The Principles of Mathematics

Vol. I

First published in 1903 by Cambridge University Press. This online edition (version 0.12: 28 May 2017) is based on that
printing, with various typographical corrections. Missing here is the Introduction to the
1937 second edition, which is not yet in the public domain. Rather than publishing a second
volume, Russell and his co-author A. N. Whitehead published the three volumes of
*Principia Mathematica* in 1910–1913.

The present work has two main objects. One of these, the proof that all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles, is undertaken in Parts II–VII of this Volume, and will be established by strict symbolic reasoning in Volume II. The demonstration of this thesis has, if I am not mistaken, all the certainty and precision of which mathematical demonstrations are capable. As the thesis is very recent among mathematicians, and is almost universally denied by philosophers, I have undertaken, in this volume, to defend its various parts, as occasion arose, against such adverse theories as appeared most widely held or most difficult to disprove. I have also endeavoured to present, in language as untechnical as possible, the more important stages in the deductions by which the thesis is established.

The other object of this work, which occupies Part I, is the explanation of the fundamental concepts which mathematics accepts as indefinable. This is a purely philosophical task, and I cannot flatter myself that I have done more than indicate a vast field of inquiry, and give a sample of the methods by which the inquiry may be conducted. The discussion of indefinables—which forms the chief part of philosophical logic—is the endeavour to see clearly, and to make others see clearly, the entities concerned, in order that the mind may have that kind of acquaintance with them which it has with redness or the taste of a pineapple. Where, as in the present case, the indefinables are obtained primarily as the necessary residue in a process of analysis, it is often easier to know that there must be such entities than actually to perceive them; there is a process analogous to that which resulted in the discovery of Neptune, with the difference that the final stage—the search with a mental telescope for the entity which has been inferred—is often the most difficult part of the undertaking. In the case of classes, I must confess, I have failed to perceive any concept fulfilling the conditions requisite for the notion of class. And the contradiction discussed in Chapter x proves that something is amiss, but what this is I have hitherto failed to discover.

The second volume, in which I have had the great good fortune to secure the collaboration of Mr A. N. Whitehead, will be addressed exclusively to mathematicians; it will contain chains of deductions, from the premisses of symbolic logic through Arithmetic, finite and infinite, to Geometry, in an order similar to that adopted in the present volume; it will also contain various original developments, in which the method of Professor Peano, as supplemented by the Logic of Relations, has shown itself a powerful instrument of mathematical investigation.

The present volume, which may be regarded either as a commentary
upon, or as an introduction to, the second volume, is addressed in equal
measure to the philosopher and to the mathematician; but some parts
will be more interesting to the one, others to the other. I should advise
mathematicians, unless they are specially interested in Symbolic Logic,
to begin with Part IV, and only refer to
earlier parts as occasion arises.
The following portions are more specially philosophical: Part I
(omitting Chapter ii); Part II,
Chapters xi, xv, xvi,
xvii; Part III;
Part IV, §207, Chapters
xxvi, xxvii,
xxxi; Part V,
Chapters xli,
xlii, xliii;
Part VI, Chapters l,
li,
lii; Part VII, Chapters
liii,
liv, lv,
lvii, lviii;
and the two Appendices, which belong to Part I,
and should be read in connection with it. Professor Frege’s work, which
largely anticipates my own, was for the most part unknown to me when
the printing of the present work began; I had seen his
*Grundgesetze der Arithmetik*, but,
owing to the great difficulty of his symbolism, I had
failed to grasp its importance or to understand its contents. The only
method, at so late a stage, of doing justice to his work, was to devote
an Appendix to it; and in some points the views contained in the
Appendix differ from those in Chapter vi, especially in
§§71, 73, 74.
On questions discussed in these sections, I discovered errors after passing
the sheets for the press; these errors, of which the chief are the denial
of the null-class, and the identification of a term with the class whose
only member it is, are rectified in the Appendices. The subjects
treated are so difficult that I feel little confidence in my present
opinions, and regard any conclusions which may be advocated as
essentially hypotheses.

A few words as to the origin of the present work may serve to
show the importance of the questions discussed. About six years ago,
I began an investigation into the philosophy of Dynamics. I was
met by the difficulty that, when a particle is subject to several forces,
no one of the component accelerations actually occurs, but only
the resultant acceleration, of which they are not parts; this fact
rendered illusory such causation of particulars by particulars as is
affirmed, at first sight, by the law of gravitation. It appeared also that
the difficulty in regard to absolute motion is insoluble on a relational
theory of space. From these two questions I was led to a re-examination
of the principles of Geometry, thence to the philosophy of continuity
and infinity, and thence, with a view to discovering the meaning of the
word *any*, to Symbolic Logic. The final outcome, as regards the
philosophy of Dynamics, is perhaps rather slender; the reason of this
is, that almost all the problems of Dynamics appear to me empirical,
and therefore outside the scope of such a work as the present. Many
very interesting questions have had to be omitted, especially in Parts
VI and VII, as not relevant to my purpose, which, for fear of
misunderstandings, it may be well to explain at this stage.

When actual objects are counted, or when Geometry and Dynamics are applied to actual space or actual matter, or when, in any other way, mathematical reasoning is applied to what exists, the reasoning employed has a form not dependent upon the objects to which it is applied being just those objects that they are, but only upon their having certain general properties. In pure mathematics, actual objects in the world of existence will never be in question, but only hypothetical objects having those general properties upon which depends whatever deduction is being considered; and these general properties will always be expressible in terms of the fundamental concepts which I have called logical constants. Thus when space or motion is spoken of in pure mathematics, it is not actual space or actual motion, as we know them in experience, that are spoken of, but any entity possessing those abstract general properties of space or motion that are employed in the reasonings of geometry or dynamics. The question whether these properties belong, as a matter of fact, to actual space or actual motion, is irrelevant to pure mathematics, and therefore to the present work, being, in my opinion, a purely empirical question, to be investigated in the laboratory or the observatory. Indirectly, it is true, the discussions connected with pure mathematics have a very important bearing upon such empirical questions, since mathematical space and motion are held by many, perhaps most, philosophers to be self-contradictory, and therefore necessarily different from actual space and motion, whereas, if the views advocated in the following pages be valid, no such self-contradictions are to be found in mathematical space and motion. But extra-mathematical considerations of this kind have been almost wholly excluded from the present work.

On fundamental questions of philosophy, my position, in all its chief features, is derived from Mr G. E. Moore. I have accepted from him the non-existential nature of propositions (except such as happen to assert existence) and their independence of any knowing mind; also the pluralism which regards the world, both that of existents and that of entities, as composed of an infinite number of mutually independent entities, with relations which are ultimate, and not reducible to adjectives of their terms or of the whole which these compose. Before learning these views from him, I found myself completely unable to construct any philosophy of arithmetic, whereas their acceptance brought about an immediate liberation from a large number of difficulties which I believe to be otherwise insuperable. The doctrines just mentioned are, in my opinion, quite indispensable to any even tolerably satisfactory philosophy of mathematics, as I hope the following pages will show. But I must leave it to my readers to judge how far the reasoning assumes these doctrines, and how far it supports them. Formally, my premisses are simply assumed; but the fact that they allow mathematics to be true, which most current philosophies do not, is surely a powerful argument in their favour.

In Mathematics, my chief obligations, as is indeed evident, are to Georg Cantor and Professor Peano. If I had become acquainted sooner with the work of Professor Frege, I should have owed a great deal to him, but as it is I arrived independently at many results which he had already established. At every stage of my work, I have been assisted more than I can express by the suggestions, the criticisms, and the generous encouragement of Mr A. N. Whitehead; he also has kindly read my proofs, and greatly improved the final expression of a very large number of passages. Many useful hints I owe also to Mr W. E. Johnson; and in the more philosophical parts of the book I owe much to Mr G. E. Moore besides the general position which underlies the whole.

In the endeavour to cover so wide a field, it has been impossible to acquire an exhaustive knowledge of the literature. There are doubtless many important works with which I am unacquainted; but where the labour of thinking and writing necessarily absorbs so much time, such ignorance, however regrettable, seems not wholly avoidable.

Many words will be found, in the course of discussion, to be defined
in senses apparently departing widely from common usage. Such
departures, I must ask the reader to believe, are never wanton, but have
been made with great reluctance. In philosophical matters, they have
been necessitated mainly by two causes. First, it often happens that
two cognate notions are both to be considered, and that language has
two names for the one, but none for the other. It is then highly
convenient to distinguish between the two names commonly used as
synonyms, keeping one for the usual, the other for the hitherto nameless
sense. The other cause arises from philosophical disagreement with
received views. Where two qualities are commonly supposed inseparably
conjoined, but are here regarded as separable, the name which has
applied to their combination will usually have to be restricted to one
or other. For example, propositions are commonly regarded as (1) true
or false, (2) mental. Holding, as I do, that what is true or false is not
in general mental, I require a name for the true or false as such, and
this name can scarcely be other than *proposition*. In such a case, the
departure from usage is in no degree arbitrary. As regards mathematical
terms, the necessity for establishing the existence-theorem in each
case—*i.e.* the proof that
there are entities of the kind in question—has led to
many definitions which appear widely different from the notions usually
attached to the terms in question. Instances of this are the definitions
of cardinal, ordinal and complex numbers. In the two former of these,
and in many other cases, the definition as a class, derived from the
principle of abstraction, is mainly recommended by the fact that it
leaves no doubt as to the existence-theorem. But in many instances of
such apparent departure from usage, it may be doubted whether more
has been done than to give precision to a notion which had hitherto
been more or less vague.

For publishing a work containing so many unsolved difficulties, my apology is, that investigation revealed no near prospect of adequately resolving the contradiction discussed in Chapter x, or of acquiring a better insight into the nature of classes. The repeated discovery of errors in solutions which for a time had satisfied me caused these problems to appear such as would have been only concealed by any seemingly satisfactory theories which a slightly longer reflection might have produced; it seemed better, therefore, merely to state the difficulties, than to wait until I had become persuaded of the truth of some almost certainly erroneous doctrine.

My thanks are due to the Syndics of the University Press, and to their Secretary, Mr R. T. Wright, for their kindness and courtesy in regard to the present volume.

London,*December,* 1902

THE INDEFINABLES OF MATHEMATICS.

Definition of Pure Mathematics.

**1.** Pure Mathematics is the class of all propositions of the form
“`p` implies `q`,” where `p` and `q` are propositions containing one or more
variables, the same in the two propositions, and neither `p` nor `q` contains
any constants except logical constants. And logical constants are all
notions definable in terms of the following: Implication, the relation
of a term to a class of which it is a member, the notion of *such that*,
the notion of relation, and such further notions as may be involved
in the general notion of propositions of the above form. In addition
to these, mathematics *uses* a notion which is not a constituent of the
propositions which it considers, namely the notion of truth.

**2.** The above definition of pure mathematics is, no doubt,
somewhat unusual. Its various parts, nevertheless, appear to be capable of
exact justification—a justification which it will be the object of the
present work to provide. It will be shown that whatever has, in the
past, been regarded as pure mathematics, is included in our definition,
and that whatever else is included possesses those marks by which
mathematics is commonly though vaguely distinguished from other
studies. The definition professes to be, not an arbitrary decision to
use a common word in an uncommon signification, but rather a precise
analysis of the ideas which, more or less unconsciously, are implied in
the ordinary employment of the term. Our method will therefore be
one of analysis, and our problem may be called philosophical—in the
sense, that is to say, that we seek to pass from the complex to the
simple, from the demonstrable to its indemonstrable premisses. But
in one respect not a few of our discussions will differ from those that
are usually called philosophical. We shall be able, thanks to the labours
of the mathematicians themselves, to arrive at certainty in regard to
most of the questions with which we shall be concerned; and among
those capable of an exact solution we shall find many of the problems
which, in the past, have been involved in all the traditional uncertainty
of philosophical strife. The nature of number, of infinity, of space,
time and motion, and of mathematical inference itself, are all questions
to which, in the present work, an answer professing itself demonstrable
with mathematical certainty will be given—an answer which, however,
consists in reducing the above problems to problems in pure logic,
which last will not be found satisfactorily solved in what follows.

**3.** The Philosophy of Mathematics has been hitherto as
controversial, obscure and unprogressive as the other branches of philosophy.
Although it was generally agreed that mathematics is in some sense
true, philosophers disputed as to what mathematical propositions really
meant: although something was true, no two people were agreed as to
what it was that was true, and if something was known, no one knew
what it was that was known. So long, however, as this was doubtful,
it could hardly be said that any certain and exact knowledge was to be
obtained in mathematics. We find, accordingly, that idealists have
tended more and more to regard all mathematics as dealing with mere
appearance, while empiricists have held everything mathematical to be
approximation to some exact truth about which they had nothing to
tell us. This state of things, it must be confessed, was thoroughly
unsatisfactory. Philosophy asks of Mathematics: What does it mean?
Mathematics in the past was unable to answer, and Philosophy answered
by introducing the totally irrelevant notion of mind. But now
Mathematics is able to answer, so far at least as to reduce the whole
of its propositions to certain fundamental notions of logic. At this
point, the discussion must be resumed by Philosophy. I shall endeavour
to indicate what are the fundamental notions involved, to prove at
length that no others occur in mathematics, and to point out briefly
the philosophical difficulties involved in the analysis of these notions.
A complete treatment of these difficulties would involve a treatise on
Logic, which will not be found in the following pages.

**4.** There was, until very lately, a special difficulty in the principles
of mathematics. It seemed plain that mathematics consists of deductions,
and yet the orthodox accounts of deduction were largely or wholly
inapplicable to existing mathematics. Not only the Aristotelian
syllogistic theory, but also the modern doctrines of Symbolic Logic,
were either theoretically inadequate to mathematical reasoning, or at
any rate required such artificial forms of statement that they could not
be practically applied. In this fact lay the strength of the Kantian
view, which asserted that mathematical reasoning is not strictly formal,
but always uses intuitions, *i.e.* the *à priori* knowledge of space and
time. Thanks to the progress of Symbolic Logic, especially as treated
by Professor Peano, this part of the Kantian philosophy is now capable
of a final and irrevocable refutation. By the help of ten principles
of deduction and ten other premisses of a general logical nature
(*e.g.* “implication is a relation”), all mathematics can be strictly and
formally deduced; and all the entities that occur in mathematics can
be defined in terms of those that occur in the above twenty premisses.
In this statement, Mathematics includes not only Arithmetic and
Analysis, but also Geometry, Euclidean and non-Euclidean, rational
Dynamics, and an indefinite number of other studies still unborn or in
their infancy. The fact that all Mathematics is Symbolic Logic is one
of the greatest discoveries of our age; and when this fact has been
established, the remainder of the principles of mathematics consists in
the analysis of Symbolic Logic itself.

**5.** The general doctrine that all mathematics is deduction by
logical principles from logical principles was strongly advocated by
Leibniz, who urged constantly that axioms ought to be proved and
that all except a few fundamental notions ought to be defined. But
owing partly to a faulty logic, partly to belief in the logical necessity
of Euclidean Geometry, he was led into hopeless errors in the endeavour
to carry out in detail a view which, in its general outline, is now known
to be correct*. The actual propositions of Euclid, for example, do not
follow from the principles of logic alone; and the perception of this fact
led Kant to his innovations in the theory of knowledge. But since
the growth of non-Euclidean Geometry, it has appeared that pure
mathematics has no concern with the question whether the axioms
and propositions of Euclid hold of actual space or not: this is a question
for applied mathematics, to be decided, so far as any decision is possible,
by experiment and observation. What pure mathematics asserts is merely
that the Euclidean propositions follow from the Euclidean axioms—*i.e.* it asserts an implication: any space which has such and such properties
has also such and such other properties. Thus, as dealt with in pure
mathematics, the Euclidean and non-Euclidean Geometries are equally
true: in each nothing is affirmed except implications. All propositions
as to what actually exists, like the space we live in, belong to
experimental or empirical science, not to mathematics; when they belong to
applied mathematics, they arise from giving to one or more of the
variables in a proposition of pure mathematics some constant value
satisfying the hypothesis, and thus enabling us, for that value of the
variable, actually to assert both hypothesis and consequent instead of
asserting merely the implication. We assert always in mathematics
that if a certain assertion `p` is true of any entity `x`, or of any set of
entities `x`, `y`, `z`, … , then some other assertion `q` is true of those entities;
but we do not assert either `p` or `q` separately of our entities. We assert
a relation between the assertions `p` and `q`, which I shall call *formal implication*.

**6.** Mathematical propositions are not only characterized by the
fact that they assert implications, but also by the fact that they contain
variables. The notion of the variable is one of the most difficult with
which Logic has to deal, and in the present work a satisfactory theory
as to its nature, in spite of much discussion, will hardly be found.
For the present, I only wish to make it plain that there are variables
in all mathematical propositions, even where at first sight they might
seem to be absent. Elementary Arithmetic might be thought to form
an exception: 1 + 1 = 2 appears neither to contain variables nor to
assert an implication. But as a matter of fact, as will be shown in
Part II, the true meaning of this proposition
is: “If `x` is one and
`y` is one, and `x` differs from `y`,
then `x` and `y` are two.” And this
proposition both contains variables and asserts an implication. We
shall find always, in all mathematical propositions, that the words *any* or *some* occur; and these words are the marks of a variable and a formal
implication. Thus the above proposition may be expressed in the form:
“Any unit and any other unit are two units.” The typical proposition
of mathematics is of the form “`φ`(`x`, `y`, `z`, …) implies `ψ`(`x`, `y`, `z`, …),
whatever values `x`, `y`, `z`, … may have”; where `φ`(`x`, `y`, `z`, …) and
`ψ`(`x`, `y`, `z`, …), for every set of values of `x`, `y`, `z`, …, are propositions.
It is not asserted that `φ` is always true, nor yet that `ψ` is always true,
but merely that, in all cases, when `φ` is false as much as when `φ` is true,
`ψ` follows from it.

The distinction between a variable and a constant is somewhat
obscured by mathematical usage. It is customary, for example, to speak
of parameters as in some sense constants, but this is a usage which
we shall have to reject. A constant is to be something absolutely
definite, concerning which there is no ambiguity whatever. Thus 1, 2,
3, `e`, `π`, Socrates, are constants; and so
are *man*, and the human race,
past, present and future, considered collectively. Proposition, implication,
class, etc. are constants; but a proposition,
any proposition, some
proposition, are not constants, for these phrases do not denote one
definite object. And thus what are called parameters are simply
variables. Take, for example, the equation `a``x` + `b``y` + `c` = 0, considered
as the equation to a straight line in a plane. Here we say that `x` and `y`
are variables, while `a`, `b`, `c` are constants. But unless we are dealing
with one absolutely particular line, say the line from a particular point
in London to a particular point in Cambridge, our `a`, `b`, `c` are not
definite numbers, but stand for *any* numbers, and are thus also variables.
And in Geometry nobody does deal with actual particular lines; we
always discuss *any* line. The point is that we collect the various
couples `x`, `y` into classes of classes, each class being defined as those
couples that have a certain fixed relation to one triad (`a`, `b`, `c`). But
from class to class, `a`, `b`, `c` also vary, and are therefore properly variables.

**7.** It is customary in mathematics to regard our variables as
restricted to certain classes: in Arithmetic, for instance, they are
supposed to stand for numbers. But this only means that *if* they
stand for numbers, they satisfy some formula, *i.e.* the hypothesis that
they are numbers implies the formula. This, then, is what is really
asserted, and in this proposition it is no longer necessary that our
variables should be numbers: the implication holds equally when they
are not so. Thus, for example, the proposition “`x` and `y` are numbers
implies (`x` + `y`)^{2} = `x`^{2} + 2`x``y` + `y`^{2}” holds equally if for `x` and `y` we
substitute Socrates and Plato*: both hypothesis and consequent, in this case,
will be false, but the implication will still be true. Thus in every
proposition of pure mathematics, when fully stated, the variables have
an absolutely unrestricted field: any conceivable entity may be
substituted for any one of our variables without impairing the truth of our
proposition.

**8.** We can now understand why the constants in mathematics are
to be restricted to logical constants in the sense defined above. The
process of transforming constants in a proposition into variables leads
to what is called generalization, and gives us, as it were, the formal
essence of a proposition. Mathematics is interested exclusively in
*types* of propositions; if a proposition `p` containing only constants be proposed,
and for a certain one of its terms we imagine others to be successively
substituted, the result will in general be sometimes true and sometimes
false. Thus, for example, we have “Socrates is a man”; here we turn
Socrates into a variable, and consider “`x` is a man.” Some hypotheses
as to `x`, for example, “`x` is a Greek,” insure the truth of “`x` is a man”;
thus “`x` is a Greek” implies “`x` is a man,” and this holds for all values of
`x`. But the statement is not one of pure mathematics, because it depends
upon the particular nature of *Greek* and *man*. We may, however, vary
these too, and obtain: If `a` and `b` are classes, and `a` is contained in `b`,
then “`x` is an `a`” implies “`x` is a `b`.” Here at last we have a proposition
of pure mathematics, containing three variables and the constants *class*,
*contained in*, and those involved in the notion of formal implications with
variables. So long as any term in our proposition can be turned into
a variable, our proposition can be generalized; and so long as this is
possible, it is the business of mathematics to do it. If there are several
chains of deduction which differ only as to the meaning of the symbols,
so that propositions symbolically identical become capable of several
interpretations, the proper course, mathematically, is to form the class of
meanings which may attach to the symbols, and to assert that the
formula in question follows from the hypothesis that the symbols belong
to the class in question. In this way, symbols which stood for constants
become transformed into variables, and new constants are substituted,
consisting of classes to which the old constants belong. Cases of such
generalization are so frequent that many will occur at once to every
mathematician, and innumerable instances will be given in the present
work. Whenever two sets of terms have mutual relations of the same
type, the same form of deduction will apply to both. For example, the
mutual relations of points in a Euclidean plane are of the same type as
those of the complex numbers; hence plane geometry, considered as a
branch of pure mathematics, ought not to decide whether its variables
are points or complex numbers or some other set of entities having the
same type of mutual relations. Speaking generally, we ought to deal,
in every branch of mathematics, with any class of entities whose mutual
relations are of a specified type; thus the class, as well as the particular
term considered, becomes a variable, and the only true constants are the
types of relations and what they involve. Now a *type* of relation is to
mean, in this discussion, a class of relations characterized by the above
formal identity of the deductions possible in regard to the various
members of the class; and hence a type of relations, as will appear more
fully hereafter, if not already evident, is always a class definable in
terms of logical constants*. We may therefore define a type of relations
as a class of relations defined by some property definable in terms of
logical constants alone.

**9.** Thus pure mathematics must contain no indefinables except
logical constants, and consequently no premisses, or indemonstrable
propositions, but such as are concerned exclusively with logical constants
and with variables. It is precisely this that distinguishes pure from
applied mathematics. In applied mathematics, results which have been
shown by pure mathematics to follow from some hypothesis as to the
variable are actually asserted of some constant satisfying the hypothesis
in question. Thus terms which were variables become constant, and a
new premiss is always required, namely: this particular entity satisfies
the hypothesis in question. Thus for example Euclidean Geometry, as a
branch of pure mathematics, consists wholly of propositions having the
hypothesis “`S` is a Euclidean space.” If we go on to: “The space
that exists is Euclidean,” this enables us to assert of the space that exists
the consequents of all the hypotheticals constituting Euclidean Geometry,
where now the variable `S` is replaced by the constant *actual space*. But
by this step we pass from pure to applied mathematics.

**10.** The connection of mathematics with logic, according to the
above account, is exceedingly close. The fact that all mathematical
constants are logical constants, and that all the premisses of mathematics
are concerned with these, gives, I believe, the precise statement of what
philosophers have meant in asserting that mathematics is *à priori*. The
fact is that, when once the apparatus of logic has been accepted, all
mathematics necessarily follows. The logical constants themselves are
to be defined only by enumeration, for they are so fundamental that all
the properties by which the class of them might be defined presuppose
some terms of the class. But practically, the method of discovering the
logical constants is the analysis of symbolic logic, which will be the
business of the following chapters. The distinction of mathematics from
logic is very arbitrary, but if a distinction is desired, it may be made as
follows. Logic consists of the premisses of mathematics, together with
all other propositions which are concerned exclusively with logical
constants and with variables but do not fulfil the above definition of
mathematics (§1). Mathematics consists of all the consequences of the
above premisses which assert formal implications containing variables,
together with such of the premisses themselves as have these marks.
Thus some of the premisses of mathematics, *e.g.* the principle of the
syllogism, “if `p` implies `q` and `q` implies `r`, then `p` implies `r`,” will
belong to mathematics, while others, such as “implication is a relation,”
will belong to logic but not to mathematics. But for the desire to
adhere to usage, we might identify mathematics and logic, and define
either as the class of propositions containing only variables and logical
constants; but respect for tradition leads me rather to adhere to the
above distinction, while recognizing that certain propositions belong to
both sciences.

From what has now been said, the reader will perceive that the present work has to fulfil two objects, first, to show that all mathematics follows from symbolic logic, and secondly to discover, as far as possible, what are the principles of symbolic logic itself. The first of these objects will be pursued in the following Parts, while the second belongs to Part I. And first of all, as a preliminary to a critical analysis, it will be necessary to give an outline of Symbolic Logic considered simply as a branch of mathematics. This will occupy the following chapter.

Notes

*^{[page 5]} On this subject, cf. Couturat, *La Logique de Leibniz*, Paris, 1901.

*^{[page 7]} It is necessary to suppose arithmetical addition and multiplication defined (as
may be easily done) so that the above formula remains significant when `x` and `y` are
not numbers.

*^{[page 8]} One-one, many-one, transitive, symmetrical, are instances of types of relations with which we shall be often concerned.

Symbolic Logic.

**11.** Symbolic or Formal Logic—I shall use these terms as
synonyms—is the study of the various general types of deduction.
The word *symbolic* designates the subject by an accidental characteristic,
for the employment of mathematical symbols, here as elsewhere, is merely
a theoretically irrelevant convenience. The syllogism in all its figures
belongs to Symbolic Logic, and would be the whole subject if all
deduction were syllogistic, as the scholastic tradition supposed. It is
from the recognition of asyllogistic inferences that modern Symbolic
Logic, from Leibniz onward, has derived the motive to progress. Since
the publication of Boole’s *Laws of Thought* (1854), the subject has
been pursued with a certain vigour, and has attained to a very considerable technical development*. Nevertheless, the subject achieved almost
nothing of utility either to philosophy or to other branches of mathematics,
until it was transformed by the new methods of Professor Peano†.
Symbolic Logic has now become not only absolutely essential to every
philosophical logician, but also necessary for the comprehension of
mathematics generally, and even for the successful practice of certain
branches of mathematics. How useful it is in practice can only be
judged by those who have experienced the increase of power derived
from acquiring it; its theoretical functions must be briefly set forth in
the present chapter‡.

**12.** Symbolic Logic is essentially concerned with inference in
general*, and is distinguished from various special branches of mathematics
mainly by its generality. Neither mathematics nor symbolic
logic will study such special relations as (say) temporal priority, but
mathematics will deal explicitly with the class of relations possessing
the formal properties of temporal priority—properties which are
summed up in the notion of continuity†. And the formal properties
of a relation may be defined as those that can be expressed in terms
of logical constants, or again as those which, while they are preserved,
permit our relation to be varied without invalidating any inference in
which the said relation is regarded in the light of a variable. But
symbolic logic, in the narrower sense which is convenient, will not
investigate what inferences are possible in respect of continuous relations
(*i.e.* relations generating continuous series); this investigation belongs
to mathematics, but is still too special for symbolic logic. What
symbolic logic does investigate is the general rules by which inferences
are made, and it requires a classification of relations or propositions
only in so far as these general rules introduce particular notions. The
particular notions which appear in the propositions of symbolic logic,
and all others definable in terms of these notions, are the logical
constants. The number of indefinable logical constants is not great:
it appears, in fact, to be eight or nine. These notions alone form the
subject-matter of the whole of mathematics: no others, except such
as are definable in terms of the original eight or nine, occur anywhere
in Arithmetic, Geometry, or rational Dynamics. For the technical
study of Symbolic Logic, it is convenient to take as a single indefinable
the notion of a formal implication, *i.e.* of such propositions as “`x` is
a man implies `x` is a mortal, for all values of `x`”—propositions whose
general type is: “`φ`(`x`) implies `ψ`(`x`) for all values of `x`,” where `φ`(`x`),
`ψ`(`x`), for all values of `x`, are propositions. The analysis of this notion
of formal implication belongs to the principles of the subject, but is not
required for its formal development. In addition to this notion, we
require as indefinables the following: Implication between propositions
not containing variables, the relation of a term to a class of which it
is a member, the notion of *such that*, the notion of relation, and truth.
By means of these notions, all the propositions of symbolic logic can be
stated.

**13.** The subject of Symbolic Logic consists of three parts, the
calculus of propositions, the calculus of classes, and the calculus of
relations. Between the first two, there is, within limits, a certain
parallelism, which arises as follows: In any symbolic expression, the
letters may be interpreted as classes or as propositions, and the relation
of inclusion in the one case may be replaced by that of formal implication
in the other. Thus, for example, in the principle of the syllogism, if
`a`, `b`, `c` be classes, and `a` is contained in `b`, `b` in `c`, then `a` is contained in `c`;
but if `a`, `b`, `c` be propositions, and `a` implies `b`, `b` implies `c`, then `a` implies `c`.
A great deal has been made of this duality, and in the later editions of
the *Formulaire*, Peano appears to have sacrificed logical precision to its
preservation*. But, as a matter of fact, there are many ways in which
the calculus of propositions differs from that of classes. Consider,
for example, the following: “If `p`, `q`, `r` are propositions, and `p` implies
`q` or `r`, then `p` implies `q` or `p` implies `r`.” This proposition is true; but
its correlative is false, namely: “If `a`, `b`, `c` are classes, and `a` is contained
in `b` or `c`, then `a` is contained in `b` or `a` is contained in `c`.” For example,.
English people are all either men or women, but are not all men nor yet
all women. The fact is that the duality holds for propositions asserting
of a variable term that it belongs to a class, *i.e.* such propositions as
“`x` is a man,” provided that the implication involved be formal, *i.e.* one
which holds for all values of `x`. But “`x` is a man” is itself not a
proposition at all, being neither true nor false; and it is not with such
entities that we are concerned in the propositional calculus, but with
genuine propositions. To continue the above illustration: It is true
that, for all values of `x`, “`x` is a man or a woman” either implies “`x` is a
man” or implies “`x` is a woman.” But it is false that “`x` is a man or
woman” either implies “`x` is a man” for all values of `x`, or implies
“`x` is a woman” for all values of `x`. Thus the implication involved, which
is always one of the two, is not formal, since it does not hold for all values
of `x`, being not always the same one of the two. The symbolic affinity
of the propositional and the class logic is, in fact, something of a snare,
and we have to decide which of the two we are to make fundamental.
Mr McColl, in an important series of papers†, has contended for the
view that implication and propositions are more fundamental than
inclusion and classes; and in this opinion I agree with him. But he
does not appear to me to realize adequately the distinction between
genuine propositions and such as contain a real variable: thus he is led
to speak of propositions as sometimes true and sometimes false, which
of course is impossible with a genuine proposition. As the distinction
involved is of very great importance, I shall dwell on it before proceeding
further. A proposition, we may say, is anything that is true or that is
false. An expression such as “`x` is a man” is therefore not a proposition,
for it is neither true nor false. If we give to `x` any constant value
whatever, the expression becomes a proposition: it is thus as it were a
schematic form standing for any one of a whole class of propositions.
And when we say “`x` is a man implies `x` is a mortal for all values of `x`,”
we are not asserting a single implication, but a class of implications;
we have now a genuine proposition, in which, though the letter `x` appears,
there is no real variable: the variable is absorbed in the same kind of
way as the `x` under the integral sign in a definite integral, so that the
result is no longer a function of `x`. Peano distinguishes a variable which
appears in this way as *apparent*, since the proposition does not depend
upon the variable; whereas in “`x` is a man” there are different propositions
for different values of the variable, and the variable is what Peano
calls *real**. I shall speak of propositions exclusively where there is no
real variable: where there are one or more real variables, and for all
values of the variables the expression involved is a proposition, I shall
call the expression a *propositional function*. The study of genuine
propositions is, in my opinion, more fundamental than that of classes;
but the study of propositional functions appears to be strictly on a
par with that of classes, and indeed scarcely distinguishable therefrom.
Peano, like McColl, at first regarded propositions as more fundamental
than classes, but he, even more definitely, considered propositional functions
rather than propositions. From this criticism, Schröder is exempt:
his second volume deals with genuine propositions, and points out their
formal differences from classes.

