur still exist, though in a somewhat mutilated form.

Before enunciating the multiplicative axiom, we must first
explain the theory of selections, and the definition of multiplication
when the number of factors may be infinite.

In defining the arithmetical operations, the only correct procedure
is to construct an actual class (or relation, in the case
of relation-numbers) having the required number of terms.
This sometimes demands a certain amount of ingenuity, but
it is essential in order to prove the existence of the number
defined. Take, as the simplest example, the case of addition.
Suppose we are given a cardinal number~$\mu$, and a class~$\alpha$ which
has $\mu$~terms. How shall we define $\mu + \mu$? For this purpose
we must have \emph{two} classes having $\mu$~terms, and they must not
overlap. We can construct such classes from~$\alpha$ in various ways,
of which the following is perhaps the simplest: Form first all
the ordered couples whose first term is a class consisting of a
single member of~$\alpha$, and whose second term is the null-class;
then, secondly, form all the ordered couples whose first term is
\PageSep{118}%
the null-class and whose second term is a class consisting of a
single member of~$\alpha$. These two classes of couples have no
member in common, and the logical sum of the two classes will
have $\mu + \mu$ terms. Exactly analogously we can define $\mu + \nu$,
given that $\mu$~is the number of some class~$\alpha$ and $\nu$~is the number
of some class~$\beta$.

Such definitions, as a rule, are merely a question of a suitable
\index{Multiplication|EtSeq}%
technical device. But in the case of multiplication, where the
number of factors may be infinite, important problems arise out
of the definition.

Multiplication when the number of factors is finite offers no
difficulty. Given two classes $\alpha$ and~$\beta$, of which the first has
$\mu$~terms and the second $\nu$~terms, we can define $\mu × \nu$ as the number
of ordered couples that can be formed by choosing the first term
out of~$\alpha$ and the second out of~$\beta$. It will be seen that this definition
does not require that $\alpha$ and~$\beta$ should not overlap; it
even remains adequate when $\alpha$ and $\beta$ are identical. For example,
let $\alpha$ be the class whose members are $x_{1}$,~$x_{2}$,~$x_{3}$. Then the class
which is used to define the product $\mu × \mu$ is the class of couples:
%[** TN: Broken unevenly across two lines in the original]
\begin{gather*}
(x_{1}, x_{1}),\ (x_{1}, x_{2}),\ (x_{1}, x_{3}); \\
(x_{2}, x_{1}),\ (x_{2}, x_{2}),\ (x_{2}, x_{3}); \\
(x_{3}, x_{1}),\ (x_{3}, x_{2}),\ (x_{3}, x_{3}).
\end{gather*}
This definition remains applicable when $\mu$ or $\nu$ or both are
infinite, and it can be extended step by step to three or four or
any finite number of factors. No difficulty arises as regards
this definition, except that it cannot be extended to an \emph{infinite}
number of factors.

The problem of multiplication when the number of factors
may be infinite arises in this way: Suppose we have a class~$\kappa$
consisting of classes; suppose the number of terms in each of
these classes is given. How shall we define the product of all
these numbers? If we can frame our definition generally, it
will be applicable whether $\kappa$~is finite or infinite. It is to be
observed that the problem is to be able to deal with the case
when $\kappa$~is infinite, not with the case when its members are. If
\PageSep{119}%
$\kappa$~is not infinite, the method defined above is just as applicable
when its members are infinite as when they are finite. It is
the case when $\kappa$~is infinite, even though its members may be
finite, that we have to find a way of dealing with.

The following method of defining multiplication generally is
due to Dr~Whitehead. It is explained and treated at length in
\index{Whitehead}%
\Title{Principia Mathematica}, vol.~i.\ *~80~ff., and vol.~ii.\ *~114.

Let us suppose to begin with that $\kappa$~is a class of classes no two
of which overlap---say the constituencies in a country where
there is no plural voting, each constituency being considered
as a class of voters. Let us now set to work to choose one term
out of each class to be its \emph{representative}, as constituencies do
when they elect members of Parliament, assuming that by law
each constituency has to elect a man who is a voter in that
constituency. We thus arrive at a class of representatives, who
make up our Parliament, one being selected out of each constituency.
How many different possible ways of choosing a
Parliament are there? Each constituency can select any one
of its voters, and therefore if there are $\mu$~voters in a constituency,
it can make $\mu$~choices. The choices of the different constituencies
are independent; thus it is obvious that, when the total number
of constituencies is finite, the number of possible Parliaments
is obtained by multiplying together the numbers of voters in the
various constituencies. When we do not know whether the
number of constituencies is finite or infinite, we may take the
number of possible Parliaments as \emph{defining} the product of the
numbers of the separate constituencies. This is the method
by which infinite products are defined. We must now drop our
illustration, and proceed to exact statements.

Let $\kappa$~be a class of classes, and let us assume to begin with that
no two members of~$\kappa$ overlap, \ie\ that if $\alpha$ and~$\beta$ are two different
members of~$\kappa$, then no member of the one is a member of the
other. We shall call a class a ``selection'' from~$\kappa$ when it consists
of just one term from each member of~$\kappa$; \ie\ $\mu$~is a ``selection''
from~$\kappa$ if every member of~$\mu$ belongs to some member
\PageSep{120}%
of~$\kappa$, and if $\alpha$ be any member of~$\kappa$, $\mu$~and~$\alpha$ have exactly one term
in common. The class of all ``selections'' from~$\kappa$ we shall call
the ``multiplicative class'' of~$\kappa$. The number of terms in the
multiplicative class of~$\kappa$, \ie\ the number of possible selections
from~$\kappa$, is defined as the product of the numbers of the members
of~$\kappa$. This definition is equally applicable whether $\kappa$~is finite
or infinite.

Before we can be wholly satisfied with these definitions, we
must remove the restriction that no two members of~$\kappa$ are to
overlap. For this purpose, instead of defining first a class
called a ``selection,'' we will define first a relation which we will
call a ``selector.'' A relation~$R$ will be called a ``selector''
from~$\kappa$ if, from every member of~$\kappa$, it picks out one term as the
representative of that member, \ie\ if, given any member~$\alpha$ of~$\kappa$,
there is just one term~$x$ which is a member of~$\alpha$ and has the
relation~$R$ to~$\alpha$; and this is to be all that $R$~does. The formal
definition is:

A ``selector'' from a class of classes~$\kappa$ is a one-many relation,
having~$\kappa$ for its converse domain, and such that, if $x$~has the
relation to~$\alpha$, then $x$~is a member of~$\alpha$.

If $R$~is a selector from~$\kappa$, and $\alpha$~is a member of~$\kappa$, and $x$~is the
\index{Representatives}%
term which has the relation~$R$ to~$\alpha$, we call~$x$ the ``representative''
of~$\alpha$ in respect of the relation~$R$.

A ``selection'' from~$\kappa$ will now be defined as the domain of a
selector; and the multiplicative class, as before, will be the class
of selections.

But when the members of~$\kappa$ overlap, there may be more selectors
than selections, since a term~$x$ which belongs to two classes $\alpha$
and~$\beta$ may be selected once to represent~$\alpha$ and once to represent~$\beta$,
giving rise to different selectors in the two cases, but to the same
selection. For purposes of defining multiplication, it is the
selectors we require rather than the selections. Thus we define:

``The product of the numbers of the members of a class of
classes~$\kappa$'' is the number of selectors from~$\kappa$.

We can define exponentiation by an adaptation of the above
\index{Exponentiation}%
\PageSep{121}%
plan. We might, of course, define~$\mu^{\nu}$ as the number of selectors
from $\nu$~classes, each of which has $\mu$~terms. But there are
objections to this definition, derived from the fact that the
multiplicative axiom (of which we shall speak shortly) is unnecessarily
involved if it is adopted. We adopt instead the following
construction:---

Let $\alpha$ be a class having $\mu$~terms, and $\beta$~a class having $\nu$~terms.

Let $y$~be a member of~$\beta$, and form the class of all ordered
couples that have $y$~for their second term and a member of~$\alpha$ for
their first term. There will be $\mu$~such couples for a given~$y$, since
any member of~$\alpha$ may be chosen for the first term, and $\alpha$ has $\mu$~members.
If we now form all the classes of this sort that result
from varying~$y$, we obtain altogether $\nu$~classes, since $y$~may be
any member of~$\beta$, and $\beta$~has $\nu$~members. These $\nu$~classes are each
of them a class of couples, namely, all the couples that can be
formed of a variable member of~$\alpha$ and a fixed member of~$\beta$. We
define~$\mu^{\nu}$ as the number of selectors from the class consisting of
these $\nu$~classes. Or we may equally well define~$\mu^{\nu}$ as the number of
selections, for, since our classes of couples are mutually exclusive,
the number of selectors is the same as the number of selections.
A selection from our class of classes will be a set of ordered couples,
of which there will be exactly one having any given member of~$\beta$
for its second term, and the first term may be any member of~$\alpha$.
Thus $\mu^{\nu}$~is defined by the selectors from a certain set of $\nu$~classes
each having $\mu$~terms, but the set is one having a certain structure
and a more manageable composition than is the case in general.
The relevance of this to the multiplicative axiom will appear
shortly.

What applies to exponentiation applies also to the product of
two cardinals. We might define ``$\mu × \nu$'' as the sum of the
numbers of $\nu$~classes each having $\mu$~terms, but we prefer to define
it as the number of ordered couples to be formed consisting of a
member of~$\alpha$ followed by a member of~$\beta$, where $\alpha$~has $\mu$~terms
and $\beta$~has $\nu$~terms. This definition, also, is designed to evade the
necessity of assuming the multiplicative axiom.
\PageSep{122}%

With our definitions, we can prove the usual formal laws of
multiplication and exponentiation. But there is one thing we
cannot prove: we cannot prove that a product is only zero when
one of its factors is zero. We can prove this when the number
of factors is finite, but not when it is infinite. In other words,
we cannot prove that, given a class of classes none of which is
null, there must be selectors from them; or that, given a class
of mutually exclusive classes, there must be at least one class
consisting of one term out of each of the given classes. These
things cannot be proved; and although, at first sight, they seem
obviously true, yet reflection brings gradually increasing doubt,
until at last we become content to register the assumption and
its consequences, as we register the axiom of parallels, without
assuming that we can know whether it is true or false. The
assumption, loosely worded, is that selectors and selections exist
when we should expect them. There are many equivalent ways
of stating it precisely. We may begin with the following:---

``Given any class of mutually exclusive classes, of which none
is null, there is at least one class which has exactly one term in
common with each of the given classes.''

This proposition we will call the ``multiplicative axiom.''\footnote
  {See \Title{Principia Mathematica}, vol.~i.\ *~88. Also vol.~iii.\ *~257--258.}
We will first give various equivalent forms of the proposition,
and then consider certain ways in which its truth or falsehood
is of interest to mathematics.

The multiplicative axiom is equivalent to the proposition that
a product is only zero when at least one of its factors is zero;
\ie\ that, if any number of cardinal numbers be multiplied together,
the result cannot be~$0$ unless one of the numbers concerned is~$0$.

The multiplicative axiom is equivalent to the proposition that,
if $R$~be any relation, and $\kappa$~any class contained in the converse
domain of~$R$, then there is at least one one-many relation implying~$R$
and having~$\kappa$ for its converse domain.

The multiplicative axiom is equivalent to the assumption that
if $\alpha$~be any class, and $\kappa$~all the sub-classes of~$\alpha$ with the exception
\PageSep{123}%
of the null-class, then there is at least one selector from~$\kappa$. This
is the form in which the axiom was first brought to the notice of
the learned world by Zermelo, in his ``Beweis, dass jede Menge
\index{Zermelo}%
wohlgeordnet werden kann.''\footnote
  {\Title{Mathematische Annalen}, vol.~lix.\ pp.~514--6. In this form we shall
  speak of it as Zermelo's axiom.}
Zermelo regards the axiom as an
unquestionable truth. It must be confessed that, until he made
it explicit, mathematicians had used it without a qualm; but it
would seem that they had done so unconsciously. And the credit
due to Zermelo for having made it explicit is entirely independent
of the question whether it is true or false.

The multiplicative axiom has been shown by Zermelo, in the
above-mentioned proof, to be equivalent to the proposition that
every class can be well-ordered, \ie\ can be arranged in a series in
\index{Series!well-ordered}%
which every sub-class has a first term (except, of course, the null-class).
The full proof of this proposition is difficult, but it is not
difficult to see the general principle upon which it proceeds. It
uses the form which we call ``Zermelo's axiom,'' \ie\ it assumes
that, given any class~$\alpha$, there is at least one one-many relation~$R$
whose converse domain consists of all existent sub-classes of~$\alpha$
and which is such that, if $x$~has the relation~$R$ to~$\xi$, then $x$~is a
member of~$\xi$. Such a relation picks out a ``representative''
from each sub-class; of course, it will often happen that two
sub-classes have the same representative. What Zermelo does,
in effect, is to count off the members of~$\alpha$, one by one, by means
of~$R$ and transfinite induction. We put first the representative
of~$\alpha$; call it~$x_{1}$. Then take the representative of the class consisting
of all of~$\alpha$ except~$x_{1}$; call it~$x_{2}$. It must be different from~$x_{1}$,
because every representative is a member of its class, and $x_{1}$~is
shut out from this class. Proceed similarly to take away~$x_{2}$, and
let $x_{3}$ be the representative of what is left. In this way we first
obtain a progression $x_{1}$, $x_{2}$,~\dots\ $x_{n}$,~\dots, assuming that $\alpha$~is not
finite. We then take away the whole progression; let $x_{\omega}$ be the
representative of what is left of~$\alpha$. In this way we can go on
until nothing is left. The successive representatives will form a
\PageSep{124}%
well-ordered series containing all the members of~$\alpha$. (The above
is, of course, only a hint of the general lines of the proof.) This
proposition is called ``Zermelo's theorem.''

The multiplicative axiom is also equivalent to the assumption
that of any two cardinals which are not equal, one must be the
greater. If the axiom is false, there will be cardinals $\mu$ and~$\nu$
such that $\mu$~is neither less than, equal to, nor greater than~$\nu$. We
have seen that $\aleph_{1}$ and~$2^{\aleph_{0}}$ possibly form an instance of such a pair.

Many other forms of the axiom might be given, but the above
are the most important of the forms known at present. As to
the truth or falsehood of the axiom in any of its forms, nothing
is known at present.

The propositions that depend upon the axiom, without being
known to be equivalent to it, are numerous and important. Take
first the connection of addition and multiplication. We naturally
think that the sum of $\nu$~mutually exclusive classes, each having
$\mu$~terms, must have $\mu × \nu$ terms. When $\nu$~is finite, this can be
proved. But when $\nu$~is infinite, it cannot be proved without the
multiplicative axiom, except where, owing to some special circumstance,
the existence of certain selectors can be proved. The
way the multiplicative axiom enters in is as follows: Suppose
we have two sets of $\nu$ mutually exclusive classes, each having $\mu$~terms,
and we wish to prove that the sum of one set has as many
terms as the sum of the other. In order to prove this, we must
establish a one-one relation. Now, since there are in each case
$\nu$~classes, there is some one-one relation between the two sets of
classes; but what we want is a one-one relation between their
terms. Let us consider some one-one relation~$S$ between the
classes. Then if $\kappa$ and~$\lambda$ are the two sets of classes, and $\alpha$~is some
member of~$\kappa$, there will be a member~$\beta$ of~$\lambda$ which will be the
correlate of~$\alpha$ with respect to~$S$. Now $\alpha$~and~$\beta$ each have $\mu$~terms,
and are therefore similar. There are, accordingly, one-one correlations
of $\alpha$~and~$\beta$. The trouble is that there are so many. In
order to obtain a one-one correlation of the sum of~$\kappa$ with the
sum of~$\lambda$, we have to pick out \emph{one selection} from a set of classes
\PageSep{125}%
of correlators, one class of the set being all the one-one correlators
of $\alpha$ with~$\beta$. If $\kappa$~and~$\lambda$ are infinite, we cannot in general know
that such a selection exists, unless we can know that the multiplicative
axiom is true. Hence we cannot establish the usual
kind of connection between addition and multiplication.

This fact has various curious consequences. To begin with,
\index{Alephs}%
we know that $\aleph_{0}^{2} = \aleph_{0} × \aleph_{0} = \aleph_{0}$. It is commonly inferred from
this that the sum of $\aleph_{0}$~classes each having $\aleph_{0}$~members must
itself have $\aleph_{0}$~members, but this inference is fallacious, since we
do not know that the number of terms in such a sum is $\aleph_{0} × \aleph_{0}$,
nor consequently that it is~$\aleph_{0}$. This has a bearing upon the theory
of transfinite ordinals. It is easy to prove that an ordinal which
has $\aleph_{0}$~predecessors must be one of what Cantor calls the ``second
class,'' \ie\ such that a series having this ordinal number will have
$\aleph_{0}$~terms in its field. It is also easy to see that, if we take any
progression of ordinals of the second class, the predecessors of
their limit form at most the sum of $\aleph_{0}$~classes each having $\aleph_{0}$~terms.
It is inferred thence---fallaciously, unless the multiplicative
axiom is true---that the predecessors of the limit are $\aleph_{0}$~in
number, and therefore that the limit is a number of the ``second
class.'' That is to say, it is supposed to be proved that any progression
of ordinals of the second class has a limit which is again
an ordinal of the second class. This proposition, with the corollary
that $\omega_{1}$ (the smallest ordinal of the third class) is not the
limit of any progression, is involved in most of the recognised
theory of ordinals of the second class. In view of the way in
which the multiplicative axiom is involved, the proposition and
its corollary cannot be regarded as proved. They may be true,
or they may not. All that can be said at present is that we do
not know. Thus the greater part of the theory of ordinals of
the second class must be regarded as unproved.

Another illustration may help to make the point clearer. We
know that $2 × \aleph_{0} = \aleph_{0}$. Hence we might suppose that the sum
of $\aleph_{0}$~pairs must have $\aleph_{0}$~terms. But this, though we can prove
that it is sometimes the case, cannot be proved to happen \emph{always}
\PageSep{126}%
\index{Boots and socks}%
unless we assume the multiplicative axiom. This is illustrated
by the millionaire who bought a pair of socks whenever he bought
a pair of boots, and never at any other time, and who had such
a passion for buying both that at last he had $\aleph_{0}$~pairs of boots
and $\aleph_{0}$~pairs of socks. The problem is: How many boots had
he, and how many socks? One would naturally suppose that
he had twice as many boots and twice as many socks as he had
pairs of each, and that therefore he had $\aleph_{0}$~of each, since that
number is not increased by doubling. But this is an instance of
the difficulty, already noted, of connecting the sum of $\nu$~classes
each having $\mu$~terms with $\mu × \nu$. Sometimes this can be done,
sometimes it cannot. In our case it can be done with the boots,
but not with the socks, except by some very artificial device.
The reason for the difference is this: Among boots we can distinguish
right and left, and therefore we can make a selection of
one out of each pair, namely, we can choose all the right boots or
all the left boots; but with socks no such principle of selection
suggests itself, and we cannot be sure, unless we assume the
multiplicative axiom, that there is any class consisting of one
sock out of each pair. Hence the problem.

We may put the matter in another way. To prove that a
class has $\aleph_{0}$~terms, it is necessary and sufficient to find some way
of arranging its terms in a progression. There is no difficulty in
doing this with the boots. The \emph{pairs} are given as forming an~$\aleph_{0}$,
and therefore as the field of a progression. Within each pair,
take the left boot first and the right second, keeping the order
of the \Typo{pair}{pairs} unchanged; in this way we obtain a progression of
all the boots. But with the socks we shall have to choose arbitrarily,
with each pair, which to put first; and an infinite number
of arbitrary choices is an impossibility. Unless we can find a
\emph{rule} for selecting, \ie\ a relation which is a selector, we do not know
that a selection is even theoretically possible. Of course, in the
case of objects in space, like socks, we always can find some
principle of selection. For example, take the centres of mass
of the socks: there will be points~$p$ in space such that, with any
\PageSep{127}%
pair, the centres of mass of the two socks are not both at exactly
the same distance from~$p$; thus we can choose, from each pair,
that sock which has its centre of mass nearer to~$p$. But there is
no theoretical reason why a method of selection such as this
should always be possible, and the case of the socks, with a little
goodwill on the part of the reader, may serve to show how a
selection might be impossible.

It is to be observed that, if it \emph{were} impossible to select one out
of each pair of socks, it would follow that the socks \emph{could} not be
arranged in a progression, and therefore that there were not $\aleph_{0}$~of
them. This case illustrates that, if $\mu$~is an infinite number,
one set of $\mu$~pairs may not contain the same number of terms as
another set of $\mu$~pairs; for, given $\aleph_{0}$~pairs of boots, there are
certainly $\aleph_{0}$~boots, but we cannot be sure of this in the case of
the socks unless we assume the multiplicative axiom or fall back
upon some fortuitous geometrical method of selection such as
the above.

Another important problem involving the multiplicative
axiom is the relation of reflexiveness to non-inductiveness. It
will be remembered that in \ChapRef{VIII}.\ we pointed out that a
reflexive number must be non-inductive, but that the converse
(so far as is known at present) can only be proved if we assume
the multiplicative axiom. The way in which this comes about
is as follows:---

It is easy to prove that a reflexive class is one which contains
\index{Number!non-inductive}%
\index{Number!reflexive}%
sub-classes having $\aleph_{0}$~terms. (The class may, of course, itself
\index{Classes!reflexive}%
have $\aleph_{0}$~terms.) Thus we have to prove, if we can, that, given
any non-inductive class, it is possible to choose a progression
out of its terms. Now there is no difficulty in showing that
a non-inductive class must contain more terms than any inductive
class, or, what comes to the same thing, that if $\alpha$~is a non-inductive
class and $\nu$~is any inductive number, there are sub-classes
of~$\alpha$ that have $\nu$~terms. Thus we can form sets of finite sub-classes
of~$\alpha$: First one class having no terms, then classes having
$1$~term (as many as there are members of~$\alpha$), then classes having
\PageSep{128}%
$2$~terms, and so on. We thus get a progression of sets of sub-classes,
each set consisting of all those that have a certain given
finite number of terms. So far we have not used the multiplicative
axiom, but we have only proved that the number of collections
of sub-classes of~$\alpha$ is a reflexive number, \ie\ that, if $\mu$~is
the number of members of~$\alpha$, so that $2^{\mu}$~is the number of sub-classes
of~$\alpha$ and $2^{2^{\mu}}$~is the number of collections of sub-classes,
then, provided $\mu$~is not inductive, $2^{2^{\mu}}$~must be reflexive. But
this is a long way from what we set out to prove.

In order to advance beyond this point, we must employ the
multiplicative axiom. From each set of sub-classes let us
choose out one, omitting the sub-class consisting of the null-class
alone. That is to say, we select one sub-class containing
one term, $\alpha_{1}$,~say; one containing two terms, $\alpha_{2}$,~say; one containing
three, $\alpha_{3}$,~say; and so on. (We can do this if the multiplicative
axiom is assumed; otherwise, we do not know whether
we can always do it or not.) We have now a progression
$\alpha_{1}$,~$\alpha_{2}$, $\alpha_{3}$,~\dots\ sub-classes of~$\alpha$, instead of a progression of
collections of sub-classes; thus we are one step nearer to our
goal. We now know that, assuming the multiplicative axiom,
if $\mu$~is a non-inductive number, $2^{\mu}$~must be a reflexive number.

The next step is to notice that, although we cannot be sure
that new members of~$\alpha$ come in at any one specified stage in the
progression $\alpha_{1}$,~$\alpha_{2}$, $\alpha_{3}$,~\dots\ we can be sure that new members
keep on coming in from time to time. Let us illustrate.
The class~$\alpha_{1}$, which consists of one term, is a new beginning;
let the one term be~$x_{1}$. The class~$\alpha_{2}$, consisting of two terms,
may or may not contain~$x_{1}$; if it does, it introduces one new
term; and if it does not, it must introduce two new terms, say
$x_{2}$,~$x_{3}$. In this case it is possible that $\alpha_{3}$~consists of $x_{1}$,~$x_{2}$,~$x_{3}$,
and so introduces no new terms, but in that case $\alpha_{4}$~must introduce
a new term. The first $\nu$ classes $\alpha_{1}$,~$\alpha_{2}$, $\alpha_{3}$,~\dots\ $\alpha_{\nu}$ contain, at
the very most, $1 + 2 + 3 + \dots + \nu$ terms, \ie\ $\nu/(\nu + 1)/2$~terms;
thus it would be possible, if there were no repetitions in the
first $\nu$ classes, to go on with only repetitions from the $(\nu + 1)$\Ord{th}
\PageSep{129}%
class to the $\nu(\nu + 1)/2$\Ord{th} class. But by that time the old terms
would no longer be sufficiently numerous to form a next class
with the right number of members, \ie\ $\nu(\nu + 1)/2 + 1$, therefore
new terms must come in at this point if not sooner. It
follows that, if we omit from our progression $\alpha_{1}$,~$\alpha_{2}$, $\alpha_{3}$,~\dots\ all
those classes that are composed entirely of members that have
occurred in previous classes, we shall still have a progression.
Let our new progression be called $\beta_{1}$,~$\beta_{2}$, $\beta_{3}$\Add{,}~\dots. (We shall
have $\alpha_{1} = \beta_{1}$ and $\alpha_{2} = \beta_{2}$, because $\alpha_{1}$~and~$\alpha_{2}$ \emph{must} introduce new
terms. We may or may not have $\alpha_{3} = \beta_{3}$, but, speaking generally,
$\beta_{\nu}$~will be~$\alpha_{\nu}$, where $\nu$~is some number greater than~$\mu$; \ie\ the
$\beta$'s are \emph{some} of the~$\alpha$'s.) Now these $\beta$'s are such that any one
of them, say~$\beta_{\mu}$, contains members which have not occurred in
any of the previous~$\beta$'s. Let $\gamma_{\mu}$~be the part of~$\beta_{\mu}$ which consists
of new members. Thus we get a new progression $\gamma_{1}$,~$\gamma_{2}$, $\gamma_{3}$,~\dots\Add{.}
(Again $\gamma_{1}$~will be identical with~$\beta_{1}$ and with~$\alpha_{1}$; if $\alpha_{2}$~does not
contain the one member of~$\alpha_{1}$, we shall have $\gamma_{2} = \beta_{2} = \alpha_{2}$, but if
$\alpha_{2}$~does contain this one member, $\gamma_{2}$~will consist of the other
member of~$\alpha_{2}$.) This new progression of~$\gamma$'s consists of mutually
exclusive classes. Hence a selection from them will be a progression;
\ie\ if $x_{1}$~is the member of~$\Typo{y_{1}}{\gamma_{1}}$, $x_{2}$~is a member of~$\Typo{y_{2}}{\gamma_{2}}$, $x_{3}$~is
a member of~$\Typo{y_{3}}{\gamma_{3}}$, and so on; then $x_{1}$,~$x_{2}$, $x_{3}$,~\dots\ is a progression,
and is a sub-class of~$\alpha$. Assuming the multiplicative axiom,
such a selection can be made. Thus by twice using this axiom
we can prove that, if the axiom is true, every non-inductive
cardinal must be reflexive. This could also be deduced from
Zermelo's theorem, that, if the axiom is true, every class can be
\index{Zermelo}%
well ordered; for a well-ordered series must have either a finite
or a reflexive number of terms in its field.

There is one advantage in the above direct argument, as
against deduction from Zermelo's theorem, that the above
argument does not demand the universal truth of the multiplicative
axiom, but only its truth as applied to a set of $\aleph_{0}$~classes.
It may happen that the axiom holds for $\aleph_{0}$~classes, though not
for larger numbers of classes. For this reason it is better, when
\PageSep{130}%
it is possible, to content ourselves with the more restricted
assumption. The assumption made in the above direct argument
is that a product of $\aleph_{0}$~factors is never zero unless one of
the factors is zero. We may state this assumption in the form:
``$\aleph_{0}$~is a \emph{multipliable} number,'' where a number~$\nu$ is defined as
\index{Number!multipliable}%
``multipliable'' when a product of $\nu$~factors is never zero unless
one of the factors is zero. We can \emph{prove} that a \emph{finite} number is
always multipliable, but we cannot prove that any infinite number
is so. The multiplicative axiom is equivalent to the assumption
that \emph{all} cardinal numbers are multipliable. But in order to
identify the reflexive with the non-inductive, or to deal with the
problem of the boots and socks, or to show that any progression
of numbers of the second class is of the second class, we only
need the very much smaller assumption that $\aleph_{0}$~is multipliable.

It is not improbable that there is much to be discovered
in regard to the topics discussed in the present chapter. Cases
may be found where propositions which seem to involve the
multiplicative axiom can be proved without it. It is conceivable
that the multiplicative axiom in its general form may be shown
to be false. From this point of view, Zermelo's theorem offers
the best hope: the continuum or some still more dense series
\emph{might} be proved to be incapable of having its terms well ordered,
which would prove the multiplicative axiom false, in virtue of
Zermelo's theorem. But so far, no method of obtaining such
results has been discovered, and the subject remains wrapped in
obscurity.
\PageSep{131}%


\Chapter{XIII}{The Axiom of Infinity and Logical Types}

\First{The} axiom of infinity is an assumption which may be enunciated
\index{Infinity, axiom of|EtSeq}%
as follows:---

``If $n$ be any inductive cardinal number, there is at least one
\index{Number!inductive}%
class of individuals having $n$~terms.''

If this is true, it follows, of course, that there are many classes
of individuals having $n$~terms, and that the total number of
individuals in the world is not an inductive number. For, by
the axiom, there is at least one class having $n + 1$~terms, from which
it follows that there are many classes of $n$~terms and that $n$~is
not the number of individuals in the world. Since $n$~is \emph{any}
inductive number, it follows that the number of individuals
in the world must (if our axiom be true) exceed any inductive
number. In view of what we found in the preceding chapter,
about the possibility of cardinals which are neither inductive
nor reflexive, we cannot infer from our axiom that there are at
least $\aleph_{0}$ individuals, unless we assume the multiplicative axiom.
But we do know that there are at least $\aleph_{0}$ classes of classes,
since the inductive cardinals are classes of classes, and form a
progression if our axiom is true. The way in which the need
%[** TN: No paragraph break in the original]
for this axiom arises may be explained as follows:---One of
Peano's assumptions is that no two inductive cardinals have the
\index{Peano}%
same successor, \ie\ that we shall not have $m + 1 = n + 1$ unless
$m = n$, if $m$~and $n$ are inductive cardinals. In \ChapRef{VIII}.\ we
had occasion to use what is virtually the same as the above
assumption of Peano's, namely, that, if $n$~is an inductive cardinal,
\PageSep{132}%
$n$~is not equal to~$n + 1$. It might be thought that this could be
proved. We can prove that, if $\alpha$~is an inductive class, and $n$~is
the number of members of~$\alpha$, then $n$~is not equal to~$n + 1$.
This proposition is easily proved by induction, and might be
thought to imply the other. But in fact it does not, since there
might be no such class as~$\alpha$. What it does imply is this: If
$n$~is an inductive cardinal such that there is at least one class
having $n$~members, then $n$~is not equal to~$n + 1$. The axiom of
infinity assures us (whether truly or falsely) that there are classes
having $n$~members, and thus enables us to assert that $n$~is not
equal to~$n + 1$. But without this axiom we should be left with
the possibility that $n$ and $n + 1$ might both be the null-class.
\index{Null-class}%

Let us illustrate this possibility by an example: Suppose
there were exactly nine individuals in the world. (As to what
\index{Individuals}%
is meant by the word ``individual,'' I must ask the reader to
be patient.) Then the inductive cardinals from $0$ up to~$9$ would
be such as we expect, but $10$ (defined as $9 + 1$) would be the
null-class. It will be remembered that $n + 1$~may be defined as
follows: $n + 1$~is the collection of all those classes which have a
term~$x$ such that, when $x$~is taken away, there remains a class
of $n$~terms. Now applying this definition, we see that, in the
case supposed, $9 + 1$~is a class consisting of no classes, \ie\ it is
the null-class. The same will be true of~$9 + 2$, or generally of
$9 + n$, unless $n$~is zero. Thus $10$ and all subsequent inductive
cardinals will all be identical, since they will all be the null-class.
In such a case the inductive cardinals will not form a progression,
nor will it be true that no two have the same successor, for $9$
and~$10$ will both be succeeded by the null-class ($10$~being itself
the null-class). It is in order to prevent such arithmetical
catastrophes that we require the axiom of infinity.

As a matter of fact, so long as we are content with the arithmetic
of finite integers, and do not introduce either infinite
integers or infinite classes or series of finite integers or ratios,
it is possible to obtain all desired results without the axiom of
infinity. That is to say, we can deal with the addition, multiplication,
\PageSep{133}%
and exponentiation of finite integers and of ratios,
\index{Ratios}%
but we cannot deal with infinite integers or with irrationals.
Thus the theory of the transfinite and the theory of real numbers
fails us. How these various results come about must now be
explained.

Assuming that the number of individuals in the world is~$n$,
the number of classes of individuals will be~$2^{n}$. This is in virtue
of the general proposition mentioned in \ChapRef{VIII}.\ that the
number of classes contained in a class which has $n$~members
is~$2^{n}$. Now~$2^{n}$ is always greater than~$n$. Hence the number
of classes in the world is greater than the number of individuals.
If, now, we suppose the number of individuals to be~$9$, as we did
just now, the number of classes will be~$2^{9}$, \ie~$512$. Thus if we
take our numbers as being applied to the counting of classes
instead of to the counting of individuals, our arithmetic will
be normal until we reach~$512$: the first number to be null will
be~$513$. And if we advance to classes of classes we shall do still
better: the number of them will be~$2^{512}$, a number which is so
large as to stagger imagination, since it has about $153$~digits.
And if we advance to classes of classes of classes, we shall obtain
a number represented by $2$~raised to a power which has about
$153$~digits; the number of digits in this number will be about
three times~$10^{152}$. In a time of paper shortage it is undesirable
to write out this number, and if we want larger ones we can
obtain them by travelling further along the logical hierarchy.
In this way any assigned inductive cardinal can be made to
find its place among numbers which are not null, merely by
travelling along the hierarchy for a sufficient distance.\footnote
  {On this subject see \Title{Principia Mathematica}, vol.~ii.\ *~120~ff. On the
  corresponding problems as regards ratio, see \ibid, vol.~iii.\ *~303~ff.}

As regards ratios, we have a very similar state of affairs.
If a ratio~$\mu/\nu$ is to have the expected properties, there must
be enough objects of whatever sort is being counted to insure
that the null-class does not suddenly obtrude itself. But this
can be insured, for any given ratio~$\mu/\nu$, without the axiom of
\PageSep{134}%
infinity, by merely travelling up the hierarchy a sufficient distance.
If we cannot succeed by counting individuals, we can try counting
classes of individuals; if we still do not succeed, we can try
classes of classes, and so on. Ultimately, however few individuals
there may be in the world, we shall reach a stage where
there are many more than $\mu$~objects, whatever inductive number
$\mu$~may be. Even if there were no individuals at all, this would
still be true, for there would then be one class, namely, the null-class,
$2$~classes of classes (namely, the null-class of classes and the
class whose only member is the null-class of individuals), $4$~classes
of classes of classes, $16$~at the next stage, $65,536$~at the next
stage, and so on. Thus no such assumption as the axiom of
infinity is required in order to reach any given ratio or any given
inductive cardinal.

It is when we wish to deal with the whole class or series of
inductive cardinals or of ratios that the axiom is required. We
need the whole class of inductive cardinals in order to establish
the existence of~$\aleph_{0}$, and the whole series in order to establish
the existence of progressions: for these results, it is necessary
that we should be able to make a single class or series in which
no inductive cardinal is null. We need the whole series of ratios
in order of magnitude in order to define real numbers as segments:
this definition will not give the desired result unless the series
of ratios is compact, which it cannot be if the total number of
ratios, at the stage concerned, is finite.

It would be natural to suppose---as I supposed myself in former
days---that, by means of constructions such as we have been
considering, the axiom of infinity could be \emph{proved}. It may be
said: Let us assume that the number of individuals is~$n$, where
$n$~may be~$0$ without spoiling our argument; then if we form the
complete set of individuals, classes, classes of classes, etc., all
taken together, the number of terms in our whole set will be
\[
n + 2^{n} + 2^{2^{n}}\Add{+} \dots\ \text{\Foreign{ad inf.},}
\]
which is~$\aleph_{0}$. Thus taking all kinds of objects together, and not
\PageSep{135}%
confining ourselves to objects of any one type, we shall certainly
obtain an infinite class, and shall therefore not need the axiom
of infinity. So it might be said.

Now, before going into this argument, the first thing to observe
is that there is an air of hocus-pocus about it: something reminds
one of the conjurer who brings things out of the hat. The man
who has lent his hat is quite sure there wasn't a live rabbit in it
before, but he is at a loss to say how the rabbit got there. So
the reader, if he has a robust sense of reality, will feel convinced
that it is impossible to manufacture an infinite collection out of
a finite collection of individuals, though he may be unable to
say where the flaw is in the above construction. It would be a
mistake to lay too much stress on such feelings of hocus-pocus;
like other emotions, they may easily lead us astray. But they
afford a \Foreign{prima facie} ground for scrutinising very closely any
argument which arouses them. And when the above argument
is scrutinised it will, in my opinion, be found to be fallacious,
though the fallacy is a subtle one and by no means easy to avoid
consistently.

The fallacy involved is the fallacy which may be called ``confusion
\index{Contradictions|EtSeq}%
\index{Types, logical|EtSeq}%
of types.'' To explain the subject of ``types'' fully would
require a whole volume; moreover, it is the purpose of this book
to avoid those parts of the subjects which are still obscure and
controversial, isolating, for the convenience of beginners, those
parts which can be accepted as embodying mathematically ascertained
truths. Now the theory of types emphatically does not
belong to the finished and certain part of our subject: much of
this theory is still inchoate, confused, and obscure. But the need
of \emph{some} doctrine of types is less doubtful than the precise form
the doctrine should take; and in connection with the axiom of
infinity it is particularly easy to see the necessity of some such
doctrine.

This necessity results, for example, from the ``contradiction of
the greatest cardinal.'' We saw in \ChapRef{VIII}.\ that the number
\index{Number!maximum?}%
of classes contained in a given class is always greater than the
\PageSep{136}%
number of members of the class, and we inferred that there is
no greatest cardinal number. But if we could, as we suggested
a moment ago, add together into one class the individuals, classes
of individuals, classes of classes of individuals, etc., we should
obtain a class of which its own sub-classes would be members.
The class consisting of all objects that can be counted, of whatever
sort, must, if there be such a class, have a cardinal number which
is the greatest possible. Since all its sub-classes will be members
of it, there cannot be more of them than there are members.
Hence we arrive at a contradiction.

When I first came upon this contradiction, in the year~1901,
\index{Cantor, Georg}%
I attempted to discover some flaw in Cantor's proof that there is
no greatest cardinal, which we gave in \ChapRef{VIII}. Applying
this proof to the supposed class of all imaginable objects,
I was led to a new and simpler contradiction, namely, the
following:---

The comprehensive class we are considering, which is to embrace
everything, must embrace itself as one of its members. In other
words, if there is such a thing as ``everything,'' then ``everything''
is something, and is a member of the class ``everything.''
But normally a class is not a member of itself. Mankind, for
example, is not a man. Form now the assemblage of all classes
which are not members of themselves. This is a class: is it a
member of itself or not? If it is, it is one of those classes that
are not members of themselves, \ie\ it is not a member of itself.
If it is not, it is not one of those classes that are not members of
themselves, \ie\ it is a member of itself. Thus of the two hypotheses---that
it is, and that it is not, a member of itself---each
implies its contradictory. This is a contradiction.

There is no difficulty in manufacturing similar contradictions
\Foreign{ad~lib}. The solution of such contradictions by the theory of
%[** TN: More commas here than elsewhere]
types is set forth fully in \Title{Principia Mathematica},\footnote
  {Vol.~i., Introduction, chap.~ii., *~12 and *~20; vol\Add{.}~ii., Prefatory
  Statement.}
and also, more
briefly, in articles by the present author in the \Title{American Journal
\PageSep{137}%
of Mathematics},\footnote
  {``Mathematical Logic as based on the Theory of Types,'' vol.~xxx.,
  1908, pp.~222--262.}
and in the \Title{Revue de Metaphysique et de Morale}.\footnote
  {``Les paradoxes de la logique,'' 1906, pp.~627--650.}
For the present an outline of the solution must suffice.

The fallacy consists in the formation of what we may call
``impure'' classes, \ie\ classes which are not pure as to ``type.''
\index{Classes}%
As we shall see in a later chapter, classes are logical fictions, and
\index{Fictions, logical}%
a statement which appears to be about a class will only be significant
if it is capable of translation into a form in which no mention
is made of the class. This places a limitation upon the ways in
which what are nominally, though not really, names for classes
can occur significantly: a sentence or set of symbols in which
such pseudo-names occur in wrong ways is not false, but strictly
devoid of meaning. The supposition that a class is, or that it
is not, a member of itself is meaningless in just this way. And
more generally, to suppose that one class of individuals is a
member, or is not a member, of another class of individuals
will be to suppose nonsense; and to construct symbolically any
class whose members are not all of the same grade in the logical
hierarchy is to use symbols in a way which makes them no
longer symbolise anything.

Thus if there are $n$~individuals in the world, and $2^{n}$~classes of
individuals, we cannot form a new class, consisting of both
individuals and classes and having $n + 2^{n}$~members. In this way
the attempt to escape from the need for the axiom of infinity
breaks down. I do not pretend to have explained the doctrine
of types, or done more than indicate, in rough outline, why there
is need of such a doctrine. I have aimed only at saying just
so much as was required in order to show that we cannot \emph{prove}
the existence of infinite numbers and classes by such conjurer's
methods as we have been examining. There remain, however,
certain other possible methods which must be considered.

Various arguments professing to prove the existence of infinite
classes are given in the \Title{Principles of Mathematics}, §\;339 (p.~357).
\PageSep{138}%
In so far as these arguments assume that, if $n$~is an inductive
cardinal, $n$~is not equal to~$n + 1$, they have been already dealt
with. There is an argument, suggested by a passage in Plato's
\index{Plato}%
\Title{Parmenides}, to the effect that, if there is such a number as~$1$,
\index{Parmenides@\Title{Parmenides}}%
then $1$~has being; but $1$~is not identical with being, and therefore
$1$~and being are two, and therefore there is such a number as~$2$,
and $2$~together with~$1$ and being gives a class of three terms, and
so on. This argument is fallacious, partly because ``being'' is
not a term having any definite meaning, and still more because,
if a definite meaning were invented for it, it would be found that
numbers do not have being---they are, in fact, what are called
``logical fictions,'' as we shall see when we come to consider
the definition of classes.
\index{Classes!reflexive}%

The argument that the number of numbers from $0$ to~$n$ (both
inclusive) is~$n + 1$ depends upon the assumption that up to and
including~$n$ no number is equal to its successor, which, as we have
seen, will not be always true if the axiom of infinity is false. It
must be understood that the equation $n = n + 1$, which might be
true for a finite~$n$ if $n$~exceeded the total number of individuals
in the world, is quite different from the same equation as applied
to a reflexive number. As applied to a reflexive number, it
means that, given a class of $n$~terms, this class is ``similar'' to
that obtained by adding another term. But as applied to a
number which is too great for the actual world, it merely means
that there is no class of $n$~individuals, and no class of $n + 1$~individuals;
it does not mean that, if we mount the hierarchy of
types sufficiently far to secure the existence of a class of $n$~terms,
we shall then find this class ``similar'' to one of $n + 1$~terms, for
if $n$~is inductive this will not be the case, quite independently of
the truth or falsehood of the axiom of infinity.