A. *The Propositional Calculus.*

**14.** The propositional calculus is characterized by the fact that
all its propositions have as hypothesis and as consequent the assertion of
a material implication. Usually, the hypothesis is of the form “`p` implies `p`,” etc., which (§16) is equivalent to the assertion that the letters
which occur in the consequent are propositions. Thus the consequents
consist of propositional functions which are true of all propositions.
It is important to observe that, though the letters employed are symbols
for variables, and the consequents are true when the variables are given
values which are propositions, these values must be genuine propositions,
not propositional functions. The hypothesis “`p` is a proposition” is
not satisfied if for `p` we put “`x` is a man,” but it is satisfied if we put
“Socrates is a man” or if we put “`x` is a man implies `x` is a mortal for
all values of `x`.” Shortly, we may say that the propositions represented
by single letters in this calculus are variables, but do not contain
variables—in the case, that is to say, where the hypotheses of the
propositions which the calculus asserts are satisfied.

**15.** Our calculus studies the relation of *implication* between
propositions. This relation must be distinguished from the relation
of *formal* implication, which holds between propositional functions
when the one implies the other for all values of the variable. Formal
implication is also involved in this calculus, but is not explicitly
studied: we do not consider propositional functions in general, but
only certain definite propositional functions which occur in the propositions
of our calculus. How far formal implication is definable in
terms of implication simply, or material implication as it may be
called, is a difficult question, which will be discussed in Chapter iii.
What the difference is between the two, an illustration will explain.
The fifth proposition of Euclid follows from the fourth: if the fourth
is true, so is the fifth, while if the fifth is false, so is the fourth.
This is a case of material implication, for both propositions are absolute
constants, not dependent for their meaning upon the assigning of a
value to a variable. But each of them *states* a formal implication. The
fourth states that if `x` and `y` be triangles fulfilling certain conditions,
then `x` and `y` are triangles fulfilling certain other conditions, and that
this implication holds for all values of `x` and `y`; and the fifth states that
if `x` is an isosceles triangle, `x` has the angles at the base equal. The
formal implication involved in each of these two propositions is quite
a different thing from the material implication holding between the
propositions as wholes; both notions are required in the propositional
calculus, but it is the study of material implication which specially
distinguishes this subject, for formal implication occurs throughout the
whole of mathematics.

It has been customary, in treatises on logic, to confound the two
kinds of implication, and often to be really considering the formal kind
where the material kind only was apparently involved. For example,
when it is said that “Socrates is a man, therefore Socrates is a mortal,”
Socrates is *felt* as a variable: he is a type of humanity, and one feels that
any other man would have done as well. If, instead of *therefore*, which
implies the truth of hypothesis and consequent, we put “Socrates is a
man implies Socrates is a mortal,” it appears at once that we may
substitute not only another man, but any other entity whatever, in the
place of Socrates. Thus although what is explicitly stated, in such a
case, is a material implication, what is meant is a formal implication; and
some effort is needed to confine our imagination to material implication.

**16.** A definition of implication is quite impossible. If `p` implies
`q`, then if `p` is true `q` is true, *i.e.* `p`’s truth implies `q`’s truth; also if `q` is
false `p` is false, *i.e.* `q`’s falsehood implies `p`’s falsehood*. Thus truth and
falsehood give us merely new implications, not a definition of implication.
If `p` implies `q`, then both are false or both true, or `p` is false and `q` true;
it is impossible to have `q` false and `p` true, and it is necessary to have
`q` true or `p` false*. In fact, the assertion that `q` is true or `p` false turns
out to be strictly equivalent to “`p` implies `q`”; but as equivalence means
mutual implication, this still leaves implication fundamental, and not
definable in terms of disjunction. Disjunction, on the other hand, is
definable in terms of implication, as we shall shortly see. It follows
from the above equivalence that of any two propositions there must be
one which implies the other, that false propositions imply all propositions,
and true propositions are implied by all propositions. But these are
results to be demonstrated; the premisses of our subject deal exclusively
with rules of inference.

It may be observed that, although implication is indefinable,
*proposition* can be defined. Every proposition implies itself, and
whatever is not a proposition implies nothing. Hence to say “`p` is a
proposition” is equivalent to saying “`p` implies `p`”; and this equivalence
may be used to define propositions. As the mathematical sense of
*definition* is widely different from that current among philosophers,
it may be well to observe that, in the mathematical sense, a new
propositional function is said to be defined when it is stated to be
equivalent to (*i.e.* to imply and be implied by) a propositional function
which has either been accepted as indefinable or has been defined in
terms of indefinables. The definition of entities which are not
propositional functions is derived from such as are in ways which will
be explained in connection with classes and relations.

**17.** We require, then, in the propositional calculus, no indefinables
except the two kinds of implication—remembering, however, that formal
implication is a complex notion, whose analysis remains to be undertaken.
As regards our two indefinables, we require certain indemonstrable
propositions, which hitherto I have not succeeded in reducing to less
than ten. Some indemonstrables there must be; and some propositions,
such as the syllogism, must be of the number, since no demonstration
is possible without them. But concerning others, it may be doubted
whether they are indemonstrable or merely undemonstrated; and it
should be observed that the method of supposing an axiom false, and
deducing the consequences of this assumption, which has been found
admirable in such cases as the axiom of parallels, is here not universally
available. For all our axioms are principles of deduction; and if they
are true, the consequences which appear to follow from the employment
of an opposite principle will not really follow, so that arguments from
the supposition of the falsity of an axiom are here subject to special
fallacies. Thus the number of indemonstrable propositions may be
capable of further reduction, and in regard to some of them I know of
no grounds for regarding them as indemonstrable except that they have
hitherto remained undemonstrated.

**18.** The ten axioms are the following. (1) If `p` implies `q`, then
`p` implies `q`*; in other words, whatever `p` and `q` may be, “`p` implies `q`”
is a proposition. (2) If `p` implies `q`, then `p` implies `p`; in other words,
whatever implies anything is a proposition. (3) If `p` implies `q`, then `q`
implies `q`; in other words, whatever is implied by anything is a proposition.
(4) A true hypothesis in an implication may be dropped, and the
consequent asserted. This is a principle incapable of formal symbolic
statement, and illustrating the essential limitations of formalism—a
point to which I shall return at a later stage. Before proceeding
further, it is desirable to define the joint assertion of two propositions,
or what is called their logical product. This definition is highly artificial,
and illustrates the great distinction between mathematical and philosophical
definitions. It is as follows: If `p` implies `p`, then, if `q` implies `q`,
`p``q` (the logical product of `p` and `q`) means that if `p` implies that `q` implies
`r`, then `r` is true. In other words, if `p` and `q` are propositions, their joint
assertion is equivalent to saying that every proposition is true which is
such that the first implies that the second implies it. We cannot, with
formal correctness, state our definition in this shorter form, for the
hypothesis “`p` and `q` are propositions” is already the logical product of
“`p` is a proposition” and “`q` is a proposition.” We can now state the
six main principles of inference, to each of which, owing to its importance,
a name is to be given; of these all except the last will be found in
Peano’s accounts of the subject. (5) If `p` implies `p` and `q` implies `q`,
then `p``q` implies `p`. This is called *simplification*, and asserts merely that
the joint assertion of two propositions implies the assertion of the first
of the two. (6) If `p` implies `q` and `q` implies `r`, then `p` implies `r`. This
will be called the *syllogism*. (7) If `q` implies `q` and `r` implies `r`, and
if `p` implies that `q` implies `r`, then `p``q` implies `r`. This is the principle of
*importation*. In the hypothesis, we have a product of three propositions;
but this can of course be defined by means of the product of two.
The principle states that if `p` implies that `q` implies `r`, then `r` follows
from the joint assertion of `p` and `q`. For example: “If I call on so-and-so, then
if she is at home I shall be admitted” implies “If I call on
so-and-so and she is at home, I shall be admitted.” (8) If `p` implies
`p` and `q` implies `q`, then, if `p``q` implies `r`, then `p` implies that `q` implies `r`.
This is the converse of the preceding principle, and is called *exportation*†.
The previous illustration reversed will illustrate this principle. (9) If
`p` implies `q` and `p` implies `r`, then `p` implies `q``r`: in other words, a
proposition which implies each of two propositions implies them both.
This is called the principle of *composition*. (10) If `p` implies `p` and
`q` implies `q`, then “‘`p` implies `q`’ implies `p`” implies `p`. This is called
the principle of *reduction*; it has less self-evidence than the previous
principles, but is equivalent to many propositions that are self-evident.
I prefer it to these, because it is explicitly concerned, like its predecessors,
with implication, and has the same kind of logical character as they
have. If we remember that “`p` implies `q`” is equivalent to “`q` or not-`p`,”
we can easily convince ourselves that the above principle is true; for
“‘`p` implies `q`’ implies `p`” is equivalent to “`p` or the denial of ‘`q` or not-`p`,’” *i.e.* to “`p` or ‘`p` and not `q`,’” *i.e.* to `p`. But this way of persuading
ourselves that the principle of reduction is true involves many logical
principles which have not yet been demonstrated, and cannot be
demonstrated except by reduction or some equivalent. The principle is
especially useful in connection with negation. Without its help, by
means of the first nine principles, we can prove the law of contradiction;
we can prove, if `p` and `q` be propositions, that `p` implies not-not-`p`; that
“`p` implies not-`q`” is equivalent to “`q` implies not-`p`” and to not-`p``q`;
that “`p` implies `q`” implies “not-`q` implies not-`p`”; that `p` implies that
not-`p` implies `p`; that not-`p` is equivalent to “`p` implies not-`p`”; and that
“`p` implies not-`q`” is equivalent to “not-not-`p` implies not-`q`.” But we
cannot prove without reduction or some equivalent (so far at least as
I have been able to discover) that `p` or not-`p` must be true (the law of
excluded middle); that every proposition is equivalent to the negation
of some other proposition; that not-not-`p` implies `p`; that “not-`q` implies
not-`p`” implies “`p` implies `q`”; that “not-`p` implies `p`” implies `p`, or that
“`p` implies `q`” implies “`q` or not-`p`.” Each of these assumptions is
equivalent to the principle of reduction, and may, if we choose, be substituted
for it. Some of them—especially excluded middle and double
negation—appear to have far more self-evidence. But when we have
seen how to define disjunction and negation in terms of implication, we
shall see that the supposed simplicity vanishes, and that, for formal
purposes at any rate, reduction is simpler than any of the possible
alternatives. For this reason I retain it among my premisses in
preference to more usual and more superficially obvious propositions.

**19.** Disjunction or logical addition is defined as follows: “`p` or `q`”
is equivalent to “‘`p` implies `q`’ implies `q`.” It is easy to persuade
ourselves of this equivalence, by remembering that a false proposition
implies every other; for if `p` is false, `p` does imply `q`, and therefore,
if “`p` implies `q`” implies `q`, it follows that `q` is true. But this argument
again uses principles which have not yet been demonstrated, and is
merely designed to elucidate the definition by anticipation. From this
definition, by the help of reduction, we can prove that “`p` or `q`” is
equivalent to “`q` or `p`.” An alternative definition, deducible from the
above, is: “Any proposition implied by `p` and implied by `q` is true,” or,
in other words, “‘`p` implies `s`’ and ‘`q` implies `s`’ together imply `s`, whatever
`s` may be.” Hence we proceed to the definition of negation: not-`p` is
equivalent to the assertion that `p` implies all propositions, *i.e.* that
“`r` implies `r`” implies “`p` implies `r`” whatever `r` may be*. From this
point we can prove the laws of contradiction and excluded middle and
double negation, and establish all the formal properties of logical
multiplication and addition—the associative, commutative and distributive
laws. Thus the logic of propositions is now complete.

Philosophers will object to the above definitions of disjunction and
negation on the ground that what we *mean* by these notions is something
quite distinct from what the definitions assign as their meanings,
and that the equivalences stated in the definitions are, as a matter of
fact, significant propositions, not mere indications as to the way in
which symbols are going to be used. Such an objection is, I think,
well-founded, if the above account is advocated as giving the true philosophic
analysis of the matter. But where a purely formal purpose is to be
served, any equivalence in which a certain notion appears on one side
but not on the other will do for a definition. And the advantage of
having before our minds a strictly formal development is that it provides
the data for philosophical analysis in a more definite shape than
would be otherwise possible. Criticism of the procedure of formal logic,
therefore, will be best postponed until the present brief account has been
brought to an end.

B. *The Calculus of Classes.*

**20.** In this calculus there are very much fewer new primitive
propositions—in fact, two seem sufficient—but there are much greater
difficulties in the way of non-symbolic exposition of the ideas embedded
in our symbolism. These difficulties, as far as possible, will be postponed
to later chapters. For the present, I shall try to make an exposition
which is to be as straightforward and simple as possible.

The calculus of classes may be developed by regarding as fundamental
the notion of *class*, and also the relation of a member of a class to its
class. This method is adopted by Professor Peano, and is perhaps more
philosophically correct than a different method which, for formal purposes,
I have found more convenient. In this method we still take as
fundamental the relation (which, following Peano, I shall denote by ε)
of an individual to a class to which it belongs, *i.e.* the relation of Socrates
to the human race which is expressed by saying that Socrates is a man.
In addition to this, we take as indefinables the notion of a propositional
function and the notion of *such that*. It is these three notions that
characterize the class-calculus. Something must be said in explanation
of each of them.

**21.** The insistence on the distinction between ε and the relation of
whole and part between classes is due to Peano, and is of very great
importance to the whole technical development and the whole of the
applications to mathematics. In the scholastic doctrine of the syllogism,
and in all previous symbolic logic, the two relations are confounded,
except in the work of Frege*. The distinction is the same as that
between the relation of individual to species and that of species to
genus, between the relation of Socrates to the class of Greeks and the
relation of Greeks to men. On the philosophical nature of this distinction
I shall enlarge when I come to deal critically with the nature of
classes; for the present it is enough to observe that the relation of
whole and part is transitive, while ε is not so: we have Socrates is a
a man, and men are a class, but not Socrates is a class. It is to be
observed that the class must be distinguished from the class-concept
or predicate by which it is to be defined: thus men are a class, while
*man* is a class-concept. The relation ε must be regarded as holding
between Socrates and men considered collectively, not between Socrates
and *man*. I shall return to this point in Chapter vi. Peano holds
that all propositional functions containing only a single variable are
capable of expression in the form “`x` is an `a`,” where `a` is a constant
class; but this view we shall find reason to doubt.

**22.** The next fundamental notion is that of a propositional function. Although propositional functions occur in the calculus of propositions, they are there each defined as it occurs, so that the general
notion is not required. But in the class-calculus it is necessary to introduce the general notion explicitly. Peano does not require it, owing to
his assumption that the form “`x` is an `a`” is general for one variable, and
that extensions of the same form are available for any number of
variables. But we must avoid this assumption, and must therefore
introduce the notion of a propositional function. We may explain (but
not define) this notion as follows: `φ``x` is a propositional function if, for
every value of `x`, `φ``x` is a proposition, determinate when `x` is given.
Thus “`x` is a man” is a propositional function. In any proposition, however complicated, which contains no real variables, we may imagine one
of the terms, not a verb or adjective, to be replaced by other terms: instead
of “Socrates is a man” we may put “Plato is a man,” “the number 2
is a man,” and so on*. Thus we get successive propositions all agreeing
except as to the one variable term. Putting `x` for the variable term,
“`x` is a man” expresses the type of all such propositions. A propositional function in general will be true for some values of the variable
and false for others. The instances where it is true for *all* values of the
variable, so far as they are known to me, all express implications, such as
“`x` is a man implies `x` is a mortal”; but I know of no *à priori* reason for
asserting that no other propositional functions are true for all values of
the variable.

**23.** This brings me to the notion of *such that*. The values of `x`
which render a propositional function `φ``x` true are like the roots of an
equation—indeed the latter are a particular case of the former—and we
may consider all the values of `x` which are *such that* `φ``x` is true. In general,
these values form a *class*, and in fact a class may be defined as all
the terms satisfying some propositional function. There is, however,
some limitation required in this statement, though I have not been able to
discover precisely what the limitation is. This results from a certain
contradiction which I shall discuss at length at a later stage (Chap. x).
The reasons for defining *class* in this way are, that we require to provide
for the null-class, which prevents our defining a class as a term to
which some other has the relation ε, and that we wish to be able
to define classes by relations, *i.e.* all the terms which have to other
terms the relation `R` are to form a class, and such cases require somewhat
complicated propositional functions.

**24.** With regard to these three fundamental notions, we require
two primitive propositions. The first asserts that if `x` belongs to the
class of terms satisfying a propositional function `φ``x`, then `φ``x` is true.
The second asserts that if `φ``x` and `ψ``x` are equivalent propositions for all
values of `x`, then the class of `x`’s such that `φ``x` is true is identical with
the class of `x`’s such that `ψ``x` is true. Identity, which occurs here, is
defined as follows: `x` is identical with `y` if `y` belongs to every class to
which `x` belongs, in other words, if “`x` is a `u`” implies “`y` is a `u`” for
all values of `u`. With regard to the primitive proposition itself, it is to
be observed that it decides in favour of an extensional view of classes.
Two class-concepts need not be identical when their extensions are so:
*man* and *featherless biped* are by no means identical, and no more are *even prime* and *integer between 1 and 3*. These are class-`c``o``n``c``e``p``t``s`, and if our
axiom is to hold, it must not be of these that we are to speak in dealing
with classes. We must be concerned with the actual assemblage of
terms, not with any concept denoting that assemblage. For mathematical purposes, this is quite essential. Consider, for example, the
problem as to how many combinations can be formed of a given set
of terms taken any number at a time, *i.e.* as to how many classes are
contained in a given class. If distinct classes may have the same extension, this problem becomes utterly indeterminate. And certainly
common usage would regard a class as determined when all its terms are
given. The extensional view of classes, in some form, is thus essential to
Symbolic Logic and to mathematics, and its necessity is expressed in the
above axiom. But the axiom itself is not employed until we come to
Arithmetic; at least it need not be employed, if we choose to distinguish
the equality of classes, which is defined as mutual inclusion, from the
identity of individuals. Formally, the two are totally distinct: identity
is defined as above, equality of `a` and `b` is defined by the equivalence of
“`x` is an `a`” and “`x` is a `b`” for all values of `x`.

**25.** Most of the propositions of the class-calculus are easily
deduced from those of the propositional calculus. The logical product
or common part of two classes `a` and `b` is the class of `x`’s such that the
logical product of “`x` is an `a`” and “`x` is a `b`” is true. Similarly we define
the logical sum of two classes (`a` or `b`), and the negation of a class (not-`a`).
A new idea is introduced by the logical product and sum of a class of
classes. If `k` is a class of classes, its logical product is the class of terms belonging to each of the classes of `k`, *i.e.* the class of terms `x` such that “`u`
is a `k`” implies “`x` is a `u`” for all values of `u`. The logical sum is the class
which is contained in every class in which every class of the class `k` is
contained, *i.e.* the class of terms `x` such that, if “`u` is a `k`” implies “`u` is
contained in `c`” for all values of `u`, then, for all values of `c`, `x` is a `c`.
And we say that a class `a` is contained in a class `b` when “`x` is an `a`”
implies “`x` is a `b`” for all values of `x`. In like manner with the above
we may define the product and sum of a class of propositions. Another
very important notion is what is called the *existence* of a class—a word
which must not be supposed to mean what existence means in philosophy.
A class is said to exist when it has at least one term. A formal definition
is as follows: `a` is an existent class when and only when any
proposition is true provided “`x` is an `a`” always implies it whatever value
we may give to `x`. It must be understood that the proposition implied
must be a genuine proposition, not a propositional function of `x`. A
class `a` exists when the logical sum of all propositions of the form “`x` is
an `a`” is true, *i.e.* when not all such propositions are false.

It is important to understand clearly the manner in which propositions
in the class-calculus are obtained from those in the propositional
calculus. Consider, for example, the syllogism. We have
“`p` implies `q`” and “`q` implies `r`” imply “`p` implies `r`.” Now put “`x` is
an `a`,” “`x` is a `b`,” “`x` is a `c`” for `p`, `q`, `r`, where `x` must have some definite
value, but it is not necessary to decide what value. We then find that
if, for the value of `x` in question, `x` is an `a` implies `x` is a `b`, and `x` is a `b`
implies `x` is a `c`, then `x` is an `a` implies `x` is a `c`. Since the value of `x` is
irrelevant, we may vary `x`, and thus we find that if `a` is contained in `b`,
and `b` in `c`, then `a` is contained in `c`. This is the class-syllogism. But in
applying this process it is necessary to employ the utmost caution,
if fallacies are to be successfully avoided. In this connection it will
be instructive to examine a point upon which a dispute has arisen
between Schröder and Mr McColl*. Schröder asserts that if `p`, `q`, `r` are
propositions, “`p``q` implies `r`” is equivalent to the disjunction “`p` implies `r`
or `q` implies `r`.” Mr McColl admits that the disjunction implies the
other, but denies the converse implication. The reason for the divergence
is, that Schröder is thinking of propositions and material implication,
while Mr McColl is thinking of propositional functions and
formal implication. As regards propositions, the truth of the principle
may be easily made plain by the following considerations. If `p``q` implies
`r`, then, if either `p` or `q` be false, the one of them which is false implies `r`,
because false propositions imply all propositions. But if both be true,
`p``q` is true, and therefore `r` is true, and therefore `p` implies `r` and `q` implies `r`, because true propositions are implied by every proposition.
Thus in any case, one at least of the propositions `p` and `q` must
imply `r`. (This is not a proof, but an elucidation.) But Mr McColl
objects: Suppose `p` and `q` to be mutually contradictory, and `r` to be the
null proposition, then `p``q` implies `r` but neither `p` nor `q` implies r. Here
we are dealing with propositional functions and formal implication. A
propositional function is said to be null when it is false for all values of
`x`; and the class of `x`’s satisfying the function is called the null-class,
being in fact a class of no terms. Either the function or the class,
following Peano, I shall denote by Λ. Now let our `r` be replaced by Λ,
our `p` by `φ``x`, and our `q` by not-`φ``x`, where `φ``x` is any propositional function.
Then `p``q` is false for all values of `x`, and therefore implies Λ. But it is
not in general the case that `φ``x` is always false, nor yet that not-`φ``x` is always
false; hence neither always implies Λ. Thus the above formula can only
be truly interpreted in the propositional calculus: in the class-calculus
it is false. This may be easily rendered obvious by the following
considerations: Let `φ``x`, `ψ``x`, `χ``x` be three propositional functions. Then
“`φ``x` . `ψ``x` implies `χ``x`” implies, for all values of `x`, that either `φ``x` implies
`χ``x` or `ψ``x` implies `χ``x`. But it does not imply that either `φ``x` implies `χ``x`
for all values of `x`, or `ψ``x` implies `χ``x` for all values of `x`. The disjunction
is what I shall call a *variable* disjunction, as opposed to a constant one:
that is, in some cases one alternative is true, in others the other, whereas
in a constant disjunction there is one of the alternatives (though it is not
stated which) that is always true. Wherever disjunctions occur in regard
to propositional functions, they will only be transformable into statements
in the class-calculus in cases where the disjunction is constant. This is
a point which is both important in itself and instructive in its bearings.
Another way of stating the matter is this: In the proposition: If
`φ``x` . `ψ``x` implies `χ``x`, then either `φ``x` implies `χ``x` or `ψ``x` implies `χ``x`, the
implication indicated by *if* and *then* is formal, while the subordinate
implications are material; hence the subordinate implications do not
lead to the inclusion of one class in another, which results only from
formal implication.

The formal laws of addition, multiplication, tautology and negation
are the same as regards classes and propositions. The law of tautology
states that no change is made when a class or proposition is added to or
multiplied by itself. A new feature of the class-calculus is the null-class,
or class having no terms. This may be defined as the class of terms that
belong to every class, as the class which does not exist (in the sense
defined above), as the class which is contained in every class, as the
class Λ which is such that the propositional function “`x` is a Λ” is false
for all values of `x`, or as the class of `x`’s satisfying any propositional
function `φ``x` which is false for all values of `x`. All these definitions are
easily shown to be equivalent.

**26.** Some important points arise in connection with the theory of
identity. We have already defined two terms as identical when the
second belongs to every class to which the first belongs. It is easy to
show that this definition is symmetrical, and that identity is transitive
and reflexive (*i.e.* if `x` and `y`, `y` and `z` are identical, so are `x` and `z`; and
whatever `x` may be, `x` is identical with `x`). Diversity is defined as the
negation of identity. If `x` be any term, it is necessary to distinguish
from `x` the class whose only member is `x`: this may be defined as the
class of terms which are identical with `x`. The necessity for this
distinction, which results primarily from purely formal considerations,
was discovered by Peano; I shall return to it at a later stage. Thus
the class of even primes is not to be identified with the number 2, and
the class of numbers which are the sum of 1 and 2 is not to be identified
with 3. In what, philosophically speaking, the difference consists, is a
point to be considered in Chapter vi.

C. *The Calculus of Relations.*

**27.** The calculus of relations is a more modern subject than the
calculus of classes. Although a few hints for it are to be found in
De Morgan*, the subject was first developed by C. S. Peirce†. A careful
analysis of mathematical reasoning shows (as we shall find in the course
of the present work) that types of relations are the true subject-matter
discussed, however a bad phraseology may disguise this fact; hence the
logic of relations has a more immediate bearing on mathematics than
that of classes or propositions, and any theoretically correct and adequate
expression of mathematical truths is only possible by its means. Peirce
and Schröder have realized the great importance of the subject, but
unfortunately their methods, being based, not on Peano, but on the
older Symbolic Logic derived (with modifications) from Boole, are so
cumbrous and difficult that most of the applications which ought to be
made are practically not feasible. In addition to the defects of the old
Symbolic Logic, their method suffers technically (whether philosophically
or not I do not at present discuss) from the fact that they regard a
relation essentially as a class of couples, thus requiring elaborate
formulae of summation for dealing with single relations. This view is
derived, I think, probably unconsciously, from a philosophical error: it
has always been customary to suppose relational propositions less
ultimate than class-propositions (or subject-predicate propositions, with
which class-propositions are habitually confounded), and this has led
to a desire to treat relations as a kind of classes. However this may
be, it was certainly from the opposite philosophical belief, which I
derived from my friend Mr G. E. Moore*, that I was led to a different
formal treatment of relations. This treatment, whether more philosophically correct or not, is certainly far more convenient and far more
powerful as an engine of discovery in actual mathematics†.

**28.** If `R` be a relation, we express by `x``R``y` the propositional function
“`x` has the relation `R` to `y`.” We require a primitive (*i.e.* indemonstrable)
proposition to the effect that `x``R``y` is a proposition for all values of `x`
and `y`. We then have to consider the following classes: The class of
terms which have the relation `R` to some term or other, which I call the
class of *referents* with respect to `R`; and the class of terms to which
some term has the relation `R`, which I call the class of *relata*. Thus if
`R` be paternity, the referents will be fathers and the relata will be
children. We have also to consider the corresponding classes with
respect to particular terms or classes of terms: so-and-so’s children, or
the children of Londoners, afford illustrations.

The intensional view of relations here advocated leads to the result
that two relations may have the same extension without being identical.
Two relations `R`, `R`′ are said to be equal or equivalent, or to have the
same extension, when `x``R``y` implies and is implied by `x``R`′`y` for all values
of `x` and `y`. But there is no need here of a primitive proposition, as
there was in the case of classes, in order to obtain a relation which is
determinate when the extension is determinate. We may replace a
relation `R` by the logical sum or product of the class of relations
equivalent to `R`, *i.e.* by the assertion of some or of all such relations;
and this is identical with the logical sum or product of the class of
relations equivalent to `R`′, if `R`′ be equivalent to `R`. Here we use
the identity of two classes, which results from the primitive proposition
as to identity of classes, to establish the identity of two relations—a
procedure which could not have been applied to classes themselves
without a vicious circle.

A primitive proposition in regard to relations is that every relation
has a converse, *i.e.* that, if `R` be any relation, there is a relation `R`′ such
that `x``R``y` is equivalent to `y``R`′`x` for all values of `x` and `y`. Following
Schröder, I shall denote the converse of `R` by ˘`R`. Greater and less,
before and after, implying and implied by, are mutually converse
relations. With some relations, such as identity, diversity, equality,
inequality, the converse is the same as the original relation: such
relations are called *symmetrical*. When the converse is incompatible
with the original relation, as in such cases as greater and less, I call the
relation *asymmetrical*; in intermediate cases, *not-symmetrical*.

The most important of the primitive propositions in this subject is
that between any two terms there is a relation not holding between any
two other terms. This is analogous to the principle that any term is
the only member of some class; but whereas that could be proved,
owing to the extensional view of classes, this principle, so far as I can
discover, is incapable of proof. In this point, the extensional view of
relations has an advantage; but the advantage appears to me to be
outweighed by other considerations. When relations are considered
intensionally, it may seem possible to doubt whether the above principle
is true at all. It will, however, be generally admitted that, of any two
terms, some propositional function is true which is not true of a certain
given different pair of terms. If this be admitted, the above principle
follows by considering the logical product of all the relations that hold
between our first pair of terms. Thus the above principle may be
replaced by the following, which is equivalent to it: If `x``R``y` implies
`x`′`R``y`′, whatever `R` may be, so long as `R` is a relation, then `x` and `x`′,
`y` and `y`′ are respectively identical. But this principle introduces a
logical difficulty from which we have been hitherto exempt, namely a
variable with a restricted field; for unless `R` is a relation, `x``R``y` is not a
proposition at all, true or false, and thus `R`, it would seem, cannot take
*all* values, but only such as are relations. I shall return to the discussion
of this point at a later stage.

**29.** Other assumptions required are that the negation of a relation
is a relation, and that the logical product of a class of relations (*i.e.* the
assertion of all of them simultaneously) is a relation. Also the *relative product* of two relations must be a relation. The relative product of two
relations `R`, `S` is the relation which holds between `x` and `z` whenever
there is a term `y` to which `x` has the relation `R` and which has to `z` the
relation `S`. Thus the relation of a maternal grandfather to his grandson
is the relative product of father and mother; that of a paternal grandmother to her grandson is the relative product of mother and father;
that of grandparent to grandchild is the relative product of parent and
parent. The relative product, as these instances show, is not in general
commutative, and does not in general obey the law of tautology. The
relative product is a notion of very great importance. Since it does not
obey the law of tautology, it leads to powers of relations: the square of
the relation of parent and child is the relation of grandparent and
grandchild, and so on. Peirce and Schröder consider also what they call
the relative sum of two relations `R` and `S`, which holds between `x` and `z`,
when, if `y` be any other term whatever, either `x` has to `y` the relation `R`,
or `y` has to `z` the relation `S`. This is a complicated notion, which I have
found no occasion to employ, and which is introduced only in order to
preserve the duality of addition and multiplication. This duality has a
certain technical charm when the subject is considered as an independent
branch of mathematics; but when it is considered solely in relation to
the principles of mathematics, the duality in question appears devoid of
all philosophical importance.

**30.** Mathematics requires, so far as I know, only two other
primitive propositions, the one that material implication is a relation,
the other that ε (the relation of a term to a class to which it belongs) is
a relation*. We can now develop the whole of mathematics without
further assumptions or indefinables. Certain propositions in the logic
of relations deserve to be mentioned, since they are important, and it
might be doubted whether they were capable of formal proof. If `u`, `v`
be any two classes, there is a relation `R` the assertion of which between
any two terms `x` and `y` is equivalent to the assertion that `x` belongs to `u`
and `y` to `v`. If `u` be any class which is not null, there is a relation which
all its terms have to it, and which holds for no other pairs of terms. If
`R` be any relation, and `u` any class contained in the class of referents
with respect to `R`, there is a relation which has `u` for the class of its
referents, and is equivalent to `R` throughout that class; this relation is
the same as `R` where it holds, but has a more restricted domain. (I use
*domain* as synonymous with *class of referents*.) From this point onwards,
the development of the subject is technical: special types of relations are
considered, and special branches of mathematics result.

D. *Peano’s Symbolic Logic.*

**31.** So much of the above brief outline of Symbolic Logic is
inspired by Peano, that it seems desirable to discuss his work explicitly,
justifying by criticism the points in which I have departed from him.

The question as to which of the notions of symbolic logic are to be
taken as indefinable, and which of the propositions as indemonstrable,
is, as Professor Peano has insisted†, to some extent arbitrary. But it is
important to establish all the mutual relations of the simpler notions
of logic, and to examine the consequence of taking various notions as
indefinable. It is necessary to realize that definition, in mathematics,
does not mean, as in philosophy, an analysis of the idea to be defined
into constituent ideas. This notion, in any case, is only applicable to
concepts, whereas in mathematics it is possible to define terms which
are not concepts*. Thus also many notions are defined by symbolic
logic which are not capable of philosophical definition, since they are
simple and unanalyzable. Mathematical definition consists in pointing
out a fixed relation to a fixed term, of which one term only is capable:
this term is then defined by means of the fixed relation and the fixed
term. The point in which this differs from philosophical definition
may be elucidated by the remark that the mathematical definition does
not point out the term in question, and that only what may be called
philosophical insight reveals which it is among all the terms there are.
This is due to the fact that the term is defined by a concept which
*denotes* it unambiguously, not by actually mentioning the term denoted.
What is meant by *denoting*, as well as the different ways of denoting,
must be accepted as primitive ideas in any symbolic logic†: in this
respect, the order adopted seems not in any degree arbitrary.

**32.** For the sake of definiteness, let us now examine some one
of Professor Peano’s expositions of the subject. In his later expositions‡
he has abandoned the attempt to distinguish clearly certain ideas and
propositions as primitive, probably because of the realization that any
such distinction is largely arbitrary. But the distinction appears useful,
as introducing greater definiteness, and as showing that a certain set
of primitive ideas and propositions are sufficient; so far from being
abandoned, it ought rather to be made in every possible way. I shall,
therefore, in what follows, expound one of his earlier expositions, that
of 1897§.

The primitive notions with which Peano starts are the following:
Class, the relation of an individual to a class of which it is a member,
the notion of a term, implication where both propositions contain the
same variables, *i.e.* formal implication, the simultaneous affirmation of
two propositions, the notion of definition, and the negation of a proposition. From these notions, together with the division of a complex
proposition into parts, Peano professes to deduce all symbolic logic by
means of certain primitive propositions. Let us examine the deduction
in outline.

We may observe, to begin with, that the simultaneous affirmation
of *two* propositions might seem, at first sight, not enough to take as a
primitive idea. For although this can be extended, by successive steps,
to the simultaneous affirmation of any finite number of propositions,
yet this is not all that is wanted; we require to be able to affirm
simultaneously all the propositions of any class, finite or infinite. But
the simultaneous assertion of a class of propositions, oddly enough, is
much easier to define than that of two propositions, (see §34, (3)). If `k`
be a class of propositions, their simultaneous affirmation is the assertion
that “`p` is a `k`” implies `p`. If this holds, all propositions of the class are
true; if it fails, one at least must be false. We have seen that the
logical product of two propositions can be defined in a highly artificial
manner; but it might almost as well be taken as indefinable, since no
further property can be proved by means of the definition. We may
observe, also, that formal and material implication are combined by
Peano into one primitive idea, whereas they ought to be kept separate.

**33.** Before giving any primitive propositions, Peano proceeds to
some definitions. (1) If `a` is a class, “`x` and `y` are `a`’s” is to mean
“`x` is an `a` and `y` is an `a`.” (2) If `a` and `b` are classes, “every `a` is a `b`”
means “`x` is an `a` implies that `x` is a `b`.” If we accept formal implication
as a primitive notion, this definition seems unobjectionable; but it may
well be held that the relation of inclusion between classes is simpler than
formal implication, and should not be defined by its means. This is a
difficult question, which I reserve for subsequent discussion. A formal
implication appears to be the assertion of a whole class of material
implications. The complication introduced at this point arises from
the nature of the variable, a point which Peano, though he has done
very much to show its importance, appears not to have himself sufficiently considered. The notion of one proposition containing a variable
implying another such proposition, which he takes as primitive, is
complex, and should therefore be separated into its constituents; from
this separation arises the necessity of considering the simultaneous
affirmation of a whole class of propositions before interpreting such
a proposition as “`x` is an `a` implies that `x` is a `b`.” (3) We come next
to a perfectly worthless definition, which has been since abandoned*.
This is the definition of *such that*. The `x`’s such that `x` is an `a`, we are
told, are to mean the class `a`. But this only gives the meaning of *such that* when placed before a proposition of the type “`x` is an `a`.” Now
it is often necessary to consider an `x` such that some proposition is true
of it, where this proposition is not of the form “`x` is an `a`.” Peano holds
(though he does not lay it down as an axiom) that every proposition
containing only one variable is reducible to the form “`x` is an `a`†.”
But we shall see (Chap. x) that at least one such proposition is not
reducible to this form. And in any case, the only utility of *such that*
is to effect the reduction, which cannot therefore be assumed to be
already effected without it. The fact is that *such that* contains a primitive idea, but one which it is not easy clearly to disengage from other ideas.

In order to grasp the meaning of *such that*, it is necessary to observe,
first of all, that what Peano and mathematicians generally call *one*
proposition containing a variable is really, if the variable is apparent,
the conjunction of a certain class of propositions defined by some
constancy of form; while if the variable is real, so that we have a
propositional function, there is not a proposition at all, but merely
a kind of schematic representation of any proposition of a certain type.
“The sum of the angles of a triangle is two right angles,” for example,
when stated by means of a variable, becomes: Let `x` be a triangle; then
the sum of the angles of `x` is two right angles. This expresses the
conjunction of all the propositions in which it is said of particular
definite entities that if they are triangles, the sum of their angles is
two right angles. But a propositional function, where the variable is
real, represents *any* proposition of a certain form, not *all* such propositions (see §§59–62). There is, for each propositional function, an
indefinable relation between propositions and entities, which may be
expressed by saying that all the propositions have the same form,
but different entities enter into them. It is this that gives rise to
propositional functions. Given, for example, a constant relation and
a constant term, there is a one-one correspondence between the propositions asserting that various terms have the said relation to the said
term, and the various terms which occur in these propositions. It is
this notion which is requisite for the comprehension of *such that*. Let
`x` be a variable whose values form the class `a`, and let `f`(`x`) be a one-valued function of `x` which is a true proposition for all values of `x` within
the class `a`, and which is false for all other values of `x`. Then the terms
of `a` are the class of terms *such that* `f`(`x`) is a true proposition. This
gives an explanation of *such that*. But it must always be remembered
that the appearance of having *one* proposition `f`(`x`) satisfied by a
number of values of `x` is fallacious: `f`(`x`) is not a proposition at all,
but a propositional function. What is fundamental is the relation of
various propositions of given form to the various terms entering
severally into them as arguments or values of the variable; this
relation is equally required for interpreting the propositional function
`f`(`x`) and the notion *such that*, but is itself ultimate and inexplicable.
(4) We come next to the definition of the logical product, or
common part, of two classes. If `a` and `b` be two classes, their common
part consists of the class of terms `x` such that `x` is an `a` and `x` is a `b`.
Here already, as Padoa points out (*loc. cit.*), it is necessary to extend the
meaning of *such that* beyond the case where our proposition asserts
membership of a class, since it is only by means of the definition that
the common part is shown to be a class.

**34.** The remainder of the definitions preceding the primitive
propositions are less important, and may be passed over. Of the
primitive propositions, some appear to be merely concerned with the
symbolism, and not to express any real properties of what is symbolized;
others, on the contrary, are of high logical importance.

(1) The first of Peano’s axioms is “every class is contained in
itself.” This is equivalent to “every proposition implies itself.” There
seems no way of evading this axiom, which is equivalent to the law of
identity, except the method adopted above, of using self-implication
to define propositions. (2) Next we have the axiom that the product
of two classes is a class. This ought to have been stated, as ought also
the definition of the logical product, for a class of classes; for when
stated for only two classes, it cannot be extended to the logical product
of an infinite class of classes. If *class* is taken as indefinable, it is a
genuine axiom, which is very necessary to reasoning. But it might
perhaps be somewhat generalized by an axiom concerning the terms
satisfying propositions of a given form: *e.g.* “the terms having one
or more given relations to one or more given terms form a class.”
In Section B, above, the axiom was wholly evaded by using a generalized
form of the axiom as the definition of class. (3) We have next two
axioms which are really only one, and appear distinct only because Peano
defines the common part of two classes instead of the common part of a
class of classes. These two axioms state that, if `a`, `b` be classes, their logical
product, `a``b`, is contained in `a` and is contained in `b`. These appear as
different axioms, because, as far as the symbolism shows, `a``b` might be
different from `b``a`. It is one of the defects of most symbolisms that they
give an order to terms which intrinsically have none, or at least none
that is relevant. So in this case: if `K` be a class of classes, the logical
product of `K` consists of all terms belonging to *every* class that belongs
to `K`. With this definition, it becomes at once evident that no order
of the terms of `K` is involved. Hence if `K` has only two terms, `a` and `b`,
it is indifferent whether we represent the logical product of `K` by `a``b`
or by `b``a`, since the order exists only in the symbols, not in what is
symbolized. It is to be observed that the corresponding axiom with
regard to propositions is, that the simultaneous assertion of a class of
propositions implies any proposition of the class; and this is perhaps
the best form of the axiom. Nevertheless, though an axiom is not
required, it is necessary, here as elsewhere, to have a means of connecting
the case where we start from a class of classes or of propositions or of
relations with the case where the class results from enumeration of its
terms. Thus although no order is involved in the product of a *class* of
propositions, there is an order in the product of two definite propositions `p`, `q`, and it is significant to assert that the products `p``q` and `q``p` are
equivalent. But this can be proved by means of the axioms with which
we began the calculus of propositions (§18). It is to be observed that
this proof is prior to the proof that the class whose terms are `p` and `q` is
identical with the class whose terms are `q` and `p`. (4) We have next
two forms of syllogism, both primitive propositions. The first asserts
that, if `a`, `b` be classes, and `a` is contained in `b`, and `x` is an `a`, then `x` is
a `b`; the second asserts that if `a`, `b`, `c` be classes, and `a` is contained in `b`,
`b` in `c`, then `a` is contained in `c`. It is one of the greatest of Peano’s
merits to have clearly distinguished the relation of the individual to its
class from the relation of inclusion between classes. The difference is
exceedingly fundamental: the former relation is the simplest and most
essential of all relations, the latter a complicated relation derived from
logical implication. It results from the distinction that the syllogism
in Barbara has two forms, usually confounded: the one the time-honoured
assertion that Socrates is a man, and therefore mortal, the other the
assertion that Greeks are men, and therefore mortal. These two forms
are stated by Peano’s axioms. It is to be observed that, in virtue of the
definition of what is meant by one class being contained in another,
the first form results from the axiom that, if `p`, `q`, `r` be propositions, and
`p` implies that `q` implies `r`, then the product of `p` and `q` implies `r`. This
axiom is now substituted by Peano for the first form of the syllogism*:
it is more general and cannot be deduced from the said form. The
second form of the syllogism, when applied to propositions instead of
classes, asserts that implication is transitive. This principle is, of course,
the very life of all chains of reasoning. (5) We have next a principle
of reasoning which Peano calls *composition*: this asserts that if `a` is
contained in `b` and also in `c`, then it is contained in the common part
of both. Stating this principle with regard to propositions, it asserts
that if a proposition implies each of two others, then it implies their
joint assertion or logical product; and this is the principle which was
called *composition* above.

**35.** From this point, we advance successfully until we require the
idea of *negation*. This is taken, in the edition of the *Formulaire* we are
considering, as a new primitive idea, and disjunction is defined by its
means. By means of the negation of a proposition, it is of course easy
to define the negation of a class: for “`x` is a not-`a`” is equivalent to “`x`
is not an `a`.” But we require an axiom to the effect that *not*-`a` is a
class, and another to the effect that not-not-`a` is `a`. Peano gives also a
third axiom, namely: If `a`, `b`, `c` be classes, and `a``b` is contained in `c`, and `x`
is an `a` but not a `c`, then `x` is not a `b`. This is simpler in the form: if `p`, `q`, `r`
be propositions, and `p`, `q` together imply `r`, and `p` is true while `r` is
false, then `q` is false. This would be still further improved by being put
in the form: If `q`, `r` are propositions, and `q` implies `r`, then not-`r` implies
not-`q`; a form which Peano obtains as a deduction. By dealing with
propositions before classes or propositional functions, it is possible, as we
saw, to avoid treating negation as a primitive idea, and to replace all
axioms respecting negation by the principle of reduction.

We come next to the definition of the disjunction or logical sum of
two classes. On this subject Peano has many times changed his
procedure. In the edition we are considering, “`a` or `b`” is defined as the
negation of the logical product of not-`a` and not-`b`, *i.e.* as the class of
terms which are not both not-`a` and not-`b`. In later editions (*e.g.* *F*. 1901,
p. 19), we find a somewhat less artificial definition, namely: “`a` or `b`”
consists of all terms which belong to any class which contains `a` and
contains `b`. Either definition seems logically unobjectionable. It is to
be observed that `a` and `b` are classes, and that it remains a question for
philosophical logic whether there is not a quite different notion of the
disjunction of individuals, as *e.g.* “Brown or Jones.” I shall consider
this question in Chapter v. It will be remembered that, when we begin
by the calculus of propositions, disjunction is defined before negation;
with the above definition (that of 1897), it is plainly necessary to take
negation first.

**36.** The connected notions of the null-class and the existence of a
class are next dealt with. In the edition of 1897, a class is defined as
null when it is contained in every class. When we remember the
definition of one class `a` being contained in another `b` (“`x` is an `a`”
implies “`a` is a `b`” for all values of `x`), we see that we are to regard
the implication as holding for *all* values, and not only for those values
for which `x` really is an `a`. This is a point upon which Peano is not
explicit, and I doubt whether he has made up his mind on it. If the
implication were only to hold when `x` really is an `a`, it would not give a
definition of the null-class, for which this hypothesis is false for all values
of `x`. I do not know whether it is for this reason or for some other that
Peano has since abandoned the definition of the inclusion of classes
by means of formal implication between propositional functions: the
inclusion of classes appears to be now regarded as indefinable. Another
definition which Peano has sometimes favoured (*e.g.* *F*. 1895, Errata,
p. 116) is, that the null-class is the product of any class into its
negation—a definition to which similar remarks apply. In *R. d. M.* vii,
No. 1 (§3, Prop. 1.0), the null-class is defined as the class of those terms
that belong to every class, *i.e.* the class of terms `x` such that “`a` is a
class” implies “`x` is an `a`” for all values of `a`. There are of course no
such terms `x`; and there is a grave logical difficulty in trying to interpret
extensionally a class which has no extension. This point is one to which
I shall return in Chapter vi.

From this point onward, Peano’s logic proceeds by a smooth development. But in one respect it is still defective: it does not recognize as
ultimate relational propositions not asserting membership of a class.
For this reason, the definitions of a function* and of other essentially
relational notions are defective. But this defect is easily remedied by
applying, in the manner explained above, the principles of the
*Formulaire* to the logic of relations†.

Notes

*^{[page 10]} By far the most complete account of the non-Peanesque methods will be found in the three volumes of Schröder, *Vorlesungen über die Algebra der Logik*, Leipzig,
1890, 1891, 1895.

†^{[page 10]} See *Formulaire de Mathématiques*, Turin, 1895, with subsequent editions in later years; also *Revue de Mathématiques*, Vol. vii. No. 1 (1900). The editions of the *Formulaire* will be quoted as *F*. 1895 and so on. The *Revue de Mathématiques*, which was originally the *Rivista di Matematica*, will be referred to as *R. d. M.*

‡^{[page 10]} In what follows the main outlines are due to Professor Peano, except as regards relations; even in those cases where I depart from his views, the problems considered have been suggested to me by his works.

*^{[page 11]} I may as well say at once that I, do not distinguish between inference and
deduction. What is called induction appears to me to be either disguised deduction
or a mere method of making plausible guesses.

†^{[page 11]} See below, Part V, Chap. xxxvi.

*^{[page 12]} On the points where the duality breaks down, cf. Schröder, *op. cit.*, Vol. ii, Lecture 21.

†^{[page 12]} Cf. “The Calculus of Equivalent Statements,” *Proceedings of the London Mathematical Society*, Vol. ix and subsequent volumes; “Symbolic Reasoning,” *Mind*, Jan. 1880, Oct. 1897, and Jan. 1900; “La Logique Symbolique et ses Applications,” *Bibliothèque du Congrès International de Philosophie*, Vol. iii (Paris, 1901). I shall in future quote the proceedings of the above Congress by the title *Congrès*.

*^{[page 13]} *F*. 1901, p. 2.

*^{[page 14]} The reader is recommended to observe that the main implications in these statements are formal, *i.e.* “`p` implies `q`” formally implies “`p`’s truth implies `q`’s truth,” while the subordinate implications are material.

*^{[page 15]} I may as well state once for all that the alternatives of a disjunction will never be considered as mutually exclusive unless expressly said to be so.

*^{[page 16]} Note that the implications denoted by *if* and *then*, in these axioms, are formal, while those denoted by *implies* are material.

†^{[page 16]} (7) and (8) cannot (I think) be deduced from the definition of the logical product, because they are required for passing from “If *p* is a proposition, then ‘*q* is a proposition’ implies etc.” to “If *p* and *q* are propositions, then etc.”

*^{[page 18]} The principle that false propositions imply all propositions solves Lewis Carroll’s logical paradox in *Mind*, N. S. No. 11 (1894). The assertion made in that paradox is that, if `p`, `q`, `r` be propositions, and `q` implies `r`, while `p` implies that `q` implies not-`r`, then `p` must be false, on the supposed ground that “`q` implies `r`” and “`q` implies not-`r`” are incompatible. But in virtue of our definition of negation, if `q` be false both these implications will hold: the two together, in fact, whatever proposition `r` may be, are equivalent to not-`q`. Thus the only inference warranted by Lewis Carroll’s premisses is that if `p` be true, `q` must be false, *i.e.* that `p` implies not-`q`; and this is the conclusion, oddly enough, which common sense would have drawn in the particular case which he discusses.

*^{[page 19]} See his *Begriffsschrift*, Halle, 1879, and *Grundgesetze der Arithmetik*, Jena, 1893, p. 2.

*^{[page 20]} Verbs and adjectives occurring as such are distinguished by the fact that, if they be taken as variable, the resulting function is only a proposition for *some* values
of the variable, *i.e.* for such as are verbs or adjectives respectively. See Chap. iv.

*^{[page 22]} Schröder, *Algebra der Logik*, Vol. ii, pp. 258–9; McColl, “Calculus of Equivalent Statements,” fifth paper, *Proc. Lond. Math. Soc.* Vol. xxviii, p. 182.

*^{[page 23]} *Camb. Phil. Trans.* Vol. x, “On the Syllogism, No. iv, and on the Logic of Relations.” Cf. *ib*. Vol. ix, p. 104; also his *Formal Logic* (London, 1847), p. 50.

†^{[page 23]} See especially his articles on the Algebra of Logic, *American Journal of Mathematics*, Vols, iii and vii. The subject is treated at length by C. S. Peirce’s methods in Schröder, *op. cit.*, Vol. iii.

*^{[page 24]} See his article “On the Nature of Judgment,” *Mind*, N. S. No. 30.

†^{[page 24]} See my articles in *R. d. M.* Vol. vii. No. 2 and subsequent numbers.

*^{[page 26]} There is a difficulty in regard to this primitive proposition, discussed in §§53, 94 below.

†^{[page 26]} *E.g. F*. 1901, p. 6; *F*. 1897, Part I, pp. 62–3.

*^{[page 27]} See Chap. iv.

†^{[page 27]} See Chap. v.

‡^{[page 27]} *F*. 1901 and *R. d. M.* Vol. vii, No. 1 (1900).

§^{[page 27]} *F*. 1897, Part I.

*^{[page 28]} In consequence of the criticisms of Padoa, *R. d. M.* Vol. vi, p. 112.

†^{[page 28]} *R. d. M.* Vol. vii, No. 1, p. 25; *F*. 1901, p. 21, §2, Prop. 4.0, Note.

*^{[page 31]} See *e.g. F*. 1901, Part I, §1, Prop. 3.3 (p. 10).

*^{[page 32]} *E.g. F*. 1901, Part I, §10, Props. 1.0.01 (p. 33).

†^{[page 32]} See my article “Sur la logique des relations,” *R. d. M.* Vol. vii, 2 (1901).

Implication and Formal Implication.

**37.** In the preceding chapter I endeavoured to present, briefly and
uncritically, all the data, in the shape of formally fundamental ideas
and propositions, that pure mathematics requires. In subsequent Parts
I shall show that these are all the data by giving definitions of the
various mathematical concepts—number, infinity, continuity, the various
spaces of geometry, and motion. In the remainder of Part I, I shall
give indications, as best I can, of the philosophical problems arising in
the analysis of the data, and of the directions in which I imagine these
problems to be probably soluble. Some logical notions will be elicited
which, though they seem quite fundamental to logic, are not commonly
discussed in works on the subject; and thus problems no longer clothed
in mathematical symbolism will be presented for the consideration of
philosophical logicians.

Two kinds of implication, the material and the formal, were found to be essential to every kind of deduction. In the present chapter I wish to examine and distinguish these two kinds, and to discuss some methods of attempting to analyze the second of them.

In the discussion of inference, it is common to permit the intrusion
of a psychological element, and to consider our acquisition of new
knowledge by its means. But it is plain that where we validly infer one
proposition from another, we do so in virtue of a relation which holds
between the two propositions whether we perceive it or not: the mind,
in fact, is as purely receptive in inference as common sense supposes it to
be in perception of sensible objects. The relation in virtue of which it
is possible for us validly to infer is what I call material implication.
We have already seen that it would be a vicious circle to define this
relation as meaning that *if* one proposition is true, *then* another is true,
for *if* and *then* already involve implication. The relation holds, in fact,
when it does hold, without any reference to the truth or falsehood of the
propositions involved.

But in developing the consequences of our assumptions as to implication, we were led to conclusions which do not by any means agree with
what is commonly held concerning implication, for we found that any
false proposition implies every proposition and any true proposition is
implied by every proposition. Thus propositions are formally like a set
of lengths each of which is one inch or two, and implication is like the
relation “equal to or less than” among such lengths. It would certainly
not be commonly maintained that “2 + 2 = 4” can be deduced from
“Socrates is a man,” or that both are implied by “Socrates is a triangle.”
But the reluctance to admit such implications is chiefly due, I think, to
preoccupation with formal implication, which is a much more familiar
notion, and is really before the mind, as a rule, even where material
implication is what is explicitly mentioned. In inferences from “Socrates
is a man,” it is customary not to consider the philosopher who vexed the
Athenians, but to regard Socrates merely as a symbol, capable of being
replaced by any other man; and only a vulgar prejudice in favour of
true propositions stands in the way of replacing Socrates by a number, a
table, or a plum-pudding. Nevertheless, wherever, as in Euclid, one
particular proposition is deduced from another, material implication is
involved, though as a rule the material implication may be regarded as a
particular instance of some formal implication, obtained by giving some
constant value to the variable or variables involved in the said formal
implication. And although, while relations are still regarded with the
awe caused by unfamiliarity, it is natural to doubt whether any such
relation as implication is to be found, yet, in virtue of the general
principles laid down in Section C of the preceding chapter, there must
be a relation holding between nothing except propositions, and holding
between any two propositions of which either the first is false or the
second true. Of the various equivalent relations satisfying these
conditions, one is to be called *implication*, and if such a notion seems
unfamiliar, that does not suffice to prove that it is illusory.

**38.** At this point, it is necessary to consider a very difficult
logical problem, namely, the distinction between a proposition actually
asserted, and a proposition considered merely as a complex concept.
One of our indemonstrable principles was, it will be remembered, that
if the hypothesis in an implication is true, it may be dropped, and the
consequent asserted. This principle, it was observed, eludes formal
statement, and points to a certain failure of formalism in general. The
principle is employed whenever a proposition is said to be *proved*; for
what happens is, in all such cases, that the proposition is shown to be
implied by some true proposition. Another form in which the principle
is constantly employed is the substitution of a constant, satisfying the
hypothesis, in the consequent of a formal implication. If `φ``x` implies `ψ``x`
for all values of `x`, and if `a` is a constant satisfying `φ``x`, we can assert
`ψ``a`, dropping the true hypothesis `φ``a`. This occurs, for example, whenever any of those rules of inference which employ the hypothesis
that the variables involved are propositions, are applied to particular
propositions. The principle in question is, therefore, quite vital to any
kind of demonstration.

The independence of this principle is brought out by a consideration
of Lewis Carroll’s puzzle, “What the Tortoise said to Achilles*.” The
principles of inference which we accepted lead to the proposition that, if
`p` and `q` be propositions, then `p` together with “`p` implies `q`” implies `q`.
At first sight, it might be thought that this would enable us to assert `q`
provided `p` is true and implies `q`. But the puzzle in question shows that
this is not the case, and that, until we have some new principle, we shall
only be led into an endless regress of more and more complicated implications, without ever arriving at the assertion of `q`. We need, in fact,
the notion of *therefore*, which is quite different from the notion of *implies*,
and holds between different entities. In grammar, the distinction is that
between a verb and a verbal noun, between, say, “`A` is greater than `B`”
and “`A`’s being greater than `B`.” In the first of these, a proposition is
actually asserted, whereas in the second it is merely considered. But
these are psychological terms, whereas the difference which I desire to
express is genuinely logical. It is plain that, if I may be allowed to
use the word *assertion* in a non-psychological sense, the proposition
“`p` implies `q`” *asserts* an implication, though it does not *assert* `p` or `q`.
The `p` and the `q` which enter into this proposition are not strictly the
same as the `p` or the `q` which are separate propositions, at least, if they
are true. The question is: How does a proposition differ by being
actually true from what it would be as an entity if it were not true? It
is plain that true and false propositions alike are entities of a kind, but
that true propositions have a quality not belonging to false ones, a
quality which, in a non-psychological sense, may be called being
*asserted*. Yet there are grave difficulties in forming a consistent theory
on this point, for if assertion in any way changed a proposition, no
proposition which can possibly in any context be unasserted could be
true, since when asserted it would become a different proposition. But
this is plainly false; for in “`p` implies `q`,” `p` and `q` are not asserted, and
yet they may be true. Leaving this puzzle to logic, however, we must
insist that there is a difference of some kind between an asserted and an
unasserted proposition†. When we say *therefore*, we state a relation
which can only hold between asserted propositions, and which thus
differs from implication. Wherever *therefore* occurs, the hypothesis
may be dropped, and the conclusion asserted by itself. This seems to
be the first step in answering Lewis Carroll’s puzzle.

**39.** It is commonly said that an inference must have premisses
and a conclusion, and it is held, apparently, that two or more premisses
are necessary, if not to all inferences, yet to most. This view is borne
out, at first sight, by obvious facts: every syllogism, for example, is held
to have two premisses. Now such a theory greatly complicates the
relation of implication, since it renders it a relation which may have any
number of terms, and is symmetrical with respect to all but one of them,
but not symmetrical with respect to that one (the conclusion). This
complication is, however, unnecessary, first, because every simultaneous
assertion of a number of propositions is itself a single proposition, and
secondly, because, by the rule which we called *exportation*, it is always
possible to exhibit an implication explicitly as holding between single
propositions. To take the first point first: if `k` be a class of propositions, all the propositions of the class `k` are asserted by the single
proposition “for all values of `x`, if `x` implies `x`, then ‘`x` is a `k`’ implies
`x`,” or, in more ordinary language, “every `k` is true.” And as regards
the second point, which assumes the number of premisses to be finite,
“`p``q` implies `r`” is equivalent, if `q` be a proposition, to “`p` implies that `q`
implies `r`,” in which latter form the implications hold explicitly between
single propositions. Hence we may safely hold implication to be a
relation between two propositions, not a relation of an arbitrary number
of premisses to a single conclusion.

**40.** I come now to formal implication, which is a far more difficult
notion than material implication. In order to avoid the general notion
of propositional function, let us begin by the discussion of a particular
instance, say “`x` is a man implies `x` is a mortal for all values of `x`.”
This proposition is equivalent to “all men are mortal” “every man is
mortal” and “any man is mortal.” But it seems highly doubtful
whether it is the same proposition. It is also connected with a purely
intensional proposition in which *man* is asserted to be a complex notion
of which *mortal* is a constituent, but this proposition is quite distinct
from the one we are discussing. Indeed, such intensional propositions
are not always present where one class is included in another: in general,
either class may be defined by various different predicates, and it is by
no means necessary that every predicate of the smaller class should
contain every predicate of the larger class as a factor. Indeed, it may
very well happen that both predicates are philosophically simple: thus
*colour* and *existent* appear to be both simple, yet the class of colours is
part of the class of existents. The intensional view, derived from
predicates, is in the main irrelevant to Symbolic Logic and to Mathematics, and I shall not consider it further at present.

**41.** It may be doubted, to begin with, whether “`x` is a man
implies `x` is a mortal” is to be regarded as asserted strictly of all possible
terms, or only of such terms as are men. Peano, though he is not explicit,
appears to hold the latter view. But in this case, the hypothesis ceases
to be significant, and becomes a mere definition of `x`: `x` is to mean any
man. The hypothesis then becomes a mere assertion concerning the
meaning of the symbol `x`, and the whole of what is asserted concerning
the matter dealt with by our symbol is put into the conclusion. The
premiss says: `x` is to mean any man. The conclusion says: `x` is mortal.
But the implication is merely concerning the symbolism: since any man
is mortal, if `x` denotes any man, `x` is mortal. Thus formal implication,
on this view, has wholly disappeared, leaving us the proposition “any
man is mortal” as expressing the whole of what is relevant in the
proposition with a variable. It would now only remain to examine
the proposition “any man is mortal,” and if possible to explain this
proposition without reintroducing the variable and formal implication.
It must be confessed that some grave difficulties are avoided by this
view. Consider, for example, the simultaneous assertion of all the
propositions of some class `k`: this is not expressed by “‘`x` is a `k`’ implies
`x` for all values of `x`.” For as it stands, this proposition does not express
what is meant, since, if `x` be not a proposition, “`x` is a `k`” cannot imply
`x`; hence the range of variability of `x` must be confined to propositions,
unless we prefix (as above, §39) the hypothesis “`x` implies `x`.” This
remark applies generally, throughout the propositional calculus, to all
cases where the conclusion is represented by a single letter: unless the
letter does actually represent a proposition, the implication asserted will
be false, since only propositions can be implied. The point is that, if `x`
be our variable, `x` itself is a proposition for all values of `x` which are
propositions, but not for other values. This makes it plain what the
limitations are to which our variable is subject: it must vary only within
the range of values for which the two sides of the principal implication
are propositions, in other words, the two sides, when the variable is not
replaced by a constant, must be genuine propositional functions. If this
restriction is not observed, fallacies quickly begin to appear. It should be
noticed that there may be any number of subordinate implications which
do not require that their terms should be propositions: it is only of the
principal implication that this is required. Take, for example, the first
principle of inference: If `p` implies `q`, then `p` implies `q`. This holds
equally whether `p` and `q` be propositions or not; for if either is not a
proposition, “`p` implies `q`” becomes false, but does not cease to be a
proposition. In fact, in virtue of the definition of a proposition, our
principle states that “`p` implies `q`” is a propositional function, *i.e.* that
it is a proposition for all values of `p` and `q`. But if we apply the
principle of importation to this proposition, so as to obtain “‘`p` implies
`q`,’ together with `p`, implies `q`,” we have a formula which is only true
when `p` and `q` are propositions: in order to make it true universally, we
must preface it by the hypothesis “`p` implies `p` and `q` implies `q`.” In this
way, in many cases, if not in all, the restriction on the variability of the
variable can be removed; thus, in the assertion of the logical product of
a class of propositions, the formula “if `x` implies `x`, then ‘`x` is a `k`’
implies `x`” appears unobjectionable, and allows `x` to vary without restriction. Here the subordinate implications in the premiss and the conclusion
are material: only the principal implication is formal.

Returning now to “`x` is a man implies `x` is a mortal,” it is plain that
no restriction is required in order to insure our having a genuine
proposition, And it is plain that, although we *might* restrict the values of
`x` to men, and although this seems to be done in the proposition
“all men are mortal,” yet there is no reason, so far as the truth of our
proposition is concerned, why we should so restrict our `x`. Whether `x`
be a man or not, “`x` is a man” is always, when a constant is substituted
for `x`, a proposition implying, for that value of `x`, the proposition “`x` is
a mortal.” And unless we admit the hypothesis equally in the cases
where it is false, we shall find it impossible to deal satisfactorily with the
null-class or with null propositional functions. We must, therefore,
allow our `x`, wherever the truth of our formal implication is thereby
unimpaired, to take *all* values without exception; and where any
restriction on variability is required, the implication is not to be
regarded as formal until the said restriction has been removed by being
prefixed as hypothesis. (If `ψ``x` be a proposition whenever `x` satisfies `φ``x`,
where `φ``x` is a propositional function, and if `ψ``x`, whenever it is a
proposition, implies `χ``x`, then “`ψ``x` implies `χ``x`” is not a formal implication,
but “`φ``x` implies that `ψ``x` implies `χ``x`” is a formal implication.)