There is an argument employed by both Bolzano\footnote
  {Bolzano, \Title{Paradoxien des Unendlichen}, 13.}
\index{Bolzano|FNote}%
and Dedekind\footnote
  {Dedekind, \Title{Was sind und was sollen die Zahlen?} No.~66.}
\index{Dedekind|FNote}%
to prove the existence of reflexive classes. The argument,
in brief, is this: An object is not identical with the idea of the
\PageSep{139}%
object, but there is (at least in the realm of being) an idea of any
object. The relation of an object to the idea of it is one-one, and
ideas are only some among objects. Hence the relation ``idea
of'' constitutes a reflexion of the whole class of objects into a
part of itself, namely, into that part which consists of ideas.
Accordingly, the class of objects and the class of ideas are both
infinite. This argument is interesting, not only on its own
account, but because the mistakes in it (or what I judge to be
mistakes) are of a kind which it is instructive to note. The
main error consists in assuming that there is an idea of every
object. It is, of course, exceedingly difficult to decide what is
meant by an ``idea''; but let us assume that we know. We are
then to suppose that, starting (say) with Socrates, there is the
idea of Socrates, and so on \Foreign{ad~inf}. Now it is plain that this is not
the case in the sense that all these ideas have actual empirical
existence in people's minds. Beyond the third or fourth stage
they become mythical. If the argument is to be upheld, the
``ideas'' intended must be Platonic ideas laid up in heaven, for
certainly they are not on earth. But then it at once becomes
doubtful whether there are such ideas. If we are to know that
there are, it must be on the basis of some logical theory, proving
that it is necessary to a thing that there should be an idea of it.
We certainly cannot obtain this result empirically, or apply it,
as Dedekind does, to ``meine Gedankenwelt''---the world of my
thoughts.

If we were concerned to examine fully the relation of idea and
object, we should have to enter upon a number of psychological
and logical inquiries, which are not relevant to our main purpose.
But a few further points should be noted. If ``idea'' is to be
understood logically, it may be \emph{identical} with the object, or it
may stand for a \emph{description} (in the sense to be explained in a
\index{Descriptions}%
subsequent chapter). In the former case the argument fails,
because it was essential to the proof of reflexiveness that object
and idea should be distinct. In the second case the argument
also fails, because the relation of object and description is not
\PageSep{140}%
one-one: there are innumerable correct descriptions of any given
object. Socrates (\eg)\ may be described as ``the master of
Plato,'' or as ``the philosopher who drank the hemlock,'' or as
``the husband of Xantippe.'' If---to take up the remaining
hypothesis---``idea'' is to be interpreted psychologically, it must
be maintained that there is not any one definite psychological
entity which could be called \emph{the} idea of the object: there are innumerable
beliefs and attitudes, each of which could be called \emph{an}
idea of the object in the sense in which we might say ``my idea
of Socrates is quite different from yours,'' but there is not any
central entity (except Socrates himself) to bind together various
``ideas of Socrates,'' and thus there is not any such one-one relation
of idea and object as the argument supposes. Nor, of course,
as we have already noted, is it true psychologically that there are
ideas (in however extended a sense) of more than a tiny proportion
of the things in the world. For all these reasons, the above
argument in favour of the logical existence of reflexive classes
must be rejected.

It might be thought that, whatever may be said of \emph{logical}
\index{Particulars|EtSeq}%
arguments, the \emph{empirical} arguments derivable from space and
\index{Space}%
time, the diversity of colours, etc., are quite sufficient to prove
\index{Time}%
the actual existence of an infinite number of particulars. I do
not believe this. We have no reason except prejudice for believing
in the infinite extent of space and time, at any rate in the sense
in which space and time are physical facts, not mathematical
fictions. We naturally regard space and time as continuous, or,
at least, as compact; but this again is mainly prejudice. The
theory of ``quanta'' in physics, whether true or false, illustrates
the fact that physics can never afford proof of continuity, though
it might quite possibly afford disproof. The senses are not
sufficiently exact to distinguish between continuous motion and
rapid discrete succession, as anyone may discover in a cinema.
A world in which all motion consisted of a series of small finite
jerks would be empirically indistinguishable from one in which
motion was continuous. It would take up too much space to
\PageSep{141}%
defend these theses adequately; for the present I am merely
suggesting them for the reader's consideration. If they are valid,
it follows that there is no empirical reason for believing the
number of particulars in the world to be infinite, and that there
never can be; also that there is at present no empirical reason
to believe the number to be finite, though it is theoretically
conceivable that some day there might be evidence pointing,
though not conclusively, in that direction.

From the fact that the infinite is not self-contradictory, but is
also not demonstrable logically, we must conclude that nothing
can be known \Foreign{a~priori} as to whether the number of things
in the world is finite or infinite. The conclusion is, therefore,
to adopt a Leibnizian phraseology, that some of the possible
worlds are finite, some infinite, and we have no means of
knowing to which of these two kinds our actual world belongs.
The axiom of infinity will be true in some possible worlds
and false in others; whether it is true or false in this world,
we cannot tell.

Throughout this chapter the synonyms ``individual'' and
\index{Individuals}%
``particular'' have been used without explanation. It would be
impossible to explain them adequately without a longer disquisition
on the theory of types than would be appropriate to the
present work, but a few words before we leave this topic may
do something to diminish the obscurity which would otherwise
envelop the meaning of these words.

In an ordinary statement we can distinguish a verb, expressing
\index{Verbs}%
an attribute or relation, from the substantives which express the
subject of the attribute or the terms of the relation. ``Cæsar
lived'' ascribes an attribute to Cæsar; ``Brutus killed Cæsar''
expresses a relation between Brutus and Cæsar. Using the word
``subject'' in a generalised sense, we may call both Brutus and
Cæsar subjects of this proposition: the fact that Brutus is grammatically
subject and Cæsar object is logically irrelevant, since
the same occurrence may be expressed in the words ``Cæsar was
killed by Brutus,'' where Cæsar is the grammatical subject.
\PageSep{142}%
Thus in the simpler sort of proposition we shall have an attribute
or relation holding of or between one, two or more ``subjects''
\index{Subjects}%
in the extended sense. (A relation may have more than two
terms: \eg\ ``$A$~gives $B$ to~$C$'' is a relation of \emph{three} terms.) Now
it often happens that, on a closer scrutiny, the apparent subjects
are found to be not really subjects, but to be capable of analysis;
the only result of this, however, is that new subjects take their
places. It also happens that the verb may grammatically be
made subject: \eg\ we may say, ``Killing is a relation which
holds between Brutus and Cæsar.'' But in such cases the
grammar is misleading, and in a straightforward statement,
following the rules that should guide philosophical grammar,
Brutus and Cæsar will appear as the subjects and killing
as the verb.

We are thus led to the conception of terms which, when they
occur in propositions, can \emph{only} occur as subjects, and never in
any other way. This is part of the old scholastic definition
of \emph{substance}; but persistence through time, which belonged to
that notion, forms no part of the notion with which we are concerned.
We shall define ``proper names'' as those terms which
can only occur as \emph{subjects} in propositions (using ``subject''
in the extended sense just explained). We shall further define
``individuals'' or ``particulars'' as the objects that can be
named by proper names. (It would be better to define them
directly, rather than by means of the kind of symbols by which
they are symbolised; but in order to do that we should have
to plunge deeper into metaphysics than is desirable here.) It
is, of course, possible that there is an endless regress: that
whatever appears as a particular is really, on closer scrutiny,
a class or some kind of complex. If this be the case, the axiom
of infinity must of course be true. But if it be not the case,
it must be theoretically possible for analysis to reach ultimate
subjects, and it is these that give the meaning of ``particulars''
or ``individuals.'' It is to the number of these that the axiom
of infinity is assumed to apply. If it is true of them, it is true
\PageSep{143}%
of classes of them, and classes of classes of them, and so on;
similarly if it is false of them, it is false throughout this hierarchy.
Hence it is natural to enunciate the axiom concerning them rather
than concerning any other stage in the hierarchy. But whether
the axiom is true or false, there seems no known method of
discovering.
\PageSep{144}%


\Chapter{XIV}{Incompatibility and the Theory of Deduction}

\First{We} have now explored, somewhat hastily it is true, that part
\index{Deduction|EtSeq}%
of the philosophy of mathematics which does not demand a
critical examination of the idea of \emph{class}. In the preceding
chapter, however, we found ourselves confronted by problems
which make such an examination imperative. Before we can
undertake it, we must consider certain other parts of the philosophy
of mathematics, which we have hitherto ignored. In a
synthetic treatment, the parts which we shall now be concerned
with come first: they are more fundamental than anything
that we have discussed hitherto. Three topics will concern us
before we reach the theory of classes, namely: (1)~the theory
of deduction, (2)~propositional functions, (3)~descriptions. Of
\index{Descriptions}%
\index{Functions!propositional}%
these, the third is not logically presupposed in the theory of
classes, but it is a simpler example of the \emph{kind} of theory that
is needed in dealing with classes. It is the first topic, the theory
of deduction, that will concern us in the present chapter.

Mathematics is a deductive science: starting from certain
premisses, it arrives, by a strict process of deduction, at the
various theorems which constitute it. It is true that, in the past,
mathematical deductions were often greatly lacking in rigour;
\index{Rigour}%
it is true also that perfect rigour is a scarcely attainable ideal.
Nevertheless, in so far as rigour is lacking in a mathematical
proof, the proof is defective; it is no defence to urge that common
sense shows the result to be correct, for if we were to rely upon
that, it would be better to dispense with argument altogether,
\PageSep{145}%
rather than bring fallacy to the rescue of common sense. No
appeal to common sense, or ``intuition,'' or anything except strict
\index{Intuition}%
deductive logic, ought to be needed in mathematics after the
premisses have been laid down.

Kant, having observed that the geometers of his day could
\index{Geometry}%
\index{Kant}%
not prove their theorems by unaided argument, but required
an appeal to the figure, invented a theory of mathematical
reasoning according to which the inference is never strictly
logical, but always requires the support of what is called
``intuition.'' The whole trend of modern mathematics, with
its increased pursuit of rigour, has been against this Kantian
theory. The things in the mathematics of Kant's day which
cannot be \emph{proved}, cannot be \emph{known}---for example, the axiom of
parallels. What can be known, in mathematics and by mathematical
methods, is what can be deduced from pure logic. What
else is to belong to human knowledge must be ascertained otherwise---empirically,
through the senses or through experience in
some form, but not \Foreign{a~priori}. The positive grounds for this
thesis are to be found in \Title{Principia Mathematica}, \Foreign{passim}; a
controversial defence of it is given in the \Title{Principles of Mathematics}.
We cannot here do more than refer the reader to those
works, since the subject is too vast for hasty treatment. Meanwhile,
we shall assume that all mathematics is deductive, and
proceed to inquire as to what is involved in deduction.

In deduction, we have one or more propositions called \emph{premisses},
from which we infer a proposition called the \emph{conclusion}.
For our purposes, it will be convenient, when there are originally
several premisses, to amalgamate them into a single proposition,
so as to be able to speak of \emph{the} premiss as well as of \emph{the} conclusion.
Thus we may regard deduction as a process by which
we pass from knowledge of a certain proposition, the premiss,
to knowledge of a certain other proposition, the conclusion.
But we shall not regard such a process as \emph{logical} deduction unless
it is \emph{correct}, \ie\ unless there is such a relation between premiss
and conclusion that we have a right to believe the conclusion
\PageSep{146}%
if we know the premiss to be true. It is this relation that is
chiefly of interest in the logical theory of deduction.

In order to be able validly to infer the truth of a proposition,
we must know that some other proposition is true, and that
there is between the two a relation of the sort called ``implication,''
\index{Implication}%
\ie\ that (as we say) the premiss ``implies'' the conclusion. (We
shall define this relation shortly.) Or we may know that a certain
other proposition is false, and that there is a relation between
the two of the sort called ``disjunction,'' expressed by ``$p$~or~$q$,''\footnote
  {We shall use the letters $p$, $q$, $r$, $s$, $t$ to denote variable propositions.}
so that the knowledge that the one is false allows us to infer
that the other is true. Again, what we wish to infer may be
the \emph{falsehood} of some proposition, not its truth. This may be
inferred from the truth of another proposition, provided we know
that the two are ``incompatible,'' \ie\ that if one is true, the other
is false. It may also be inferred from the falsehood of another
proposition, in just the same circumstances in which the truth
of the other might have been inferred from the truth of the one;
\ie\ from the falsehood of~$p$ we may infer the falsehood of~$q$, when
$q$~implies~$p$. All these four are cases of inference. When our
minds are fixed upon inference, it seems natural to take ``implication''
as the primitive fundamental relation, since this is the
relation which must hold between $p$ and~$q$ if we are to be able
to infer the \emph{truth} of~$q$ from the \emph{truth} of~$p$. But for technical
reasons this is not the best primitive idea to choose. Before
proceeding to primitive ideas and definitions, let us consider
further the various functions of propositions suggested by the
above-mentioned relations of propositions.

The simplest of such functions is the negative, ``not-$p$.''
This is that function of~$p$ which is true when $p$~is false, and false
when $p$~is true. It is convenient to speak of the truth of a proposition,
or its falsehood, as its ``truth-value''\footnotemark;
  \footnotetext{This term is due to Frege.}
\index{Frege|FNote}%
\index{Truth-value}%
\ie\ \emph{truth} is
the ``truth-value'' of a true proposition, and \emph{falsehood} of a false
one. Thus not-$p$ has the opposite truth-value to~$p$.
\PageSep{147}%

We may take next \emph{disjunction}, ``$p$~or~$q$.'' This is a function
\index{Disjunction}%
whose truth-value is truth when $p$~is true and also when $q$~is true,
but is falsehood when both $p$~and~$q$ are false.

Next we may take \emph{conjunction}, ``$p$~and~$q$'' This has truth
\index{Conjunction}%
for its truth-value when $p$~and~$q$ are both true; otherwise it
has falsehood for its truth-value.

Take next \emph{incompatibility}, \ie\ ``$p$~and $q$ are not both true.''
\index{Incompatibility|EtSeq}%
This is the negation of conjunction; it is also the disjunction
of the negations of $p$~and~$q$, \ie\ it is ``not-$p$ or not-$q$.'' Its truth-value
is truth when $p$~is false and likewise when $q$~is false; its
truth-value is falsehood when $p$~and~$q$ are both true.

Last take \emph{implication}, \ie\ ``$p$~implies~$q$,'' or ``if~$p$, then~$q$.''
This is to be understood in the widest sense that will allow us
to infer the truth of~$q$ if we know the truth of~$p$. Thus we interpret
it as meaning: ``Unless $p$~is false, $q$~is true,'' or ``either
$p$~is false or $q$~is true.'' (The fact that ``implies'' is capable
of other meanings does not concern us; this is the meaning which
is convenient for us.) That is to say, ``$p$~implies~$q$'' is to mean
``not-$p$ or~$q$'': its truth-value is to be truth if $p$~is false, likewise
if $q$~is true, and is to be falsehood if $p$~is true and $q$~is false.

We have thus five functions: negation, disjunction, conjunction,
incompatibility, and implication. We might have added others,
for example, joint falsehood, ``not-$p$ and not-$q$,'' but the above
five will suffice. Negation differs from the other four in being
a function of \emph{one} proposition, whereas the others are functions
of \emph{two}. But all five agree in this, that their truth-value depends
only upon that of the propositions which are their arguments.
Given the truth or falsehood of~$p$, or of $p$~and~$q$ (as the case may
be), we are given the truth or falsehood of the negation, disjunction,
conjunction, incompatibility, or implication. A function of
propositions which has this property is called a ``truth-function.''
\index{Truth-function}%

The whole meaning of a truth-function is exhausted by the
statement of the circumstances under which it is true or false.
``Not-$p$,'' for example, is simply that function of~$p$ which is true
when $p$~is false, and false when $p$~is true: there is no further
\PageSep{148}%
meaning to be assigned to it. The same applies to ``$p$~or~$q$''
and the rest. It follows that two truth-functions which have
the same truth-value for all values of the argument are indistinguishable.
For example, ``$p$~and~$q$'' is the negation of
``not-$p$ or not-$q$'' and \Foreign{vice versa}; thus either of these may be
\emph{defined} as the negation of the other. There is no further meaning
in a truth-function over and above the conditions under which
it is true or false.

It is clear that the above five truth-functions are not all independent.
We can define some of them in terms of others. There
is no great difficulty in reducing the number to two; the two
chosen in \Title{Principia Mathematica} are negation and disjunction.
Implication is then defined as ``not-$p$ or~$q$''; incompatibility
as ``not-$p$ or not-$q$''; conjunction as the negation of incompatibility.
\index{Sheffer}%
But it has been shown by Sheffer\footnote
  {\Title{Trans.\ Am.\ Math.\ Soc.}, vol.~xiv.\ pp.~481--488.}
that we can be content
\index{Nicod}%
with \emph{one} primitive idea for all five, and by Nicod\footnote
  {\Title{Proc.\ Camb.\ Phil.\ Soc.}, vol.~xix., i., January~1917.}
that this enables
us to reduce the primitive propositions required in the theory
of deduction to two non-formal principles and one formal one.
For this purpose, we may take as our one indefinable either
incompatibility or joint falsehood. We will choose the former.

Our primitive idea, now, is a certain truth-function called
``incompatibility,'' which we will denote by~$p/q$. Negation
can be at once defined as the incompatibility of a proposition
with itself, \ie\ ``not-$p$'' is defined as~``$p/p$.'' Disjunction is
the incompatibility of not-$p$ and not-$q$, \ie\ it is $(p/p) | (q/q)$.
Implication is the incompatibility of~$p$ and not-$q$, \ie\ $p | (q/q)$.
Conjunction is the negation of incompatibility, \ie\ it is $(p/q) | (p/q)$.
Thus all our four other functions are defined in terms
of incompatibility.

It is obvious that there is no limit to the manufacture of truth-functions,
\index{Inference|EtSeq}%
either by introducing more arguments or by repeating
arguments. What we are concerned with is the connection of
this subject with inference.
\PageSep{149}%

If we know that $p$~is true and that $p$ implies~$q$, we can proceed
to assert~$q$. There is always unavoidably \emph{something} psychological
about inference: inference is a method by which we arrive
at new knowledge, and what is not psychological about it is the
relation which allows us to infer correctly; but the actual passage
from the assertion of~$p$ to the assertion of~$q$ is a psychological
process, and we must not seek to represent it in purely logical
terms.

In mathematical practice, when we infer, we have always
some expression containing variable propositions, say $p$~and~$q$,
which is known, in virtue of its form, to be true for all values
of $p$~and~$q$; we have also some other expression, part of the former,
which is also known to be true for all values of $p$~and~$q$; and in
virtue of the principles of inference, we are able to drop this part
of our original expression, and assert what is left. This somewhat
abstract account may be made clearer by a few examples.

Let us assume that we know the five formal principles of
deduction enumerated in \Title{Principia Mathematica}. (M.~Nicod has
\index{Nicod}%
reduced these to one, but as it is a complicated proposition,
we will begin with the five.) These five propositions are as
follows:---

(1) ``$p$~or~$p$'' implies~$p$---\ie\ if either $p$~is true or $p$~is true,
then $p$~is true.

(2) $q$~implies ``$p$~or~$q$''---\ie\ the disjunction ``$p$~or~$q$'' is true
when one of its alternatives is true.

(3) ``$p$~or~$q$'' implies ``$q$~or~$p$.'' This would not be required
if we had a theoretically more perfect notation, since in the
conception of disjunction there is no order involved, so that
``$p$~or~$q$'' and ``$q$~or~$p$'' should be identical. But since our
symbols, in any convenient form, inevitably introduce an order,
we need suitable assumptions for showing that the order is
irrelevant.

(4) If either $p$~is true or ``$q$~or~$r$'' is true, then either $q$~is true
or ``$p$~or~$r$'' is true. (The twist in this proposition serves to
increase its deductive power.)
\PageSep{150}%

(5) If $q$~implies~$r$, then ``$p$~or~$q$'' implies ``$p$~or~$r$.''