**42.** It is to be observed that “`x` is a man implies `x` is a mortal”
is not a relation of two propositional functions, but is itself a single
propositional function having the elegant property of being always
true. For “`x` is a man” is, as it stands, not a proposition at all,
and does not imply anything; and we must not first vary our `x` in
“`x` is a man,” and then independently vary it in “`x` is a mortal,”
for this would lead to the proposition that “everything is a man”
implies “everything is a mortal,” which, though true, is not what was
meant. This proposition would have to be expressed, if the language
of variables were retained, by two variables, as “`x` is a man implies
`y` is a mortal.” But this formula too is unsatisfactory, for its natural
meaning would be: “If anything is a man, then everything is a mortal.”
The point to be emphasized is, of course, that our `x`, though variable,
must be the same on both sides of the implication, and this requires
that we should not obtain our formal implication by first varying (say)
Socrates in “Socrates is a man,” and then in “Socrates is a mortal,”
but that we should start from the whole proposition “Socrates is a
man implies Socrates is a mortal,” and vary Socrates in this proposition
as a whole. Thus our formal implication asserts a class of implications,
not a single implication at all. We do not, in a word, have one implication
containing a variable, but rather a variable implication. We
have a class of implications, no one of which contains a variable, and
we assert that every member of this class is true. This is a first step
towards the analysis of the mathematical notion of the variable.

But, it may be asked, how comes it that Socrates may be varied
in the proposition “Socrates is a man implies Socrates is mortal”? In
virtue of the fact that true propositions are implied by all others, we
have “Socrates is a man implies Socrates is a philosopher”; but in this
proposition, alas, the variability of Socrates is sadly restricted. This
seems to show that formal implication involves something over and
above the relation of implication, and that some additional relation
must hold where a term can be varied. In the case in question, it is
natural to say that what is involved is the relation of inclusion between
the classes *men* and *mortals*—the very relation which was to be defined
and explained by our formal implication. But this view is too simple
to meet all cases, and is therefore not required in any case. A larger
number of cases, though still not all cases, can be dealt with by the
notion of what I shall call *assertions*. This notion must now be briefly
explained, leaving its critical discussion to Chapter vii.

**43.** It has always been customary to divide propositions into
subject and predicate; but this division has the defect of omitting the
verb. It is true that a graceful concession is sometimes made by loose
talk about the copula, but the verb deserves far more respect than is
thus paid to it. We may say, broadly, that every proposition may be
divided, some in only one way, some in several ways, into a term (the
subject) and something which is said about the subject, which something
I shall call the *assertion*. Thus “Socrates is a man” may be divided
into *Socrates* and `i``s` `a` `m``a``n`. The verb, which is the distinguishing mark
of propositions, remains with the assertion; but the assertion itself,
being robbed of its subject, is neither true nor false. In logical discussions,
the notion of assertion often occurs, but as the word *proposition*
is used for it, it does not obtain separate consideration. Consider, for
example, the best statement of the identity of indiscernibles: “If `x` and `y`
be any two diverse entities, some assertion holds of `x` which does not
hold of `y`.” But for the word *assertion*, which would ordinarily be
replaced by *proposition*, this statement is one which would commonly
pass unchallenged. Again, it might be said: “Socrates was a philosopher, and the same is true of Plato.” Such statements require the
analysis of a proposition into an assertion and a subject, in order that
there may be something identical which can be said to be affirmed of
two subjects.

**44.** We can now see how, where the analysis into subject and
assertion is legitimate, to distinguish implications in which there is a
term which can be varied from others in which this is not the case. Two
ways of making the distinction may be suggested, and we shall have to
decide between them. It may be said that there is a relation between
the two assertions “is a man” and “is a mortal,” in virtue of which,
when the one holds, so does the other. Or again, we may analyze the
whole proposition “Socrates is a man implies Socrates is a mortal” into
Socrates and an assertion about him, and say that the assertion in
question holds of all terms. Neither of these theories replaces the above
analysis of “`x` is a man implies `x` is a mortal” into a class of material
implications; but whichever of the two is true carries the analysis one
step further. The first theory suffers from the difficulty that it is
essential to the relation of assertions involved that both assertions
should be made of the *same* subject, though it is otherwise irrelevant
what subject we choose. The second theory appears objectionable on
the ground that the suggested analysis of “Socrates is a man implies
Socrates is a mortal” seems scarcely possible. The proposition in
question consists of two terms and a relation, the terms being “Socrates
is a man” and “Socrates is a mortal”; and it would seem that when a
relational proposition is analyzed into a subject and an assertion, the
subject must be one of the terms of the relation which is asserted. This
objection seems graver than that against the former view; I shall
therefore, at any rate for the present, adopt the former view, and regard
formal implication as derived from a relation between assertions.

We remarked above that the relation of inclusion between classes is
insufficient. This results from the irreducible nature of relational
propositions. Take *e.g.* “Socrates is married implies Socrates had a
father.” Here it is affirmed that because Socrates has one relation,
he must have another. Or better still, take “`A` is before `B` implies `B` is
after `A`.” This is a formal implication, in which the assertions are
(superficially at least) concerning different subjects; the only way to
avoid this is to say that both propositions have both `A` and `B` as
subjects, which, by the way, is quite different from saying that they
have the one subject “`A` and `B`.” Such instances make it plain that
the notion of a propositional function, and the notion of an assertion,
are more fundamental than the notion of *class*, and that the latter is
not adequate to explain all cases of formal implication. I shall not
enlarge upon this point now, as it will be abundantly illustrated in
subsequent portions of the present work.

It is important to realize that, according to the above analysis of
formal implication, the notion of `e``v``e``r``y` `t``e``r``m` is indefinable and ultimate.
A formal implication is one which holds of every term, and therefore
*every* cannot be explained by means of formal implication. If `a` and `b`
be classes, we can explain “every `a` is a `b`” by means of “`x` is an `a`
implies `x` is a `b`”; but the *every* which occurs here is a derivative and
subsequent notion, presupposing the notion of *every term*. It seems
to be the very essence of what may be called a *formal* truth, and of
formal reasoning generally, that some assertion is affirmed to hold of
every term; and unless the notion of *every term* is admitted, formal
truths are impossible.

**45.** The fundamental importance of formal implication is brought
out by the consideration that it is involved in all the rules of inference.
This shows that we cannot hope wholly to define it in terms of material
implication, but that some further element or elements must be involved.
We may observe, however, that, in a particular inference, the rule
according to which the inference proceeds is not required as a premiss.
This point has been emphasized by Mr Bradley*; it is closely connected
with the principle of dropping a true premiss, being again a respect
in which formalism breaks down. In order to apply a rule of inference,
it is formally necessary to have a premiss asserting that the present
case is an instance of the rule; we shall then need to affirm the rule by
which we can go from the rule to an instance, and also to affirm that here
we have an instance of this rule, and so on into an endless process.
The fact is, of course, that any implication warranted by a rule of
inference does actually hold, and is not merely implied by the rule.
This is simply an instance of the non-formal principle of dropping a
true premiss: if our rule implies a certain implication, the rule may be
dropped and the implication asserted. But it remains the case that the
fact that our rule does imply the said implication, if introduced at all,
must be simply perceived, and is not guaranteed by any formal deduction;
and often it is just as easy, and consequently just as legitimate, to perceive
immediately the implication in question as to perceive that it is implied
by one or more of the rules of inference.

To sum up our discussion of formal implication: a formal implication,
we said, is the affirmation of *every* material implication of a certain
class; and the class of material implications involved is, in simple cases,
the class of all propositions in which a given fixed assertion, made concerning a certain subject or subjects, is affirmed to imply another given
fixed assertion concerning the same subject or subjects. Where a formal
implication holds, we agreed to regard it, wherever possible, as due to
some relation between the assertions concerned. This theory raises many
formidable logical problems, and requires, for its defence, a thorough
analysis of the constituents of propositions. To this task we must now
address ourselves.

Notes

*^{[page 35]} *Mind*, N. S. Vol. iv, p. 278.

†^{[page 35]} Frege (*loc. cit.*) has a special symbol to denote assertion.

*^{[page 41]} *Logic*, Book II, Part I, Chap. ii (p. 227).

Proper Names, Adjectives, and Verbs.

**46.** In the present chapter, certain questions are to be discussed
belonging to what may be called philosophical grammar. The study
of grammar, in my opinion, is capable of throwing far more light on
philosophical questions than is commonly supposed by philosophers.
Although a grammatical distinction cannot be uncritically assumed to
correspond to a genuine philosophical difference, yet the one is *primâ facie* evidence of the other, and may often be most usefully employed
as a source of discovery. Moreover, it must be admitted, I think, that
every word occurring in a sentence must have *some* meaning: a perfectly
meaningless sound could not be employed in the more or less fixed
way in which language employs words. The correctness of our philosophical analysis of a proposition may therefore be usefully checked
by the exercise of assigning the meaning of each word in the sentence
expressing the proposition. On the whole, grammar seems to me to
bring us much nearer to a correct logic than the current opinions of
philosophers; and in what follows, grammar, though not our master,
will yet be taken as our guide*.

Of the parts of speech, three are specially important: substantives,
adjectives, and verbs. Among substantives, some are derived from
adjectives or verbs, as humanity from human, or sequence from *follows*.
(I am not speaking of an etymological derivation, but of a logical one.)
Others, such as proper names, or space, time, and matter, are not
derivative, but appear primarily as substantives. What we wish to
obtain is a classification, not of words, but of ideas; I shall therefore
call adjectives or predicates all notions which are capable of being such,
even in a form in which grammar would call them substantives. The
fact is, as we shall see, that *human* and *humanity* denote precisely
the same concept, these words being employed respectively according to
the kind of relation in which this concept stands to the other constituents
of a proposition in which it occurs. The distinction which we require
is not identical with the grammatical distinction between substantive
and adjective, since one single concept may, according to circumstances,
be either substantive or adjective: it is the distinction between proper
and general names that we require, or rather between the objects indicated by such names. In every proposition, as we saw in Chapter iii,
we may make an analysis into something asserted and something about
which the assertion is made. A proper name, when it occurs in a
proposition, is always, at least according to one of the possible ways
of analysis (where there are several), the subject that the proposition
or some subordinate constituent proposition is about, and not what is
said about the subject. Adjectives and verbs, on the other hand,
are capable of occurring in propositions in which they cannot be
regarded as subject, but only as parts of the assertion. Adjectives
are distinguished by capacity for *denoting*—a term which I intend
to use in a technical sense to be discussed in Chapter v. Verbs
are distinguished by a special kind of connection, exceedingly hard
to define, with truth and falsehood, in virtue of which they distinguish an asserted proposition from an unasserted one, *e.g.* “Caesar
died” from “the death of Caesar.” These distinctions must now be
amplified, and I shall begin with the distinction between general and
proper names.

**47.** Philosophy is familiar with a certain set of distinctions, all
more or less equivalent: I mean, the distinctions of subject and predicate, substance and attribute, substantive and adjective, *this* and
*what**. I wish now to point out briefly what appears to me to be the
truth concerning these cognate distinctions. The subject is important,
since the issues between monism and monadism, between idealism and
empiricism, and between those who maintain and those who deny that
all truth is concerned with what exists, all depend, in whole or in part,
upon the theory we adopt in regard to the present question. But the
subject is treated here only because it is essential to any doctrine of
number or of the nature of the variable. Its bearings on general
philosophy, important as they are, will be left wholly out of account.

Whatever may be an object of thought, or may occur in any true
or false proposition, or can be counted as *one*, I call a *term*. This,
then, is the widest word in the philosophical vocabulary. I shall use
as synonymous with it the words unit, individual, and entity. The
first two emphasize the fact that every term is *one*, while the third is
derived from the fact that every term has being, *i.e.* *is* in some sense.
A man, a moment, a number, a class, a relation, a chimaera, or anything
else that can be mentioned, is sure to be a term; and to deny that such
and such a thing is a term must always be false.

It might perhaps be thought that a word of such extreme generality
could not be of any great use. Such a view, however, owing to certain
wide-spread philosophical doctrines, would be erroneous. A term is,
in fact, possessed of all the properties commonly assigned to substances
or substantives. Every term, to begin with, is a logical subject: it is,
for example, the subject of the proposition that itself is one. Again
every term is immutable and indestructible. What a term is, it is, and
no change can be conceived in it which would not destroy its identity
and make it another term*. Another mark which belongs to terms
is numerical identity with themselves and numerical diversity from all
other terms†. Numerical identity and diversity are the source of unity
and plurality; and thus the admission of many terms destroys monism.
And it seems undeniable that every constituent of every proposition can
be counted as one, and that no proposition contains less than two
constituents. *Term* is, therefore, a useful word, since it marks dissent
from various philosophies, as well as because, in many statements, we
wish to speak of *any* term or *some* term.

**48.** Among terms, it is possible to distinguish two kinds, which
I shall call respectively *things* and *concepts*. The former are the terms
indicated by proper names, the latter those indicated by all other words.
Here proper names are to be understood in a somewhat wider sense than
is usual, and things also are to be understood as embracing all particular points and instants, and many other entities not commonly called
things. Among concepts, again, two kinds at least must be distinguished,
namely those indicated by adjectives and those indicated by verbs. The
former kind will often be called predicates or class-concepts; the latter
are always or almost always relations. (In intransitive verbs, the notion
expressed by the verb is complex, and usually asserts a definite relation
to an indefinite relatum, as in “Smith breathes.”)

In a large class of propositions, we agreed, it is possible, in one or
more ways, to distinguish a subject and an assertion about the subject.
The assertion must always contain a verb, but except in this respect,
assertions appear to have no universal properties. In a relational
proposition, say “`A` is greater than `B`,” we may regard `A` as the subject,
and “is greater than `B`” as the assertion, or `B` as the subject and “`A` is
greater than” as the assertion. There are thus, in the case proposed,
two ways of analyzing the proposition into subject and assertion.
Where a relation has more than two terms, as in “`A` is here now,‡”
there will be more than two ways of making the analysis. But in
some propositions, there is only a single way: these are the subject-predicate propositions, such as “Socrates is human.” The proposition
“humanity belongs to Socrates,” which is equivalent to “Socrates is
human,” is an assertion about humanity; but it is a distinct proposition. In “Socrates is human,” the notion expressed by *human* occurs
in a different way from that in which it occurs when it is called
*humanity*, the difference being that in the latter case, but not in the
former, the proposition is *about* this notion. This indicates that
humanity is a concept, not a thing. I shall speak of the *terms* of a
proposition as those terms, however numerous, which occur in a proposition and may be regarded as subjects about which the proposition is.
It is a characteristic of the terms of a proposition that any one of
them may be replaced by any other entity without our ceasing to have
a proposition. Thus we shall say that “Socrates is human” is a
proposition having only one term; of the remaining components of
the proposition, one is the verb, the other is a *predicate*. With the sense
which *is* has in this proposition, we no longer have a proposition at all
if we replace *human* by something other than a predicate. Predicates,
then, are concepts, other than verbs, which occur in propositions having
only one term or subject. Socrates is a thing, because Socrates can
never occur otherwise than as term in a proposition: Socrates is not
capable of that curious twofold use which is involved in *human* and
*humanity*. Points, instants, bits of matter, particular states of mind,
and particular existents generally, are things in the above sense, and
so are many terms which do not exist, for example, the points in a
non-Euclidean space and the pseudo-existents of a novel. All classes,
it would seem, as numbers, men, spaces, etc., when taken as single terms,
are things; but this is a point for Chapter vi.

Predicates are distinguished from other terms by a number of very
interesting properties, chief among which is their connection with what
I shall call *denoting*. One predicate always gives rise to a host of
cognate notions: thus in addition to *human* and *humanity*, which
only differ grammatically, we have *man*, *a man*, *some man*, *any man*,
*every man*, *all men**, all of which appear to be genuinely distinct one
from another. The study of these various notions is absolutely vital
to any philosophy of mathematics; and it is on account of them that
the theory of predicates is important.

**49.** It might be thought that a distinction ought to be made
between a concept as such and a concept used as a term, between,
*e.g.*, such pairs as *is* and *being*, *human* and *humanity*, *one* in such a
proposition as “this is one” and 1 in “1 is a number.” But inextricable
difficulties will envelop us if we allow such a view. There is,
of course, a grammatical difference, and this corresponds to a difference
as regards relations. In the first case, the concept in question is used
as a concept, that is, it is actually predicated of a term or asserted to
relate two or more terms; while in the second case, the concept is
itself said to have a predicate or a relation. There is, therefore,
no difficulty in accounting for the grammatical difference. But what
I wish to urge is, that the difference lies solely in external relations,
and not in the intrinsic nature of the terms. For suppose that *one*
as adjective differed from 1 as term. In this statement, *one* as
adjective has been made into a term; hence either it has become
1, in which case the supposition is self-contradictory; or there is some
other difference between *one* and 1 in addition to the fact that the
first denotes a concept not a term while the second denotes a concept
which is a term. But in this latter hypothesis, there must be propositions concerning *one* as term, and we shall still have to maintain
propositions concerning *one* as adjective as opposed to *one* as term;
yet all such propositions must be false, since a proposition about *one*
as adjective makes *one* the subject, and is therefore really about *one*
as term. In short, if there were any adjectives which could not be
made into substantives without change of meaning, all propositions
concerning such adjectives (since they would necessarily turn them into
substantives) would be false, and so would the proposition that all
such propositions are false, since this itself turns the adjectives into
substantives. But this state of things is self-contradictory.

The above argument proves that we were right in saying that terms
embrace everything that can occur in a proposition, with the possible
exception of complexes of terms of the kind denoted by *any* and cognate
words*. For if `A` occurs in a proposition, then, in this statement,
`A` is the subject; and we have just seen that, if `A` is ever not the
subject, it is exactly and numerically the same `A` which is not subject
in one proposition and is subject in another. Thus the theory that
there are adjectives or attributes or ideal things, or whatever they may
be called, which are in some way less substantial, less self-subsistent,
less self-identical, than true substantives, appears to be wholly erroneous,
and to be easily reduced to a contradiction. Terms which are concepts
differ from those which are not, not in respect of self-subsistence, but
in virtue of the fact that, in certain true or false propositions, they
occur in a manner which is different in an indefinable way from the
manner in which subjects or terms of relations occur.

**50.** Two concepts have, in addition to the numerical diversity
which belongs to them as terms, another special kind of diversity
which may be called conceptual. This may be characterized by the
fact that two propositions in which the concepts occur otherwise than
as terms, even if, in all other respects, the two propositions are identical,
yet differ in virtue of the fact that the concepts which occur in
them are conceptually diverse. Conceptual diversity implies numerical
diversity, but the converse implication does not hold, since not all
terms are concepts. Numerical diversity, as its name implies, is the
source of plurality, and conceptual diversity is less important to
mathematics. But the whole possibility of making different assertions
about a given term or set of terms depends upon conceptual diversity,
which is therefore fundamental in general logic.

**51.** It is interesting and not unimportant to examine very briefly
the connection of the above doctrine of adjectives with certain traditional
views on the nature of propositions. It is customary to regard all
propositions as having a subject and a predicate, *i.e.* as having an
immediate *this*, and a general concept attached to it by way of description.
This is, of course, an account of the theory in question which will strike
its adherents as extremely crude; but it will serve for a general indication
of the view to be discussed. This doctrine develops by internal logical
necessity into the theory of Mr Bradley’s Logic, that all words stand for
ideas having what he calls *meaning*, and that in every judgment there
is a something, the true subject of the judgment, which is not an idea
and does not have meaning. To have meaning, it seems to me, is a
notion confusedly compounded of logical and psychological elements.
*Words* all have meaning, in the simple sense that they are symbols
which stand for something other than themselves. But a proposition,
unless it happens to be linguistic, does not itself contain words: it
contains the entities indicated by words. Thus meaning, in the sense
in which words have meaning, is irrelevant to logic. But such concepts
as *a man* have meaning in another sense: they are, so to speak, symbolic
in their own logical nature, because they have the property which I call
*denoting*. That is to say, when *a man* occurs in a proposition (*e.g.*
“I met a man in the street”), the proposition is not about the concept
*a man*, but about something quite different, some actual biped denoted
by the concept. Thus concepts of this kind have meaning in a non-psychological sense. And in this sense, when we say “this is a man,”
we are making a proposition in which a concept is in some sense
attached to what is not a concept. But when meaning is thus understood, the entity indicated by *John* does not have meaning, as Mr Bradley
contends*; and even among concepts, it is only those that denote that
have meaning. The confusion is largely due, I believe, to the notion
that *words* occur in propositions, which in turn is due to the notion that
propositions are essentially mental and are to be identified with cognitions.
But these topics of general philosophy must be pursued no further in
this work.

**52.** It remains to discuss the verb, and to find marks by which
it is distinguished from the adjective. In regard to verbs also, there is
a twofold grammatical form corresponding to a difference in merely
external relations. There is the verb in the form which it has as verb
(the various inflexions of this form may be left out of account), and
there is the verbal noun, indicated by the infinitive or (in English) the
present participle. The distinction is that between “Felton killed
Buckingham” and “Killing no murder.” By analyzing this difference,
the nature and function of the verb will appear.

It is plain, to begin with, that the concept which occurs in the verbal
noun is the very same as that which occurs as verb. This results from
the previous argument, that every constituent of every proposition must,
on pain of self-contradiction, be capable of being made a logical subject.
If we say “*kills* does not mean the same as to *to kill*,” we have already
made *kills* a subject, and we cannot say that the concept expressed by
the word *kills* cannot be made a subject. Thus the very verb which
occurs as verb can occur also as subject. The question is: What logical
difference is expressed by the difference of grammatical form? And it
is plain that the difference must be one in external relations. But
in regard to verbs, there is a further point. By transforming the verb,
as it occurs in a proposition, into a verbal noun, the whole proposition
can be turned into a single logical subject, no longer asserted, and no
longer containing in itself truth or falsehood. But here too, there seems
to be no possibility of maintaining that the logical subject which results
is a different entity from the proposition. “Caesar died” and “the
death of Caesar” will illustrate this point. If we ask: What is asserted
in the proposition “Caesar died”? the answer must be “the death of
Caesar is asserted.” In that case, it would seem, it is the death of Caesar
which is true or false; and yet neither truth nor falsity belongs to
a mere logical subject. The answer here seems to be that the death of
Caesar has an external relation to truth or falsehood (as the case may
be), whereas “Caesar died” in some way or other contains its own truth
or falsehood as an element. But if this is the correct analysis, it is
difficult to see how “Caesar died” differs from “the truth of Caesar’s
death” in the case where it is true, or “the falsehood of Caesar’s death”
in the other case. Yet it is quite plain that the latter, at any rate, is
never equivalent to “Caesar died.” There appears to be an ultimate
notion of assertion, given by the verb, which is lost as soon as we
substitute a verbal noun, and is lost when the proposition in question
is made the subject of some other proposition. This does not depend
upon grammatical form; for if I say “*Caesar died* is a proposition,”
I do not assert that Caesar did die, and an element which is present in
“Caesar died” has disappeared. Thus the contradiction which was to
have been avoided, of an entity which cannot be made a logical subject,
appears to have here become inevitable. This difficulty, which seems to
be inherent in the very nature of truth and falsehood, is one with which
I do not know how to deal satisfactorily. The most obvious course
would be to say that the difference between an asserted and an unasserted
proposition is not logical, but psychological. In the sense in which
false propositions may be asserted, this is doubtless true. But there
is another sense of assertion, very difficult to bring clearly before the
mind, and yet quite undeniable, in which only true propositions are
asserted. True and false propositions alike are in some sense entities,
and are in some sense capable of being logical subjects; but when
a proposition happens to be true, it has a further quality, over and
above that which it shares with false propositions, and it is this further
quality which is what I mean by assertion in a logical as opposed to
a psychological sense. The nature of truth, however, belongs no more
to the principles of mathematics than to the principles of everything
else. I therefore leave this question to the logicians with the above
brief indication of a difficulty.

**53.** It may be asked whether everything that, in the logical sense
we are concerned with, is a verb, expresses a relation or not. It seems
plain that, if we were right in holding that “Socrates is human” is a
proposition having only one term, the *is* in this proposition cannot
express a relation in the ordinary sense. In fact, subject-predicate
propositions are distinguished by just this non-relational character.
Nevertheless, a relation between Socrates and humanity is certainly
*implied*, and it is very difficult to conceive the proposition as expressing
no relation at all. We may perhaps say that it is a relation, although
it is distinguished from other relations in that it does not permit itself
to be regarded as an assertion concerning either of its terms indifferently,
but only as an assertion concerning the referent. A similar remark may
apply to the proposition “`A` is,” which holds of every term without
exception. The *is* here is quite different from the *is* in “Socrates is
human”; it may be regarded as complex, and as really predicating
Being of `A`. In this way, the true logical verb in a proposition may be
always regarded as asserting a relation. But it is so hard to know
exactly what is meant by *relation* that the whole question is in danger
of becoming purely verbal.

**54.** The twofold nature of the verb, as actual verb and as verbal
noun, may be expressed, if all verbs are held to be relations, as the
difference between a relation in itself and a relation actually relating.
Consider, for example, the proposition “`A` differs from `B`.” The
constituents of this proposition, if we analyze it, appear to be only `A`,
difference, `B`. Yet these constituents, thus placed side by side, do not
reconstitute the proposition. The difference which occurs in the
proposition actually relates `A` and `B`, whereas the difference after
analysis is a notion which has no connection with `A` and `B`. It may
be said that we ought, in the analysis, to mention the relations which
difference has to `A` and `B`, relations which are expressed by *is* and *from*
when we say “`A` is different from `B`.” These relations consist in the
fact that `A` is referent and `B` relatum with respect to difference. But
“`A`, referent, difference, relatum, `B`” is still merely a list of terms, not
a proposition. A proposition, in fact, is essentially a unity, and when
analysis has destroyed the unity, no enumeration of constituents will
restore the proposition. The verb, when used as a verb, embodies the
unity of the proposition, and is thus distinguishable from the verb considered as a term, though I do not know how to give a clear account of
the precise nature of the distinction.

**55.** It may be doubted whether the general concept *difference*
occurs at all in the proposition “`A` differs from `B`,” or whether there is
not rather a specific difference of `A` and `B`, and another specific difference
of `C` and `D`, which are respectively affirmed in “`A` differs from `B`” and
“`C` differs from `D`.” In this way, *difference* becomes a class-concept of
which there are as many instances as there are pairs of different terms;
and the instances may be said, in Platonic phrase, to partake of the
nature of difference. As this point is quite vital in the theory of
relations, it may be well to dwell upon it. And first of all, I must
point out that in “`A` differs from `B`” I intend to consider the bare
numerical difference in virtue of which they are two, not difference in
this or that respect.

Let us first try the hypothesis that a difference is *a* complex notion,
compounded of difference together with some special quality distinguishing
a particular difference from every other particular difference. So far as
the relation of difference itself is concerned, we are to suppose that
no distinction can be made between different cases; but there are to be
different associated qualities in different cases. But since cases are
distinguished by their terms, the quality must be primarily associated
with the terms, not with difference. If the quality be not a relation, it
can have no special connection with the difference of `A` and `B`, which it
was to render distinguishable from bare difference, and if it fails in this
it becomes irrelevant. On the other hand, if it be a new relation
between `A` and `B`, over and above difference, we shall have to hold that
any two terms have two relations, difference and a specific difference, the
latter not holding between any other pair of terms. This view is a
combination of two others, of which the first holds that the abstract
general relation of difference itself holds between `A` and `B`, while the
second holds that when two terms differ they have, corresponding to
this fact, a specific relation of difference, unique and unanalyzable and
not shared by any other pair of terms. Either of these views may be
held with either the denial or the affirmation of the other. Let us see
what is to be said for and against them.

Against the notion of specific differences, it may be urged that, if differences differ, their differences from each other must also differ, and thus we are led into an endless process. Those who object to endless processes will see in this a proof that differences do not differ. But in the present work, it will be maintained that there are no contradictions peculiar to the notion of infinity, and that an endless process is not to be objected to unless it arises in the analysis of the actual meaning of a proposition. In the present case, the process is one of implications, not one of analysis; it must therefore be regarded as harmless.

Against the notion that the abstract relation of difference holds
between `A` and `B`, we have the argument derived from the analysis of
“`A` differs from `B`,” which gave rise to the present discussion. It is to
be observed that the hypothesis which combines the general and the
specific difference must suppose that there are two distinct propositions,
the one affirming the general, the other the specific difference. Thus if
there cannot be a general difference between `A` and `B`, this mediating
hypothesis is also impossible. And we saw that the attempt to avoid
the failure of analysis by including in the meaning of “`A` differs from `B`”
the relations of difference to `A` and `B` was vain. This attempt, in fact,
leads to an endless process of the inadmissible kind; for we shall have to
include the relations of the said relations to `A` and `B` and difference, and
so on, and in this continually increasing complexity we are supposed
to be only analyzing the *meaning* of our original proposition. This
argument establishes a point of very great importance, namely, that
when a relation holds between two terms, the relations of the relation to
the terms, and of these relations to the relation and the terms, and so
on *ad infinitum*, though all implied by the proposition affirming the
original relation, form no part of the *meaning* of this proposition.

But the above argument does not suffice to prove that the relation
of `A` to `B` cannot be abstract difference: it remains tenable that, as
was suggested to begin with, the true solution lies in regarding every
proposition as having a kind of unity which analysis cannot preserve,
and which is lost even though it be mentioned by analysis as an element
in the proposition. This view has doubtless its own difficulties, but the
view that no two pairs of terms can have the same relation both contains
difficulties of its own and fails to solve the difficulty for the sake of which
it was invented. For, even if the difference of `A` and `B` be absolutely
peculiar to `A` and `B`, still the three terms `A`, `B`, difference of `A` from `B`,
do not reconstitute the proposition “`A` differs from `B`,” any more than
`A` and `B` and difference did. And it seems plain that, even if differences
did differ, they would still have to have something in common. But
the most general way in which two terms can have something in common
is by both having a given relation to a given term. Hence if no two
pairs of terms can have the same relation, it follows that no two terms
can have anything in common, and hence different differences will not
be in any definable sense *instances* of difference*. I conclude, then, that
the relation affirmed between `A` and `B` in the proposition “`A` differs
from `B`” is the general relation of difference, and is precisely and
numerically the same as the relation affirmed between `C` and `D` in
“`C` differs from `D`.” And this doctrine must be held, for the same
reasons, to be true of all other relations; relations do not have instances,
but are strictly the same in all propositions in which they occur.

We may now sum up the main points elicited in our discussion of the verb. The verb, we saw, is a concept which, like the adjective, may occur in a proposition without being one of the terms of the proposition, though it may also be made into a logical subject. One verb, and one only, must occur as verb in every proposition; but every proposition, by turning its verb into a verbal noun, can be changed into a single logical subject, of a kind which I shall call in future a propositional concept. Every verb, in the logical sense of the word, may be regarded as a relation; when it occurs as verb, it actually relates, but when it occurs as verbal noun it is the bare relation considered independently of the terms which it relates. Verbs do not, like adjectives, have instances, but are identical in all the cases of their occurrence. Owing to the way in which the verb actually relates the terms of a proposition, every proposition has a unity which renders it distinct from the sum of its constituents. All these points lead to logical problems, which, in a treatise on logic, would deserve to be fully and thoroughly discussed.

Having now given a general sketch of the nature of verbs and adjectives, I shall proceed, in the next two chapters, to discussions arising out of the consideration of adjectives, and in Chapter vii to topics connected with verbs. Broadly speaking, classes are connected with adjectives, while propositional functions involve verbs. It is for this reason that it has been necessary to deal at such length with a subject which might seem, at first sight, to be somewhat remote from the principles of mathematics.

Notes

*^{[page 42]} The excellence of grammar as a guide is proportional to the paucity of
inflexions, *i.e.* to the degree of analysis effected by the language considered.

*^{[page 43]} This last pair of terms is due to Mr Bradley.

*^{[page 44]} The notion of a term here set forth is a modification of Mr G. E. Moore’s notion of a *concept* in his article “On the Nature of Judgment,” *Mind*, N. S. No. 30, from which notion, however, it differs in some important respects.

†^{[page 44]} On identity, see Mr G. E. Moore’s article in the *Proceedings of the Aristotelian Society*, 1900–1901.

‡^{[page 44]} This proposition means “`A` is in this place at this time.” It will he shown in Part VII that the relation expressed is not reducible to a two-term relation.

*^{[page 45]} I use *all men* as collective *i.e.* as nearly synonymous with the *human race*, but differing therefrom by being many and not one. I shall always use *all* collectively, confining myself to *every* for the distributive sense. Thus I shall say “every man is mortal,” not “all men are mortal.”

*^{[page 46]} See the next chapter.