These are the \emph{formal} principles of deduction employed in
\Title{Principia Mathematica}. A formal principle of deduction has a
double use, and it is in order to make this clear that we have
cited the above five propositions. It has a use as the premiss
of an inference, and a use as establishing the fact that the premiss
implies the conclusion. In the schema of an inference
we have a proposition~$p$, and a proposition ``$p$~implies~$q$,'' from
which we infer~$q$. Now when we are concerned with the principles
of deduction, our apparatus of primitive propositions has
to yield both the~$p$ and the ``$p$~implies~$q$'' of our inferences.
That is to say, our rules of deduction are to be used, not \emph{only} as
\emph{rules}, which is their use for establishing ``$p$~implies~$q$'' but \emph{also}
as substantive premisses, \ie\ as the $p$ of our schema. Suppose,
for example, we wish to prove that if $p$~implies~$q$, then if $q$~implies~$r$
it follows that $p$~implies~$r$. We have here a relation of
three propositions which state implications. Put
\begin{Prop}
$p_{1} = p$ implies $q$,
$p_{2} = q$ implies $r$, and
$p_{3} = p$ implies $r$.
\end{Prop}
Then we have to prove that $p_{1}$~implies that $p_{2}$~implies~$p_{3}$. Now
take the fifth of our above principles, substitute not-$p$ for~$p$,
and remember that ``not-$p$ or~$q$'' is by definition the same as
``$p$~implies~$q$.'' Thus our fifth principle yields:
\begin{Prop}
``If $q$ implies~$r$, then `$p$~implies~$q$' implies `$p$~implies~$r$,'\,''
  \ie\ ``$p_{2}$~implies that $p_{1}$~implies~$p_{3}$.'' Call this proposition~$A$.
\end{Prop}
But the fourth of our principles, when we substitute not-$p$,
not-$q$, for $p$~and~$q$, and remember the definition of implication,
becomes:
\begin{Prop}
  ``If $p$~implies that $q$~implies~$r$, then $q$~implies that $p$~implies~$r$.''
\end{Prop}
Writing $p_{2}$~in place of~$p$, $p_{1}$~in place of~$q$, and $p_{3}$~in place of~$r$, this
becomes:
\begin{Prop}
  ``If $p_{2}$~implies that $p_{1}$~implies~$p_{3}$, then $p_{1}$~implies that $p_{2}$~implies~$p_{3}$.''
  Call this~$B$.
\end{Prop}
\PageSep{151}%

Now we proved by means of our fifth principle that
\begin{Prop}
  ``$p_{2}$~implies that $p_{1}$~implies~$p_{3}$,'' which was what we called~$A$.
\end{Prop}
Thus we have here an instance of the schema of inference,
since $A$~represents the~$p$ of our scheme, and $B$~represents the
``$p$~implies~$q$.'' Hence we arrive at~$q$, namely,
\begin{Prop}
``$p_{1}$~implies that $p_{2}$~implies~$p_{3}$,''
\end{Prop}
which was the proposition to be proved. In this proof, the
adaptation of our fifth principle, which yields~$A$, occurs as a
substantive premiss; while the adaptation of our fourth principle,
which yields~$B$, is used to give the \emph{form} of the inference. The
formal and material employments of premisses in the theory
of deduction are closely intertwined, and it is not very important
to keep them separated, provided we realise that they are in
theory distinct.

The earliest method of arriving at new results from a premiss
is one which is illustrated in the above deduction, but which
itself can hardly be called deduction. The primitive propositions,
whatever they may be, are to be regarded as asserted for all
possible values of the variable propositions $p$,~$q$,~$r$ which occur
in them. We may therefore substitute for (say)~$p$ any expression
whose value is always a proposition, \eg\ not-$p$, ``$s$~implies~$t$,''
and so on. By means of such substitutions we really obtain
sets of special cases of our original proposition, but from a practical
point of view we obtain what are virtually new propositions.
The legitimacy of substitutions of this kind has to be insured by
means of a non-formal principle of inference.\footnote
  {No such principle is enunciated in \Title{Principia Mathematica}, or in M.~Nicod's
\index{Nicod|FNote}%
  article mentioned above. But this would seem to be an omission.}

We may now state the one formal principle of inference to
which M.~Nicod has reduced the five given above. For this
purpose we will first show how certain truth-functions can be
defined in terms of incompatibility. We saw already that
\[
\text{$p | (q/q)$ means ``$p$~implies~$q$.''}
\]
\PageSep{152}%
We now observe that
\[
\text{$p | (q/r)$ means ``$p$~implies both $q$ and~$r$.''}
\]
For this expression means ``$p$~is incompatible with the incompatibility
of $q$ and~$r$,'' \ie\ ``$p$~implies that $q$~and~$r$ are not incompatible,''
\ie\ ``$p$~implies that $q$~and~$r$ are both true''---for, as
we saw, the conjunction of $q$ and~$r$ is the negation of their
incompatibility.

Observe next that $t | (t/t)$ means ``$t$~implies itself.'' This is a
particular case of $p | (q/q)$.

Let us write $\Neg{p}$ for the negation of~$p$; thus $\Neg{p/s}$~will mean the
negation of~$p/s$, \ie\ it will mean the conjunction of $p$ and~$s$. It
follows that
\[
(s/q) | \Neg{p/s}
\]
expresses the incompatibility of~$s/q$ with the conjunction of
$p$ and~$s$; in other words, it states that if $p$~and $s$ are both true,
$s/q$~is false, \ie\ $s$~and~$q$ are both true; in still simpler words,
it states that $p$~and~$s$ jointly imply $s$~and~$q$ jointly.

%[** TN: Formatted as intertext in the original]
Now, put
\begin{align*}
  P &amp;= p | (q/r), \\
  \pi &amp;= t | (t/t), \\
  Q &amp;= (s/q) | \Neg{p/s}.
\end{align*}
Then M.~Nicod's sole formal principle of deduction is
\[
P | \pi/Q,
\]
in other words, $P$~implies both $\pi$ and~$Q$.

He employs in addition one non-formal principle belonging
to the theory of types (which need not concern us), and one
corresponding to the principle that, given~$p$, and given that
$p$~implies~$q$, we can assert~$q$. This principle is:
%[** TN: Proposition set in-line in the original]
\begin{Prop}
  ``If $p | (r/q)$ is true, and $p$~is true, then $q$~is true.''
\end{Prop}
From
this apparatus the whole theory of deduction follows, except
in so far as we are concerned with deduction from or to the
existence or the universal truth of ``propositional functions,''
which we shall consider in the next chapter.

There is? if I am not mistaken, a certain confusion in the
\PageSep{153}%
minds of some authors as to the relation, between propositions,
in virtue of which an inference is valid. In order that it may
be \emph{valid} to infer $q$ from~$p$, it is only necessary that $p$~should be
true and that the proposition ``not-$p$ or~$q$'' should be true.
Whenever this is the case, it is clear that $q$~must be true. But
inference will only in fact take place when the proposition ``not-$p$
or~$q$'' is \emph{known} otherwise than through knowledge of not-$p$ or
knowledge of~$q$. Whenever $p$~is false, ``not-$p$ or~$q$'' is true,
but is useless for inference, which requires that $p$~should be true.
Whenever $q$~is already known to be true, ``not-$p$ or~$q$'' is of
course also known to be true, but is again useless for inference,
since $q$~is already known, and therefore does not need to be
inferred. In fact, inference only arises when ``not-$p$ or~$q$''
can be known without our knowing already which of the two
alternatives it is that makes the disjunction true. Now, the
circumstances under which this occurs are those in which certain
relations of form exist between $p$ and~$q$. For example, we know
that if $r$~implies the negation of~$s$, then $s$~implies the negation
of~$r$. Between ``$r$~implies not-$s$'' and ``$s$~implies not-$r$'' there
is a formal relation which enables us to \emph{know} that the first implies
the second, without having first to know that the first is false
or to know that the second is true. It is under such circumstances
that the relation of implication is practically useful for
\index{Implication}%
drawing inferences.

But this formal relation is only required in order that we may
be able to \emph{know} that either the premiss is false or the conclusion
is true. It is the truth of ``not-$p$ or~$q$'' that is required for
the \emph{validity} of the inference; what is required further is only
required for the practical feasibility of the inference. Professor
\index{Lewis, C. I.}%
C.~I. Lewis\footnote
  {See \Title{Mind}, vol.~xxi., 1912, pp.~522--531; and vol.~xxiii., 1914, pp.~240--247.}
has especially studied the narrower, formal relation
which we may call ``formal deducibility.'' He urges that the
wider relation, that expressed by ``not-$p$ or~$q$'' should not be
called ``implication.'' That is, however, a matter of words.
\PageSep{154}%
Provided our use of words is consistent, it matters little how we
define them. The essential point of difference between the
theory which I advocate and the theory advocated by Professor
Lewis is this: He maintains that, when one proposition~$q$ is
\index{Lewis, C. I.}%
``formally deducible'' from another~$p$, the relation which we
perceive between them is one which he calls ``strict implication,''
which is not the relation expressed by ``not-$p$ or~$q$'' but a narrower
relation, holding only when there are certain formal connections
between $p$ and~$q$. I maintain that, whether or not there be
such a relation as he speaks of, it is in any case one that mathematics
does not need, and therefore one that, on general grounds
of economy, ought not to be admitted into our apparatus of
fundamental notions; that, whenever the relation of ``formal
deducibility'' holds between two propositions, it is the case that
we can see that either the first is false or the second true, and that
nothing beyond this fact is necessary to be admitted into our
premisses; and that, finally, the reasons of detail which Professor
Lewis adduces against the view which I advocate can all be met
in detail, and depend for their plausibility upon a covert and
unconscious assumption of the point of view which I reject.
I conclude, therefore, that there is no need to admit as a fundamental
notion any form of implication not expressible as a
truth-function.
\PageSep{155}%


\Chapter{XV}{Propositional Functions}

\First{When}, in the preceding chapter, we were discussing propositions,
\index{Functions!propositional|EtSeq}%
\index{Propositions}%
we did not attempt to give a definition of the word ``proposition.''
But although the word cannot be formally defined, it is necessary
to say something as to its meaning, in order to avoid the very
common confusion with ``propositional functions,'' which are to
be the topic of the present chapter.

We mean by a ``proposition'' primarily a form of words which
expresses what is either true or false. I say ``primarily,''
because I do not wish to exclude other than verbal symbols, or
even mere thoughts if they have a symbolic character. But I
think the word ``proposition'' should be limited to what may,
in some sense, be called ``symbols,'' and further to such symbols
as give expression to truth and falsehood. Thus ``two and two
are four'' and ``two and two are five'' will be propositions,
and so will ``Socrates is a man'' and ``Socrates is not a man.''
The statement: ``Whatever numbers $a$~and~$b$ may be, $(a + b)^{2} = a^{2} + 2ab + b^{2}$''
is a proposition; but the bare formula ``$(a + b)^{2} = a^{2} + 2ab + b^{2}$''
alone is not, since it asserts nothing definite unless
we are further told, or led to suppose, that $a$~and~$b$ are to have
all possible values, or are to have such-and-such values. The
former of these is tacitly assumed, as a rule, in the enunciation
of mathematical formulæ, which thus become propositions;
but if no such assumption were made, they would be ``propositional
functions.'' A ``propositional function,'' in fact, is an
expression containing one or more undetermined constituents,
\PageSep{156}%
such that, when values are assigned to these constituents, the
expression becomes a proposition. In other words, it is a function
whose values are propositions. But this latter definition must
be used with caution. A descriptive function, \eg\ ``the hardest
proposition in $A$'s mathematical treatise,'' will not be a propositional
function, although its values are propositions. But in
such a case the propositions are only described: in a propositional
function, the values must actually \emph{enunciate} propositions.

Examples of propositional functions are easy to give: ``$x$~is
human'' is a propositional function; so long as $x$~remains
undetermined, it is neither true nor false, but when a value
is assigned to~$x$ it becomes a true or false proposition. Any
mathematical equation is a propositional function. So long as
the variables have no definite value, the equation is merely an
expression awaiting determination in order to become a true or
false proposition. If it is an equation containing one variable,
it becomes true when the variable is made equal to a root
of the equation, otherwise it becomes false; but if it is an
``identity'' it will be true when the variable is any number.
The equation to a curve in a plane or to a surface in space is a
propositional function, true for values of the co-ordinates belonging
to points on the curve or surface, false for other values.
Expressions of traditional logic such as ``all $A$ is~$B$'' are propositional
functions: $A$~and~$B$ have to be determined as definite
classes before such expressions become true or false.

The notion of ``cases'' or ``instances'' depends upon propositional
\index{Instances}%
functions. Consider, for example, the kind of process
suggested by what is called ``generalisation,'' and let us take
\index{Generalisation}%
some very primitive example, say, ``lightning is followed by
thunder.'' We have a number of ``instances'' of this, \ie\ a
number of propositions such as: ``this is a flash of lightning
and is followed by thunder.'' What are these occurrences
``instances'' of? They are instances of the propositional
function: ``If $x$~is a flash of lightning, $x$~is followed by thunder.''
The process of generalisation (with whose validity we are fortunately
\PageSep{157}%
not concerned) consists in passing from a number of such
instances to the \emph{universal} truth of the propositional function:
``If $x$~is a flash of lightning, $x$~is followed by thunder.'' It will
be found that, in an analogous way, propositional functions
are always involved whenever we talk of instances or cases or
examples.

We do not need to ask, or attempt to answer, the question:
``What \emph{is} a propositional function?'' A propositional function
standing all alone may be taken to be a mere schema, a mere
shell, an empty receptacle for meaning, not something already
significant. We are concerned with propositional functions,
broadly speaking, in two ways: first, as involved in the notions
``true in all cases'' and ``true in some cases''; secondly, as
involved in the theory of classes and relations. The second of
these topics we will postpone to a later chapter; the first must
occupy us now.

When we say that something is ``always true'' or ``true in
all cases,'' it is clear that the ``something'' involved cannot be
a proposition. A proposition is just true or false, and there
is an end of the matter. There are no instances or cases of
``Socrates is a man'' or ``Napoleon died at St~Helena.'' These
are propositions, and it would be meaningless to speak of their
being true ``in all cases.'' This phrase is only applicable to
propositional \emph{functions}. Take, for example, the sort of thing
that is often said when causation is being discussed. (We are
net concerned with the truth or falsehood of what is said, but
only with its logical analysis.) We are told that $A$~is, in every
instance, followed by~$B$. Now if there are ``instances'' of~$A$,
$A$~must be some general concept of which it is significant to say
``$x_{1}$~is~$A$,'' ``$x_{2}$~is~$A$,'' ``$x_{3}$~is~$A$,'' and so on, where $x_{1}$,~$x_{2}$,~$x_{3}$ are
particulars which are not identical one with another. This
applies, \eg\, to our previous case of lightning. We say that
lightning~($A$) is followed by thunder~($B$). But the separate
flashes are particulars, not identical, but sharing the common
property of being lightning. The only way of expressing a
\PageSep{158}%
common property generally is to say that a common property
of a number of objects is a propositional function which becomes
true when any one of these objects is taken as the value of the
variable. In this case all the objects are ``instances'' of the
truth of the propositional function---for a propositional function,
though it cannot itself be true or false, is true in certain instances
and false in certain others, unless it is ``always true'' or ``always
false.'' When, to return to our example, we say that $A$~is in
every instance followed by~$B$, we mean that, whatever $x$~may be,
if $x$~is an~$A$, it is followed by a~$B$; that is, we are asserting that
a certain propositional function is ``always true.''

Sentences involving such words as ``all,'' ``every,'' ``a,''
\index{All@\emph{All}|EtSeq}%
%[** TN: Entry not italicized in the original]
\index{Some@\emph{Some}|EtSeq}%
``the,'' ``some'' require propositional functions for their interpretation.
The way in which propositional functions occur
can be explained by means of two of the above words, namely,
``all'' and ``some.''

There are, in the last analysis, only two things that can be
done with a propositional function: one is to assert that it is
true in \emph{all} cases, the other to assert that it is true in at least one
case, or in \emph{some} cases (as we shall say, assuming that there is
to be no necessary implication of a plurality of cases). All the
other uses of propositional functions can be reduced to these two.
When we say that a propositional function is true ``in all cases,''
or ``always'' (as we shall also say, without any temporal suggestion),
we mean that all its values are true. If ``$\phi x$''~is the
function, and $a$~is the right sort of object to be an argument to~``$\phi x$,''
then $\phi a$~is to be true, however $a$~may have been chosen.
For example, ``if $a$~is human, $a$~is mortal'' is true whether $a$~is
human or not; in fact, every proposition of this form is true.
Thus the propositional function ``if $x$~is human, $x$~is mortal''
is ``always true,'' or ``true in all cases.'' Or, again, the statement
``there are no unicorns'' is the same as the statement
``the propositional function `$x$~is not a unicorn' is true in all
cases.'' The assertions in the preceding chapter about propositions,
\eg\ ``\,`$p$~or~$q$' implies `$q$~or~$p$,'\,'' are really assertions
\PageSep{159}%
that certain propositional functions are true in all cases. We do
not assert the above principle, for example, as being true only
of this or that particular $p$ or~$q$, but as being true of \emph{any} $p$ or~$q$
concerning which it can be made significantly. The condition
that a function is to be \emph{significant} for a given argument is the same
as the condition that it shall have a value for that argument,
either true or false. The study of the conditions of significance
belongs to the doctrine of types, which we shall not pursue
beyond the sketch given in the preceding chapter.

Not only the principles of deduction, but all the primitive
propositions of logic, consist of assertions that certain propositional
\index{Logic}%
functions are always true. If this were not the case, they
would have to mention particular things or concepts---Socrates,
or redness, or east and west, or what not,---and clearly it is not
the province of logic to make assertions which are true concerning
one such thing or concept but not concerning another. It is
part of the definition of logic (but not the whole of its definition)
that all its propositions are completely general, \ie\ they all
consist of the assertion that some propositional function containing
no constant terms is always true. We shall return in
our final chapter to the discussion of propositional functions
containing no constant terms. For the present we will proceed
to the other thing that is to be done with a propositional function,
namely, the assertion that it is ``sometimes true,'' \ie\ true in at
least one instance.

When we say ``there are men,'' that means that the propositional
function ``$x$~is a man'' is sometimes true. When we
say ``some men are Greeks,'' that means that the propositional
function ``$x$~is a man and a Greek'' is sometimes true. When we
say ``cannibals still exist in Africa,'' that means that the propositional
function ``$x$~is a cannibal now in Africa'' is sometimes
true, \ie\ is true for some values of~$x$. To say ``there are at least
$n$~individuals in the world'' is to say that the propositional
function ``$\alpha$~is a class of individuals and a member of the cardinal
number~$n$'' is sometimes true, or, as we may say, is true for certain
\PageSep{160}%
values of~$\alpha$. This form of expression is more convenient when it
is necessary to indicate which is the variable constituent which
we are taking as the argument to our propositional function.
For example, the above propositional function, which we may
shorten to ``$\alpha$~is a class of $n$~individuals,'' contains two variables,
$\alpha$ and~$n$. The axiom of infinity, in the language of propositional
functions, is: ``The propositional function `if $n$~is an inductive
number, it is true for some values of~$\alpha$ that $\alpha$~is a class of $n$~individuals' is true for all possible values of~$n$.'' Here there is a
subordinate function, ``$\alpha$~is a class of $n$~individuals,'' which is
said to be, in respect of~$\alpha$, \emph{sometimes} true; and the assertion
that this happens if $n$~is an inductive number is said to be, in
respect of~$n$, \emph{always} true.

The statement that a function~$\phi x$ is always true is the negation
of the statement that not-$\phi x$ is sometimes true, and the statement
that $\phi x$~is sometimes true is the negation of the statement
that not-$\phi x$ is always true. Thus the statement ``all
men are mortals'' is the negation of the statement that the
function ``$x$~is an immortal man'' is sometimes true. And the
statement ``there are unicorns'' is the negation of the statement
that the function ``$x$~is not a unicorn'' is always true.\footnote
  {The method of deduction is given in \Title{Principia Mathematica},
  vol.~i.\ *~9.}
We say that $\phi x$~is ``never true'' or ``always false'' if not-$\phi x$ is
always true. We can, if we choose, take one of the pair ``always,''
``sometimes'' as a primitive idea, and define the other by means
of the one and negation. Thus if we choose ``sometimes'' as
our primitive idea, we can define: ``\,`$\phi x$~is always true' is to
mean `it is false that not-$\phi x$ is sometimes true.'\,''\footnote
  {For linguistic reasons, to avoid suggesting either the plural or the
  singular, it is often convenient to say ``$\Chg{\varphi x}{\phi x}$~is not always false'' rather
  than ``$\Chg{\varphi x}{\phi x}$~sometimes'' or ``$\Chg{\varphi x}{\phi x}$~is sometimes true.''}
But for
reasons connected with the theory of types it seems more correct
to take both ``always'' and ``sometimes'' as primitive ideas,
and define by their means the negation of propositions in which
they occur. That is to say, assuming that we have already
\PageSep{161}%
defined (or adopted as a primitive idea) the negation of propositions
of the type to which $x$~belongs, we define: ``The
negation of `$\phi x$~always' is `not-$\phi x$ sometimes'; and the negation
of `$\phi x$~sometimes' is `not-$\phi x$ always.'\,'' In like manner
we can re-define disjunction and the other truth-functions,
as applied to propositions containing apparent variables, in
\index{Variables}%
terms of the definitions and primitive ideas for propositions
containing no apparent variables. Propositions containing no
apparent variables are called ``elementary propositions.'' From
\index{Propositions!elementary}%
these we can mount up step by step, using such methods as have
just been indicated, to the theory of truth-functions as applied
to propositions containing one, two, three\Add{,}~\dots\ variables, or any
number up to~$n$, where $n$~is any assigned finite number.

The forms which are taken as simplest in traditional formal
logic are really far from being so, and all involve the assertion
of all values or some values of a compound propositional function.
Take, to begin with, ``all $S$ is~$P$.'' We will take it that $S$~is
defined by a propositional function~$\phi x$, and $P$~by a propositional
function~$\psi x$. \Eg\, if $S$~is \emph{men}, $\phi x$~will be ``$x$~is human''; if $P$~is
\emph{mortals}, $\psi x$~will be ``there is a time at which $x$~dies.'' Then
``all $S$ is~$P$'' means: ``\,`$\phi x$~implies~$\psi x$' is always true.'' 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. Suppose we come across an~$x$ of which
we do not know whether it is an~$S$ or not; still, our statement
``all $S$ is~$P$'' tells us something about~$x$, namely, that if $x$~is an~$S$,
then $x$~is a~$P$. And this is every bit as true when $x$~is not an~$S$ as
when $x$~is an~$S$. If it were not equally true in both cases, the
\Foreign{reductio ad absurdum} would not be a valid method; for the
essence of this method consists in using implications in cases
where (as it afterwards turns out) the hypothesis is false. We may
put the matter another way. 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
\PageSep{162}
by ``all $S$ is~$P$,'' however little we may 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, \ie\ all the terms that are~$S$'s, together with all the
terms that are not~$S$'s---\ie\ the whole of the appropriate logical
``type.'' What applies to statements about \emph{all} applies also to
statements about \emph{some}. ``There are men,'' \eg\, means that
``$x$~is human'' is true for \emph{some} values of~$x$. Here \emph{all} values of~$x$
(\ie\ all values for which ``$x$~is human'' is significant, whether
true or false) are relevant, and not only those that in fact are
human. (This becomes obvious if we consider how we could
prove such a statement to be \emph{false}.) Every assertion about
``all'' or ``some'' thus involves not only the arguments that
make a certain function true, but all that make it significant,
\ie\ all for which it has a value at all, whether true or false.

We may now proceed with our interpretation of the traditional
forms of the old-fashioned formal logic. We assume that $S$~is
those terms~$x$ for which $\phi x$~is true, and $P$~is those for which $\psi x$~is
true. (As we shall see in a later chapter, all classes are derived
in this way from propositional functions.) Then:
\begin{Prop}
``All $S$ is~$P$'' means ``\,`$\phi x$ implies~$\psi x$' is always true.''

``Some $S$ is~$P$'' means ``\,`$\phi x$~and~$\psi x$' is sometimes true.''

``No $S$ is~$P$'' means  ``\,`$\phi x$ implies not-$\psi x$' is always true.''

``Some $S$ is not~$P$'' means  ``\,`$\phi x$~and not-$\psi x$' is sometimes true.''
\end{Prop}
It will be observed that the propositional functions which are
here asserted for all or some values are not $\phi x$ and~$\psi x$ themselves,
but truth-functions of $\phi x$ and~$\psi x$ for the \emph{same} argument~$x$.
The easiest way to conceive of the sort of thing that is
intended is to start not from $\phi x$ and $\psi x$ in general, but from
$\phi a$ and~$\psi a$, where $a$~is some constant. Suppose we are considering
\Typo{all ``men are mortal''}{``all men are mortal''}: we will begin with
\begin{Prop}
``If Socrates is human, Socrates is mortal,''
\end{Prop}
\PageSep{163}%
and then we will regard ``Socrates'' as replaced by a variable~$x$
wherever ``Socrates'' occurs. The object to be secured is that,
although $x$~remains a variable, without any definite value, yet
it is to have the same value in~``$\phi x$'' as in~``$\psi x$'' when we are
asserting that ``$\phi x$~implies~$\psi x$'' is always true. This requires
that we shall start with a function whose values are such as
``$\phi a$~implies~$\psi a$,'' rather than with two separate functions $\phi x$
and~$\psi x$; for if we start with two separate functions we can
never secure that the~$x$, while remaining undetermined, shall
have the same value in both.

For brevity we say ``$\phi x$~always implies~$\psi x$'' when we
\index{Implication!formal}%
mean that ``$\phi x$~implies~$\psi x$'' is always true. Propositions
of the form ``$\phi x$~always implies~$\psi x$'' are called ``formal
implications''; this name is given equally if there are several
variables.

The above definitions show how far removed from the simplest
forms are such propositions as ``all $S$ is~$P$,'' with which traditional
logic begins. It is typical of the lack of analysis involved
that traditional logic treats ``all $S$ is~$P$'' as a proposition of
the same form as ``$x$~is~$P$''---\eg\, it treats ``all men are mortal''
as of the same form as ``Socrates is mortal.'' As we have just
seen, the first is of the form ``$\phi x$~always implies~$\psi x$,'' while the
second is of the form~``$\psi x$.'' The emphatic separation of these
two forms, which was effected by Peano and Frege, was a very
\index{Peano}%
vital advance in symbolic logic.

It will be seen that ``all $S$ is~$P$'' and ``no $S$ is~$P$'' do not
really differ in form, except by the substitution of not-$\psi x$ for~$\psi x$,
and that the same applies to ``some $S$ is~$P$'' and ``some $S$ is
not~$P$.'' It should also be observed that the traditional rules
of conversion are faulty, if we adopt the view, which is the only
technically tolerable one, that such propositions as ``all $S$ is~$P$''
do not involve the ``existence'' of~$S$'s, \ie\ do not require that
there should be terms which are~$S$'s. The above definitions
lead to the result that, if $\phi x$~is always false, \ie\ if there are no~$S$'s,
then ``all $S$ is~$P$'' and ``no $S$ is~$P$'' will both be true, whatever
\PageSep{164}%
$P$~may be. For, according to the definition in the last
chapter, ``$\phi x$~implies~$\psi x$'' means ``not-$\phi x$ or~$\psi x$'' which is
always true if not-$\phi x$ is always true. At the first moment,
this result might lead the reader to desire different definitions,
but a little practical experience soon shows that any different
definitions would be inconvenient and would conceal the important
ideas. The proposition ``$\phi x$~always implies~$\psi x$, and $\phi x$~is
sometimes true'' is essentially composite, and it would be
very awkward to give this as the definition of ``all $S$ is~$P$,''
for then we should have no language left for ``$\phi x$~always implies~$\psi x$,''
which is needed a hundred times for once that the other is
needed. But, with our definitions, ``all $S$ is~$P$'' does not imply
``some $S$ is~$P$,'' since the first allows the non-existence of~$S$ and
the second does not; thus conversion \Foreign{per accidens} becomes
invalid, and some moods of the syllogism are fallacious, \eg\
Darapti: ``All $M$ is~$S$, all $M$ is~$P$, therefore some $S$ is~$P$,'' which
fails if there is no~$M$.

The notion of ``existence'' has several forms, one of which
\index{Existence}%
will occupy us in the next chapter; but the fundamental form
is that which is derived immediately from the notion of ``sometimes
true.'' We say that an argument~$a$ ``satisfies'' a function~$\phi x$
if $\phi a$~is true; this is the same sense in which the roots of an
equation are said to satisfy the equation. Now if $\phi x$~is sometimes
true, we may say there are~$x$'s for which it is true, or we may say
``arguments satisfying~$\phi x$ \emph{exist}'' This is the fundamental meaning
of the word ``existence.'' Other meanings are either derived
from this, or embody mere confusion of thought. We may
correctly say ``men exist,'' meaning that ``$x$~is a man'' is sometimes
true. But if we make a pseudo-syllogism: ``Men exist,
Socrates is a man, therefore Socrates exists,'' we are talking
nonsense, since ``Socrates'' is not, like ``men,'' merely an undetermined
argument to a given propositional function. The
fallacy is closely analogous to that of the argument: ``Men are
numerous, Socrates is a man, therefore Socrates is numerous.''
In this case it is obvious that the conclusion is nonsensical, but
\PageSep{165}%
in the case of existence it is not obvious, for reasons which will
appear more fully in the next chapter. For the present let us
merely note the fact that, though it is correct to say ``men exist,''
it is incorrect, or rather meaningless, to ascribe existence to a
given particular~$x$ who happens to be a man. Generally, ``terms
satisfying~$\phi x$ exist'' means ``$\phi x$~is sometimes true''; but ``$a$~exists''
(where $a$~is a term satisfying~$\phi x$) is a mere noise or shape,
devoid of significance. It will be found that by bearing in mind
this simple fallacy we can solve many ancient philosophical
puzzles concerning the meaning of existence.

Another set of notions as to which philosophy has allowed
itself to fall into hopeless confusions through not sufficiently
separating propositions and propositional functions are the
notions of ``modality'': \emph{necessary}, \emph{possible}, and \emph{impossible}.
\index{Modality}%
(Sometimes \emph{contingent} or \emph{assertoric} is used instead of \emph{possible}.)
The traditional view was that, among true propositions, some
were necessary, while others were merely contingent or assertoric;
while among false propositions some were impossible, namely,
those whose contradictories were necessary, while others merely
happened not to be true. In fact, however, there was never
any clear account of what was added to truth by the conception
of necessity. In the case of propositional functions, the three-fold
\index{Necessity}%
division is obvious. If ``$\phi x$''~is an undetermined value of a
certain propositional function, it will be \emph{necessary} if the function
is always true, \emph{possible} if it is sometimes true, and \emph{impossible} if
it is never true. This sort of situation arises in regard to probability,
for example. Suppose a ball~$x$ is drawn from a bag
which contains a number of balls: if all the balls are white,
``$x$~is white'' is necessary; if some are white, it is possible;
if none, it is impossible. Here all that is \emph{known} about~$x$ is that
it satisfies a certain propositional function, namely, ``$x$~was a
ball in the bag.'' This is a situation which is general in probability
problems and not uncommon in practical life---\eg\ when
a person calls of whom we know nothing except that he brings
a letter of introduction from our friend so-and-so. In all such
\PageSep{166}%
cases, as in regard to modality in general, the propositional
function is relevant. For clear thinking, in many very diverse
directions, the habit of keeping propositional functions sharply
separated from propositions is of the utmost importance, and
the failure to do so in the past has been a disgrace to
philosophy.
\PageSep{167}%


\Chapter{XVI}{Descriptions}

\First{We} dealt in the preceding chapter with the words \emph{all} and \emph{some};
\index{Descriptions|EtSeq}%
\index{The@\emph{The}}%
in this chapter we shall consider the word \emph{the} in the singular,
and in the next chapter we shall consider the word \emph{the} in the
plural. It may be thought excessive to devote two chapters
to one word, but to the philosophical mathematician it is a
word of very great importance: like Browning's Grammarian
with the enclitic~$\delta\epsilon$, I would give the doctrine of this word if I
were ``dead from the waist down'' and not merely in a prison.

We have already had occasion to mention ``descriptive
functions,'' \ie\ such expressions as ``the father of~$x$'' or ``the sine
of~$x$.'' These are to be defined by first defining ``descriptions.''

A ``description'' may be of two sorts, definite and indefinite
(or ambiguous). An indefinite description is a phrase of the
form ``a so-and-so,'' and a definite description is a phrase of
the form ``the so-and-so'' (in the singular). Let us begin with
the former.

``Who did you meet?'' ``I met a man.'' ``That is a very
indefinite description.'' We are therefore not departing from
usage in our terminology. Our question is: What do I really
assert when I assert ``I met a man''? Let us assume, for the
moment, that my assertion is true, and that in fact I met Jones.
It is clear that what I assert is \emph{not} ``I met Jones.'' I may say
``I met a man, but it was not Jones''; in that case, though I lie,
I do not contradict myself, as I should do if when I say I met a
\PageSep{168}%
man I really mean that I met Jones. It is clear also that the
person to whom I am speaking can understand what I say, even
if he is a foreigner and has never heard of Jones.

But we may go further: not only Jones, but no actual man,
enters into my statement. This becomes obvious when the statement
is false, since then there is no more reason why Jones
should be supposed to enter into the proposition than why anyone
else should. Indeed the statement would remain significant,
though it could not possibly be true, even if there were no man
at all. ``I met a unicorn'' or ``I met a sea-serpent'' is a
perfectly significant assertion, if we know what it would be to
be a unicorn or a sea-serpent, \ie\ what is the definition of these
fabulous monsters. Thus it is only what we may call the \emph{concept}
that enters into the proposition. In the case of ``unicorn,''
for example, there is only the concept: there is not also, somewhere
among the shades, something unreal which may be called
``a unicorn.'' Therefore, since it is significant (though false)
to say ``I met a unicorn,'' it is clear that this proposition, rightly
analysed, does not contain a constituent ``a unicorn,'' though
it does contain the concept ``unicorn.''

The question of ``unreality,'' which confronts us at this
\index{Unreality}%
point, is a very important one. Misled by grammar, the great
majority of those logicians who have dealt with this question
have dealt with it on mistaken lines. They have regarded
grammatical form as a surer guide in analysis than, in fact,
it is. And they have not known what differences in grammatical
form are important. ``I met Jones'' and ``I met a
man'' would count traditionally as propositions of the same form,
but in actual fact they are of quite different forms: the first
names an actual person, Jones; while the second involves a
propositional function, and becomes, when made explicit: ``The
function `I met~$x$ and $x$~is human' is sometimes true.'' (It
will be remembered that we adopted the convention of using
``sometimes'' as not implying more than once.) This proposition
is obviously not of the form ``I met~$x$,'' which accounts
\PageSep{169}%
for the existence of the proposition ``I met a unicorn'' in spite
of the fact that there is no such thing as ``a unicorn.''

For want of the apparatus of propositional functions, many
logicians have been driven to the conclusion that there are
\index{Meinong}%
unreal objects. It is argued, \eg\ by Meinong,\footnote
  {\Title{Untersuchungen zur Gegenstandstheorie und Psychologie}, 1904.}
that we can
speak about ``the golden mountain,'' ``the round square,''
and so on; we can make true propositions of which these are
the subjects; hence they must have some kind of logical being,
since otherwise the propositions in which they occur would be
meaningless. In such theories, it seems to me, there is a failure
of that feeling for reality which ought to be preserved even in
the most abstract studies. Logic, I should maintain, must no
\index{Logic}%
more admit a unicorn than zoology can; for logic is concerned
with the real world just as truly as zoology, though with its
more abstract and general features. To say that unicorns have
an existence in heraldry, or in literature, or in imagination,
is a most pitiful and paltry evasion. What exists in heraldry
is not an animal, made of flesh and blood, moving and breathing
of its own initiative. What exists is a picture, or a description
in words. Similarly, to maintain that Hamlet, for example,
exists in his own world, namely, in the world of Shakespeare's
imagination, just as truly as (say) Napoleon existed in the
ordinary world, is to say something deliberately confusing, or
else confused to a degree which is scarcely credible. There is
only one world, the ``real'' world: Shakespeare's imagination
is part of it, and the thoughts that he had in writing Hamlet
are real. So are the thoughts that we have in reading the play.
But it is of the very essence of fiction that only the thoughts,
feelings, etc., in Shakespeare and his readers are real, and that
there is not, in addition to them, an objective Hamlet. When
you have taken account of all the feelings roused by Napoleon
in writers and readers of history, you have not touched the actual
man; but in the case of Hamlet you have come to the end of
him. If no one thought about Hamlet, there would be nothing
\PageSep{170}%
left of him; if no one had thought about Napoleon, he would
have soon seen to it that some one did. The sense of reality is
vital in logic, and whoever juggles with it by pretending that
Hamlet has another kind of reality is doing a disservice to
thought. A robust sense of reality is very necessary in framing
a correct analysis of propositions about unicorns, golden mountains,
round squares, and other such pseudo-objects.

In obedience to the feeling of reality, we shall insist that,
in the analysis of propositions, nothing ``unreal'' is to be
admitted. But, after all, if there \emph{is} nothing unreal, how, it
may be asked, \emph{could} we admit anything unreal? The reply
is that, in dealing with propositions, we are dealing in the first
instance with symbols, and if we attribute significance to groups
of symbols which have no significance, we shall fall into the
error of admitting unrealities, in the only sense in which this is
possible, namely, as objects described. In the proposition
``I met a unicorn,'' the whole four words together make a significant
proposition, and the word ``unicorn'' by itself is significant,
in just the same sense as the word ``man.'' But the \emph{two} words
``a unicorn'' do not form a subordinate group having a meaning
of its own. Thus if we falsely attribute meaning to these two
words, we find ourselves saddled with ``a unicorn,'' and with
the problem how there can be such a thing in a world where
there are no unicorns. ``A unicorn'' is an indefinite description
which describes nothing. It is not an indefinite description
which describes something unreal. Such a proposition as
``$x$~is unreal'' only has meaning when ``$x$''~is a description,
definite or indefinite; in that case the proposition will be true
if ``$x$''~is a description which describes nothing. But whether
the description~``$x$'' describes something or describes nothing,
it is in any case not a constituent of the proposition in which it
occurs; like ``a unicorn'' just now, it is not a subordinate group
having a meaning of its own. All this results from the fact that,
when ``$x$''~is a description, ``$x$~is unreal'' or ``$x$~does not exist''
is not nonsense, but is always significant and sometimes true.
\PageSep{171}%

We may now proceed to define generally the meaning of
propositions which contain ambiguous descriptions. Suppose
we wish to make some statement about ``a so-and-so,'' where
``so-and-so's'' are those objects that have a certain property~$\phi$,
\ie\ those objects~$x$ for which the propositional function~$\phi x$ is
true. (\Eg\ if we take ``a man'' as our instance of ``a so-and-so,''
$\phi x$~will be ``$x$~is human.'') Let us now wish to assert the property~$\psi$
of ``a so-and-so,'' \ie\ we wish to assert that ``a so-and-so'' has
that property which $x$~has when $\psi x$~is true. (\Eg\ in the case
of ``I met a man,'' $\psi x$~will be ``I met~$x$.'') Now the proposition
that ``a so-and-so'' has the property~$\psi$ is \emph{not} a proposition of
the form~``$\psi x$.'' If it were, ``a so-and-so'' would have to be
identical with~$x$ for a suitable~$x$; and although (in a sense) this
may be true in some cases, it is certainly not true in such a case
as ``a unicorn.'' It is just this fact, that the statement that a
so-and-so has the property~$\psi$ is not of the form~$\psi x$, which makes
it possible for ``a so-and-so'' to be, in a certain clearly definable
sense, ``unreal.'' The definition is as follows:---

The statement that ``an object having the property~$\phi$ has
the property~$\psi$'' \\
means:

``The joint assertion of $\phi x$ and $\psi x$ is not always false.''

So far as logic goes, this is the same proposition as might
be expressed by ``some $\phi$'s are~$\psi$'s''; but rhetorically there is
a difference, because in the one case there is a suggestion of
singularity, and in the other case of plurality. This, however,
is not the important point. The important point is that, when
rightly analysed, propositions verbally about ``a so-and-so''
are found to contain no constituent represented by this phrase.
And that is why such propositions can be significant even when
there is no such thing as a so-and-so.

The definition of \emph{existence}, as applied to ambiguous descriptions,
\index{Existence}%
results from what was said at the end of the preceding
chapter. We say that ``men exist'' or ``a man exists'' if the
\PageSep{172}%
propositional function ``$x$~is human'' is sometimes true; and
generally ``a so-and-so'' exists if ``$x$~is so-and-so'' is sometimes
true. We may put this in other language. The proposition
``Socrates is a man'' is no doubt \emph{equivalent} to ``Socrates is
human,'' but it is not the very same proposition. The \emph{is} of
``Socrates is human'' expresses the relation of subject and
predicate; the \emph{is} of ``Socrates is a man'' expresses identity.
It is a disgrace to the human race that it has chosen to employ
the same word ``is'' for these two entirely different ideas---a
disgrace which a symbolic logical language of course remedies.
The identity in ``Socrates is a man'' is identity between an
object named (accepting ``Socrates'' as a name, subject to
qualifications explained later) and an object ambiguously
described. An object ambiguously described will ``exist'' when
at least one such proposition is true, \ie\ when there is at least
one true proposition of the form ``$x$~is a so-and-so,'' where ``$x$''~is
a name. It is characteristic of ambiguous (as opposed to
definite) descriptions that there may be any number of true
propositions of the above form---Socrates is a man, Plato is a
man, etc. Thus ``a man exists'' follows from Socrates, or
Plato, or anyone else. With definite descriptions, on the other
hand, the corresponding form of proposition, namely, ``$x$~is the
so-and-so'' (where ``$x$''~is a name), can only be true for one
value of~$x$ at most. This brings us to the subject of definite
descriptions, which are to be defined in a way analogous to
that employed for ambiguous descriptions, but rather more
complicated.

We come now to the main subject of the present chapter,
\index{The@\emph{The}|EtSeq}%
namely, the definition of the word \emph{the} (in the singular). One
very important point about the definition of ``a so-and-so''
applies equally to ``the so-and-so''; the definition to be sought
is a definition of propositions in which this phrase occurs, not a
definition of the phrase itself in isolation. In the case of ``a
so-and-so,'' this is fairly obvious: no one could suppose that
``a man'' was a definite object, which could be defined by itself.
\PageSep{173}%
Socrates is a man, Plato is a man, Aristotle is a man, but we
cannot infer that ``a man'' means the same as ``Socrates''
means and also the same as ``Plato'' means and also the same
as ``Aristotle'' means, since these three names have different
\index{Names}%
meanings. Nevertheless, when we have enumerated all the
men in the world, there is nothing left of which we can say,
``This is a man, and not only so, but it is \emph{the} `a man,' the quintessential
entity that is just an indefinite man without being anybody
in particular.'' It is of course quite clear that whatever
there is in the world is definite: if it is a man it is one definite
man and not any other. Thus there cannot be such an entity
as ``a man'' to be found in the world, as opposed to specific
man. And accordingly it is natural that we do not define ``a
man'' itself, but only the propositions in which it occurs.