*^{[page 47]} *Logic*, Book I, Chap. i, §§17, 18 (pp. 58–60).

*^{[page 51]} The above argument appears to prove that Mr Moore’s theory of universals with numerically diverse instances in his paper on Identity (*Proceedings of the Aristotelian Society*, 1900–1901) must not be applied to all concepts. The relation of an instance to its universal, at any rate, must be actually and numerically the same in all cases where it occurs.

Denoting.

**56.** The notion of denoting, like most of the notions of logic, has
been obscured hitherto by an undue admixture of psychology. There is
a sense in which *we* denote, when we point or describe, or employ words
as symbols for concepts; this, however, is not the sense that I wish to
discuss. But the fact that description is possible—that we are able, by
the employment of concepts, to designate a thing which is not a concept
—is due to a logical relation between some concepts and some terms, in
virtue of which such concepts inherently and logically *denote* such terms.
It is this sense of denoting which is here in question. This notion lies
at the bottom (I think) of all theories of substance, of the subject-predicate logic, and of the opposition between things and ideas,
discursive thought and immediate perception. These various developments, in the main, appear to me mistaken, while the fundamental fact
itself, out of which they have grown, is hardly ever discussed in its
logical purity.

A concept *denotes* when, if it occurs in a proposition, the proposition
is not about the concept, but about a term connected in a certain
peculiar way with the concept. If I say “I met a man,” the proposition
is not about *a man*: this is a concept which does not walk the streets,
but lives in the shadowy limbo of the logic-books. What I met was a
thing, not a concept, an actual man with a tailor and a bank-account or
a public-house and a drunken wife. Again, the proposition “any finite
number is odd or even” is plainly true; yet the *concept* “any finite
number” is neither odd nor even. It is only particular numbers that are
odd or even; there is not, in addition to these, another entity, *any number*, which is either odd or even, and if there were, it is plain that it
could not be odd and could not be even. Of the concept “any number,”
almost all the propositions that contain the *phrase* “any number” are
false. If we wish to speak of the concept, we have to indicate the fact by
italics or inverted commas. People often assert that man is mortal;
but what is mortal will die, and yet we should be surprised to find in the
“Times” such a notice as the following: “Died at his residence of
Camelot, Gladstone Road, Upper Tooting, on the 18th of June 19—,
Man, eldest son of Death and Sin.” *Man*, in fact, does not die; hence
if “man is mortal” were, as it appears to be, a proposition about *man*,
it would be simply false. The fact is, the proposition is about men;
and here again, it is not about the concept *men*, but about what this
concept denotes. The whole theory of definition, of identity, of classes,
of symbolism, and of the variable is wrapped up in the theory of
denoting. The notion is a fundamental notion of logic, and, in spite
of its difficulties, it is quite essential to be as clear about it as possible.

**57.** The notion of denoting may be obtained by a kind of logical
genesis from subject-predicate propositions, upon which it seems more or
less dependent. The simplest of propositions are those in which one
predicate occurs otherwise than as a term, and there is only one term of
which the predicate in question is asserted. Such propositions may be
called subject-predicate propositions. Instances are: `A` is, `A` is one,
`A` is human. Concepts which are predicates might also be called class-concepts, because they give rise to classes, but we shall find it necessary
to distinguish between the words *predicate* and *class-concept*. Propositions
of the subject -predicate type always imply and are implied by other propositions of the type which asserts that an individual belongs to a class.
Thus the above instances are equivalent to: `A` is an entity, `A` is a unit,
`A` is a man. These new propositions are not identical with the previous
ones, since they have an entirely different form. To begin with, *is* is now
the only concept not used as a term. *A man*, we shall find, is neither
a concept nor a term, but a certain kind of combination of certain terms,
namely of those which are human. And the relation of Socrates to
`a` `m``a``n` is quite different from his relation to humanity; indeed “Socrates
is human” must be held, if the above view is correct, to be not, in the
most usual sense, a judgment of relation between Socrates and humanity,
since this view would make *human* occur as term in “Socrates is human.”
It is, of course, undeniable that a relation to humanity is implied by
“Socrates is human,” namely the relation expressed by “Socrates has
humanity”; and this relation conversely implies the subject-predicate
proposition. But the two propositions can be clearly distinguished, and
it is important to the theory of classes that this should be done. Thus
we have, in the case of every predicate, three types of propositions
which imply one another, namely, “Socrates is human,” “Socrates has
humanity,” and “Socrates is a man.” The first contains a term and
a predicate, the second two terms and a relation (the second term being
identical with the predicate of the first proposition)*, while the third
contains a term, a relation, and what I shall call a disjunction (a term
which will be explained shortly)†. The class-concept differs little, if at
all, from the predicate, while the class, as opposed to the class-concept, is
the sum or conjunction of all the terms which have the given predicate.
The relation which occurs in the second type (Socrates has humanity) is
characterized completely by the fact that it implies and is implied by a
proposition with only one term, in which the other term of the relation
has become a predicate. A class is a certain combination of terms, a
class-concept is closely akin to a predicate, and the terms whose combination forms the class are determined by the class-concept. Predicates
are, in a certain sense, the simplest type of concepts, since they occur in
the simplest type of proposition.

**58.** There is, connected with every predicate, a great variety of
closely allied concepts, which, in so far as they are distinct, it is
important to distinguish. Starting, for example, with *human*, we have
man, men, all men, every man, any man, the human race, of which all
except the first are twofold, a denoting concept and an object denoted;
we have also, less closely analogous, the notions “a man” and “some
man,” which again denote objects* other than themselves. This vast
apparatus connected with every predicate must be borne in mind, and
an endeavour must be made to give an analysis of all the above notions.
But for the present, it is the property of denoting, rather than the
various denoting concepts, that we are concerned with.

The combination of concepts as such to form new concepts, of greater
complexity than their constituents, is a subject upon which writers on
logic have said many things. But the combination of terms as such,
to form what by analogy may be called complex terms, is a subject
upon which logicians, old and new, give us only the scantiest discussion.
Nevertheless, the subject is of vital importance to the philosophy of
mathematics, since the nature both of number and of the variable turns
upon just this point. Six words, of constant occurrence in daily life,
are also characteristic of mathematics: these are the words *all*, *every*,
*any*, *a*, *some* and *the*. For correctness of reasoning, it is essential that
these words should be sharply distinguished one from another; but
the subject bristles with difficulties, and is almost wholly neglected by
logicians†.

It is plain, to begin with, that a phrase containing one of the above
six words always denotes. It will be convenient, for the present
discussion, to distinguish a class-concept from a predicate: I shall call
*human* a predicate, and *man* a class-concept, though the distinction is
perhaps only verbal. The characteristic of a class-concept, as distinguished from terms in general, is that “`x` is a `u`” is a propositional
function when, and only when, `u` is a class-concept. It must be held that
when `u` is not a class-concept, we do not have a false proposition, but
simply no proposition at all, whatever value we may give to `x`. This
enables us to distinguish a class-concept belonging to the null-class, for
which all propositions of the above form are false, from a term which is
not a class-concept at all, for which there are no propositions of the
above form. Also it makes it plain that a class-concept is not a term
in the proposition “`x` is a `u`,” for `u` has a restricted variability if the
formula is to remain a proposition. A denoting phrase, we may now say,
consists always of a class-concept preceded by one of the above six words
or some synonym of one of them.

**59.** The question which first meets us in regard to denoting is
this: Is there one way of denoting six different kinds of objects, or are
the ways of denoting different? And in the latter case, is the object
denoted the same in all six cases, or does the object differ as well as the
way of denoting it? In order to answer this question, it will be first
necessary to explain the differences between the six words in question.
Here it will be convenient to omit the word *the* to begin with, since this
word is in a different position from the others, and is liable to limitations
from which they are exempt.

In cases where the class defined by a class-concept has only a finite
number of terms, it is possible to omit the class-concept wholly, and
indicate the various objects denoted by enumerating the terms and
connecting them by means of *and* or *or* as the case may be. It will
help to isolate a part of our problem if we first consider this case,
although the lack of subtlety in language renders it difficult to grasp the
difference between objects indicated by the same form of words.

Let us begin by considering two terms only, say Brown and Jones.
The objects denoted by *all*, *every*, *any*, *a* and *some** are respectively
involved in the following five propositions. (1) Brown and Jones are
two of Miss Smith’s suitors; (2) Brown and Jones are paying court to
Miss Smith; (3) if it was Brown or Jones you met, it was a very ardent
lover; (4) if it was one of Miss Smith’s suitors, it must have been
Brown or Jones; (5) Miss Smith will marry Brown or Jones. Although
only two forms of words, *Brown and Jones* and *Brown or Jones*, are
involved in these propositions, I maintain that five different combinations
are involved. The distinctions, some of which are rather subtle, may be
brought out by the following considerations. In the first proposition, it
is Brown *and* Jones who are two, and this is not true of either separately;
nevertheless it is not the whole composed of Brown and Jones which is
two, for this is only one. The two are a genuine combination of Brown
with Jones, the kind of combination which, as we shall see in the next
chapter, is characteristic of classes. In the second proposition, on the
contrary, what is asserted is true of Brown and Jones severally; the
proposition is equivalent to, though not (I think) identical with, “Brown
is paying court to Miss Smith and Jones is paying court to Miss Smith.”
Thus the combination indicated by *and* is not the same here as in the
first case: the first case concerned *all* of them collectively, while the
second concerns *all* distributively, *i.e.* each or every one of them. For
the sake of distinction, we may call the first a *numerical* conjunction,
since it gives rise to number, the second a *propositional* conjunction,
since the proposition in which it occurs is equivalent to a conjunction of
propositions. (It should be observed that the conjunction of propositions in question is of a wholly different kind from any of the combinations we are considering, being in fact of the kind which is called
the logical product. The propositions are combined *quâ* propositions,
not *quâ* terms.)

The third proposition gives the kind of conjunction by which *any* is
defined. There is some difficulty about this notion, which seems half-way
between a conjunction and a disjunction. This notion may be further
explained as follows. Let `a` and `b` be two different propositions,
each of which implies a third proposition `c`. Then the disjunction
“`a` or `b`” implies `c`. Now let `a` and `b` be propositions assigning the
same predicate to two different subjects, then there is a combination
of the two subjects to which the given predicate may be assigned so
that the resulting proposition is equivalent to the disjunction “`a` or `b`.”
Thus suppose we have “if you met Brown, you met a very ardent lover,”
and “if you met Jones, you met a very ardent lover.” Hence we infer
“if you met Brown or if you met Jones, you met a very ardent lover,”
and we regard this as equivalent to “if you met Brown or Jones, etc.”
The combination of Brown and Jones here indicated is the same as that
indicated by *either* of them. It differs from a disjunction by the fact
that it implies and is implied by a statement concerning *both*; but in
some more complicated instances, this mutual implication fails. The
method of combination is, in fact, different from that indicated by *both*,
and is also different from both forms of disjunction. I shall call it the
*variable* conjunction. The first form of disjunction is given by (4): this
is the form which I shall denote by *a* suitor. Here, although it must
have been Brown or Jones, it is not true that it must have been Brown
nor yet that it must have been Jones. Thus the proposition is not
equivalent to the disjunction of propositions “it must have been Brown
or it must have been Jones.” The proposition, in fact, is not capable of
statement either as a disjunction or as a conjunction of propositions,
except in the very roundabout form: “if it was not Brown, it was
Jones, and if it was not Jones, it was Brown,” a form which rapidly
becomes intolerable when the number of terms is increased beyond two,
and becomes theoretically inadmissible when the number of terms is
infinite. Thus this form of disjunction denotes a variable term, that
is, whichever of the two terms we fix upon, it does not denote this term,
and yet it does denote one or other of them. This form accordingly I
shall call the *variable* disjunction. Finally, the second form of disjunction
is given by (5). This is what I shall call the *constant* disjunction, since
here either Brown is denoted, or Jones is denoted, but the alternative
is undecided. That is to say, our proposition is now equivalent to a
disjunction of propositions, namely “Miss Smith will marry Brown, or
she will marry Jones.” She will marry *some* one of the two, and the
disjunction denotes a particular one of them, though it may denote
either particular one. Thus all the five combinations are distinct.

It is to be observed that these five combinations yield neither terms
nor concepts, but strictly and only combinations of terms. The first
yields many terms, while the others yield something absolutely peculiar,
which is neither one nor many. The combinations are combinations of
terms, effected without the use of relations. Corresponding to each
combination there is, at least if the terms combined form a class, a
perfectly definite concept, which *denotes* the various terms of the combination combined in the specified manner. To explain this, let us repeat
our distinctions in a case where the terms to be combined are not
enumerated, as above, but are defined as the terms of a certain class.

**60.** When a class-concept *a* is given, it must be held that the
various terms belonging to the class are also given. That is to say, any
term being proposed, it can be decided whether or not it belongs to the
class. In this way, a collection of terms can be given otherwise than by
enumeration. Whether a collection can be given otherwise than by
enumeration or by a class-concept, is a question which, for the present,
I leave undetermined. But the possibility of giving a collection by a
class-concept is highly important, since it enables us to deal with infinite
collections, as we shall see in Part V. For the present, I wish to examine
the meaning of such phrases as *all a*’s, *every a*, *any a*, *an a*, and *some a*.
*All a*’s, to begin with, denotes a numerical conjunction; it is definite as
soon as `a` is given. The concept `a``l``l` `a`’s is a perfectly definite single
concept, which denotes the terms of `a` taken all together. The terms
so taken have a number, which may thus be regarded, if we choose, as
a property of the class-concept, since it is determinate for any given
class-concept. `E``v``e``r``y` `a`, on the contrary, though it still denotes all the
`a`’s, denotes them in a different way, *i.e.* severally instead of collectively.
*Any a* denotes only one `a`, but it is wholly irrelevant which it denotes,
and what is said will be equally true whichever it may be. Moreover,
*any a* denotes a variable `a`, that is, whatever particular `a` we may fasten
upon, it is certain that *any a* does not denote that one; and yet of that
one any proposition is true which is true of any `a`. *An a* denotes a
variable disjunction: that is to say, a proposition which holds of *an a*
may be false concerning each particular `a`, so that it is not reducible to
a disjunction of propositions. For example, a point lies between any
point and any other point; but it would not be true of any one
particular point that it lay between any point and any other point,
since there would be many pairs of points between which it did not lie.
This brings us finally to *some a*, the constant disjunction. This denotes
just one term of the class `a`, but the term it denotes may be any term
of the class. Thus “some moment does not follow any moment” would
mean that there was a first moment in time, while “a moment precedes
any moment” means the exact opposite, namely, that every moment has
predecessors.

**61.** In the case of a class `a` which has a finite number of terms—say `a`_{1}, `a`_{2}, `a`_{3}, … `a`_{n}, we can illustrate these various notions as follows:

(1) *All a*’s denotes `a`_{1} and `a`_{2} and … and `a`_{n}.

(2) *Every a* denotes `a`_{1} and denotes `a`_{2} and … and denotes `a`_{n}.

(3) *Any a* denotes `a`_{1} or `a`_{2} or … or `a`_{n}, where *or* has the meaning
that it is irrelevant which we take.

(4) *An a* denotes `a`_{1} or `a`_{2} or … or `a`_{n}, where *or* has the meaning
that no one in particular must be taken, just as in *all a*’s we must not
take any one in particular.

(5) *Some a* denotes `a`_{1} or denotes `a`_{2} … or denotes `a`_{n}, where it is
not irrelevant which is taken, but on the contrary some one particular `a`
must be taken.

As the nature and properties of the various ways of combining terms are of vital importance to the principles of mathematics, it may be well to illustrate their properties by the following important examples.

(`α`) Let `a` be a class, and `b` a class of classes. We then obtain
in all six possible relations of `a` to `b` from various combinations of *any*,
*a* and *some*. *All* and *every* do not, in this case, introduce anything new.
The six cases are as follows.

(1) *Any a* belongs to any class belonging to `b`, in other words, the
class `a` is wholly contained in the common part or logical product of
the various classes belonging to `b`.

(2) Any `a` belongs to a `b`, *i.e.* the class `a` is contained in any
class which contains all the `b`’s, or, is contained in the logical sum of
all the `b`’s.

(3) Any `a` belongs to some `b`, *i.e.* there is a class belonging to `b`,
in which the class `a` is contained. The difference between this case and
the second arises from the fact that here there is one `b` to which every
`a` belongs, whereas before it was only decided that every `a` belonged to
a `b`, and different `a`’s might belong to different `b`’s.

(4) An `a` belongs to any `b`, *i.e.* whatever `b` we take, it has a part
in common with `a`.

(5) An `a` belongs to a `b`, *i.e.* there is a `b` which has a part in common
with `a`. This is equivalent to “some (or an) `a` belongs to some `b`.”

(6) Some `a` belongs to any `b`, *i.e.* there is an `a` which belongs to
the common part of all the `b`’s, or `a` and all the `b`’s have a common part.
These are all the cases that arise here.

(`β`) It is instructive, as showing the generality of the type of
relations here considered, to compare the above case with the following.
Let `a`, `b` be two series of real numbers; then six precisely analogous
cases arise.

(1) Any `a` is less than any `b`, or, the series `a` is contained among
numbers less than every `b`.

(2) Any `a` is less than a `b`, or, whatever `a` we take, there is a `b`
which is greater, or, the series `a` is contained among numbers less than
a (variable) term of the series `b`. It does not follow that some term of
the series `b` is greater than all the `a`’s.

(3) Any `a` is less than some `b`, or, there is a term of `b` which is
greater than all the `a`’s. This case is not to be confounded with (2).

(4) An `a` is less than any `b`, *i.e.* whatever `b` we take, there is an
`a` which is less than it.

(5) An `a` is less than a `b`, *i.e.* it is possible to find an `a` and a `b`
such that the `a` is less than the `b`. This merely denies that any `a` is
greater than any `b`.

(6) Some `a` is less than any `b`, *i.e.* there is an `a` which is less than
all the `b`’s. This was not implied in (4), where the `a` was variable,
whereas here it is constant.

In this case, actual mathematics have compelled the distinction between the variable and the constant disjunction. But in other cases, where mathematics have not obtained sway, the distinction has been neglected; and the mathematicians, as was natural, have not investigated the logical nature of the disjunctive notions which they employed.

(`γ`) I shall give one other instance, as it brings in the difference
between *any* and *every*, which has not been relevant in the previous
cases. Let `a` and `b` be two classes of classes; then twenty different
relations between them arise from different combinations of the terms
of their terms. The following technical terms will be useful. If `a` be
a class of classes, its logical sum consists of all terms belonging to any
`a`, *i.e.* all terms such that there is an `a` to which they belong, while
its logical product consists of all terms belonging to every `a`, *i.e.* to the
common part of all the `a`’s. We have then the following cases.

(1) Any term of any `a` belongs to every `b`, *i.e.* the logical sum of
`a` is contained in the logical product of `b`.

(2) Any term of any `a` belongs to a `b`, *i.e.* the logical sum of `a`
is contained in the logical sum of `b`.

(3) Any term of any `a` belongs to some `b`, *i.e.* there is a `b` which
contains the logical sum of `a`.

(4) Any term of some (or an) *a* belongs to every *b*, *i.e.* there is an
*a* which is contained in the product of *b*.

(5) Any term of some (or an) `a` belongs to a `b`, *i.e.* there is an `a`
which is contained in the sum of `b`.

(6) Any term of some (or an) `a` belongs to some `b`, *i.e.* there is a
`b` which contains one class belonging to `a`.

(7) A term of any `a` belongs to any `b`, *i.e.* any class of `a` and any
class of `b` have a common part.

(8) A term of any `a` belongs to a `b`, *i.e.* any class of `a` has a part
in common with the logical sum of `b`.

(9) A term of any `a` belongs to some `b`, *i.e.* there is a `b` with which
any `a` has a part in common.

(10) A term of an `a` belongs to every `b`, *i.e.* the logical sum of `a`
and the logical product of `b` have a common part.

(11) A term of an `a` belongs to any `b`, *i.e.* given any `b`, an `a` can
be found with which it has a common part.

(12) A term of an `a` belongs to a `b`, *i.e.* the logical sums of `a` and
of `b` have a common part.

(13) Any term of every `a` belongs to every `b`, *i.e.* the logical
product of `a` is contained in the logical product of `b`.

(14) Any term of every `a` belongs to a `b`, *i.e.* the logical product
of `a` is contained in the logical sum of `b`.

(15) Any term of every `a` belongs to some `b`, *i.e.* there is a term
of `b` in which the logical product of `a` is contained.

(16) A (or some) term of every `a` belongs to every `b`, *i.e.* the logical
products of `a` and of `b` have a common part.

(17) A (or some) term of every `a` belongs to a `b`, *i.e.* the logical
product of `a` and the logical sum of `b` have a common part.

(18) Some term of any `a` belongs to every `b`, *i.e.* any `a` has a part
in common with the logical product of `b`.

(19) A term of some `a` belongs to any `b`, *i.e.* there is some term
of `a` with which any `b` has a common part.

(20) A term of every `a` belongs to any `b`, *i.e.* any `b` has a part in
common with the logical product of `a`.

The above examples show that, although it may often happen that
there is a mutual implication (which has not always been stated) of
corresponding propositions concerning *some* and *a*, or concerning *any*
and *every*, yet in other cases there is no such mutual implication. Thus
the five notions discussed in the present chapter are genuinely distinct,
and to confound them may lead to perfectly definite fallacies.

**62.** It appears from the above discussion that, whether there are
different ways of denoting or not, the objects denoted by *all men*, *every man*, etc. are certainly distinct. It seems therefore legitimate to say
that the whole difference lies in the objects, and that denoting itself is
the same in all cases. There are, however, many difficult problems
connected with the subject, especially as regards the nature of the
objects denoted. *All men*, which I shall identify with the class of men,
seems to be an unambiguous object, although grammatically it is plural.
But in the other cases the question is not so simple: we may doubt
whether an ambiguous object is unambiguously denoted, or a definite
object ambiguously denoted. Consider again the proposition “I met
a man.” It is quite certain, and is implied by this proposition, that
what I met was an unambiguous perfectly definite man: in the technical
language which is here adopted, the proposition is expressed by “I met
some man.” But the actual man whom I met forms no part of the
proposition in question, and is not specially denoted by *some man*.
Thus the concrete event which happened is not asserted in the proposition. What is asserted is merely that some one of a class of concrete
events took place. The whole human race is involved in my assertion:
if any man who ever existed or will exist had not existed or been going
to exist, the purport of my proposition would have been different. Or,
to put the same point in more intensional language, if I substitute for
*man* any of the other class-concepts applicable to the individual whom
I had the honour to meet, my proposition is changed, although the
individual in question is just as much denoted as before. What this
proves is, that *some man* must not be regarded as actually denoting
Smith and actually denoting Brown, and so on: the whole procession
of human beings throughout the ages is always relevant to every proposition in which *some man* occurs, and what is denoted is essentially
not each separate man, but a kind of combination of all men. This
is more evident in the case of *every*, *any*, and *a*. There is, then, a
definite something, different in each of the five cases, which must, in
a sense, be an object, but is characterized as a set of terms combined
in a certain way, which something is denoted by *all men*, *every man*,
*any man*, *a man* or *some man*; and it is with this very paradoxical
object that propositions are concerned in which the corresponding
concept is used as denoting.

**63.** It remains to discuss the notion of *the*. This notion has
been symbolically emphasized by Peano, with very great advantage to
his calculus; but here it is to be discussed philosophically. The use
of identity and the theory of definition are dependent upon this notion,
which has thus the very highest philosophical importance.

The word *the*, in the singular, is correctly employed only in relation
to a class-concept of which there is only one instance. We speak of
*the* King, *the* Prime Minister, and so on (understanding *at the present time*); and in such cases there is a method of denoting one single definite
term by means of a concept, which is not given us by any of our other five
words. It is owing to this notion that mathematics can give definitions
of terms which are not concepts—a possibility which illustrates the
difference between mathematical and philosophical definition. Every
term is the only instance of *some* class-concept, and thus every term,
theoretically, is capable of definition, provided we have not adopted
a system in which the said term is one of our indefinables. It is a
curious paradox, puzzling to the symbolic mind, that definitions, theoretically, are nothing but statements of symbolic abbreviations, irrelevant
to the reasoning and inserted only for practical convenience, while yet,
in the development of a subject, they always require a very large amount
of thought, and often embody some of the greatest achievements of
analysis. This fact seems to be explained by the theory of denoting.
An object may be present to the mind, without our knowing any concept
of which the said object is *the* instance; and the discovery of such a
concept is not a mere improvement in notation. The reason why this
appears to be the case is that, as soon as the definition is found, it
becomes wholly unnecessary to the reasoning to remember the actual
object defined, since only concepts are relevant to our deductions. In
the moment of discovery, the definition is seen to be *true*, because the
object to be defined was already in our thoughts; but as part of our
reasoning it is not true, but merely symbolic, since what the reasoning
requires is not that it should deal with *that* object, but merely that
it should deal with the object denoted by the definition.

In most actual definitions of mathematics, what is defined is a *class*
of entities, and the notion of *the* does not then explicitly appear. But
even in this case, what is really defined is *the* class satisfying certain
conditions; for a class, as we shall see in the next chapter, is always
a term or conjunction of terms and never a concept. Thus the notion of
*the* is always relevant in definitions; and we may observe generally that
the adequacy of concepts to deal with things is wholly dependent upon
the unambiguous denoting of a single term which this notion gives.

**64.** The connection of denoting with the nature of identity is
important, and helps, I think, to solve some rather serious problems.
The question whether identity is or is not a relation, and even whether
there is such a concept at all, is not easy to answer. For, it may be
said, identity cannot be a relation, since, where it is truly asserted,
we have only one term, whereas two terms are required for a relation.
And indeed identity, an objector may urge, cannot be anything at all:
two terms plainly are not identical, and one term cannot be, for what
is it identical with? Nevertheless identity must be something. We
might attempt to remove identity from terms to relations, and say that
two terms are identical in some respect when they have a given relation
to a given term. But then we shall have to hold either that there is
strict identity between the two cases of the given relation, or that the
two cases have identity in the sense of having a given relation to a given
term; but the latter view leads to an endless process of the illegitimate
kind. Thus identity must be admitted, and the difficulty as to the
two terms of a relation must be met by a sheer denial that two different
terms are necessary. There must always be a referent and a relatum,
but these need not be distinct; and where identity is affirmed, they are
not so*.

But the question arises: Why is it ever worth while to affirm
identity? This question is answered by the theory of denoting. If
we say “Edward VII is the King,” we assert an identity; the reason
why this assertion is worth making is, that in the one case the actual
term occurs, while in the other a denoting concept takes its place.
(For purposes of discussion, I ignore the fact that Edwards form a
class, and that seventh Edwards form a class having only one term.
Edward VII is practically, though not formally, a proper name.) Often
two denoting concepts occur, and the term itself is not mentioned, as
in the proposition “the present Pope is the last survivor of his generation.” When a term is given, the assertion of its identity with itself,
though true, is perfectly futile, and is never made outside the logic-books; but where denoting concepts are introduced, identity is at once
seen to be significant. In this case, of course, there is involved, though
not asserted, a relation of the denoting concept to the term, or of the
two denoting concepts to each other. But the *is* which occurs in such
propositions does not itself state this further relation, but states pure
identity†.

**65.** To sum up. When a class-concept, preceded by one of the
six words *all*, *every*, *any*, *a*, *some*, *the*, occurs in a proposition, the
proposition is, as a rule, not *about* the concept formed of the two words
together, but about an object quite different from this, in general not
a concept at all, but a term or complex of terms. This may be seen by
the fact that propositions in which such concepts occur are in general
false concerning the concepts themselves. At the same time, it is
possible to consider and make propositions about the concepts themselves, but these are not the natural propositions to make in employing
the concepts. “Any number is odd or even” is a perfectly natural proposition, whereas “*Any number* is a variable conjunction” is a proposition
only to be made in a logical discussion. In such cases, we say that the
concept in question *denotes*. We decided that denoting is a perfectly
definite relation, the same in all six cases, and that it is the nature of
the denoted object and the denoting concept which distinguishes the
cases. We discussed at some length the nature and the differences of
the denoted objects in the five cases in which these objects are combinations of terms. In a full discussion, it would be necessary also to
discuss the denoting concepts: the actual meanings of these concepts, as
opposed to the nature of the objects they denote, have not been discussed
above. But I do not know that there would be anything further to say
on this topic. Finally, we discussed *the*, and showed that this notion
is essential to what mathematics calls definition, as well as to the
possibility of uniquely determining a term by means of concepts; the
actual use of identity, though not its meaning, was also found to depend
upon this way of denoting a single term. From this point we can
advance to the discussion of classes, thereby continuing the development
of the topics connected with adjectives.

Notes

*^{[page 54]} Cf. §49.

†^{[page 54]} There are two allied propositions expressed by the same words, namely “Socrates is a-man” and “Socrates is-a man.” The above remarks apply to the former; but in future, unless the contrary is indicated by a hyphen or otherwise, the latter will always be in question. The former expresses the identity of Socrates with an ambiguous individual; the latter expresses a relation of Socrates to the class-concept *man*.

*^{[page 55]} I shall use the word *object* in a wider sense than *term*, to cover both singular and plural, and also cases of ambiguity, such as “a man.” The fact that a word can be framed with a wider meaning than *term* raises grave logical problems. Cf. §47.

†^{[page 55]} On the indefinite article, some good remarks are made by Meinong,
“Abstrahiren und Vergleichen,” *Zeitschrift für Psychologie und Physiologie der Sinnesorgane*, Vol. xxiv, p. 63.

*^{[page 56]} I intend to distinguish between *a* and *some * in a way not warranted by language; the distinction of *all* and *every* is also a straining of usage. Both are necessary to avoid circumlocution.

*^{[page 64]} On relations of terms to themselves, *v. inf.* Chap. ix, §95.

†^{[page 64]} The word *is* is terribly ambiguous, and great care is necessary in order not to confound its various meanings. We have (1) the sense in which it asserts Being, as in “`A` is”; (2) the sense of identity; (3) the sense of predication, in “`A` is human”; (4) the sense of “`A` is a-man” (cf. p. 54, note), which is very like identity. In addition to these there are less common uses, as “to be good is to be happy,” where a relation of assertions is meant, that relation, in fact, which, where it exists, gives rise to formal implication. Doubtless there are further meanings which have not occurred to me. On the meanings of *is*, cf. De Morgan, *Formal Logic*, pp. 49, 50.

Classes.

**66.** To bring clearly before the mind what is meant by *class*, and
to distinguish this notion from all the notions to which it is allied, is
one of the most difficult and important problems of mathematical
philosophy. Apart from the fact that *class* is a very fundamental
concept, the utmost care and nicety is required in this subject on
account of the contradiction to be discussed in Chapter x. I must
ask the reader, therefore, not to regard as idle pedantry the apparatus
of somewhat subtle discriminations to be found in what follows.

It has been customary, in works on logic, to distinguish two standpoints, that of extension and that of intension. Philosophers have usually regarded the latter as more fundamental, while Mathematics has been held to deal specially with the former. M. Couturat, in his admirable work on Leibniz, states roundly that Symbolic Logic can only be built up from the standpoint of extension*; and if there really were only these two points of view, his statement would be justified. But as a matter of fact, there are positions intermediate between pure intension and pure extension, and it is in these intermediate regions that Symbolic Logic has its lair. It is essential that the classes with which we are concerned should be composed of terms, and should not be predicates or concepts, for a class must be definite when its terms are given, but in general there will be many predicates which attach to the given terms and to no others. We cannot of course attempt an intensional definition of a class as the class of predicates attaching to the terms in question and to no others, for this would involve a vicious circle; hence the point of view of extension is to some extent unavoidable. On the other hand, if we take extension pure, our class is defined by enumeration of its terms, and this method will not allow us to deal, as Symbolic Logic does, with infinite classes. Thus, our classes must in general be regarded as objects denoted by concepts, and to this extent the point of view of intension is essential. It is owing to this consideration that the theory of denoting is of such great importance. In the present chapter we have to specify the precise degree in which extension and intension respectively enter into the definition and employment of classes; and throughout the discussion, I must ask the reader to remember that whatever is said has to be applicable to infinite as well as to finite classes.

**67.** When an object is unambiguously denoted by a concept, I shall
speak of the concept as a concept (or sometimes, loosely, as *the* concept)
of the object in question. Thus it will be necessary to distinguish the
concept of a class from a class-concept. We agreed to call *man* a class-concept, but *man* does not, in its usual employment, denote anything.
On the other hand, *men* and *all men* (which I shall regard as synonyms) do
denote, and I shall contend that what they denote is the class composed
of all men. Thus *man* is the class-concept, *men* (the concept) is the
concept of the class, and men (the object denoted by the concept *men*)
are the class. It is no doubt confusing, at first, to use *class-concept* and
*concept of a class* in different senses; but so many distinctions are
required that some straining of language seems unavoidable. In
the phraseology of the preceding chapter, we may say that a class is a
numerical conjunction of terms. This is the thesis which is to be
established.

**68.** In Chapter ii we regarded classes as derived from assertions,
*i.e.* as all the entities satisfying some assertion, whose form was left
wholly vague. I shall discuss this view critically in the next chapter;
for the present, we may confine ourselves to classes as they are derived
from predicates, leaving open the question whether every assertion is
equivalent to a predication. We may, then, imagine a kind of genesis
of classes, through the successive stages indicated by the typical propositions “Socrates is human,” “Socrates has humanity,” “Socrates is a
man,” “Socrates is one among men.” Of these propositions, the last
only, we should say, explicitly contains the class as a constituent; but
every subject-predicate proposition gives rise to the other three equivalent
propositions, and thus every predicate (provided it can be sometimes
truly predicated) gives rise to a class. This is the genesis of classes from
the intensional standpoint.

On the other hand, when mathematicians deal with what they call a
manifold, aggregate, *Menge*, *ensemble*, or some equivalent name, it is
common, especially where the number of terms involved is finite, to regard
the object in question (which is in fact a class) as defined by the enumeration of its terms, and as consisting possibly of a single term, which in
that case *is* the class. Here it is not predicates and denoting that are
relevant, but terms connected by the word *and*, in the sense in which
this word stands for a *numerical* conjunction. Thus Brown and Jones
are a class, and Brown singly is a class. This is the extensional genesis
of classes.

**69.** The best formal treatment of classes in existence is that of
Peano*. But in this treatment a number of distinctions of great
philosophical importance are overlooked. Peano, not I think quite
consciously, identifies the class with the class-concept; thus the relation
of an individual to its class is, for him, expressed by *is a*. For him,
“2 is a number” is a proposition in which a term is said to belong to
the class *number*. Nevertheless, he identifies the equality of classes,
which consists in their having the same terms, with identity—a proceeding which is quite illegitimate when the class is regarded as the
class-concept. In order to perceive that *man* and *featherless biped* are
not identical, it is quite unnecessary to take a hen and deprive the poor
bird of its feathers. Or, to take a less complex instance, it is plain that
*even prime* is not identical with *integer next after 1*. Thus when we
identify the class with the class-concept, we must admit that two classes
may be equal without being identical. Nevertheless, it is plain that
when two class-concepts are equal, some identity is involved, for we say
that they have the *same* terms. Thus there is some object which is
positively identical when two class-concepts are equal; and this object,
it would seem, is more properly called the class. Neglecting the plucked
hen, the class of featherless bipeds, every one would say, is the *same* as
the class of men; the class of even primes is the *same* as the class of
integers next after 1. Thus we must not identify the class with the
class-concept, or regard “Socrates is a man” as expressing the relation
of an individual to a class of which it is a member. This has two
consequences (to be established presently) which prevent the philosophical
acceptance of certain points in Peano’s formalism. The first consequence
is, that there is no such thing as the null-class, though there are null
class-concepts. The second is, that a class having only one term is to
be identified, contrary to Peano’s usage, with that one term. I should
not propose, however, to alter his practice or his notation in consequence
of either of these points; rather I should regard them as proofs that
Symbolic Logic ought to concern itself, as far as notation goes, with
class-concepts rather than with classes.

**70.** A class, we have seen, is neither a predicate nor a class-concept, for different predicates and different class-concepts may correspond to the same class. A class also, in one sense at least, is distinct
from the whole composed of its terms, for the latter is only and essentially
one, while the former, where it has many terms, is, as we shall see later,
the very kind of object of which *many* is to be asserted. The distinction
of a class as many from a class as a whole is often made by language:
space and points, time and instants, the army and the soldiers, the navy
and the sailors, the Cabinet and the Cabinet Ministers, all illustrate the
distinction. The notion of a whole, in the sense of a pure aggregate
which is here relevant, is, we shall find, not always applicable where the
notion of the class as many applies (see Chapter x). In such cases,
though terms may be said to belong to the class, the class must not be
treated as itself a single logical subject*. But this case never arises
where a class can be generated by a predicate. Thus we may for the
present dismiss this complication from our minds. In a class as many,
the component terms, though they have some kind of unity, have less
than is required for a whole. They have, in fact, just so much unity
as is required to make them many, and not enough to prevent them from
remaining many. A further reason for distinguishing wholes from
classes as many is that a class as one may be one of the terms of itself
as many, as in “classes are one among classes” (the extensional equivalent of “class is a class-concept”), whereas a complex whole can never
be one of its own constituents.

**71.** *Class* may be defined either extensionally or intensionally.
That is to say, we may define the kind of object which is a class, or the
kind of concept which denotes a class: this is the precise meaning of
the opposition of extension and intension in this connection. But
although the general notion can be defined in this two-fold manner,
particular classes, except when they happen to be finite, can only be
defined intensionally, *i.e.* as the objects denoted by such and such concepts. I believe this distinction to be purely psychological: logically,
the extensional definition appears to be equally applicable to infinite
classes, but practically, if we were to attempt it, Death would cut short
our laudable endeavour before it had attained its goal. Logically,
therefore, extension and intension seem to be on a par. I will begin
with the extensional view.

When a class is regarded as defined by the enumeration of its terms,
it is more naturally called a *collection*. I shall for the moment adopt
this name, as it will not prejudge the question whether the objects
denoted by it are truly classes or not. By a collection I mean what is
conveyed by “`A` and `B`” or “`A` and `B` and `C`,” or any other enumeration
of definite terms. The collection is defined by the actual mention of
the terms, and the terms are connected by *and*. It would seem that
*and* represents a fundamental way of combining terms, and that just
this way of combination is essential if anything is to result of which a
number other than 1 can be asserted. Collections do not presuppose
numbers, since they result simply from the terms together with *and*:
they could only presuppose numbers in the particular case where the
terms of the collection themselves presupposed numbers. There is a
grammatical difficulty which, since no method exists of avoiding it,
must be pointed out and allowed for. A collection, grammatically, is
singular, whereas `A` and `B`, `A` and `B` and `C`, etc. are essentially plural.
This grammatical difficulty arises from the logical fact (to be discussed
presently) that whatever is many in general forms a whole which is
one; it is, therefore, not removable by a better choice of technical
terms.

The notion of *and* was brought into prominence by Bolzano*. In
order to understand what infinity is, he says, “we must go back to one
of the simplest conceptions of our understanding, in order to reach an
agreement concerning the word that we are to use to denote it. This is
the conception which underlies the conjunction *and*, which, however, if
it is to stand out as clearly as is required, in many cases, both by the
purposes of mathematics and by those of philosophy, I believe to be best
expressed by the words: ‘A system (*Inbegriff*) of certain things,’ or
‘a whole consisting of certain parts.’ But we must add that every
arbitrary object `A` can be combined in a system with any others
`B`, `C`, `D`, …, or (speaking still more correctly) already forms a system
by itself†, of which some more or less important truth can be enunciated,
provided only that each of the presentations `A`, `B`, `C`, `D`, … in fact
represents a *different* object, or in so far as none of the propositions
‘`A` is the same as `B`,’ ‘`A` is the same as `C`,’ ‘`A` is the same as `D`,’ etc.,
is true. For if *e.g.* `A` is the same as `B`, then it is certainly unreasonable
to speak of a system of the things `A` and `B`.”

The above passage, good as it is, neglects several distinctions which
we have found necessary. First and foremost, it does not distinguish
the many from the whole which they form. Secondly, it does not appear
to observe that the method of enumeration is not practically applicable
to infinite systems. Thirdly, and this is connected with the second point,
it does not make any mention of intensional definition nor of the notion
of a class. What we have to consider is the difference, if any, of a class
from a collection on the one hand, and from the whole formed of the
collection on the other. But let us first examine further the notion
of *and*.

Anything of which a finite number other than 0 or 1 can be asserted
would be commonly said to be many, and many, it might be said, are
always of the form “`A` and `B` and `C` and …” Here `A`, `B`, `C`, … are
each one and are all different. To say that `A` is one seems to amount
to much the same as to say that `A` is not of the form “`A`_{1} and `A`_{2} and
`A`_{3} and ….” To say that `A`, `B`, `C`, … are all different seems to amount
only to a condition as regards the symbols: it should be held that
“`A` and `A`” is meaningless, so that diversity is implied by *and*, and need
not be specially stated.

A term `A` which is one may be regarded as a particular case of a
collection, namely as a collection of one term. Thus every collection
which is many presupposes many collections which are each one: `A` *and*
`B` presupposes `A` and presupposes `B`. Conversely some collections of
one term presuppose many, namely those which are complex: thus
“`A` differs from `B`” is one, but presupposes `A` *and difference and* `B`.
But there is not symmetry in this respect, for the ultimate presuppositions of anything are always simple terms.

Every pair of terms, without exception, can be combined in the
manner indicated by `A` *and* `B`, and if neither `A` nor `B` be many, then
`A` and `B` are two. `A` and `B` may be any conceivable entities, any
possible objects of thought, they may be points or numbers or true or
false propositions or events or people, in short anything that can be
counted. A teaspoon and the number 3, or a chimaera and a four-dimensional space, are certainly two. Thus no restriction whatever is
to be placed on `A` and `B`, except that neither is to be many. It should
be observed that `A` and `B` need not exist, but must, like anything that
can be mentioned, have Being. The distinction of Being and existence
is important, and is well illustrated by the process of counting. What
can be counted must be something, and must certainly *be*, though it
need by no means be possessed of the further privilege of existence.
Thus what we demand of the terms of our collection is merely that each
should be an entity.

The question may now be asked: What is meant by `A` *and* `B`?
Does this mean anything more than the juxtaposition of `A` with `B`?
That is, does it contain any element over and above that of `A` and that
of `B`? Is *and* a separate concept, which occurs besides `A`, `B`? To
either answer there are objections. In the first place, *and*, we might
suppose, cannot be a new concept, for if it were, it would have to be
some kind of relation between `A` and `B`; `A` *and* `B` would then be a
proposition, or at least a propositional concept, and would be one, not
two. Moreover, if there are two concepts, there *are* two, and no third
mediating concept seems necessary to make them two. Thus *and* would
seem meaningless. But it is difficult to maintain this theory. To begin
with, it seems rash to hold that any word is meaningless. When we use
the word *and*, we do not seem to be uttering mere idle breath, but some
idea seems to correspond to the word. Again some kind of combination
seems to be implied by the fact that `A` *and* `B` are two, which is not true
of either separately. When we say “`A` and `B` are yellow,” we can replace
the proposition by “`A` is yellow” and “`B` is yellow”; but this cannot
be done for “`A` and `B` are two”; on the contrary, `A` is *one* and `B` is *one*.
Thus it seems best to regard *and* as expressing a definite unique kind of
combination, not a relation, and not combining `A` and `B` into a whole,
which would be one. This unique kind of combination will in future be
called *addition of individuals*. It is important to observe that it applies
to terms, and only applies to numbers in consequence of their being
terms. Thus for the present, 1 and 2 are two, and 1 and 1 is
meaningless.

As regards what is meant by the combination indicated by *and*, it is
indistinguishable from what we before called a numerical conjunction.
That is, `A` *and* `B` is what is denoted by the concept of a class of which
`A` and `B` are the only members. If `u` be a class-concept of which the
propositions “`A` is a `u`” “`B` is a `u`” are true, but of which all other
propositions of the same form are false, then “all `u`’s” is the concept of
a class whose only terms are `A` and `B`; this concept *denotes* the terms
`A`, `B` combined in a certain way, and “`A` and `B`” *are* those terms combined in just that way. Thus “`A` and `B`” are the class, but are distinct
from the class-concept and from the concept of the class.

The notion of *and*, however, does not enter into the *meaning* of a
class, for a single term is a class, although it is not a numerical
conjunction. If `u` be a class-concept, and only one proposition of the
form “`x` is a `u`” be true, then “all `u`’s” is a concept denoting a single
term, and this term is the class of which “all `u`’s” is a concept. Thus
what seems essential to a class is not the notion of *and*, but the being
denoted by some concept of a class. This brings us to the intensional
view of classes.

**72.** We agreed in the preceding chapter that there are not
different ways of denoting, but only different kinds of denoting concepts
and correspondingly different kinds of denoted objects. We have
discussed the kind of denoted object which constitutes a class; we have
now to consider the kind of denoting concept.

The consideration of classes which results from denoting concepts
is more general than the extensional consideration, and that in two
respects. In the first place it allows, what the other *practically*
excludes, the admission of infinite classes; in the second place it
introduces the null concept of a class. But, before discussing these
matters, there is a purely logical point of some importance to be
examined.

If `u` be a class-concept, is the concept “all `u`’s” analyzable into two
constituents, *all* and `u`, or is it a new concept, defined by a certain
relation to `u`, and no more complex than `u` itself? We may observe,
to begin with, that “all `u`’s” is synonymous with “`u`’s,” at least according
to a very common use of the plural. Our question is, then, as to the
meaning of the plural. The word *all* has certainly some definite
meaning, but it seems highly doubtful whether it means more than
the indication of a relation. “All men” and “all numbers” have in
common the fact that they both have a certain relation to a class-concept, namely to *man* and *number* respectively. But it is very difficult
to isolate any further element of *all-ness* which both share, unless we
take as this element the mere fact that both are concepts of classes.
It would seem, then, that “all `u`’s” is not validly analyzable into *all*
and `u`, and that language, in this case as in some others, is a misleading
guide. The same remark will apply to *every*, *any*, *some*, *a*, and *the*.

It might perhaps be thought that a class ought to be considered, not merely as a numerical conjunction of terms, but as a numerical conjunction denoted by the concept of a class. This complication, however, would serve no useful purpose, except to preserve Peano’s distinction between a single term and the class whose only term it is—a distinction which is easy to grasp when the class is identified with the class-concept, but which is inadmissible in our view of classes. It is evident that a numerical conjunction considered as denoted is either the same entity as when not so considered, or else is a complex of denoting together with the object denoted; and the object denoted is plainly what we mean by a class.

With regard to infinite classes, say the class of numbers, it is to be
observed that the concept *all numbers*, though not itself infinitely
complex, yet denotes an infinitely complex object. This is the inmost
secret of our power to deal with infinity. An infinitely complex
concept, though there may be such, can certainly not be manipulated
by the human intelligence; but infinite collections, owing to the notion
of denoting, can be manipulated without introducing any concepts of
infinite complexity. Throughout the discussions of infinity in later
Parts of the present work, this remark should be borne in mind: if
it is forgotten, there is an air of magic which causes the results obtained
to seem doubtful.

**73.** Great difficulties are associated with the null-class, and
generally with the idea of *nothing*. It is plain that there is such a
concept as *nothing*, and that in some sense nothing is something. In
fact, the proposition “nothing is not nothing” is undoubtedly capable
of an interpretation which makes it true—a point which gives rise to
the contradictions discussed in Plato’s *Sophist*. In Symbolic Logic
the null-class is the class which has no terms at all; and symbolically
it is quite necessary to introduce some such notion. We have to
consider whether the contradictions which naturally arise can be
avoided.

It is necessary to realize, in the first place, that a concept may denote although it does not denote anything. This occurs when there are propositions in which the said concept occurs, and which are not about the said concept, but all such propositions are false. Or rather, the above is a first step towards the explanation of a denoting concept which denotes nothing. It is not, however, an adequate explanation. Consider, for example, the proposition “chimaeras are animals” or “even primes other than 2 are numbers.” These propositions appear to be true, and it would seem that they are not concerned with the denoting concepts, but with what these concepts denote; yet that is impossible, for the concepts in question do not denote anything. Symbolic Logic says that these concepts denote the null-class, and that the propositions in question assert that the null-class is contained in certain other classes. But with the strictly extensional view of classes propounded above, a class which has no terms fails to be anything at all: what is merely and solely a collection of terms cannot subsist when all the terms are removed. Thus we must either find a different interpretation of classes, or else find a method of dispensing with the null-class.

The above imperfect definition of a concept which denotes, but
does not denote anything, may be amended as follows. All denoting
concepts, as we saw, are derived from class-concepts; and `a` is a class-concept when “`x` is an `a`” is a propositional function. The denoting
concepts associated with `a` will not denote anything when and only
when “`x` is an `a`” is false for all values of `x`. This is a complete
definition of a denoting concept which does not denote anything; and
in this case we shall say that `a` is a null class-concept, and that “all `a`’s”
is a null concept of a class. Thus for a system such as Peano’s, in
which what are called classes are really class-concepts, technical difficulties
need not arise; but for us a genuine logical problem remains.

The proposition “chimaeras are animals” may be easily interpreted
by means of formal implication, as meaning “`x` is a chimaera implies
`x` is an animal for all values of `x`.” But in dealing with classes we
have been assuming that propositions containing *all* or *any* or *every*,
though equivalent to formal implications, were yet distinct from them,
and involved ideas requiring independent treatment. Now in the case
of chimaeras, it is easy to substitute the pure intensional view, according
to which what is really stated is a relation of predicates: in the case in
question the adjective *animal* is part of the definition of the adjective
*chimerical* (if we allow ourselves to use this word, contrary to usage,
to denote the defining predicate of chimaeras). But here again it is
fairly plain that we are dealing with a proposition which implies that
chimaeras are animals, but is not the same proposition—indeed, in the
present case, the implication is not even reciprocal. By a negation
we can give a kind of extensional interpretation: nothing is denoted
by *a chimaera* which is not denoted by *an animal*. But this is a very
roundabout interpretation. On the whole, it seems most correct to
reject the proposition altogether, while retaining the various other
propositions that would be equivalent to it if there were chimaeras.
By symbolic logicians, who have experienced the utility of the null-class, this will be felt as a reactionary view. But I am not at present
discussing what should be done in the logical calculus, where the
established practice appears to me the best, but what is the philosophical truth concerning the null-class. We shall say, then, that,
of the bundle of normally equivalent interpretations of logical symbolic
formulae, the class of interpretations considered in the present chapter,
which are dependent upon actual classes, fail where we are concerned
with null class-concepts, on the ground that there is no actual null-class.

We may now reconsider the proposition “nothing is not nothing”—a proposition plainly true, and yet, unless carefully handled, a source of
apparently hopeless antinomies. *Nothing* is a denoting concept, which
denotes nothing. The concept which denotes is of course not nothing,
*i.e.* it is not denoted by itself. The proposition which looks so paradoxical means no more than this: *Nothing*, the denoting concept, is
not nothing, *i.e.* is not what itself denotes. But it by no means follows
from this that there is an actual null-class: only the null class-concept
and the null concept of a class are to be admitted.

But now a new difficulty has to be met. The equality of class-concepts, like all relations which are reflexive, symmetrical, and transitive,
indicates an underlying identity, *i.e.* it indicates that every class-concept
has to some term a relation which all equal class-concepts also have to
that term—the term in question being different for different sets of
equal class-concepts, but the same for the various members of a single
set of equal class-concepts. Now for all class-concepts which are not
null, this term is found in the corresponding class; but where are we
to find it for null class-concepts? To this question several answers may
be given, any of which may be adopted. For we now know what a
class is, and we may therefore adopt as our term the class of all null
class-concepts or of all null propositional functions. These are not null-classes, but genuine classes, and to either of them all null class-concepts
have the same relation. If we then wish to have an entity analogous
to what is elsewhere to be called a class, but corresponding to null
class-concepts, we shall be forced, wherever it is necessary (as in counting
classes) to introduce a term which is identical for equal class-concepts,
to substitute everywhere the class of class-concepts equal to a given
class-concept for the class corresponding to that class-concept. The
class corresponding to the class-concept remains logically fundamental,
but need not be actually employed in our symbolism. The null-class,
in fact, is in some ways analogous to an irrational in Arithmetic: it
cannot be interpreted on the same principles as other classes, and if
we wish to give an analogous interpretation elsewhere, we must substitute
for classes other more complicated entities—in the present case, certain
correlated classes. The object of such a procedure will be mainly
technical; but failure to understand the procedure will lead to inextricable difficulties in the interpretation of the symbolism. A very
closely analogous procedure occurs constantly in Mathematics, for
example with every generalization of number; and so far as I know,
no single case in which it occurs has been rightly interpreted either by
philosophers or by mathematicians. So many instances will meet us
in the course of the present work that it is unnecessary to linger longer
over the point at present. Only one possible misunderstanding must
be guarded against. No vicious circle is involved in the above account
of the null-class; for the general notion of *class* is first laid down, is
found to involve what is called existence, is then symbolically, not
philosophically, replaced by the notion of a class of equal class-concepts,
and is found, in this new form, to be applicable to what corresponds to
null class-concepts, since what corresponds is now a class which is not
null. Between classes *simpliciter* and classes of equal class-concepts
there is a one-one correlation, which breaks down in the sole case of the
class of null class-concepts, to which no null-class corresponds; and this
fact is the reason for the whole complication.

**74.** A question which is very fundamental in the philosophy of
Arithmetic must now be discussed in a more or less preliminary fashion. Is
a class which has many terms to be regarded as itself one or many? Taking
the class as equivalent simply to the numerical conjunction “`A` and `B`
and `C` and etc.,” it seems plain that it is many; yet it is quite necessary
that we should be able to count classes as one each, and we do habitually
speak of *a* class. Thus classes would seem to be one in one sense and
many in another.

There is a certain temptation to identify the class as many and the
class as one, *e.g.*, *all men* and *the human race*. Nevertheless, wherever
a class consists of more than one term, it can be proved that no such
identification is permissible. A concept of a class, if it denotes a class
as one, is not the same as any concept of the class which it denotes.
That is to say, *classes of all rational animals*, which denotes the human
race as one term, is different from *men*, which denotes men, *i.e.* the
human race as many. But if the human race were identical with men,
it would follow that whatever denotes the one must denote the other,
and the above difference would be impossible. We might be tempted
to infer that Peano’s distinction, between a term and a class of which
the said term is the only member, must be maintained, at least when the
term in question is a class*. But it is more correct, I think, to infer an
ultimate distinction between a class as many and a class as one, to
hold that the many are only many, and are not also one. The class as
one may be identified with the whole composed of the terms of the class,
*i.e.*, in the case of men, the class as one will be the human race.

But can we now avoid the contradiction always to be feared,
where there is something that cannot be made a logical subject?
I do not myself see any way of eliciting a precise contradiction in this
case. In the case of concepts, we were dealing with what was plainly
one entity; in the present case, we are dealing with a complex essentially
capable of analysis into units. In such a proposition as “`A` and `B` are
two,” there is no logical subject: the assertion is not about `A`, nor
about `B`, nor about the whole composed of both, but strictly and only
about `A` and `B`. Thus it would seem that assertions are not necessarily
*about* single subjects, but may be about many subjects; and this removes
the contradiction which arose, in the case of concepts, from the impossibility of making assertions about them unless they were turned
into subjects. This impossibility being here absent, the contradiction
which was to be feared does not arise.

**75.** We may ask, as suggested by the above discussion, what is to be
said of the objects denoted by *a man*, *every man*, *some man*, and *any man*.
Are these objects one or many or neither? Grammar treats them all as
one. But to this view, the natural objection is, which one? Certainly
not Socrates, nor Plato, nor any other particular person. Can we
conclude that no one is denoted? As well might we conclude that
every one is denoted, which in fact is true of the concept *every man*.
I think one is denoted in every case, but in an impartial distributive
manner. *Any number* is neither 1 nor 2 nor any other particular number,
whence it is easy to conclude that *any number* is not any one number,
a proposition at first sight contradictory, but really resulting from an
ambiguity in *any*, and more correctly expressed by “*any number* is not
*some* one number.” There are, however, puzzles in this subject which
I do not yet know how to solve.

A logical difficulty remains in regard to the nature of the whole composed of all the terms of a class. Two propositions appear self-evident: (1) Two wholes composed of different terms must be different; (2) A whole composed of one term only is that one term. It follows that the whole composed of a class considered as one term is that class considered as one term, and is therefore identical with the whole composed of the terms of the class; but this result contradicts the first of our supposed self-evident principles. The answer in this case, however, is not difficult. The first of our principles is only universally true when all the terms composing our two wholes are simple. A given whole is capable, if it has more than two parts, of being analyzed in a plurality of ways; and the resulting constituents, so long as analysis is not pushed as far as possible, will be different for different ways of analyzing. This proves that different sets of constituents may constitute the same whole, and thus disposes of our difficulty.

**76.** Something must be said as to the relation of a term to a class
of which it is a member, and as to the various allied relations. One of
the allied relations is to be called ε, and is to be fundamental in Symbolic
Logic. But it is to some extent optional which of them we take as
symbolically fundamental.

Logically, the fundamental relation is that of subject and predicate,
expressed in “Socrates is human”—a relation which, as we saw in
Chapter iv, is peculiar in that the relatum cannot be regarded as a term
in the proposition. The first relation that grows out of this is the one
expressed by “Socrates has humanity,” which is distinguished by the
fact that here the relation is a term. Next comes “Socrates is a
man.” This proposition, considered as a relation between Socrates and
the concept man, is the one which Peano regards as fundamental; and
his ε expresses the relation *is a* between Socrates and *man*. So long
as we use class-concepts for classes in our symbolism, this practice is
unobjectionable; but if we give ε this meaning, we must not assume
that two symbols representing equal class-concepts both represent one
and the same entity. We may go on to the relation between Socrates
and the human race, *i.e.* between a term and its class considered as
a whole; this is expressed by “Socrates belongs to the human race.”
This relation might equally well be represented by ε. It is plain that,
since a class, except when it has one term, is essentially many, it cannot
be *as such* represented by a single letter: hence in any possible Symbolic
Logic the letters which do duty for classes cannot represent the classes
*as many*, but must represent either class-concepts, or the wholes composed of classes, or some other allied single entities. And thus ε cannot
represent the relation of a term to its class as many; for this would be
a relation of one term to many terms, not a two-term relation such as
we want. This relation might be expressed by “Socrates is one among
men”; but this, in any case, cannot be taken to be the meaning of ε.

**77.** A relation which, before Peano, was almost universally confounded with ε, is the relation of inclusion between classes, as *e.g.*
between men and mortals. This is a time-honoured relation, since
it occurs in the traditional form of the syllogism: it has been a battleground between intension and extension, and has been so much discussed that it is astonishing how much remains to be said about it.
Empiricists hold that such propositions mean an actual enumeration
of the terms of the contained class, with the assertion, in each case,
of membership of the containing class. They must, it is to be inferred, regard it as doubtful whether all primes are integers, since they
will scarcely have the face to say that they have examined all primes
one by one. Their opponents have usually held, on the contrary, that
what is meant is a relation of whole and part between the defining
predicates, but turned in the opposite sense from the relation between
the classes: *i.e.* the defining predicate of the larger class is part of that
of the smaller. This view seems far more defensible than the other;
and wherever such a relation does hold between the defining predicates,
the relation of inclusion follows. But two objections may be made,
first, that in some cases of inclusion there is no such relation between
the defining predicates, and secondly, that in any case what is *meant*
is a relation between the classes, not a relation of their defining
predicates. The first point may be easily established by instances.
The concept *even prime* does not contain as a constituent the concept
*integer between 1 and 10*; the concept “English King whose head was
cut off” does not contain the concept “people who died in 1649”; and
so on through innumerable obvious cases. This might be met by saying
that, though the relation of the defining predicates is not one of whole
and part, it is one more or less analogous to implication, and is always
what is really meant by propositions of inclusion. Such a view represents, I think, what is said by the better advocates of intension, and
I am not concerned to deny that a relation of the kind in question does
always subsist between defining predicates of classes one of which is
contained in the other. But the second of the above points remains
valid as against any intensional interpretation. When we say that
men are mortals, it is evident that we are saying something about men,
not about the concept *man* or the predicate *human*. The question is,
then, what exactly are we saying?

Peano held, in earlier editions of his *Formulaire*, that what is
asserted is the formal implication “`x` is a man implies `x` is a mortal.”
This is certainly implied, but I cannot persuade myself that it is the
same proposition. For in this proposition, as we saw in Chapter iii,
it is essential that *x* should take *all* values, and not only such as are
men. But when we say “all men are mortals,” it seems plain that we
are only speaking of men, and not of all other imaginable terms. We
may, if we wish for a genuine relation of classes, regard the assertion
as one of whole and part between the two classes each considered as
a single term. Or we may give a still more purely extensional form
to our proposition, by making it mean: Every (or any) man is a mortal.
This proposition raises very interesting questions in the theory of
denoting: for it appears to assert an identity, yet it is plain that what
is denoted by *every man* is different from what is denoted by *a mortal*.
These questions, however, interesting as they are, cannot be pursued
here. It is only necessary to realize clearly what are the various
equivalent propositions involved where one class is included in another.
The form most relevant to Mathematics is certainly the one with formal
implication, which will receive a fresh discussion in the following
chapter.

Finally, we must remember that classes are to be derived, by means
of the notion of *such that*, from other sources than subject-predicate
propositions and their equivalents. Any propositional function in
which a fixed assertion is made of a variable term is to be regarded,
as was explained in Chapter ii, as giving rise to a class of values
satisfying it. This topic requires a discussion of assertions; but one
strange contradiction, which necessitates the care in discrimination
aimed at in the present chapter, may be mentioned at once.

**78.** Among predicates, most of the ordinary instances cannot be
predicated of themselves, though, by introducing negative predicates,
it will be found that there are just as many instances of predicates which
are predicable of themselves. One at least of these, namely predicability,
or the property of being a predicate, is not negative: predicability, as
is evident, is predicable, *i.e.* it is a predicate of itself. But the most
common instances are negative: thus non-humanity is non-human, and
so on. The predicates which are not predicable of themselves are,
therefore, only a selection from among predicates, and it is natural to
suppose that they form a class having a defining predicate. But if so,
let us examine whether this defining predicate belongs to the class or
not. If it belongs to the class, it is not predicable of itself, for that is
the characteristic property of the class. But if it is not predicable
of itself, then it does not belong to the class whose defining predicate
it is, which is contrary to the hypothesis. On the other hand, if it
does not belong to the class whose defining predicate it is, then it is not
predicable of itself, *i.e.* it *is* one of those predicates that are not predicable of themselves, and therefore it does belong to the class whose
defining predicate it is—again contrary to the hypothesis. Hence from
either hypothesis we can deduce its contradictory. I shall return to
this contradiction in Chapter x; for the present, I have introduced
it merely as showing that no subtlety in distinguishing is likely to be
excessive.

**79.** To sum up the above somewhat lengthy discussion. A class,
we agreed, is essentially to be interpreted in extension; it is either
a single term, or that kind of combination of terms which is indicated
when terms are connected by the word *and*. But practically, though
not theoretically, this purely extensional method can only be applied
to finite classes. All classes, whether finite or infinite, can be obtained
as the objects denoted by the plurals of class-concepts—men, numbers,
points, etc. Starting with predicates, we distinguished two kinds of
proposition, typified by “Socrates is human” and “Socrates has
humanity,” of which the first uses *human* as predicate, the second
as a term of a relation. These two classes of propositions, though
very important logically, are not so relevant to Mathematics as their
derivatives. Starting from *human*, we distinguished (1) the class-concept
*man*, which differs slightly, if at all, from *human*; (2) the various
denoting concepts *all men*, *every man*, *any man*, *a man* and *some man*;
(3) the objects denoted by these concepts, of which the one denoted by
*all men* was called the *class as many*, so that *all men* (the concept) was
called the *concept of the class*; (4) the class as one, *i.e.* the human race.
We had also a classification of propositions about Socrates, dependent
upon the above distinctions, and approximately parallel with them:
(1) “Socrates is-a man” is nearly, if not quite, identical with “Socrates
has humanity”; (2) “Socrates is a-man” expresses identity between
Socrates and one of the terms denoted by *a man*; (3) “Socrates is one among men,” a proposition which raises difficulties owing to the plurality
of men; (4) “Socrates belongs to the human race,” which alone expresses
a relation of an individual to its class, and, as the possibility of relation
requires, takes the class as one, not as many. We agreed that the null-class, which has no terms, is a fiction, though there are null class-concepts.
It appeared throughout that, although any symbolic treatment must
work largely with class-concepts and intension, classes and extension are
logically more fundamental for the principles of Mathematics; and this
may be regarded as our main general conclusion in the present chapter.

Notes

*^{[page 66]} *La Logique de Leibniz*, Paris, 1901, p. 387.

*^{[page 68]} Neglecting Frege, who is discussed in the Appendix.

*^{[page 69]} A plurality of terms is not the logical subject when a number is asserted of it:
such propositions have not one subject, but many subjects. See end of §74.