In the case of ``the so-and-so'' this is equally true, though
at first sight less obvious. We may demonstrate that this must
be the case, by a consideration of the difference between a \emph{name}
and a \emph{definite description}. Take the proposition, ``Scott is the
author of \Title{Waverley}.'' We have here a name, ``Scott,'' and a
description, ``the author of \Title{Waverley},'' which are asserted to
apply to the same person. The distinction between a name and
all other symbols may be explained as follows:---

A name is a simple symbol whose meaning is something that
\index{Particulars}%
can only occur as subject, \ie\ something of the kind that, in
\ChapRef{XIII}., we defined as an ``individual'' or a ``particular.''
And a ``simple'' symbol is one which has no parts that are
symbols. Thus ``Scott'' is a simple symbol, because, though it
has parts (namely, separate letters), these parts are not symbols.
On the other hand, ``the author of \Title{Waverley}'' is not a simple
symbol, because the separate words that compose the phrase
are parts which are symbols. If, as may be the case, whatever
\emph{seems} to be an ``individual'' is really capable of further analysis,
we shall have to content ourselves with what may be called
``relative individuals,'' which will be terms that, throughout
\index{Individuals}%
the context in question, are never analysed and never occur
\PageSep{174}%
otherwise than as subjects. And in that case we shall have
correspondingly to content ourselves with ``relative names.''
From the standpoint of our present problem, namely, the definition
of descriptions, this problem, whether these are absolute
names or only relative names, may be ignored, since it concerns
different stages in the hierarchy of ``types,'' whereas we
have to compare such couples as ``Scott'' and ``the author of
\Title{Waverley},'' which both apply to the same object, and do not
raise the problem of types. We may, therefore, for the moment,
treat names as capable of being absolute; nothing that we shall
have to say will depend upon this assumption, but the wording
may be a little shortened by it.

We have, then, two things to compare: (1)~a \emph{name}, which
is a simple symbol, directly designating an individual which
is its meaning, and having this meaning in its own right, independently
of the meanings of all other words; (2)~a \emph{description},
which consists of several words, whose meanings are already
fixed, and from which results whatever is to be taken as the
``meaning'' of the description.

A proposition containing a description is not identical with
what that proposition becomes when a name is substituted,
even if the name names the same object as the description
describes. ``Scott is the author of \Title{Waverley}'' is obviously a
different proposition from ``Scott is Scott'': the first is a fact
in literary history, the second a trivial truism. And if we put
anyone other than Scott in place of ``the author of \Title{Waverley},''
our proposition would become false, and would therefore certainly
no longer be the same proposition. But, it may be said, our
proposition is essentially of the same form as (say) ``Scott is
Sir~Walter,'' in which two names are said to apply to the same
person. The reply is that, if ``Scott is Sir~Walter'' really means
``the person named `Scott' is the person named `Sir~Walter,'\,''
then the names are being used as descriptions: \ie\ the individual,
instead of being named, is being described as the person having
that name. This is a way in which names are frequently used
\PageSep{175}%
in practice, and there will, as a rule, be nothing in the phraseology
to show whether they are being used in this way or \emph{as} names.
When a name is used directly, merely to indicate what we are
speaking about, it is no part of the \emph{fact} asserted, or of the falsehood
if our assertion happens to be false: it is merely part of the
symbolism by which we express our thought. What we want
to express is something which might (for example) be translated
into a foreign language; it is something for which the actual
words are a vehicle, but of which they are no part. On the other
hand, when we make a proposition about ``the person called
`Scott,'\,'' the actual name ``Scott'' enters into what we are
asserting, and not merely into the language used in making the
assertion. Our proposition will now be a different one if we
substitute ``the person called `Sir~Walter.'\,'' But so long as
we are using names \emph{as} names, whether we say ``Scott'' or whether
we say ``Sir~Walter'' is as irrelevant to what we are asserting
as whether we speak English or French. Thus so long as names
are used as names, ``Scott is Sir~Walter'' is the same trivial
proposition as ``Scott is Scott.'' This completes the proof that
``Scott is the author of \Title{Waverley}'' is not the same proposition
as results from substituting a name for ``the author of \Title{Waverley},''
no matter what name may be substituted.

When we use a variable, and speak of a propositional function,
$\phi x$~say, the process of applying general statements about~$x$ to
particular cases will consist in substituting a name for the letter~``$x$,''
assuming that $\phi$~is a function which has individuals for its
arguments. Suppose, for example, that $\phi x$~is ``always true'';
let it be, say, the ``law of identity,'' $x = x$. Then we may substitute
for~``$x$'' any name we choose, and we shall obtain a true
proposition. Assuming for the moment that ``Socrates,''
``Plato,'' and ``Aristotle'' are names (a very rash assumption),
we can infer from the law of identity that Socrates is Socrates,
Plato is Plato, and Aristotle is Aristotle. But we shall commit
a fallacy if we attempt to infer, without further premisses, that
the author of \Title{Waverley} is the author of \Title{Waverley}. This results
\PageSep{176}%
from what we have just proved, that, if we substitute a name for
``the author of \Title{Waverley}'' in a proposition, the proposition
we obtain is a different one. That is to say, applying the result
to our present case: If ``$x$''~is a name, ``$x = x$'' is not the same
proposition as ``the author of \Title{Waverley} is the author of \Title{Waverley},''
no matter what name ``$x$''~may be. Thus from the fact that
all propositions of the form ``$x = x$'' are true we cannot infer,
without more ado, that the author of \Title{Waverley} is the author of
\Title{Waverley}. In fact, propositions of the form ``the so-and-so
is the so-and-so'' are not always true: it is necessary that the
so-and-so should \emph{exist} (a term which will be explained shortly).
It is false that the present King of France is the present King of
France, or that the round square is the round square. When we
substitute a description for a name, propositional functions
which are ``always true'' may become false, if the description
describes nothing. There is no mystery in this as soon as we
realise (what was proved in the preceding paragraph) that when
we substitute a description the result is not a value of the
propositional function in question.

We are now in a position to define propositions in which a
definite description occurs. The only thing that distinguishes
``the so-and-so'' from ``a so-and-so'' is the implication of
uniqueness. We cannot speak of ``\emph{the} inhabitant of London,''
because inhabiting London is an attribute which is not unique.
We cannot speak about ``the present King of France,'' because
there is none; but we can speak about ``the present King of
England.'' Thus propositions about ``the so-and-so'' always
imply the corresponding propositions about ``a so-and-so,''
with the addendum that there is not more than one so-and-so.
Such a proposition as ``Scott is the author of \Title{Waverly}'' could
not be true if \Title{Waverly} had never been written, or if several
people had written it; and no more could any other proposition
resulting from a propositional function~$x$ by the substitution
of ``the author of \Title{Waverly}'' for~``$x$.'' We may say that ``the
author of \Title{Waverly}'' means ``the value of~$x$ for which `$x$~wrote
\PageSep{177}%
\Title{Waverly}' is true.'' Thus the proposition ``the author of
\Title{Waverly} was Scotch,'' for example, involves:
\begin{List}
\item[(1)] ``$x$~wrote \Title{Waverly}'' is not always false;

\item[(2)] ``if $x$~and~$y$ wrote \Title{Waverly}, $x$~and~$y$ are identical'' is
always true;

\item[(3)] ``if $x$~wrote \Title{Waverly}, $x$~was Scotch'' is always true.
\end{List}
These three propositions, translated into ordinary language,
state:
\begin{List}
\item[(1)] at least one person wrote \Title{Waverly};

\item[(2)] at most one person wrote \Title{Waverly};

\item[(3)] whoever wrote \Title{Waverly} was Scotch.
\end{List}
All these three are implied by ``the author of \Title{Waverly} was
Scotch.'' Conversely, the three together (but no two of them)
imply that the author of \Title{Waverly} was Scotch. Hence the
three together may be taken as defining what is meant by the
proposition ``the author of \Title{Waverly} was Scotch.''

We may somewhat simplify these three propositions. The
first and second together are equivalent to: ``There is a term~$c$
such that `$x$~wrote \Title{Waverly}' is true when $x$ is~$c$ and is false
when $x$~is not~$c$.'' In other words, ``There is a term~$c$ such that
`$x$~wrote \Title{Waverly}' is always equivalent to `$x$~is~$c$.'\,'' (Two
propositions are ``equivalent'' when both are true or both are
false.) We have here, to begin with, two functions of~$x$, ``$x$~wrote
\Title{Waverly}'' and ``$x$~is~$c$,'' and we form a function of~$c$ by
considering the equivalence of these two functions of~$x$ for all
values of~$x$; we then proceed to assert that the resulting function
of~$c$ is ``sometimes true,'' \ie\ that it is true for at least one value
of~$c$. (It obviously cannot be true for more than one value of~$c$.)
These two conditions together are defined as giving the meaning
of ``the author of \Title{Waverly} exists.''
\index{Existence}%

We may now define ``the term satisfying the function~$\phi x$
exists.'' This is the general form of which the above is a particular
case. ``The author of \Title{Waverly}'' is ``the term satisfying
the function `$x$~wrote \Title{Waverly}.'\,'' And ``the so-and-so'' will
\PageSep{178}%
always involve reference to some propositional function, namely,
that which defines the property that makes a thing a so-and-so.
Our definition is as follows:---

``The term satisfying the function~$\phi x$ exists'' means:

``There is a term~$c$ such that $\phi x$~is always equivalent to `$x$~is~$c$.'\,''

In order to define ``the author of \Title{Waverly} was Scotch,''
we have still to take account of the third of our three propositions,
namely, ``Whoever wrote \Title{Waverly} was Scotch.'' This
will be satisfied by merely adding that the~$c$ in question is to
be Scotch. Thus ``the author of \Title{Waverly} was Scotch'' is:
\begin{Prop}
``There is a term~$c$ such that (1)~`$x$~wrote \Title{Waverly}' is always
equivalent to `$x$~is~$c$,' (2)~$c$~is Scotch.''
\end{Prop}
And generally: ``the term satisfying~$\phi x$ satisfies~$\psi x$'' is
defined as meaning:
\begin{Prop}
``There is a term~$c$ such that (1)~$\phi x$~is always equivalent to
`$x$~is~$c$,' (2)~$\psi c$~is true.''
\end{Prop}
This is the definition of propositions in which descriptions occur.

It is possible to have much knowledge concerning a term
described, \ie\ to know many propositions concerning ``the so-and-so,''
without actually knowing what the so-and-so is, \ie\
without knowing any proposition of the form ``$x$~is the so-and-so,''
where ``$x$''~is a name. In a detective story propositions about
``the man who did the deed'' are accumulated, in the hope
that ultimately they will suffice to demonstrate that it was~$A$
who did the deed. We may even go so far as to say that,
in all such knowledge as can be expressed in words---with the
exception of ``this'' and ``that'' and a few other words of
which the meaning varies on different occasions---no names,
in the strict sense, occur, but what seem like names are really
descriptions. We may inquire significantly whether Homer
existed, which we could not do if ``Homer'' were a name. The
proposition ``the so-and-so exists'' is significant, whether
true or false; but if $a$~is the so-and-so (where ``$a$''~is a name),
the words ``$a$~exists'' are meaningless. It is only of descriptions---definite
\PageSep{179}%
or indefinite---that existence can be significantly
asserted; for, if ``$a$''~is a name, it \emph{must} name something: what
does not name anything is not a name, and therefore, if intended
to be a name, is a symbol devoid of meaning, whereas a description,
like ``the present King of France,'' does not become incapable
of occurring significantly merely on the ground that it
describes nothing, the reason being that it is a \emph{complex} symbol,
of which the meaning is derived from that of its constituent
symbols. And so, when we ask whether Homer existed, we are
using the word ``Homer'' as an abbreviated description: we
may replace it by (say) ``the author of the \Title{Iliad} and the \Title{Odyssey}.''
The same considerations apply to almost all uses of what look
like proper names.

When descriptions occur in propositions, it is necessary to
\index{Occurrences, primary and secondary}%
distinguish what may be called ``primary'' and ``secondary''
occurrences. The abstract distinction is as follows. A description
has a ``primary'' occurrence when the proposition in
which it occurs results from substituting the description for~``$x$''
in some propositional function~$\phi x$; a description has a
``secondary'' occurrence when the result of substituting the
description for~$x$ in~$\phi x$ gives only \emph{part} of the proposition concerned.
An instance will make this clearer. Consider ``the
present King of France is bald.'' Here ``the present King of
France'' has a primary occurrence, and the proposition is false.
Every proposition in which a description which describes nothing
has a primary occurrence is false. But now consider ``the
present King of France is not bald.'' This is ambiguous. If
we are first to take ``$x$~is bald,'' then substitute ``the present
King of France'' for~``$x$'' and then deny the result, the occurrence
of ``the present King of France'' is secondary and our proposition
is true; but if we are to take ``$x$~is not bald'' and substitute
``the present King of France'' for~``$x$'' then ``the present
King of France'' has a primary occurrence and the proposition
is false. Confusion of primary and secondary occurrences is a
ready source of fallacies where descriptions are concerned.
\PageSep{180}%

Descriptions occur in mathematics chiefly in the form of
\emph{descriptive functions}, \ie\ ``the term having the relation~$R$ to~$y$,''
\index{Functions!descriptive}%
or ``the~$R$ of~$y$'' as we may say, on the analogy of ``the
father of~$y$'' and similar phrases. To say ``the father of~$y$ is
rich,'' for example, is to say that the following propositional
function of~$c$: ``$c$~is rich, and `$x$~begat~$y$' is always equivalent
to `$x$~is~$c$,'\,'' is ``sometimes true,'' \ie\ is true for at least one
value of~$c$. It obviously cannot be true for more than one
value.

The theory of descriptions, briefly outlined in the present
chapter, is of the utmost importance both in logic and in theory
of knowledge. But for purposes of mathematics, the more
philosophical parts of the theory are not essential, and have
therefore been omitted in the above account, which has confined
itself to the barest mathematical requisites.
\PageSep{181}%


\Chapter{XVII}{Classes}

\First{In} the present chapter we shall be concerned with \emph{the} in the
\index{Classes|EtSeq}%
plural: the inhabitants of London, the sons of rich men, and
so on. In other words, we shall be concerned with \emph{classes}. We
saw in \ChapRef{II}.\ that a cardinal number is to be defined as a
class of classes, and in \ChapRef{III}.\ that the number~$1$ is to be
defined as the class of all unit classes, \ie\ of all that have just
one member, as we should say but for the vicious circle. Of
course, when the number~$1$ is defined as the class of all unit
classes, ``unit classes'' must be defined so as not to assume
that we know what is meant by ``one''; in fact, they are defined
in a way closely analogous to that used for descriptions, namely:
A class~$\alpha$ is said to be a ``unit'' class if the propositional function
``\,`$x$~is an~$\alpha$' is always equivalent to `$x$~is~$c$'\,'' (regarded as a
function of~$c$) is not always false, \ie\, in more ordinary language,
if there is a term~$c$ such that $x$~will be a member of~$\alpha$ when $x$~is~$c$
but not otherwise. This gives us a definition of a unit class if we
already know what a class is in general. Hitherto we have, in
dealing with arithmetic, treated ``class'' as a primitive idea.
But, for the reasons set forth in \ChapRef{XIII}., if for no others,
we cannot accept ``class'' as a primitive idea. We must seek a
definition on the same lines as the definition of descriptions,
\ie\ a definition which will assign a meaning to propositions in
whose verbal or symbolic expression words or symbols apparently
representing classes occur, but which will assign a meaning that
altogether eliminates all mention of classes from a right analysis
\PageSep{182}%
of such propositions. We shall then be able to say that the
symbols for classes are mere conveniences, not representing
objects called ``classes,'' and that classes are in fact, like descriptions,
logical fictions, or (as we say) ``incomplete symbols.''
\index{Incomplete symbols}%

The theory of classes is less complete than the theory of descriptions,
and there are reasons (which we shall give in outline)
for regarding the definition of classes that will be suggested as
not finally satisfactory. Some further subtlety appears to be
required; but the reasons for regarding the definition which
will be offered as being approximately correct and on the right
lines are overwhelming.

The first thing is to realise why classes cannot be regarded
as part of the ultimate furniture of the world. It is difficult
to explain precisely what one means by this statement, but one
consequence which it implies may be used to elucidate its meaning.
If we had a complete symbolic language, with a definition for
everything definable, and an undefined symbol for everything
indefinable, the undefined symbols in this language would represent
symbolically what I mean by ``the ultimate furniture of
the world.'' I am maintaining that no symbols either for ``class''
in general or for particular classes would be included in this
apparatus of undefined symbols. On the other hand, all the
particular things there are in the world would have to have
names which would be included among undefined symbols.
\index{Names}%
We might try to avoid this conclusion by the use of descriptions.
Take (say) ``the last thing Cæsar saw before he died.'' This
is a description of some particular; we might use it as (in one
perfectly legitimate sense) a \emph{definition} of that particular. But
if ``$a$''~is a \emph{name} for the same particular, a proposition in which
``$a$''~occurs is not (as we saw in the preceding chapter) identical
with what this proposition becomes when for~``$a$'' we substitute
``the last thing Cæsar saw before he died.'' If our language
does not contain the name~``$a$'' or some other name for the same
particular, we shall have no means of expressing the proposition
which we expressed by means of~``$a$'' as opposed to the one that
\PageSep{183}%
we expressed by means of the description. Thus descriptions
would not enable a perfect language to dispense with names for
all particulars. In this respect, we are maintaining, classes
differ from particulars, and need not be represented by undefined
symbols. Our first business is to give the reasons for this opinion.

We have already seen that classes cannot be regarded as a
species of individuals, on account of the contradiction about
classes which are not members of themselves (explained in
\ChapRef{XIII}.), and because we can prove that the number of
classes is greater than the number of individuals.

We cannot take classes in the \emph{pure} extensional way as simply
heaps or conglomerations. If we were to attempt to do that,
we should find it impossible to understand how there can be such
a class as the null-class, which has no members at all and cannot
be regarded as a ``heap''; we should also find it very hard to
understand how it comes about that a class which has only one
member is not identical with that one member. I do not mean
to assert, or to deny, that there are such entities as ``heaps.''
As a mathematical logician, I am not called upon to have an
opinion on this point. All that I am maintaining is that, if there
are such things as heaps, we cannot identify them with the classes
composed of their constituents.

We shall come much nearer to a satisfactory theory if we
try to identify classes with propositional functions. Every
class, as we explained in \ChapRef{II}., is defined by some propositional
function which is true of the members of the class
and false of other things. But if a class can be defined by one
propositional function, it can equally well be defined by any
other which is true whenever the first is true and false whenever
the first is false. For this reason the class cannot be identified
with any one such propositional function rather than with
any other---and given a propositional function, there are always
many others which are true when it is true and false when it is
\index{Equivalence}%
false. We say that two propositional functions are ``formally
equivalent'' when this happens. Two \emph{propositions} are ``equivalent''
\PageSep{184}%
when both are true or both false; two propositional
functions $\phi x$,~$\psi x$ are ``formally equivalent'' when $\phi x$~is always
equivalent to~$\psi x$. It is the fact that there are other functions
formally equivalent to a given function that makes it impossible
to identify a class with a function; for we wish classes to be such
that no two distinct classes have exactly the same members,
and therefore two formally equivalent functions will have to
determine the same class.

When we have decided that classes cannot be things of the
same sort as their members, that they cannot be just heaps or
aggregates, and also that they cannot be identified with propositional
functions, it becomes very difficult to see what they
can be, if they are to be more than symbolic fictions. And if
we can find any way of dealing with them as symbolic fictions,
we increase the logical security of our position, since we avoid
the need of assuming that there are classes without being compelled
to make the opposite assumption that there are no classes.
We merely abstain from both assumptions. This is an example
of Occam's razor, namely, ``entities are not to be multiplied
\index{Occam}%
without necessity.'' But when we refuse to assert that there
are classes, we must not be supposed to be asserting dogmatically
that there are none. We are merely agnostic as regards them:
like Laplace, we can say, ``\Foreign{je n'ai pas besoin de cette hypothèse.}''

Let us set forth the conditions that a symbol must fulfil if
it is to serve as a class. I think the following conditions will
be found necessary and sufficient:---

(1) Every propositional function must determine a class,
consisting of those arguments for which the function is true.
Given any proposition (true or false), say about Socrates, we
can imagine Socrates replaced by Plato or Aristotle or a gorilla
or the man in the moon or any other individual in the world.
In general, some of these substitutions will give a true proposition
and some a false one. The class determined will consist of all
those substitutions that give a true one. Of course, we have
still to decide what we mean by ``all those which, etc.'' All that
\PageSep{185}%
we are observing at present is that a class is rendered determinate
by a propositional function, and that every propositional function
determines an appropriate class.

(2) Two formally equivalent propositional functions must
determine the same class, and two which are not formally equivalent
must determine different classes. That is, a class is determined
by its membership, and no two different classes can have
the same membership. (If a class is determined by a function~$\phi x$,
we say that $a$~is a ``member'' of the class if $\phi a$~is true.)

(3) We must find some way of defining not only classes, but
classes of classes. We saw in \ChapRef{II}.\ that cardinal numbers
are to be defined as classes of classes. The ordinary phrase
of elementary mathematics, ``The combinations of $n$~things
$m$~at a time'' represents a class of classes, namely, the class of
all classes of $m$~terms that can be selected out of a given class
of $n$~terms. Without some symbolic method of dealing with
classes of classes, mathematical logic would break down.

(4) It must under all circumstances be meaningless (not false)
to suppose a class a member of itself or not a member of itself.
This results from the contradiction which we discussed in
\ChapRef{XIII}.

(5) Lastly---and this is the condition which is most difficult
\index{Induction, mathematical}%
\index{Types, logical}%
of fulfilment,---it must be possible to make propositions about
\emph{all} the classes that are composed of individuals, or about \emph{all} the
classes that are composed of objects of any one logical ``type.''
If this were not the case, many uses of classes would go astray---for
example, mathematical induction. In defining the posterity
of a given term, we need to be able to say that a member of the
posterity belongs to \emph{all} hereditary classes to which the given
term belongs, and this requires the sort of totality that is in
question. The reason there is a difficulty about this condition
is that it can be proved to be impossible to speak of \emph{all} the propositional
functions that can have arguments of a given type.