*^{[page 70]} *Paradoxien des Unendlichen*, Leipzig, 1854 (2nd ed., Berlin, 1889), §3.

†^{[page 70]} *i.e.* the combination of `A` with `B`, `C`, `D`, … already forms a system.

*^{[page 76]} This conclusion is actually drawn by Frege from an analogous argument: *Archiv für syst. Phil.* i, p. 444. See Appendix.

Propositional Functions.

**80.** In the preceding chapter an endeavour was made to indicate
the kind of object that is to be called a class, and for purposes of
discussion classes were considered as derived from subject-predicate
propositions. This did not affect our view as to the notion of *class*
itself; but if adhered to, it would greatly restrict the extension of
the notion. It is often necessary to recognize as a class an object
not defined by means of a subject-predicate proposition. The explanation of this necessity is to be sought in the theory of assertions and
*such that*.

The general notion of an assertion has been already explained in
connection with formal implication. In the present chapter its scope
and legitimacy are to be critically examined, and its connection with
classes and *such that* is to be investigated. The subject is full of
difficulties, and the doctrines which I intend to advocate are put forward
with a very limited confidence in their truth.

The notion of *such that* might be thought, at first sight, to be
capable of definition; Peano used, in fact, to define the notion by the
proposition “the `x`’s such that `x` is an `a` are the class `a`.” Apart from
further objections, to be noticed immediately, it is to be observed that
the class as obtained from *such that* is the genuine class, taken in
extension and as many, whereas the `a` in “`x` is an `a`” is not the class,
but the class-concept. Thus it is formally necessary, if Peano’s procedure is to be permissible, that we should substitute for “`x`’s such that
so-and-so” the genuine class-concept “`x` such that so-and-so,” which
may be regarded as obtained from the predicate “such that so-and-so”
or rather, “being an `x` such that so-and-so,” the latter form being
necessary because so-and-so is a propositional function containing `x`.
But when this purely formal emendation has been made the point
remains that *such that* must often be put before such propositions as
*xRa*, where *R* is a given relation and `a` a given term. We cannot
reduce this proposition to the form “`x` is an `a`′” without using *such that*;
for if we ask what `a`′ must be, the answer is: `a`′ must be such that each
of its terms, and no other terms, have the relation `R` to `a`. To take
examples from daily life: the children of Israel are a class defined by
a certain relation to Israel, and the class can only be defined as the
terms such that they have this relation. *Such that* is roughly equivalent
to *who* or *which*, and represents the general notion of satisfying a
propositional function. But we may go further: given a class `a`, we
cannot define, in terms of `a`, the class of propositions “`x` is an `a`” for
different values of `x`. It is plain that there is a relation which each
of these propositions has to the `x` which occurs in it, and that the
relation in question is determinate when `a` is given. Let us call the
relation `R`. Then any entity which is a referent with respect to `R`
is a proposition of the type “`x` is an `a`.” But here the notion of
*such that* is already employed. And the relation `R` itself can only be
defined as the relation which holds between “`x` is an `a`” and `x` for all
values of `x`, and does not hold between any other pairs of terms. Here
*such that* again appears. The point which is chiefly important in these
remarks is the indefinability of propositional functions. When these
have been admitted, the general notion of one-valued functions is easily
defined. Every relation which is many-one, *i.e.* every relation for which
a given referent has only one relatum, defines a function: the relatum
is that function of the referent which is defined by the relation in
question. But where the function is a proposition, the notion involved
is presupposed in the symbolism, and cannot be defined by means of it
without a vicious circle: for in the above general definition of a function
propositional functions already occur. In the case of propositions of
the type “`x` is an `a`,” if we ask *what* propositions are of this type,
we can only answer “all propositions in which a term is said to be `a`”;
and here the notion to be defined reappears.

**81.** Can the indefinable element involved in propositional functions be identified with assertion together with the notion of *every*
proposition containing a given assertion, or an assertion made concerning
*every* term? The only alternative, so far as I can see, is to accept the
general notion of a propositional function itself as indefinable, and for
formal purposes this course is certainly the best; but philosophically,
the notion appears at first sight capable of analysis, and we have to
examine whether or not this appearance is deceptive.

We saw in discussing verbs, in Chapter iv, that when a proposition is completely analyzed into its simple constituents, these constituents taken together do not reconstitute it. A less complete analysis of propositions into subject and assertion has also been considered; and this analysis does much less to destroy the proposition. A subject and an assertion, if simply juxtaposed, do not, it is true, constitute a proposition; but as soon as the assertion is actually asserted of the subject, the proposition reappears. The assertion is everything that remains of the proposition when the subject is omitted: the verb remains an asserted verb, and is not turned into a verbal noun; or at any rate the verb retains that curious indefinable intricate relation to the other terms of the proposition which distinguishes a relating relation from the same relation abstractly considered. It is the scope and legitimacy of this notion of assertion which is now to be examined. Can every proposition be regarded as an assertion concerning any term occurring in it, or are limitations necessary as to the form of the proposition and the way in which the term enters into it?

In some simple cases, it is obvious that the analysis into subject
and assertion is legitimate. In “Socrates is a man,” we can plainly
distinguish Socrates and something that is asserted about him; we
should admit unhesitatingly that the same thing may be said about
Plato or Aristotle. Thus we can consider a class of propositions
containing this assertion, and this will be the class of which a typical
number is represented by “`x` is a man.” It is to be observed that the
assertion must appear *as* assertion, not as term: thus “to be a man
is to suffer” contains the same assertion, but used as term, and this
proposition does not belong to the class considered. In the case of
propositions asserting a fixed relation to a fixed term, the analysis
seems equally undeniable. To be more than a yard long, for example,
is a perfectly definite assertion, and we may consider the class of
propositions in which this assertion is made, which will be represented
by the propositional function “`x` is more than a yard long.” In such
phrases as “snakes which are more than a yard long,” the assertion
appears very plainly; for it is here explicitly referred to a variable
subject, not asserted of any one definite subject. Thus if `R` be a fixed
relation and `a` a fixed term, … `R``a` is a perfectly definite assertion.
(I place dots before the `R`, to indicate the place where the subject
must be inserted in order to make a proposition.) It may be doubted
whether a relational proposition can be regarded as an assertion concerning the relatum. For my part, I hold that this can be done except
in the case of subject-predicate propositions; but this question is better
postponed until we have discussed relations*.

**82.** More difficult questions must now be considered. Is such
a proposition as “Socrates is a man implies Socrates is a mortal,” or
“Socrates has a wife implies Socrates has a father,” an assertion concerning Socrates or not? It is quite certain that, if we replace Socrates
by a variable, we obtain a propositional function; in fact, the truth
of this function for all values of the variable is what is asserted in the
corresponding formal implication, which does not, as might be thought
at first sight, assert a relation between two propositional functions.
Now it was our intention, if possible, to explain propositional functions
by means of assertions; hence, if our intention can be carried out, the
above propositions must be assertions concerning Socrates. There is,
however, a very great difficulty in so regarding them. An assertion was
to be obtained from a proposition by simply omitting one of the terms
occurring in the proposition. But when we omit Socrates, we obtain
“… is a man implies … is a mortal.” In this formula it is essential
that, in restoring the proposition, the same term should be substituted
in the two places where dots indicate the necessity of a term. It does
not matter what term we choose, but it must be identical in both places.
Of this requisite, however, no trace whatever appears in the would-be
assertion, and no trace can appear, since all mention of the term to be
inserted is necessarily omitted. When an `x` is inserted to stand for
the variable, the identity of the term to be inserted is indicated by the
repetition of the letter `x`; but in the assertional form no such method is
available. And yet, at first sight, it seems very hard to deny that the
proposition in question tells us a fact *about* Socrates, and that the *same*
fact is true about Plato or a plum-pudding or the number 2. It is
certainly undeniable that “Plato is a man implies Plato is a mortal”
is, in some sense or other, the *same* function of Plato as our previous
proposition is of Socrates. The natural interpretation of this statement
would be that the one proposition has to Plato the same relation as the
other has to Socrates. But this requires that we should regard the
propositional function in question as definable by means of its relation
to the variable. Such a view, however, requires a propositional function
more complicated than the one we are considering. If we represent
“`x` is a man implies `x` is a mortal” by `φ``x`, the view in question maintains
that `φ``x` is the term having to `x` the relation `R`, where `R` is some definite
relation. The formal statement of this view is as follows: For all values
of `x` and `y`, “`y` is identical with `φ``x`” is equivalent to “`y` has the relation
`R` to `x`.” It is evident that this will not do as an explanation, since it
has far greater complexity than what it was to explain. It would seem
to follow that propositions may have a certain constancy of form, expressed in the fact that they are instances of a given propositional
function, without its being possible to analyze the propositions into a
constant and a variable factor. Such a view is curious and difficult:
constancy of form, in all other cases, is reducible to constancy of relations, but the constancy involved here is presupposed in the notion
of constancy of relation, and cannot therefore be explained in the
usual way.

The same conclusion, I think, will result from the case of two
variables. The simplest instance of this case is `x``R``y`, where `R` is a
constant relation, while `x` and `y` are independently variable. It seems
evident that this is a propositional function of two independent variables:
there is no difficulty in the notion of the class of all propositions of the
form `x``R``y`. This class is involved—or at least all those members of
the class that are true are involved—in the notion of the classes of
referents and relata with respect to `R`, and these classes are unhesitatingly admitted in such words as parents and children, masters and
servants, husbands and wives, and innumerable other instances from
daily life, as also in logical notions such as premisses and conclusions,
causes and effects, and so on. All such notions depend upon the class
of propositions typified by `x``R``y`, where `R` is constant while `x` and `y` are
variable. Yet it is very difficult to regard `x``R``y` as analyzable into the
assertion `R` concerning `x` and `y`, for the very sufficient reason that this
view destroys the sense of the relation, *i.e.* its direction from `x` to `y`,
leaving us with some assertion which is symmetrical with respect to
`x` and `y`, such as “the relation `R` holds between `x` and `y`.” Given a
relation and its terms, in fact, two distinct propositions are possible.
Thus if we take `R` itself to be an assertion, it becomes an ambiguous
assertion: in supplying the terms, if we are to avoid ambiguity, we
must decide which is referent and which relatum. We may quite
legitimately regard … `R``y` as an assertion, as was explained before; but
here `y` has become constant. We may then go on to vary `y`, considering
the class of assertions … `R``y` for different values of `y`; but this process
does not seem to be identical with that which is indicated by the
independent variability of `x` and `y` in the propositional function `x``R``y`.
Moreover, the suggested process requires the variation of an element
in an assertion, namely of `y` in … `R``y`, and this is in itself a new and difficult notion.

A curious point arises, in this connection, from the consideration,
often essential in actual Mathematics, of a relation of a term to itself.
Consider the propositional function `x``R``x`, where `R` is a constant relation.
Such functions are required in considering, *e.g.*, the class of suicides or
of self-made men; or again, in considering the values of the variable
for which it is equal to a certain function of itself, which may often be
necessary in ordinary Mathematics. It seems exceedingly evident, in
this case, that the proposition contains an element which is lost when
it is analyzed into a term `x` and an assertion `R`. Thus here again, the
propositional function must be admitted as fundamental.

**83.** A difficult point arises as to the variation of the concept in a
proposition. Consider, for example, all propositions of the type `a``R``b`,
where `a` and `b` are fixed terms, and `R` is a variable relation. There
seems no reason to doubt that the class-concept “relation between `a`
and `b`” is legitimate, and that there is a corresponding class; but this
requires the admission of such propositional functions as `a``R``b`, which,
moreover, are frequently required in actual Mathematics, as, for example,
in counting the number of many-one relations whose referents and relata
are given classes. But if our variable is to have, as we normally
require, an unrestricted field, it is necessary to substitute the propositional function “`R` is a relation implies `a``R``b`.” In this proposition
the implication involved is material, not formal. If the implication were
formal, the proposition would not be a function of `R`, but would be
equivalent to the (necessarily false) proposition: “All relations hold
between `a` and `b`.” Generally we have some such proposition as “`a``R``b`
implies `φ`(`R`) provided `R` is a relation,” and we wish to turn this into a
formal implication. If `φ`(`R`) is a proposition for all values of `R`, our
object is effected by substituting “If ‘`R` is a relation’ implies ‘`a``R``b`,’
then `φ`(`R`).” Here `R` can take all values*, and the *if* and *then* is a formal
implication, while the *implies* is a material implication. If `φ`(`R`) is not
a propositional function, but is a proposition only when `R` satisfies `ψ`(`R`),
where `ψ`(`R`) is a propositional function implied by “`R` is a relation” for
all values of `R`, then our formal implication can be put in the form “If
‘`R` is a relation’ implies `a``R``b`, then, for all values of `R`, `ψ`(`R`) implies
`φ`(`R`),” where both the subordinate implications are material. As regards
the material implication “‘`R` is a relation’ implies `a``R``b`,” this is always
a proposition, whereas `a``R``b` is only a proposition when `R` is a relation.
The new propositional function will only be true when `R` is a relation
which does hold between `a` and `b`: when `R` is not a relation, the antecedent is false and the consequent is not a proposition, so that the
implication is false; when `R` is a relation which does not hold between
`a` and `b`, the antecedent is true and the consequent false, so that again
the implication is false; only when both are true is the implication true.
Thus in defining the class of relations holding between `a` and `b`, the
formally correct course is to define them as the values satisfying “`R`
is a relation implies `a``R``b`”—an implication which, though it contains a
variable, is not formal, but material, being satisfied by some only of the
possible values of `R`. The variable `R` in it is, in Peano’s language, real
and not apparent.

The general principle involved is: If `φ``x` is only a proposition for
some values of `x`, then “‘`φ``x` implies `φ``x`’ implies `φ``x`” is a proposition
for *all* values of `x`, and is true when and only when `φ``x` is true. (The
implications involved are both material.) In some cases, “`φ``x` implies `φ``x`”
will be equivalent to some simpler propositional function `ψ``x` (such as “`R` is
a relation” in the above instance), which may then be substituted for it†.

Such a propositional function as “`R` is a relation implies `a``R``b`”
appears even less capable than previous instances of analysis into `R` and
an assertion about `R`, since we should have to assign a meaning to “`a` … `b`,”
where the blank space may be filled by anything, not necessarily by a
relation. There is here, however, a suggestion of an entity which has
not yet been considered, namely the couple with sense. It may be
doubted whether there is any such entity, and yet such phrases as
“`R` is a relation holding from `a` to `b`” seem to show that its rejection
would lead to paradoxes. This point, however, belongs to the theory
of relations, and will be resumed in Chapter ix (§98).

From what has been said, it appears that propositional functions must be accepted as ultimate data. It follows that formal implication and the inclusion of classes cannot be generally explained by means of a relation between assertions, although, where a propositional function asserts a fixed relation to a fixed term, the analysis into subject and assertion is legitimate and not unimportant.

**84.** It only remains to say a few words concerning the derivation
of classes from propositional functions. When we consider the `x`’s *such that* `φ``x`, where `φ``x` is a propositional function, we are introducing a
notion of which, in the calculus of propositions, only a very shadowy use
is made—I mean the notion of *truth*. We are considering, among
all the propositions of the type `φ``x`, those that are true: the corresponding values of `x` give the class defined by the function `φ``x`. It must
be held, I think, that every propositional function which is not null
defines a class, which is denoted by “`x`’s such that `φ``x`.” There is thus
always a concept of the class, and the class-concept corresponding will
be the singular, “`x` such that `φ``x`.” But it may be doubted—indeed the
contradiction with which I ended the preceding chapter gives reason for
doubting—whether there is always a defining predicate of such classes.
Apart from the contradiction in question, this point might appear to be
merely verbal: “being an `x` such that `φ``x`,” it might be said, may always
be taken to be a predicate. But in view of our contradiction, all
remarks on this subject must be viewed with caution. This subject,
however, will be resumed in Chapter x.

**85.** It is to be observed that, according to the theory of propositional functions here advocated, the `φ` in `φ``x` is not a separate and
distinguishable entity: it lives in the propositions of the form `φ``x`, and
cannot survive analysis. I am highly doubtful whether such a view does
not lead to a contradiction, but it appears to be forced upon us, and it
has the merit of enabling us to avoid a contradiction arising from the
opposite view. If `φ` were a distinguishable entity, there would be a
proposition asserting `φ` of itself, which we may denote by `φ`(`φ`); there
would also be a proposition not-`φ`(`φ`), denying `φ`(`φ`). In this proposition we may regard `φ` as variable; we thus obtain a propositional
function. The question arises: Can the assertion in this propositional
function be asserted of itself? The assertion is non-assertibility of self,
hence if it can be asserted of itself, it cannot, and if it cannot, it can.
This contradiction is avoided by the recognition that the functional
part of a propositional function is not an independent entity. As the
contradiction in question is closely analogous to the other, concerning
predicates net predicable of themselves, we may hope that a similar
solution will apply there also.

Notes

*^{[page 84]} See §96.

*^{[page 87]} It is necessary to assign some meaning (other than a proposition) to `a``R``b` when `R` is not a relation.

†^{[page 87]} A propositional function, though for every value of the variable it is true or false, is not itself true or false, being what is denoted by “any proposition of the type in question,” which is not itself a proposition.

The Variable.

**86.** The discussions of the preceding chapter elicited the fundamental nature of the variable; no apparatus of assertions enables us to
dispense with the consideration of the varying of one or more elements
in a proposition while the other elements remain unchanged. The
variable is perhaps the most distinctively mathematical of all notions;
it is certainly also one of the most difficult to understand. The attempt,
if not the deed, belongs to the present chapter.

The theory as to the nature of the variable, which results from our
previous discussions, is in outline the following. When a given term
occurs as term in a proposition, that term may be replaced by any other
while the remaining terms are unchanged. The class of propositions
so obtained have what may be called constancy of form, and this constancy of form must be taken as a primitive idea. The notion of a class
of propositions of constant form is more fundamental than the general
notion of class, for the latter can be defined in terms of the former,
but not the former in terms of the latter. Taking *any* term, a certain
member of any class of propositions of constant form will contain that
term. Thus `x`, the variable, is what is denoted by *any term*, and `φ``x`,
the propositional function, is what is denoted by *the* proposition of the
form `φ` in which `x` occurs. We may say that `x` is *the* `x` is *any* `φ``x`, where
`φ``x` denotes the class of propositions resulting from different values of `x`.
Thus in addition to propositional functions, the notions of *any* and of
denoting are presupposed in the notion of the variable. This theory,
which, I admit, is full of difficulties, is the least objectionable that I
have been able to imagine. I shall now set it forth more in detail.

**87.** Let us observe, to begin with, that the explicit mention of
*any*, *some*, etc., need not occur in Mathematics: formal implication will
express all that is required. Let us recur to an instance already discussed in connection with denoting, where `a` is a class and `b` a class
of classes. We have

“Any `a` belongs to any `b`” is equivalent to “‘`x` is an `a`’ implies that
‘`u` is a `b`’ implies ‘`x` is a `u`’”;

“Any `a` belongs to a `b`” is equivalent to “‘`x` is an `a`’ implies ‘there
is a `b`, say `u`, such that `x` is a `u`’”*;

“Any `a` belongs to some `b`” is equivalent to “there is a `b`, say `u`, such
that ‘`x` is an `a`’ implies ‘`x` is a `u`’”;

and so on for the remaining relations considered in Chapter v. The
question arises: How far do these equivalences constitute definitions of
*any*, *a*, *some*, and how far are these notions involved in the symbolism
itself?

The variable is, from the formal standpoint, *the* characteristic notion
of Mathematics. Moreover it is *the* method of stating general theorems,
which always *mean* something different from the intensional propositions
to which such logicians as Mr Bradley endeavour to reduce them. That
the meaning of an assertion about all men or any man is different from
the meaning of an equivalent assertion about the concept *man*, appears
to me, I must confess, to be a self-evident truth—as evident as the fact
that propositions about John are not about the *name* John. This point,
therefore, I shall not argue further. That the variable characterizes
Mathematics will be generally admitted, though it is not generally
perceived to be present in elementary Arithmetic. Elementary Arithmetic, as taught to children, is characterized by the fact that the *numbers*
occurring in it are constants; the answer to any schoolboy’s sum is
obtainable without propositions concerning *any* number. But the fact
that this is the case can only be proved by the help of propositions
about *any* number, and thus we are led from schoolboy’s Arithmetic to
the Arithmetic which uses letters for numbers and proves general
theorems. How very different this subject is from childhood’s enemy may
be seen at once in such works as those of Dedekind† and Stolz‡. Now
the difference consists simply in this, that our numbers have now become
variables instead of being constants. We now prove theorems concerning `n`, not concerning 3 or 4 or any other particular number. Thus it is
absolutely essential to any theory of Mathematics to understand the
nature of the variable.

Originally, no doubt, the variable was conceived dynamically, as
something which changed with the lapse of time, or, as is said, as something which successively assumed all values of a certain class. This
view cannot be too soon dismissed. If a theorem is proved concerning
`n`, it must not be supposed that `n` is a kind of arithmetical Proteus,
which is 1 on Sundays and 2 on Mondays, and so on. Nor must it be
supposed that `n` simultaneously assumes all its values. If `n` stands for
any integer, we cannot say that `n` is 1, nor yet that it is 2, nor yet that
it is any other particular number. In fact, `n` just denotes *any* number,
and this is something quite distinct from each and all of the numbers.
It is not true that 1 is any number, though it is true that whatever
holds of any number holds of 1. The variable, in short, requires the
indefinable notion of `a``n``y` which was explained in Chapter v.

**88.** We may distinguish what may be called the true or formal
variable from the restricted variable. *Any term* is a concept denoting
the true variable; if `u` be a class not containing all terms, *any u* denotes
a restricted variable. The terms included in the object denoted by the
defining concept of a variable are called the *values* of the variable: thus
every value of a variable is a constant. There is a certain difficulty
about such propositions as “any number is a number.” Interpreted by
formal implication, they offer no difficulty, for they assert merely that
the propositional function “`x` is a number implies `x` is a number” holds
for all values of `x`. But if “any number” be taken to be a definite
object, it is plain that it is not identical with 1 or 2 or 3 or any number
that may be mentioned. Yet these are all the numbers there are, so
that “any number” cannot be a number at all. The fact is that the
concept “any number” does denote one number, but not a particular
one. This is just the distinctive point about *any*, that it denotes a term
of a class, but in an impartial distributive manner, with no preference
for one term over another. Thus although *x* is a number, and no one
number is `x`, yet there is here no contradiction, so soon as it is recognized
that `x` is not one definite term.

The notion of the restricted variable can be avoided, except in regard
to propositional functions, by the introduction of a suitable hypothesis,
namely the hypothesis expressing the restriction itself. But in respect
of propositional functions this is not possible. The `x` in `φ``x`, where `φ``x`
is a propositional function, is an unrestricted variable; but the `φ``x` itself
is restricted to the class which we may call `φ`. (It is to be remembered
that the *class* is here fundamental, for we found it impossible, without a
vicious circle, to discover any common characteristic by which the class
could be defined, since the statement of any common characteristic is
itself a propositional function.) By making our `x` always an unrestricted
variable, we can speak of *the* variable, which is conceptually identical in
Logic, Arithmetic, Geometry, and all other formal subjects. The *terms*
dealt with are always *all* terms; only the complex concepts that occur
distinguish the various branches of Mathematics.

**89.** We may now return to the apparent definability of *any*, *some*,
and *a*, in terms of formal implication. Let `a` and `b` be class-concepts,
and consider the proposition “any `a` is a `b`.” This is to be interpreted
as meaning “`x` is an `a` implies `x` is a `b`.” It is plain that, to begin with,
the two propositions do not *mean* the same thing: for *any a* is a concept
denoting only `a`’s, whereas in the formal implication `x` need not be an `a`.
But we might, in Mathematics, dispense altogether with “any `a` is a `b`,”
and content ourselves with the formal implication: this is, in fact,
symbolically the best course. The question to be examined, therefore,
is: How far, if at all, do *any* and *some* and *a* enter into the formal
implication? (The fact that the indefinite article appears in “`x` is
an `a`” and “`x` is a `b`” is irrelevant, for these are merely taken as typical
propositional functions.) We have, to begin with, a class of true
propositions, each asserting of some constant term that if it is an `a` it is
a `b`. We then consider the restricted variable, “any proposition of this
class.” We assert the truth of any term included among the values of
this restricted variable. But in order to obtain the suggested formula,
it is necessary to transfer the variability from the proposition as a whole
to its variable term. In this way we obtain “`x` is an `a` implies `x` is `b`.”
But the genesis remains essential, for we are not here expressing a
relation of two propositional functions “`x` is an `a`” and “`x` is a `b`.” If
this were expressed, we should not require the *same* `x` both times. Only
one propositional function is involved, namely the whole formula. Each
proposition of the class expresses a relation of one term of the propositional function “`x` is an `a`” to one of “`x` is a `b`”; and we may say,
if we choose, that the whole formula expresses a relation of *any* term of
“`x` is an `a`” to some term of “`x` is a `b`.” We do not so much have
an implication containing a variable as a variable implication. Or
again, we may say that the first `x` is *any* term, but the second is *some*
term, namely the first `x`. We have a class of implications not containing
variables, and we consider *any* member of this class. If *any* member
is true, the fact is indicated by introducing a typical implication containing a variable. This typical implication is what is called a *formal*
implication: it is *any* member of a class of material implications. Thus
it would seem that *any* is presupposed in mathematical formalism, but
that *some* and *a* may be legitimately replaced by their equivalents in
terms of formal implications.

**90.** Although *some may* be replaced by its equivalent in terms of
*any*, it is plain that this does not give the meaning of *some*. There is,
in fact, a kind of duality of *any* and *some*: given a certain propositional
function, if *all* terms belonging to the propositional function are asserted,
we have *any*, while if one at least is asserted (which gives what is called
an existence-theorem), we get *some*. The proposition `φ``x` asserted without comment, as in “`x` is a man implies `x` is a mortal,” is to be taken
to mean that `φ``x` is true for *all* values of `x` (or for *any* value), but it
might equally well have been taken to mean that `φ``x` is true for *some*
value of `x`. In this way we might construct a calculus with two kinds
of variable, the conjunctive and the disjunctive, in which the latter
would occur wherever an existence-theorem was to be stated. But this
method does not appear to possess any practical advantages.

**91.** It is to be observed that what is fundamental is not particular
propositional functions, but the class-concept *propositional function*. A
propositional function is the class of all propositions which arise from
the variation of a single term, but this is not to be considered as a
definition, for reasons explained in the preceding chapter.

**92.** From propositional functions all other classes can be derived
by definition, with the help of the notion of *such that*. Given a propositional function `φ``x`, the terms such that, when `x` is identified with
any one of them, `φ``x` is true, are the class defined by `φ``x`. This is the
class as many, the class in extension. It is not to be assumed that every
class so obtained has a defining predicate: this subject will be discussed
afresh in Chapter x. But it must be assumed, I think, that a class in
extension is defined by any propositional function, and in particular
that *all* terms form a class, since many propositional functions (*e.g.*
all formal implications) are true of *all* terms. Here, as with formal
implications, it is necessary that the whole propositional function whose
truth defines the class should be kept intact, and not, even where this
is possible for every value of `x`, divided into separate propositional
functions. For example, if `a` and `b` be two classes, defined by `φ``x` and `ψ``x`
respectively, their common part is defined by the product `φ``x` . `ψ``x`, where
the product has to be made for every value of `x`, and then `x` varied
afterwards. If this is not done, we do not necessarily have the *same*
`x` in `φ``x` and `ψ``x`. Thus we do not multiply propositional functions, but
propositions: the new propositional function is the class of products
of corresponding propositions belonging to the previous functions, and
is by no means the product of `φ``x` and `ψ``x`. It is only in virtue of
a definition that the logical product of the classes defined by `φ``x` and `ψ``x`
is the class defined by `φ``x` . `ψ``x`. And wherever a proposition containing
an apparent variable is asserted, what is asserted is the truth, for all
values of the variable or variables, of the propositional function corresponding to the whole proposition, and is never a relation of propositional
functions.

**93.** It appears from the above discussion that the variable is a
very complicated logical entity, by no means easy to analyze correctly.
The following appears to be as nearly correct as any analysis I can make.
Given any proposition (not a propositional function), let `a` be one of
its terms, and let us call the proposition `φ`(`a`). Then in virtue of the
primitive idea of a propositional function, if `x` be any term, we can
consider the proposition `φ`(`x`), which arises from the substitution of `x`
in place of `a`. We thus arrive at the class of all propositions `φ`(`x`).
If all are true, `φ`(`x`) is asserted simply: `φ`(`x`) may then be called a
*formal* truth. In a formal implication, `φ`(`x`), *for every value of* `x`, states
an implication, and the assertion of `φ`(`x`) is the assertion of a *class* of
implications, not of a single implication. If `φ`(`x`) is sometimes true,
the values of `x` which make it true form a class, which is the class defined
by `φ`(`x`): the class is said to *exist* in this case. If `φ`(`x`) is false for all
values of `x`, the class defined by `φ`(`x`) is said not to exist, and as a
matter of fact, as we saw in Chapter vi, there is no such class, if classes
are taken in extension. Thus `x` is, in some sense, the object denoted by
*any term*; yet this can hardly be strictly maintained, for different
variables may occur in a proposition, yet the object denoted by *any term*, one would suppose, is unique. This, however, elicits a new point
in the theory of denoting, namely that *any term* does not denote,
properly speaking, an assemblage of terms, but denotes one term, only
not one particular definite term. Thus *any term* may denote different
terms in different places. We may say: any term has some relation to
any term; and this is quite a different proposition from: any term has
some relation to itself. Thus variables have a kind of individuality.
This arises, as I have tried to show, from propositional functions.
When a propositional function has two variables, it must be regarded
as obtained by successive steps. If the propositional function `φ`(`x`, `y`)
is to be asserted for all values of `x` and `y`, we must consider the assertion,
for all values of `y`, of the propositional function `φ`(`a`, `y`) where `a` is
a constant. This does not involve `y`, and may be represented by `ψ`(`a`).
We then vary `a`, and assert `ψ`(`x`) for all values of `x`. The process is
analogous to double integration; and it is necessary to prove formally
that the order in which the variations are made makes no difference
to the result. The individuality of variables appears to be thus explained. A variable is not *any term* simply, but any term as entering
into a propositional function. We may say, if `φ``x` be a propositional
function, that `x` is *the* term in *any* proposition of the class of propositions whose type is `φ``x`. It thus appears that, as regards propositional
functions, the notions of class, of denoting, and of *any*, are fundamental,
being presupposed in the symbolism employed. With this conclusion,
the analysis of formal implication, which has been one of the principal
problems of Part I, is carried as far as I am able to carry it. May
some reader succeed in rendering it more complete, and in answering the
many questions which I have had to leave unanswered.

Notes

*^{[page 90]} Here “there is a `c`,” where `c` is any class, is defined as equivalent to “If `p` implies `p`, and ‘`x` is a `c`’ implies `p` for all values of `x`, then `p` is true.”

†^{[page 90]} *Was sind und was sollen die Zahlen?* Brunswick, 1893.

‡^{[page 90]} *Allgemeine Arithmetik*, Leipzig, 1886.

Relations.

**94.** Next after subject-predicate propositions come two types of
propositions which appear equally simple. These are the propositions
in which a relation is asserted between two terms, and those in which
two terms are said to be two. The latter class of propositions will be
considered hereafter; the former must be considered at once. It has
often been held that every proposition can be reduced to one of the
subject-predicate type, but this view we shall, throughout the present
work, find abundant reason for rejecting. It might be held, however,
that all propositions not of the subject-predicate type, and not asserting
numbers, could be reduced to propositions containing two terms and
a relation. This opinion would be more difficult to refute, but this too,
we shall find, has no good grounds in its favour*. We may therefore
allow that there are relations having more than two terms; but as these
are more complex, it will be well to consider first such as have two
terms only.