We will, to begin with, ignore this last condition and the
problems which it raises. The first two conditions may be
\PageSep{186}%
taken together. They state that there is to be one class, no
more and no less, for each group of formally equivalent propositional
functions; \eg\ the class of men is to be the same as
that of featherless bipeds or rational animals or Yahoos or whatever
other characteristic may be preferred for defining a human
being. Now, when we say that two formally equivalent propositional
functions may be not identical, although they define
the same class, we may prove the truth of the assertion by pointing
out that a statement may be true of the one function and
false of the other; \eg\ ``I believe that all men are mortal''
may be true, while ``I believe that all rational animals are
mortal'' may be false, since I may believe falsely that the
Phoenix is an immortal rational animal. Thus we are led to
consider \emph{statements about functions}, or (more correctly) \emph{functions
of functions}.

Some of the things that may be said about a function may
be regarded as said about the class defined by the function,
whereas others cannot. The statement ``all men are mortal''
involves the functions ``$x$~is human'' and ``$x$~is mortal''; or,
if we choose, we can say that it involves the classes \emph{men} and
\emph{mortals}. We can interpret the statement in either way, because
its truth-value is unchanged if we substitute for ``$x$~is human''
or for ``$x$~is mortal'' any formally equivalent function. But,
as we have just seen, the statement ``I believe that all men are
mortal'' cannot be regarded as being about the class determined
by either function, because its truth-value may be changed
by the substitution of a formally equivalent function (which
leaves the class unchanged). We will call a statement involving
\index{Functions!intensional and extensional}%
a function~$\phi x$ an ``extensional'' function of the function~$\phi x$, if
it is like ``all men are mortal,'' \ie\ if its truth-value is unchanged
by the substitution of any formally equivalent function; and
when a function of a function is not extensional, we will call it
``intensional,'' so that ``I believe that all men are mortal''
is an intensional function of ``$x$~is human'' or ``$x$~is mortal.''
Thus \emph{extensional} functions of a function~$x$ may, for practical
\PageSep{187}%
purposes, be regarded as functions of the class determined by~$x$,
while \emph{intensional} functions cannot be so regarded.

It is to be observed that all the \emph{specific} functions of functions
that we have occasion to introduce in mathematical logic are
extensional. Thus, for example, the two fundamental functions
of functions are: ``$\phi x$~is always true'' and ``$\phi x$~is sometimes
true.'' Each of these has its truth-value unchanged if any
formally equivalent function is substituted for~$\phi x$. In the
language of classes, if $\alpha$~is the class determined by~$\phi x$, ``$\phi x$~is
always true'' is equivalent to ``everything is a member of~$\alpha$,''
and ``$\phi x$~is sometimes true'' is equivalent to ``$\alpha$~has members''
or (better) ``$\alpha$~has at least one member.'' Take, again, the
condition, dealt with in the preceding chapter, for the existence
of ``the term satisfying~$\phi x$.'' The condition is that there is a
term~$c$ such that $\phi x$~is always equivalent to ``$x$~is~$c$.'' This
is obviously extensional. It is equivalent to the assertion
that the class defined by the function~$\phi x$ is a unit class, \ie\ a
class having one member; in other words, a class which is a
member of~$1$.

Given a function of a function which may or may not be
extensional, we can always derive from it a connected and
certainly extensional function of the same function, by the
following plan: Let our original function of a function be one
which attributes to~$\phi x$ the property~$f$; then consider the assertion
``there is a function having the property~$f$ and formally
equivalent to~$\phi x$.'' This is an extensional function of~$\phi x$; it
is true when our original statement is true, and it is formally
equivalent to the original function of~$\phi x$ if this original function
is extensional; but when the original function is intensional,
the new one is more often true than the old one. For example,
consider again ``I believe that all men are mortal,'' regarded
as a function of ``$x$~is human.'' The derived extensional function
is: ``There is a function formally equivalent to `$x$~is human'
and such that I believe that whatever satisfies it is mortal.''
This remains true when we substitute ``$x$~is a rational animal''
\PageSep{188}
for ``$x$~is human,'' even if I believe falsely that the Phoenix is
rational and immortal.

We give the name of ``derived extensional function'' to the
function constructed as above, namely, to the function: ``There
is a function having the property~$f$ and formally equivalent to~$\phi x$,''
where the original function was ``the function~$\phi x$ has
the property~$f$.''

We may regard the derived extensional function as having
for its argument the class determined by the function~$\phi x$, and
as asserting~$f$ of this class. This may be taken as the definition
of a proposition about a class. \Ie\ we may define:

To assert that ``the class determined by the function~$\phi x$
has the property~$f$'' is to assert that $\phi x$~satisfies the extensional
function derived from~$f$.

This gives a meaning to any statement about a class which
can be made significantly about a function; and it will be
found that technically it yields the results which are required
in order to make a theory symbolically satisfactory.\footnote
  {See \Title{Principia Mathematica}, vol.~i.\ pp.~75--84 and *~20.}

What we have said just now as regards the definition of
classes is sufficient to satisfy our first four conditions. The
way in which it secures the third and fourth, namely, the possibility
of classes of classes, and the impossibility of a class being
or not being a member of itself, is somewhat technical; it is
explained in \Title{Principia Mathematica}, but may be taken for
granted here. It results that, but for our fifth condition, we
might regard our task as completed. But this condition---at
once the most important and the most difficult---is not fulfilled
in virtue of anything we have said as yet. The difficulty is
\index{Types, logical}%
connected with the theory of types, and must be briefly discussed.\footnote
  {The reader who desires a fuller discussion should consult \Title{Principia
  Mathematica}, Introduction, chap.~ii.; also *~12.}

We saw in \ChapRef{XIII}.\ that there is a hierarchy of logical
types, and that it is a fallacy to allow an object belonging to
one of these to be substituted for an object belonging to another.
\PageSep{189}%
Now it is not difficult to show that the various functions which
\index{Functions!predicative}%
can take a given object~$a$ as argument are not all of one type.
Let us call them all $a$-functions. We may take first those among
them which do not involve reference to any collection of functions;
these we will call ``predicative $a$-functions.'' If we now proceed
to functions involving reference to the totality of predicative
$a$-functions, we shall incur a fallacy if we regard these as of the
same type as the predicative $a$-functions. Take such an everyday
statement as ``$a$~is a typical Frenchman.'' How shall
we define a ``typical'' Frenchman? We may define him as
one ``possessing all qualities that are possessed by most French
men.'' But unless we confine ``all qualities'' to such as do not
involve a reference to any totality of qualities, we shall have to
observe that most Frenchmen are \emph{not} typical in the above sense,
and therefore the definition shows that to be not typical is
essential to a typical Frenchman. This is not a logical contradiction,
since there is no reason why there should be any typical
Frenchmen; but it illustrates the need for separating off
qualities that involve reference to a totality of qualities from
those that do not.

Whenever, by statements about ``all'' or ``some'' of the
values that a variable can significantly take, we generate a
new object, this new object must not be among the values which
our previous variable could take, since, if it were, the totality
of values over which the variable could range would only be
definable in terms of itself, and we should be involved in a vicious
circle. For example, if I say ``Napoleon had all the qualities
that make a great general,'' I must define ``qualities'' in such a
way that it will not include what I am now saying, \ie\ ``having
all the qualities that make a great general'' must not be itself a
quality in the sense supposed. This is fairly obvious, and is
the principle which leads to the theory of types by which vicious-circle
paradoxes are avoided. As applied to $a$-functions, we
may suppose that ``qualities'' is to mean ``predicative functions.''
Then when I say ``Napoleon had all the qualities, etc.,'' I mean
\PageSep{190}%
``Napoleon satisfied all the predicative functions, etc.'' This
statement attributes a property to Napoleon, but not a predicative
property; thus we escape the vicious circle. But
wherever ``all functions which'' occurs, the functions in question
must be limited to one type if a vicious circle is to be avoided;
and, as Napoleon and the typical Frenchman have shown, the
type is not rendered determinate by that of the argument. It
would require a much fuller discussion to set forth this point
fully, but what has been said may suffice to make it clear that
the functions which can take a given argument are of an infinite
series of types. We could, by various technical devices, construct
a variable which would run through the first~$n$ of these
types, where $n$~is finite, but we cannot construct a variable which
will run through them all, and, if we could, that mere fact would
at once generate a new type of function with the same arguments,
and would set the whole process going again.

We call predicative $a$-functions the \emph{first} type of $a$-functions;
$a$-functions involving reference to the totality of the first type
we call the \emph{second} type; and so on. No variable $a$-function
can run through all these different types: it must stop short at
some definite one.

These considerations are relevant to our definition of the
derived extensional function. We there spoke of ``a function
formally equivalent to~$\phi x$.'' It is necessary to decide upon
the type of our function. Any decision will do, but some decision
is unavoidable. Let us call the supposed formally equivalent
function~$\psi$. Then $\psi$~appears as a variable, and must be of
some determinate type. All that we know necessarily about
the type of~$\phi$ is that it takes arguments of a given type---that
it is (say) an $a$-function. But this, as we have just seen, does
not determine its type. If we are to be able (as our fifth requisite
demands) to deal with \emph{all} classes whose members are of the same
type as~$a$, we must be able to define all such classes by means of
functions of some one type; that is to say, there must be some
type of $a$-function, say the~$n$\Ord{th}, such that any $a$-function is formally
\PageSep{191}%
equivalent to some $a$-function of the $n$\Ord{th}~type. If this is the case,
then any extensional function which holds of all $a$-functions
of the $n$\Ord{th}~type will hold of any $a$-function whatever. It is chiefly
as a technical means of embodying an assumption leading to
this result that classes are useful. The assumption is called the
\index{Reducibility, axiom of}%
``axiom of reducibility,'' and may be stated as follows:---

``There is a type ($\Typo{r}{\tau}$~say) of $a$-functions such that, given any
$a$-function, it is formally equivalent to some function of the type
in question.''

If this axiom is assumed, we use functions of this type in
defining our associated extensional function. Statements about
all $a$-classes (\ie\ all classes defined by $a$-functions) can be reduced
to statements about all $a$-functions of the type~$\tau$. So long as
only extensional functions of functions are involved, this gives
us in practice results which would otherwise have required the
impossible notion of ``all $a$-functions.'' One particular region
where this is vital is mathematical induction.

The axiom of reducibility involves all that is really essential
in the theory of classes. It is therefore worth while to ask
whether there is any reason to suppose it true.

This axiom, like the multiplicative axiom and the axiom
of infinity, is necessary for certain results, but not for the bare
existence of deductive reasoning. The theory of deduction,
as explained in \ChapRef{XIV}., and the laws for propositions
involving ``all'' and ``some,'' are of the very texture of mathematical
reasoning: without them, or something like them,
we should not merely not obtain the same results, but we should
not obtain any results at all. We cannot use them as hypotheses,
and deduce hypothetical consequences, for they are
rules of deduction as well as premisses. They must be absolutely
true, or else what we deduce according to them does not even
follow from the premisses. On the other hand, the axiom of
reducibility, like our two previous mathematical axioms, could
perfectly well be stated as an hypothesis whenever it is used,
instead of being assumed to be actually true. We can deduce
\PageSep{192}%
its consequences hypothetically; we can also deduce the consequences
of supposing it false. It is therefore only convenient,
not necessary. And in view of the complication of the theory
of types, and of the uncertainty of all except its most general
principles, it is impossible as yet to say whether there may
not be some way of dispensing with the axiom of reducibility
altogether. However, assuming the correctness of the theory
outlined above, what can we say as to the truth or falsehood of
the axiom?

The axiom, we may observe, is a generalised form of Leibniz's
\index{Leibniz}%
identity of indiscernibles. Leibniz assumed, as a logical principle,
\index{Indiscernibles}%
that two different subjects must differ as to predicates. Now
predicates are only some among what we called ``predicative
functions,'' which will include also relations to given terms,
and various properties not to be reckoned as predicates. Thus
Leibniz's assumption is a much stricter and narrower one than
ours. (Not, of course, according to \emph{his} logic, which regarded
\emph{all} propositions as reducible to the subject-predicate form.)
But there is no good reason for believing his form, so far as I can
see. There might quite well, as a matter of abstract logical
possibility, be two things which had exactly the same predicates,
in the narrow sense in which we have been using the word ``predicate.''
How does our axiom look when we pass beyond predicates
in this narrow sense? In the actual world there seems
no way of doubting its empirical truth as regards particulars,
owing to spatio-temporal differentiation: no two particulars
have exactly the same spatial and temporal relations to all other
particulars. But this is, as it were, an accident, a fact about
the world in which we happen to find ourselves. Pure logic,
and pure mathematics (which is the same thing), aims at being
true, in Leibnizian phraseology, in all possible worlds, not only
in this higgledy-piggledy job-lot of a world in which chance has
imprisoned us. There is a certain lordliness which the logician
should preserve: he must not condescend to derive arguments
from the things he sees about him.
\PageSep{193}%

Viewed from this strictly logical point of view, I do not see
any reason to believe that the axiom of reducibility is logically
necessary, which is what would be meant by saying that it is
true in all possible worlds. The admission of this axiom into
a system of logic is therefore a defect, even if the axiom is empirically
true. It is for this reason that the theory of classes cannot
be regarded as being as complete as the theory of descriptions.
There is need of further work on the theory of types, in the hope
of arriving at a doctrine of classes which does not require such a
dubious assumption. But it is reasonable to regard the theory
outlined in the present chapter as right in its main lines, \ie\ in
its reduction of propositions nominally about classes to propositions
about their defining functions. The avoidance of
classes as entities by this method must, it would seem, be sound
in principle, however the detail may still require adjustment.
It is because this seems indubitable that we have included the
theory of classes, in spite of our desire to exclude, as far as possible,
whatever seemed open to serious doubt.

The theory of classes, as above outlined, reduces itself to one
axiom and one definition. For the sake of definiteness, we will
here repeat them. The axiom is:

\begin{Definition}[\itshape]
There is a type~$\tau$ such that if $\phi$~is a function which can take a
%[** TN: "a" unitalicized in the original]
given object~$a$ as argument, then there is a function~$\psi$ of the type~$\tau$
which is formally equivalent to~$\phi$.
\end{Definition}

The definition is:

\begin{Definition}[\itshape]
If $\phi$~is a function which can take a given object~$a$ as argument,
and $\tau$~the type mentioned in the above axiom, then to say that
the class determined by~$\phi$ has the property~$f$ is to say that there
is a function of type~$\tau$, formally equivalent to~$\phi$, and having the
property~$f$.
\end{Definition}
\PageSep{194}%


\Chapter{XVIII}{Mathematics and Logic}

\First{Mathematics} and logic, historically speaking, have been entirely
\index{Logic|EtSeq}%
\index{Mathematics|EtSeq}%
distinct studies. Mathematics has been connected with science,
logic with Greek. But both have developed in modern times:
logic has become more mathematical and mathematics has
become more logical. The consequence is that it has now become
wholly impossible to draw a line between the two; in fact, the
two are one. They differ as boy and man: logic is the youth
of mathematics and mathematics is the manhood of logic. This
view is resented by logicians who, having spent their time in
the study of classical texts, are incapable of following a piece
of symbolic reasoning, and by mathematicians who have learnt
a technique without troubling to inquire into its meaning or
justification. Both types are now fortunately growing rarer.
So much of modern mathematical work is obviously on the
border-line of logic, so much of modern logic is symbolic and
formal, that the very close relationship of logic and mathematics
has become obvious to every instructed student. The proof
of their identity is, of course, a matter of detail: starting with
premisses which would be universally admitted to belong to
logic, and arriving by deduction at results which as obviously
belong to mathematics, we find that there is no point at which
a sharp line can be drawn, with logic to the left and mathematics
to the right. If there are still those who do not admit
the identity of logic and mathematics, we may challenge them
to indicate at what point, in the successive definitions and
\PageSep{195}%
deductions of \Title{Principia Mathematica}, they consider that logic
ends and mathematics begins. It will then be obvious that any
answer must be quite arbitrary.

In the earlier chapters of this book, starting from the natural
numbers, we have first defined ``cardinal number'' and shown
how to generalise the conception of number, and have then
analysed the conceptions involved in the definition, until we found
ourselves dealing with the fundamentals of logic. In a synthetic,
deductive treatment these fundamentals come first, and the
natural numbers are only reached after a long journey. Such
treatment, though formally more correct than that which we
have adopted, is more difficult for the reader, because the ultimate
logical concepts and propositions with which it starts are remote
and unfamiliar as compared with the natural numbers. Also
they represent the present frontier of knowledge, beyond which
is the still unknown; and the dominion of knowledge over them
is not as yet very secure.

It used to be said that mathematics is the science of ``quantity.''
\index{Quantity}%
``Quantity'' is a vague word, but for the sake of argument
we may replace it by the word ``number.'' The statement
that mathematics is the science of number would be untrue
in two different ways. On the one hand, there are recognised
branches of mathematics which have nothing to do with number---all
geometry that does not use co-ordinates or measurement,
for example: projective and descriptive geometry, down to
the point at which co-ordinates are introduced, does not have
to do with number, or even with quantity in the sense of \emph{greater}
and \emph{less}. On the other hand, through the definition of cardinals,
through the theory of induction and ancestral relations, through
the general theory of series, and through the definitions of the
arithmetical operations, it has become possible to generalise much
that used to be proved only in connection with numbers. The
result is that what was formerly the single study of Arithmetic
has now become divided into numbers of separate studies, no
one of which is specially concerned with numbers. The most
\PageSep{196}%
elementary properties of numbers are concerned with one-one
relations, and similarity between classes. Addition is concerned
with the construction of mutually exclusive classes respectively
similar to a set of classes which are not known to be mutually
exclusive. Multiplication is merged in the theory of ``selections,''
\ie\ of a certain kind of one-many relations. Finitude
is merged in the general study of ancestral relations, which yields
the whole theory of mathematical induction. The ordinal
properties of the various kinds of number-series, and the elements
of the theory of continuity of functions and the limits of functions,
can be generalised so as no longer to involve any essential reference
to numbers. It is a principle, in all formal reasoning, to generalise
to the utmost, since we thereby secure that a given process of
deduction shall have more widely applicable results; we are,
therefore, in thus generalising the reasoning of arithmetic,
merely following a precept which is universally admitted in
mathematics. And in thus generalising we have, in effect,
created a set of new deductive systems, in which traditional
arithmetic is at once dissolved and enlarged; but whether any
one of these new deductive systems---for example, the theory of
selections---is to be said to belong to logic or to arithmetic is
entirely arbitrary, and incapable of being decided rationally.

We are thus brought face to face with the question: What
is this subject, which may be called indifferently either mathematics
or logic? Is there any way in which we can define it?

Certain characteristics of the subject are clear. To begin
with, we do not, in this subject, deal with particular things or
particular properties: we deal formally with what can be said
about \emph{any} thing or \emph{any} property. We are prepared to say that
one and one are two, but not that Socrates and Plato are two,
because, in our capacity of logicians or pure mathematicians,
we have never heard of Socrates and Plato. A world in which
there were no such individuals would still be a world in which
one and one are two. It is not open to us, as pure mathematicians
or logicians, to mention anything at all, because, if we do so,
\PageSep{197}%
we introduce something irrelevant and not formal. We may
make this clear by applying it to the case of the syllogism.
\index{Syllogism}%
Traditional logic says: ``All men are mortal, Socrates is a man,
therefore Socrates is mortal.'' Now it is clear that what we
\emph{mean} to assert, to begin with, is only that the premisses imply
the conclusion, not that premisses and conclusion are actually
true; even the most traditional logic points out that the actual
truth of the premisses is irrelevant to logic. Thus the first
change to be made in the above traditional syllogism is to state
it in the form: ``If all men are mortal and Socrates is a man,
then Socrates is mortal.'' We may now observe that it is intended
to convey that this argument is valid in virtue of its \emph{form}, not
in virtue of the particular terms occurring in it. If we had
omitted ``Socrates is a man'' from our premisses, we should
have had a non-formal argument, only admissible because
Socrates is in fact a man; in that case we could not have generalised
the argument. But when, as above, the argument is \emph{formal},
nothing depends upon the terms that occur in it. Thus we may
substitute $\alpha$~for \emph{men}, $\beta$~for \emph{mortals}, and $x$~for Socrates, where
$\alpha$~and~$\beta$ are any classes whatever, and $x$~is any individual. We
then arrive at the statement: ``No matter what possible values
$x$~and $\alpha$ and~$\beta$ may have, if all~$\alpha$'s are~$\beta$'s and $x$~is an~$\alpha$, then $x$~is
a~$\beta$''; in other words, ``the propositional function `if all~$\alpha$'s
are~$\beta$ and $x$ is~an~$\alpha$, then $x$~is a~$\beta$' is always true.'' Here at last
we have a proposition of logic---the one which is only \emph{suggested} by
the traditional statement about Socrates and men and mortals.

It is clear that, if \emph{formal} reasoning is what we are aiming at,
we shall always arrive ultimately at statements like the above,
in which no actual things or properties are mentioned; this
will happen through the mere desire not to waste our time proving
in a particular case what can be proved generally. It would be
ridiculous to go through a long argument about Socrates, and then
go through precisely the same argument again about Plato. If
our argument is one (say) which holds of all men, we shall prove
it concerning~``$x$,'' with the hypothesis ``if $x$~is a man.'' With
\PageSep{198}%
this hypothesis, the argument will retain its hypothetical validity
even when $x$~is not a man. But now we shall find that our argument
would still be valid if, instead of supposing $x$ to be a man,
we were to suppose him to be a monkey or a goose or a Prime
Minister. We shall therefore not waste our time taking as our
premiss ``$x$~is a man'' but shall take ``$x$~is an~$\alpha$,'' where $\alpha$~is any
class of individuals, or~``$\phi x$'' where $\phi$~is any propositional
function of some assigned type. Thus the absence of all mention
of particular things or properties in logic or pure mathematics
is a necessary result of the fact that this study is, as we say,
``purely formal.''

At this point we find ourselves faced with a problem which
is easier to state than to solve. The problem is: ``What are
the constituents of a logical proposition?'' I do not know the
answer, but I propose to explain how the problem arises.

Take (say) the proposition ``Socrates was before Aristotle.''
Here it seems obvious that we have a relation between two terms,
and that the constituents of the proposition (as well as of the
corresponding fact) are simply the two terms and the relation,
\ie\ Socrates, Aristotle, and \emph{before}. (I ignore the fact that
Socrates and Aristotle are not simple; also the fact that what
appear to be their names are really truncated descriptions.
Neither of these facts is relevant to the present issue.) We may
represent the general form of such propositions by~``$xRy$,''
\index{Form}%
which may be read ``$x$~has the relation~$R$ to~$y$.'' This general
form may occur in logical propositions, but no particular instance
of it can occur. Are we to infer that the general form itself is a
constituent of such logical propositions?

Given a proposition, such as ``Socrates is before Aristotle,''
we have certain constituents and also a certain form. But the
form is not itself a new constituent; if it were, we should need a
new form to embrace both it and the other constituents. We
can, in fact, turn \emph{all} the constituents of a proposition into
variables, while keeping the form unchanged. This is what we
do when we use such a schema as~``$xRy$,'' which stands for any
\PageSep{199}%
one of a certain class of propositions, namely, those asserting
relations between two terms. We can proceed to general assertions,
such as ``$xRy$~is sometimes true''---\ie\ there are cases
where dual relations hold. This assertion will belong to logic
(or mathematics) in the sense in which we are using the word.
But in this assertion we do not mention any particular things
or particular relations; no particular things or relations can
ever enter into a proposition of pure logic. We are left with pure
\emph{forms} as the only possible constituents of logical propositions.

I do not wish to assert positively that pure forms---\eg\ the
form~``$xRy$''---do actually enter into propositions of the kind
we are considering. The question of the analysis of such propositions
is a difficult one, with conflicting considerations on the
one side and on the other. We cannot embark upon this question
now, but we may accept, as a first approximation, the view
that \emph{forms} are what enter into logical propositions as their
constituents. And we may explain (though not formally define)
what we mean by the ``form'' of a proposition as follows:---

The ``form'' of a proposition is that, in it, that remains unchanged
when every constituent of the proposition is replaced
by another.

Thus ``Socrates is earlier than Aristotle'' has the same form
as ``Napoleon is greater than Wellington,'' though every constituent
of the two propositions is different.

We may thus lay down, as a necessary (though not sufficient)
characteristic of logical or mathematical propositions, that they
are to be such as can be obtained from a proposition containing
no variables (\ie\ no such words as \emph{all}, \emph{some}, \emph{a}, \emph{the}, etc.)\ by turning
\index{Variables}%
every constituent into a variable and asserting that the result
is always true or sometimes true, or that it is always true in
respect of some of the variables that the result is sometimes true
in respect of the others, or any variant of these forms. And
another way of stating the same thing is to say that logic (or
mathematics) is concerned only with \emph{forms}, and is concerned
with them only in the way of stating that they are always or
\PageSep{200}%
sometimes true---with all the permutations of ``always'' and
``sometimes'' that may occur.

There are in every language some words whose sole function is
to indicate form. These words, broadly speaking, are commonest
in languages having fewest inflections. Take ``Socrates is
human.'' Here ``is'' is not a constituent of the proposition,
but merely indicates the subject-predicate form. Similarly
in ``Socrates is earlier than Aristotle,'' ``is'' and ``than''
merely indicate form; the proposition is the same as ``Socrates
precedes Aristotle,'' in which these words have disappeared
and the form is otherwise indicated. Form, as a rule, \emph{can} be
indicated otherwise than by specific words: the order of the
words can do most of what is wanted. But this principle
must not be pressed. For example, it is difficult to see how we
could conveniently express molecular forms of propositions
(\ie\ what we call ``truth-functions'') without any word at all.
We saw in \ChapRef{XIV}.\ that one word or symbol is enough for
this purpose, namely, a word or symbol expressing \emph{incompatibility}.
\index{Incompatibility}%
But without even one we should find ourselves in difficulties.
This, however, is not the point that is important for
our present purpose. What is important for us is to observe
that form may be the one concern of a general proposition,
even when no word or symbol in that proposition designates
the form. If we wish to speak about the form itself, we must
have a word for it; but if, as in mathematics, we wish to speak
about all propositions that have the form, a word for the form
will usually be found not indispensable; probably in theory it
is \emph{never} indispensable.

Assuming---as I think we may---that the forms of propositions
can be represented by the forms of the propositions in which
they are expressed without any special word for forms, we should
arrive at a language in which everything formal belonged to
syntax and not to vocabulary. In such a language we could
express \emph{all} the propositions of mathematics even if we did not
know one single word of the language. The language of mathematical
\PageSep{201}%
logic, if it were perfected, would be such a language.
\index{Logic!mathematical}%
We should have symbols for variables, such as ``$x$'' and ``$R$''
and~``$y$,'' arranged in various ways; and the way of arrangement
would indicate that something was being said to be true of
all values or some values of the variables. We should not need
to know any words, because they would only be needed for giving
values to the variables, which is the business of the applied
mathematician, not of the pure mathematician or logician.
It is one of the marks of a proposition of logic that, given a
suitable language, such a proposition can be asserted in such a
language by a person who knows the syntax without knowing
a single word of the vocabulary.

But, after all, there are words that express form, such as ``is''
and ``than.'' And in every symbolism hitherto invented for
mathematical logic there are symbols having constant formal
meanings. We may take as an example the symbol for incompatibility
which is employed in building up truth-functions.
Such words or symbols may occur in logic. The question is:
How are we to define them?

Such words or symbols express what are called ``logical
constants.'' Logical constants may be defined exactly as
we defined forms; in fact, they are in essence the same thing.
A fundamental logical constant will be that which is in common
among a number of propositions, any one of which can result
from any other by substitution of terms one for another. For
example, ``Napoleon is greater than Wellington'' results from
``Socrates is earlier than Aristotle'' by the substitution of
``Napoleon'' for ``Socrates,'' ``Wellington'' for ``Aristotle,''
and ``greater'' for ``earlier.'' Some propositions can be obtained
in this way from the prototype ``Socrates is earlier than Aristotle''
and some cannot; those that can are those that are of
the form~``$xRy$,'' \ie\ express dual relations. We cannot obtain
from the above prototype by term-for-term substitution such
propositions as ``Socrates is human'' or ``the Athenians gave
the hemlock to Socrates,'' because the first is of the subject-predicate
\PageSep{202}%
form and the second expresses a three-term relation.
If we are to have any words in our pure logical language, they
must be such as express ``logical constants,'' and ``logical
\index{Constants}%
constants'' will always either be, or be derived from, what is in
common among a group of propositions derivable from each
other, in the above manner, by term-for-term substitution. And
this which is in common is what we call ``form.''

In this sense all the ``constants'' that occur in pure mathematics
are logical constants. The number~$1$, for example, is
derivative from propositions of the form: ``There is a term~$c$
such that $\phi x$~is true when, and only when, $x$~is~$c$.'' This is a
function of~$\phi$, and various different propositions result from
giving different values to~$\phi$. We may (with a little omission
of intermediate steps not relevant to our present purpose) take
the above function of~$\phi$ as what is meant by ``the class determined
by~$\phi$ is a unit class'' or ``the class determined by~$\phi$ is a
member of~$1$'' ($1$~being a class of classes). In this way, propositions
in which $1$~occurs acquire a meaning which is derived from
a certain constant logical form. And the same will be found
to be the case with all mathematical constants: all are logical
constants, or symbolic abbreviations whose full use in a proper
context is defined by means of logical constants.

But although all logical (or mathematical) propositions can
\index{Primitive ideas and propositions}%
be expressed wholly in terms of logical constants together with
variables, it is not the case that, conversely, all propositions
that can be expressed in this way are logical. We have found
so far a necessary but not a sufficient criterion of mathematical
propositions. We have sufficiently defined the character of the
primitive \emph{ideas} in terms of which all the ideas of mathematics
can be \emph{defined}, but not of the primitive \emph{propositions} from which
all the propositions of mathematics can be \emph{deduced}. This is a
more difficult matter, as to which it is not yet known what the
full answer is.

We may take the axiom of infinity as an example of a proposition
\index{Infinity, axiom of}%
which, though it can be enunciated in logical terms,
\PageSep{203}%
cannot be asserted by logic to be true. All the propositions of
logic have a characteristic which used to be expressed by saying
that they were analytic, or that their contradictories were self-contradictory.
This mode of statement, however, is not satisfactory.
The law of contradiction is merely one among logical
propositions; it has no special pre-eminence; and the proof
that the contradictory of some proposition is self-contradictory
is likely to require other principles of deduction besides the
law of contradiction. Nevertheless, the characteristic of logical
propositions that we are in search of is the one which was felt,
and intended to be defined, by those who said that it consisted
in deducibility from the law of contradiction. This characteristic,
which, for the moment, we may call \emph{tautology}, obviously
\index{Tautology}%
does not belong to the assertion that the number of individuals
in the universe is~$n$, whatever number $n$~may be. But for the
diversity of types, it would be possible to prove logically that
there are classes of $n$~terms, where $n$~is any finite integer; or even
that there are classes of $\aleph_{0}$~terms. But, owing to types, such
proofs, as we saw in \ChapRef{XIII}., are fallacious. We are left
to empirical observation to determine whether there are as many
as $n$~individuals in the world. Among ``possible'' worlds,
in the Leibnizian sense, there will be worlds having one, two,
three,~\dots\ individuals. There does not even seem any logical
necessity why there should be even one individual\footnotemark---why, in
\footnotetext{The primitive propositions in \Title{Principia Mathematica} are such as to
  allow the inference that at least one individual exists. But I now view
  this as a defect in logical purity.}
fact, there should be any world at all. The ontological proof
\index{Ontological proof}%
of the existence of God, if it were valid, would establish the
logical necessity of at least one individual. But it is generally
recognised as invalid, and in fact rests upon a mistaken view of
existence---\ie\ it fails to realise that existence can only be asserted
of something described, not of something named, so that it is
meaningless to argue from ``this is the so-and-so'' and ``the
so-and-so exists'' to ``this exists.'' If we reject the ontological
\PageSep{204}%
argument, we seem driven to conclude that the existence of a
world is an accident---\ie\ it is not logically necessary. If that
be so, no principle of logic can assert ``existence'' except under
a hypothesis, \ie\ none can be of the form ``the propositional
function so-and-so is sometimes true.'' Propositions of this
\index{Propositions!analytic}%
form, when they occur in logic, will have to occur as hypotheses
or consequences of hypotheses, not as complete asserted propositions.
The complete asserted propositions of logic will all
be such as affirm that some propositional function is \emph{always} true.
For example, it is always true that if $p$~implies~$q$ and $q$~implies~$r$
then $p$~implies~$r$, or that, if all~$\alpha$'s are~$\beta$'s and $x$~is an~$\alpha$ then
$x$~is a~$\beta$. Such propositions may occur in logic, and their truth
is independent of the existence of the universe. We may lay
it down that, if there were no universe, \emph{all} general propositions
would be true; for the contradictory of a general proposition
(as we saw in \ChapRef{XV}.)\ is a proposition asserting existence,
and would therefore always be false if no universe existed.

Logical propositions are such as can be known \Foreign{a~priori}, without
study of the actual world. We only know from a study of
empirical facts that Socrates is a man, but we know the correctness
of the syllogism in its abstract form (\ie\ when it is stated
in terms of variables) without needing any appeal to experience.
This is a characteristic, not of logical propositions in themselves,
but of the way in which we know them. It has, however, a
bearing upon the question what their nature may be, since there
are some kinds of propositions which it would be very difficult
to suppose we could know without experience.

It is clear that the definition of ``logic'' or ``mathematics''
must be sought by trying to give a new definition of the old
notion of ``analytic'' propositions. Although we can no longer
be satisfied to define logical propositions as those that follow
from the law of contradiction, we can and must still admit that
they are a wholly different class of propositions from those that
we come to know empirically. They all have the characteristic
which, a moment ago, we agreed to call ``tautology.'' This,
\PageSep{205}%
combined with the fact that they can be expressed wholly in terms
of variables and logical constants (a logical constant being something
which remains constant in a proposition even when \emph{all}
its constituents are changed)---will give the definition of logic
or pure mathematics. For the moment, I do not know how to
\index{Tautology}%
define ``tautology.''\footnote
  {The importance of ``tautology'' for a definition of mathematics was
  pointed out to me by my former pupil Ludwig Wittgenstein, who was
\index{Wittgenstein|FNote}%
  working on the problem. I do not know whether he has solved it, or even
  whether he is alive or dead.}
It would be easy to offer a definition
which might seem satisfactory for a while; but I know of none
that I feel to be satisfactory, in spite of feeling thoroughly
familiar with the characteristic of which a definition is wanted.
At this point, therefore, for the moment, we reach the frontier
of knowledge on our backward journey into the logical foundations
of mathematics.

We have now come to an end of our somewhat summary introduction
to mathematical philosophy. It is impossible to convey
adequately the ideas that are concerned in this subject so long
as we abstain from the use of logical symbols. Since ordinary
language has no words that naturally express exactly what we
wish to express, it is necessary, so long as we adhere to ordinary
language, to strain words into unusual meanings; and the reader
is sure, after a time if not at first, to lapse into attaching the usual
meanings to words, thus arriving at wrong notions as to what is
intended to be said. Moreover, ordinary grammar and syntax
is extraordinarily misleading. This is the case, \eg\, as regards
numbers; ``ten men'' is grammatically the same form as
``white men,'' so that $10$~might be thought to be an adjective
qualifying ``men.'' It is the case, again, wherever propositional
functions are involved, and in particular as regards existence and
descriptions. Because language is misleading, as well as because
it is diffuse and inexact when applied to logic (for which it was
never intended), logical symbolism is absolutely necessary to
any exact or thorough treatment of our subject. Those readers,
\PageSep{206}%
therefore, who wish to acquire a mastery of the principles of
mathematics, will, it is to be hoped, not shrink from the labour
of mastering the symbols---a labour which is, in fact, much less
than might be thought. As the above hasty survey must have
made evident, there are innumerable unsolved problems in the
subject, and much work needs to be done. If any student is
led into a serious study of mathematical logic by this little
\index{Logic!mathematical}%
book, it will have served the chief purpose for which it has been
written.
\PageSep{207}
\BackMatter
\printindex

%
\iffalse

INDEX

Aggregates 12

Alephs 83, 92, 97, 125

Aliorelatives 32

All@\emph{All}|EtSeq 158

Analysis 4

Ancestors 25, 33

Argument of a function 47, 108

Arithmetising of mathematics 4

Associative law 58, 94

Axioms 1

Between|EtSeq 38

Between 58

Bolzano|FNote 138

Boots and socks 126

Boundary 70, 98, 99

Cantor, Georg 77, 79, 86, 89, 95, 102, 136

Cantor, Georg|FNote 85

Classes 12, 137
  reflexive 80, 127, 138
  similar 15, 16

Classes|EtSeq 181

Clifford, W. K. 76

Collections, infinite 13

Commutative law 58, 94

Conjunction 147

Consecutiveness 37, 38, 81

Constants 202

Construction, method of 73

Continuity 86
  Cantorian|EtSeq 102
  Dedekindian 101
  in philosophy 105
  of functions|EtSeq 106

Continuity|EtSeq 97

Contradictions|EtSeq 135

Convergence 115

Converse 16, 32, 49

Correlators 54

Counterparts, objective 61

Counting 14, 16

Dedekind 69, 99

Dedekind|FNote 138

Deduction|EtSeq 144

Definition 3
  extensional and intensional 12

Derivatives 100

Descriptions 139, 144

Descriptions|EtSeq 167

Dimensions 29

Disjunction 147

Distributive law 58, 94

Diversity 87

Domain 16, 32, 49

Equivalence 183

Euclid 67

Existence 164, 171, 177

Exponentiation 94, 120

Extension of a relation 60

Fictions, logical|FNote 14

Fictions, logical 45, 137

Field of a relation 32, 53

Finite 27

Flux 105

Form 198

Fractions 37, 64

Frege 7, 10, 77, 95

Frege|FNote 25, 146

Functions 46
  descriptive 46, 180
  intensional and extensional 186
  predicative 189
  propositional 46, 144
  propositional|EtSeq 155

Gap, Dedekindian 99

Gap, Dedekindian|EtSeq 70

Generalisation 156

Geometry 29, 59, 67, 74, 100, 145
  analytical 4, 86

Greater and less 65, 90

Hegel 107

Hereditary properties 21

Implication 146, 153
  formal 163

Incommensurables 4, 66

Incompatibility 200

Incompatibility|EtSeq 147

Incomplete symbols 182

Indiscernibles 192

Individuals 132, 141, 173

Induction, mathematical 87, 93, 185

Induction, mathematical|EtSeq 20

Inductive properties 21

Inference|EtSeq 148

Infinite 28
  Cantorian 65

%[** TN: Headings under "infinite" in the original]
Infinity
  of rationals 65
  of cardinals|EtSeq 77
  and series and ordinals|EtSeq 89

Infinity, axiom of 77, 202

Infinity, axiom of|EtSeq 131

Infinity, axiom of|FNote 66

Instances 156

Integers, positive and negative 64

Intervals 115

Intuition 145

Irrationals 66, 72
\PageSep{208}%

Kant 145

Leibniz 80, 107, 192

Lewis, C. I.#Lewis 153, 154

Likeness 52

Limit|EtSeq 69, 97

Limit 29
  of functions|EtSeq 106

Limiting points 99

Logic|EtSeq 194

Logic 159, 169
  mathematical v, 201, 206

Logicising of mathematics 7

Maps|EtSeq 60

Maps 52, 80

Mathematics|EtSeq 194

Maximum 70, 98

Median class 104

Meinong 169

Method vi

Minimum 70, 98

Modality 165

Multiplication|EtSeq 118

Multiplicative axiom 92

Multiplicative axiom|EtSeq 117

Names 173, 182

Necessity 165

Neighbourhood 109

Nicod|FNote 151

Nicod 148, 149

Null-class 23, 132

Number
  cardinal 56, 95
  cardinal|EtSeq 10, 77
  complex|EtSeq 74
  finite|EtSeq 20
  inductive 27, 78, 131
  infinite|EtSeq 77
  irrational 66, 72
  maximum? 135
  multipliable 130
  natural 22
  natural|EtSeq 2
  non-inductive 88, 127
  real 66, 72, 84
  reflexive 80, 127
  relation 56, 94
  serial 57

Occam 184

Occurrences, primary and secondary 179

Ontological proof 203

Order|EtSeq 29

Order
  cyclic 40

Oscillation, ultimate 111

Parmenides@\emph{Parmenides} 138

Particulars|EtSeq 140

Particulars 173

Peano|EtSeq 5

Peano 23, 24, 78, 81, 131, 163

Peirce|FNote 32

Permutations 50

Philosophy, mathematical v, 1

Plato 138

Plurality 10

Poincaré 27

Points 59

Posterity|EtSeq 32

Posterity
  proper 36

Postulates 71, 73

Precedent 98

Premisses of arithmetic 5

Primitive ideas and propositions 5, 202

Progressions 8

Progressions|EtSeq 81

Propositions 155
  analytic 204
  elementary 161

Pythagoras 4, 67

Quantity 97, 195

Ratios 64, 71, 84, 133

Reducibility, axiom of 191

Referent@\emph{Referent} 48

Relation numbers|EtSeq 56

Relations
  asymmetrical 31, 42
  connected 32
  many-one 15
  one-many 15, 45
  one-one 15, 47, 79
  reflexive 16
  serial 34
  similar|EtSeq 52
  squares of 32
  symmetrical 16, 44
  transitive 16, 32

Relatum@\emph{Relatum} 48

Representatives 120

Rigour 144

Royce 80

Section
  Dedekindian|EtSeq 69
  ultimate 111

Segments 72, 98

Selections|EtSeq 117

Sequent 98

Series|EtSeq 29

Series
  closed 103
  compact 66, 93, 100
  condensed in itself 102
  Dedekindian 71, 73, 101
  generation of 41
  infinite|EtSeq 89
  perfect 102, 103
  well-ordered 92, 123

Sheffer 148

Similarity
  of classes|EtSeq 15
  of relations 83
  of relations|EtSeq 52

Some|EtSeq 158

Space 61, 86, 140

Structure|EtSeq 60

Sub-classes|EtSeq 84

Subjects 142

Subtraction 87

Successor of a number 23, 35

Syllogism 197

Tautology 203, 205

The@\emph{The} 167

The@\emph{The}|EtSeq 172

Time 61, 86, 140

Truth-function 147

Truth-value 146

Types, logical 53, 185, 188

Types, logical|EtSeq 135

Unreality 168

Value of a function 47, 108

Variables 10, 161, 199

Veblen 58

Verbs 141

Weierstrass 97, 107

Wells, H. G. 114

Whitehead 64, 76, 107, 119

Wittgenstein|FNote 205

Zermelo 123, 129

Zero 65
\fi

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