A relation between two terms is a concept which occurs in a
proposition in which there are two terms not occurring as concepts†,
and in which the interchange of the two terms gives a different proposition. This last mark is required to distinguish a relational
proposition from one of the type “`a` and `b` are two,” which is identical
with “`b` and `a` are two.” A relational proposition may be symbolized
by `a``R``b`, where `R` is the relation and `a` and `b` are the terms; and `a``R``b`
will then always, provided `a` and `b` are not identical, denote a different
proposition from `b``R``a`. That is to say, it is characteristic of a relation
of two terms that it proceeds, so to speak, *from* one *to* the other. This
is what may be called the *sense* of the relation, and is, as we shall find,
the source of order and series. It must be held as an axiom that *aRb*
implies and is implied by a relational proposition `b``R`′`a`, in which the
relation `R`′ proceeds from `b` to `a`, and may or may not be the same
relation as `R`. But even when `a``R``b` implies and is implied by `b``R``a`,
it must be strictly maintained that these are different propositions.
We may distinguish the term *from* which the relation proceeds as the
*referent*, and the term *to* which it proceeds as the *relatum*. The sense
of a relation is a fundamental notion, which is not capable of definition.
The relation which holds between `b` and `a` whenever `R` holds between
`a` and `b` will be called the *converse* of `R`, and will be denoted (following
Schröder) by ˘`R`. The relation of `R` to ˘`R` is the relation of oppositeness,
or difference of sense; and this must not be defined (as would seem at
first sight legitimate) by the above mutual implication in any single
case, but only by the fact of its holding for all cases in which the given
relation occurs. The grounds for this view are derived from certain
propositions in which terms are related to themselves not-symmetrically,
*i.e.* by a relation whose converse is not identical with itself. These
propositions must now be examined.

**95.** There is a certain temptation to affirm that no term can he
related to itself; and there is a still stronger temptation to affirm that,
if a term can be related to itself, the relation must be symmetrical,
*i.e.* identical with its converse. But both these temptations must be
resisted. In the first place, if no term were related to itself, we should
never be able to assert self-identity, since this is plainly a relation.
But since there is such a notion as identity, and since it seems undeniable
that every term is identical with itself, we must allow that a term may
be related to itself. Identity, however, is still a symmetrical relation,
and may be admitted without any great qualms. The matter becomes
far worse when we have to admit not-symmetrical relations of terms
to themselves. Nevertheless the following propositions seem undeniable;
Being is, or has being; 1 is one, or has unity; concept is conceptual;
term is a term; class-concept is a class-concept. All these are of one
of the three equivalent types which we distinguished at the beginning of
Chapter v, which may be called respectively subject-predicate propositions, propositions asserting the relation of predication, and propositions
asserting membership of a class. What we have to consider is, then,
the fact that a predicate may be predicable of itself. It is necessary, for
our present purpose, to take our propositions in the second form (Socrates
has humanity), since the subject-predicate form is not in the above sense
relational. We may take, as the type of such propositions, “unity has
unity.” Now it is certainly undeniable that the relation of predication
is asymmetrical, since subjects cannot in general be predicated of their
predicates. Thus “unity has unity” asserts one relation of unity to
itself, and implies another, namely the converse relation: unity has
to itself both the relation of subject to predicate, and the relation of
predicate to subject. Now if the referent and the relatum are identical,
it is plain that the relatum has to the referent the same relation as the
referent has to the relatum. Hence if the converse of a relation in
a particular case were defined by mutual implication in that particular
case, it would appear that, in the present case, our relation has two
converses, since two different relations of relatum to referent are implied
by “unity has unity.” We must therefore define the converse of a
relation by the fact that `a``R``b` implies and is implied by `b`˘`R``a` whatever
`a` and `b` may be, and whether or not the relation `R` holds between them.
That is to say, `a` and `b` are here essentially variables, and if we give
them any constant value, we may find that `a``R``b` implies and is implied
by `b``R`′`a`, where `R`′ is some relation other than ˘`R`.

Thus three points must be noted with regard to relations of two
terms: (1) they all have sense, so that, provided `a` and `b` are not
identical, we can distinguish `a``R``b` from `b``R``a`; (2) they all have a
converse, *i.e.* a relation ˘`R` such that `a``R``b` implies and is implied by
`b`˘`R``a`, whatever `a` and `b` may be; (3) some relations hold between a
term and itself, and such relations are not necessarily symmetrical,
*i.e.* there may be two different relations, which are each other’s converses, and which both hold between a term and itself.

**96.** For the general theory of relations, especially in its mathematical developments, certain axioms relating classes and relations are
of great importance. It is to be held that to have a given relation to a
given term is a predicate, so that all terms having this relation to this
term form a class. It is to be held further that to have a given relation
at all is a predicate, so that all referents with respect to a given relation
form a class. It follows, by considering the converse relation, that all
relata also form a class. These two classes I shall call respectively the
*domain* and the *converse domain* of the relation; the logical sum of the
two I shall call the *field* of the relation.

The axiom that all referents with respect to a given relation form a class seems, however, to require some limitation, and that on account of the contradiction mentioned at the end of Chapter vi. This contradiction may be stated as follows. We saw that some predicates can be predicated of themselves. Consider now those of which this is not the case. These are the referents (and also the relata) in what seems like a complex relation, namely the combination of non-predicability with identity. But there is no predicate which attaches to all of them and to no other terms. For this predicate will either be predicable or not predicable of itself. If it is predicable of itself, it is one of those referents by relation to which it was defined, and therefore, in virtue of their definition, it is not predicable of itself. Conversely, if it is not predicable of itself, then again it is one of the said referents, of all of which (by hypothesis) it is predicable, and therefore again it is predicable of itself. This is a contradiction, which shows that all the referents considered have no exclusive common predicate, and therefore, if defining predicates are essential to classes, do not form a class.

The matter may be put otherwise. In defining the would-be class of
predicates, all those not predicable of themselves have been used up.
The common predicate of all these predicates cannot be one of them,
since for each of them there is at least one predicate (namely itself) of
which it is not predicable. But again, the supposed common predicate
cannot be any other predicate, for if it were, it would be predicable of
itself, *i.e.* it would be a member of the supposed class of predicates, since
these were defined as those of which it is predicable. Thus no predicate
is left over which could attach to all the predicates considered.

It follows from the above that not every definable collection of terms forms a class defined by a common predicate. This fact must be borne in mind, and we must endeavour to discover what properties a collection must have in order to form such a class. The exact point established by the above contradiction may be stated as follows: A proposition apparently containing only one variable may not be equivalent to any proposition asserting that the variable in question has a certain predicate. It remains an open question whether every class must have a defining predicate.

That all terms having a given relation to a given term form a class
defined by an exclusive common predicate results from the doctrine of
Chapter vii, that the proposition `a``R``b` can be analyzed into the subject
`a` and the assertion `R``b`. To be a term of which `R``b` can be asserted
appears to be plainly a predicate. But it does not follow, I think,
that to be a term of which, for some value of `y`, `R``y` can be asserted, is
a predicate. The doctrine of propositional functions requires, however,
that all terms having the latter property should form a class. This
class I shall call the *domain* of the relation `R` as well as the class of
referents. The domain of the converse relation will be also called the
converse domain, as well as the class of relata. The two domains
together will be called the *field* of the relation—a notion chiefly important as regards series. Thus if paternity be the relation, fathers form
its domain, children its converse domain, and fathers and children
together its field.

It may be doubted whether a proposition `a``R``b` can be regarded as
asserting `a``R` of `b`, or whether only ˘`R``a` can be asserted of `b`. In other
words, is a relational proposition only an assertion concerning the
referent, or also an assertion concerning the relatum? If we take the
latter view, we shall have, connected with (say) “`a` is greater than `b`,”
four assertions, namely “is greater than `b`,” “`a` is greater than,” “is less
than `a`” and “`b` is less than.” I am inclined myself to adopt this view,
but I know of no argument on either side.

**97.** We can form the logical sum and product of two relations or
of a class of relations exactly as in the case of classes, except that here
we have to deal with double variability. In addition to these ways of
combination, we have also the relative product, which is in general non-commutative, and therefore requires that the number of factors should
be finite. If `R`, `S` be two relations, to say that their relative product
`R``S` holds between two terms `x`, `z` is to say that there is a term `y` to
which `x` has the relation `R`, and which itself has the relation `S` to `z`. Thus
brother-in-law is the relative product of wife and brother or of sister
and husband: father-in-law is the relative product of wife and father,
whereas the relative product of father and wife is mother or step-mother.

**98.** There is a temptation to regard a relation as definable in
extension as a class of couples. This has the formal advantage that it
avoids the necessity for the primitive proposition asserting that every
couple has a relation holding between no other pair of terms. But it is
necessary to give sense to the couple, to distinguish the referent from the
relatum: thus a couple becomes essentially distinct from a class of two
terms, and must itself be introduced as a primitive idea. It would seem,
viewing the matter philosophically, that sense can only be derived from
some relational proposition, and that the assertion that `a` is referent and
`b` relatum already involves a purely relational proposition in which `a` and
`b` are terms, though the relation asserted is only the general one of
referent to relatum. There are, in fact, concepts such as *greater*, which
occur otherwise than as terms in propositions having two terms (§§48, 54);
and no doctrine of couples can evade such propositions. It seems therefore more correct to take an intensional view of relations, and to identify
them rather with class-concepts than with classes. This procedure is
formally more convenient, and seems also nearer to the logical facts.
Throughout Mathematics there is the same rather curious relation of
intensional and extensional points of view: the symbols other than
variable terms (*i.e.* the variable class-concepts and relations) stand for
intensions, while the actual objects dealt with are always extensions.
Thus in the calculus of relations, it is classes of couples that are relevant,
but the symbolism deals with them by means of relations. This is
precisely similar to the state of things explained in relation to classes,
and it seems unnecessary to repeat the explanations at length.

**99.** Mr Bradley, in *Appearance and Reality*, Chapter iii, has based
an argument against the reality of relations upon the endless regress
arising from the fact that a relation which relates two terms must
be related to each of them. The endless regress is undeniable, if
relational propositions are taken to be ultimate, but it is very doubtful
whether it forms any logical difficulty. We have already had occasion
(§55) to distinguish two kinds of regress, the one proceeding merely to
perpetually new implied propositions, the other in the meaning of a
proposition itself; of these two kinds, we agreed that the former, since
the solution of the problem of infinity, has ceased to be objectionable,
while the latter remains inadmissible. We have to inquire which kind
of regress occurs in the present instance. It may be urged that it is
part of the very meaning of a relational proposition that the relation
involved should have to the terms the relation expressed in saying that
it relates them, and that this is what makes the distinction, which we
formerly (§54) left unexplained, between a relating relation and a relation
in itself. It may be urged, however, against this view, that the assertion
of a relation between the relation and the terms, though implied, is no
part of the original proposition, and that a relating relation is distinguished from a relation in itself by the indefinable element of assertion
which distinguishes a proposition from a concept. Against this it
might be retorted that, in the concept “difference of `a` and `b`,” difference
relates `a` and `b` just as much as in the proposition “`a` and `b` differ”; but
to this it may be rejoined that we found the difference of `a` and `b`, except
in so far as some specific point of difference may be in question, to be
indistinguishable from bare difference. Thus it seems impossible to
prove that the endless regress involved is of the objectionable kind.
We may distinguish, I think, between “`a` exceeds `b`” and “`a` is greater
than `b`,” though it would be absurd to deny that people usually mean
the same thing by these two propositions. On the principle, from which
I can see no escape, that every genuine word must have some meaning,
the *is* and *than* must form part of “`a` is greater than `b`,” which thus
contains more than two terms and a relation. The *is* seems to state
that `a` has to *greater* the relation of referent, while the *than* states
similarly that `b` has to *greater* the relation of relatum. But “`a` exceeds
`b`” may be held to express solely the relation of `a` to `b`, without including any of the implications of further relations. Hence we shall
have to conclude that a relational proposition `a``R``b` does not include
in its *meaning* any relation of `a` or `b` to `R`, and that the endless regress,
though undeniable, is logically quite harmless. With these remarks,
we may leave the further theory of relations to later Parts of the present
work.

Notes

*^{[page 95]} See *inf*., Part IV, Chap. xxv, §200.

†^{[page 95]} This description, as we saw above (§48), excludes the pseudo-relation of subject to predicate.

The Contradiction.

**100.** Before taking leave of fundamental questions, it is necessary
to examine more in detail the singular contradiction, already mentioned,
with regard to predicates not predicable of themselves. Before attempting to solve this puzzle, it will be well to make some deductions connected
with it, and to state it in various different forms. I may mention that I
was led to it in the endeavour to reconcile Cantor’s proof that there can
be no greatest cardinal number with the very plausible supposition that
the class of all terms (which we have seen to be essential to all formal
propositions) has necessarily the greatest possible number of members*.

Let `w` be a class concept which can be asserted of itself, *i.e.* such that
“`w` is a `w`.” Instances are *class-concept*, and the negations of ordinary
class-concepts, *e.g.* not-man. Then (`α`) if `w` be contained in another class `v`,
since `w` is a `w`, `w` is a `v`; consequently there is a term of `v` which is
a class-concept that can be asserted of itself. Hence by contraposition,
(`β`) if `u` be a class-concept none of whose members are class-concepts
that can be asserted of themselves, no class-concept contained in `u` can
be asserted of itself. Hence further, (`γ`) if `u` be any class-concept whatever, and `u`′ the class-concept of those members of `u` which are not
predicable of themselves, this class-concept is contained in itself, and
none of its members are predicable of themselves; hence by (`β`) `u`′ is not
predicable of itself. Thus `u`′ is not a `u`′, and is therefore not a `u`; for
the terms of `u` that are not terms of `u`′ are all predicable of themselves,
which `u`′ is not. Thus (`δ`) if `u` be any class-concept whatever, there is a
class-concept contained in `u` which is not a member of `u`, and is also one
of those class-concepts that are not predicable of themselves. So far, our
deductions seem scarcely open to question. But if we now take the last
of them, and admit the class of those class-concepts that cannot be
asserted of themselves, we find that this class must contain a class-concept
not a member of itself and yet not belonging to the class in question.

We may observe also that, in virtue of what we have proved in (`β`), the
class of class-concepts which cannot be asserted of themselves, which we
will call `w`, contains as members of itself all its sub-classes, although it is
easy to prove that every class has more sub-classes than terms. Again,
if `y` be any term of `w`, and `w`′ be the whole of `w` except `y`, then `w`′, being
a sub-class of `w`, is not a `w`′ but is a `w`, and therefore is `y`. Hence each
class-concept which is a term of `w` has all other terms of `w` as its
extension. It follows that the concept *bicycle* is a teaspoon, and *teaspoon*
is a bicycle. This is plainly absurd, and any number of similar
absurdities can be proved.

**101.** Let us leave these paradoxical consequences, and attempt the
exact statement of the contradiction itself. We have first the statement
in terms of predicates, which has been given already. If `x` be a predicate,
`x` may or may not be predicable of itself. Let us assume that “not-predicable of oneself” is a predicate. Then to suppose either that this
predicate is, or that it is not, predicable of itself, is self-contradictory.
The conclusion, in this case, seems obvious: “not-predicable of oneself”
is not a predicate.

Let us now state the same contradiction in terms of class-concepts.
A class-concept may or may not be a term of its own extension. “Class-concept which is not a term of its own extension” appears to be a class-concept. But if it is a term of its own extension, it is a class-concept
which is not a term of its own extension, and *vice versâ*. Thus we must
conclude, against appearances, that “class-concept which is not a term of
its own extension” is not a class-concept.

In terms of classes the contradiction appears even more extraordinary.
A class as one may be a term of itself as many. Thus the class of all
classes is a class; the class of all the terms that are not men is not a man,
and so on. Do all the classes that have this property form a class? If
so, is it as one a member of itself as many or not? If it is, then it is
one of the classes which, as ones, are not members of themselves as many,
and *vice versâ*. Thus we must conclude again that the classes which as
ones are not members of themselves as many do not form a class—or
rather, that they do not form a class as one, for the argument cannot
show that they do not form a class as many.

**102.** A similar result, which, however, does not lead to a contradiction, may be proved concerning any relation. Let `R` be a relation, and
consider the class `w` of terms which do not have the relation `R` to themselves. Then it is impossible that there should be any term `a` to which
all of them and no other terms have the relation `R`. For, if there were
such a term, the propositional function “`x` does not have the relation `R`
to `x`” would be equivalent to “`x` has the relation `R` to `a`.” Substituting
`a` for `x` throughout, which is legitimate since the equivalence is formal,
we find a contradiction. When in place of `R` we put ε, the relation of
a term to a class-concept which can be asserted of it, we get the above
contradiction. The reason that a contradiction emerges here is that
we have taken it as an axiom that any propositional function containing
only one variable is equivalent to asserting membership of a class defined
by the propositional function. Either this axiom, or the principle that
every class can be taken as one term, is plainly false, and there is no
fundamental objection to dropping either. But having dropped the
former, the question arises: Which propositional functions define classes
which are single terms as well as many, and which do not? And with
this question our real difficulties begin.

Any method by which we attempt to establish a one-one or many-one correlation of all terms and all propositional functions must omit at
least one propositional function. Such a method would exist if all
propositional functions could be expressed in the form … ε`u`, since this
form correlates `u` with … ε`u`. But the impossibility of any such correlation is proved as follows. Let `φ`` _{x}` be a propositional function correlated
with

**103.** The first method which suggests itself is to seek an ambiguity
in the notion of ε. But in Chapter vi we distinguished the various
meanings as far as any distinction seemed possible, and we have just
seen that with each meaning the same contradiction emerges. Let us,
however, attempt to state the contradiction throughout in terms of
propositional functions. Every propositional function which is not null,
we supposed, defines a class, and every class can certainly be defined by
a propositional function. Thus to say that a class as one is not a
member of itself as many is to say that the class as one does not satisfy
the function by which itself as many is defined. Since all propositional
functions except such as are null define classes, all will be used up, in
considering all classes having the above property, except such as do not
have the above property. If any propositional function were satisfied
by every class having the above property, it would therefore necessarily
be one satisfied also by the class `w` of all such classes considered as a
single term. Hence the class `w` does not itself belong to the class `w`,
and therefore there must be some propositional function satisfied by the
terms of `w` but not by `w` itself. Thus the contradiction re-emerges, and
we must suppose, either that there is no such entity as `w`, or that there
is no propositional function satisfied by its terms and by no others.

It might be thought that a solution could be found by denying the
legitimacy of variable propositional functions. If we denote by `k`_{φ}, for
the moment, the class of values satisfying `φ`, our propositional function
is the denial of `φ`(`k`_{φ}) where `φ` is the variable. The doctrine of
Chapter vii, that `φ` is not a separable entity, might make such a variable
seem illegitimate; but this objection can be overcome by substituting for `φ` the class of propositions `φ``x`, or the relation of `φ``x` to `x`.
Moreover it is impossible to exclude variable propositional functions
altogether. Wherever a variable class or a variable relation occurs,
we have admitted a variable propositional function, which is thus
essential to assertions about every class or about every relation. The
definition of the domain of a relation, for example, and all the general
propositions which constitute the calculus of relations, would be swept
away by the refusal to allow this type of variation. Thus we require
some further characteristic by which to distinguish two kinds of variation. This characteristic is to be found, I think, in the independent
variability of the function and the argument. In general, `φ``x` is itself
a function of two variables, `φ` and `x`; of these, either may be given a
constant value, and either may be varied without reference to the other.
But in the type of propositional functions we are considering in this
Chapter, the argument is itself a function of the propositional function:
instead of `φ``x`, we have `φ`{`f`(`φ`)}, where `f`(`φ`) is defined as a function of
`φ`. Thus when `φ` is varied, the argument of which `φ` is asserted is
varied too. Thus “`x` is an `x`” is equivalent to: “`φ` can be asserted of
the class of terms satisfying `φ`,” this class of terms being `x`. If here
`φ` is varied, the argument is varied at the same time in a manner
dependent upon the variation of `φ`. For this reason, `φ`{`f`(`φ`)}, though
it is a definite proposition when `x` is assigned, is not a propositional
function, in the ordinary sense, when `x` is variable. Propositional
functions of this doubtful type may be called *quadratic forms*, because
the variable enters into them in a way somewhat analogous to that in
which, in Algebra, a variable appears in an expression of the second
degree.

**104.** Perhaps the best way to state the suggested solution is to say
that, if a collection of terms can only be defined by a variable propositional function, then, though a class as many may be admitted,
a class as one must be denied. When so stated, it appears that propositional functions may be varied, provided the resulting collection is
never itself made into the subject in the original propositional function.
In such cases there is only a class as many, not a class as one. We took
it as axiomatic that the class as one is to be found wherever there is
a class as many; but this axiom need not be universally admitted,
and appears to have been the source of the contradiction. By denying
it, therefore, the whole difficulty will be overcome.

A class as one, we shall say, is an object of the same *type* as its
terms; *i.e.* any propositional function `φ`(`x`) which is significant when one
of the terms is substituted for `x` is also significant when the class as one
is substituted. But the class as one does not always exist, and the class
as many is of a different type from the terms of the class, even when the
class has only one term, *i.e.* there are propositional functions `φ`(`u`) in
which `u` may be the class as many, which are meaningless if, for `u`, we
substitute one of the terms of the class. And so “`x` is one among `x`’s”
is not a proposition at all if the relation involved is that of a term to its
class as many; and this is the only relation of whose presence a propositional function always assures us. In this view, a class as many may
be a logical subject, but in propositions of a different kind from those in
which its terms are subjects; of any object other than a single term, the
question whether it is one or many will have different answers according
to the proposition in which it occurs. Thus we have “Socrates is one
among men,” in which men are plural; but “men are one among species
of animals,” in which men are singular. It is the distinction of logical
types that is the key to the whole mystery*.

**105.** Other ways of evading the contradiction, which might be
suggested, appear undesirable, on the ground that they destroy too
many quite necessary kinds of propositions. It might be suggested
that identity is introduced in “`x` is not an `x`” in a way which is not
permissible. But it has been already shown that relations of terms
to themselves are unavoidable, and it may be observed that suicides
or self-made men or the heroes of Smiles’s *Self-Help* are all defined
by relations to themselves. And generally, identity enters in a very
similar way into formal implication, so that it is quite impossible to
reject it.

A natural suggestion for escaping from the contradiction would be
to demur to the notion of *all* terms or of *all* classes. It might be
urged that no such sum-total is conceivable; and if *all* indicates a whole,
our escape from the contradiction requires us to admit this. But we
have already abundantly seen that if this view were maintained against
*any* term, all formal truth would be impossible, and Mathematics, whose
characteristic is the statement of truths concerning *any* term, would be
abolished at one stroke. Thus the correct statement of formal truths
requires the notion of *any* term or *every* term, but not the collective
notion of *all* terms.

It should be observed, finally, that no peculiar philosophy is involved in the above contradiction, which springs directly from common sense, and can only be solved by abandoning some common-sense assumption. Only the Hegelian philosophy, which nourishes itself on contradictions, can remain indifferent, because it finds similar problems everywhere. In any other doctrine, so direct a challenge demands an answer, on pain of a confession of impotence. Fortunately, no other similar difficulty, so far as I know, occurs in any other portion of the Principles of Mathematics.

**106.** We may now briefly review the conclusions arrived at in
Part I. Pure Mathematics was defined as the class of propositions
asserting formal implications and containing no constants except logical
constants. And logical constants are: Implication, the relation of a
term to a class of which it is a member, the notion of *such that*, the
notion of relation, and such further notions as are involved in formal
implication, which we found (§93) to be the following: propositional
function, class*, denoting, and *any* or *every* term. This definition brought
Mathematics into very close relation to Logic, and made it practically
identical with Symbolic Logic. An examination of Symbolic Logic justified the above enumeration of mathematical indefinables. In Chapter iii
we distinguished implication and formal implication. The former holds
between any two propositions provided the first be false or the second
true. The latter is not a relation, but the assertion, for every value
of the variable or variables, of a propositional function which, for every
value of the variable or variables, asserts an implication. Chapter iv
distinguished what may be called *things* from predicates and relations
(including the *is* of predications among relations for this purpose). It
was shown that this distinction is connected with the doctrine of
substance and attributes, but does not lead to the traditional results.
Chapters v and vi developed the theory of predicates. In the former
of these chapters it was shown that certain concepts, derived from
predicates, occur in propositions not *about* themselves, but about combinations of terms, such as are indicated by *all*, *every*, *any*, *a*, *some*, and
*the*. Concepts of this kind, we found, are fundamental in Mathematics,
and enable us to deal with infinite classes by means of propositions of
finite complexity. In Chapter vi we distinguished predicates, class-concepts, concepts of classes, classes as many, and classes as one. We
agreed that single terms, or such combinations as result from *and*, are
classes, the latter being classes as many; and that classes as many
are the objects denoted by concepts of classes, which are the plurals
of class-concepts. But in the present chapter we decided that it is
necessary to distinguish a single term from the class whose only member
it is, and that consequently the null-class may be admitted.

In Chapter vii we resumed the study of the verb. Subject-predicate
propositions, and such as express a fixed relation to a fixed term, could
be analyzed, we found, into a subject and an assertion; but this analysis
becomes impossible when a given term enters into a proposition in a
more complicated manner than as referent of a relation. Hence it
became necessary to take *propositional function* as a primitive notion.
A propositional function of one variable is any proposition of a set
defined by the variation of a single term, while the other terms remain
constant. But in general it is impossible to define or isolate the
constant element in a propositional function, since what remains, when
a certain term, wherever it occurs, is left out of a proposition, is in
general no discoverable kind of entity. Thus the term in question
must be not simply omitted, but replaced by a *variable*.

The notion of the variable, we found, is exceedingly complicated.
The `x` is not simply *any* term, but any term with a certain individuality;
for if not, any two variables would be indistinguishable. We agreed
that a variable is any term *quâ* term in a certain propositional function,
and that variables are distinguished by the propositional functions in
which they occur, or, in the case of several variables, by the place they
occupy in a given multiply variable propositional function. A variable,
we said, is *the* term in *any* proposition of the set denoted by a given
propositional function.

Chapter ix pointed out that relational propositions are ultimate,
and that they all have *sense*: *i.e.* the relation being the concept as such
in a proposition with two terms, there is another proposition containing
the same terms and the same concept as such, as in “`A` is greater
than `B`” and “`B` is greater than `A`.” These two propositions, though
different, contain precisely the same constituents. This is a characteristic
of relations, and an instance of the loss resulting from analysis. Relations, we agreed, are to be taken intensionally, not as classes of couples*.

Finally, in the present chapter, we examined the contradiction resulting from the apparent fact that, if `w` be the class of all classes which
as single terms are not members of themselves as many, then `w` as one
can be proved both to be and not to be a member of itself as many.
The solution suggested was that it is necessary to distinguish various
types of objects, namely terms, classes of terms, classes of classes, classes
of couples of terms, and so on; and that a propositional function `φ``x` in
general requires, if it is to have any meaning, that `x` should belong to
some one type. Thus `x`ε`x` was held to be meaningless, because ε requires
that the relatum should be a class composed of objects which are of the
type of the referent. The class as one, where it exists, is, we said, of the
same type as its constituents; but a quadratic propositional function in
general appears to define only a class as many, and the contradiction
proves that the class as one, if it ever exists, is certainly sometimes
absent.

Notes

*^{[page 101]} See Part V, Chap. xliii, §344ff.

*^{[page 105]} On this subject, see Appendix.

*^{[page 106]} The notion of *class* in general, we decided, could be replaced, as an indefinable, by that of the class of propositions defined by a propositional function.

*^{[page 107]} On this point, however, see Appendix.

NUMBER.

Definition of Cardinal Numbers.

**107.** We have now briefly reviewed the apparatus of general logical
notions with which Mathematics operates. In the present Part, it is to
be shown how this apparatus suffices, without new indefinables or new
postulates, to establish the whole theory of cardinal integers as a special
branch of Logic*. No mathematical subject has made, in recent years,
greater advances than the theory of Arithmetic. The movement in
favour of correctness in deduction, inaugurated by Weierstrass, has been
brilliantly continued by Dedekind, Cantor, Frege, and Peano, and attains
what seems its final goal by means of the logic of relations. As the
modern mathematical theory is but imperfectly known even by most
mathematicians, I shall begin this Part by four chapters setting forth
its outlines in a non-symbolic form. I shall then examine the process
of deduction from a philosophical standpoint, in order to discover, if
possible, whether any unperceived assumptions have covertly intruded
themselves in the course of the argument.

**108.** It is often held that both number and particular numbers are
indefinable. Now definability is a word which, in Mathematics, has a
precise sense, though one which is relative to some given set of notions†.
Given any set of notions, a term is definable by means of these notions
when, and only when, it is the only term having to certain of these
notions a certain relation which itself is one of the said notions. But
philosophically, the word *definition* has not, as a rule, been employed in
this sense; it has, in fact, been restricted to the analysis of an idea
into its constituents. This usage is inconvenient and, I think, useless;
moreover it seems to overlook the fact that wholes are *not*, as a
rule, determinate when their constituents are given, but are themselves
new entities (which may be in some sense simple), defined, in the
mathematical sense, by certain relations to their constituents. I shall,
therefore, in future, ignore the philosophical sense, and speak only of
mathematical definability. I shall, however, restrict this notion more
than is done by Professor Peano and his disciples. They hold that the
various branches of Mathematics have various indefinables, by means of
which the remaining ideas of the said subjects are defined. I hold—and it is an important part of my purpose to prove—that all Pure
Mathematics (including Geometry and even rational Dynamics) contains
only one set of indefinables, namely the fundamental logical concepts
discussed in Part I. When the various logical constants have been
enumerated, it is somewhat arbitrary which of them we regard as
indefinable, though there are apparently some which must be indefinable
in any theory. But my contention is, that the indefinables of Pure
Mathematics are all of this kind, and that the presence of any other
indefinables indicates that our subject belongs to Applied Mathematics.
Moreover, of the three kinds of definition admitted by Peano—the
nominal definition, the definition by postulates, and the definition by
abstraction*—I recognize only the nominal: the others, it would seem,
are only necessitated by Peano’s refusal to regard relations as part of the
fundamental apparatus of logic, and by his somewhat undue haste in
regarding as an individual what is really a class. These remarks will be
best explained by considering their application to the definition of
cardinal numbers.

**109.** It has been common in the past, among those who regarded
numbers as definable, to make an exception as regards the number 1,
and to define the remainder by its means. Thus 2 was 1 + 1, 3 was
2 + 1, and so on. This method was only applicable to finite numbers,
and made a tiresome difference between 1 and other numbers; moreover
the meaning of + was commonly not explained. We are able now-a-days to improve greatly upon this method. In the first place, since
Cantor has shown how to deal with the infinite, it has become both
desirable and possible to deal with the fundamental properties of numbers
in a way which is equally applicable to finite and infinite numbers. In
the second place, the logical calculus has enabled us to give an exact
definition of arithmetical addition; and in the third place, it has become
as easy to define 0 and 1 as to define any other number. In order to
explain how this is done, I shall first set forth the definition of numbers
by abstraction; I shall then point out formal defects in this definition,
and replace it by a nominal definition.

Numbers are, it will be admitted, applicable essentially to classes.
It is true that, where the number is finite, individuals may be enumerated
to make up the given number, and may be counted one by one without
any mention of a class-concept. But all finite collections of individuals
form classes, so that what results is after all the number of a class.
And where the number is infinite, the individuals cannot be enumerated,
but must be defined by intension, *i.e.* by some common property in
virtue of which they form a class. Thus when any class-concept is
given, there is a certain number of individuals to which this class-concept
is applicable, and the number may therefore be regarded as a property
of the class. It is this view of numbers which has rendered possible the
whole theory of infinity, since it relieves us of the necessity of enumerating the individuals whose number is to be considered. This view
depends fundamentally upon the notion of *all*, the numerical conjunction
as we agreed to call it (§59). *All men*, for example, denotes men conjoined in a certain way; and it is as thus denoted that they have a
number. Similarly *all numbers* or *all points* denotes numbers or points
conjoined in a certain way, and as thus conjoined numbers or points have
a number. Numbers, then, are to be regarded as properties of classes.

The next question is: Under what circumstances do two classes have
the same number? The answer is, that they have the same number
when their terms can be correlated one to one, so that any one term of
either corresponds to one and only one term of the other. This requires
that there should be some one-one relation whose domain is the one
class and whose converse domain is the other class. Thus, for example,
if in a community all the men and all the women are married, and
polygamy and polyandry are forbidden, the number of men must be the
same as the number of women. It might be thought that a one-one
relation could not be defined except by reference to the number 1. But
this is not the case. A relation is one-one when, if `x` and `x`′ have the
relation in question to `y`, then `x` and `x`′ are identical; while if `x` has the
relation in question to `y` and `y`′, then `y` and `y`′ are identical. Thus it is
possible, without the notion of unity, to define what is meant by a one-one relation. But in order to provide for the case of two classes which
have no terms, it is necessary to modify slightly the above account of
what is meant by saying that two classes have the same number. For if
there are no terms, the terms cannot be correlated one to one. We
must say: Two classes have the same number when, and only when, there
is a one-one relation whose domain includes the one class, and which is
such that the class of correlates of the terms of the one class is identical
with the other class. From this it appears that two classes having no
terms have always the same number of terms; for if we take any one-one relation whatever, its domain includes the null-class, and the class
of correlates of the null-class is again the null-class. When two classes
have the same number, they are said to be *similar*.

Some readers may suppose that a definition of what is meant by saying that two classes have the same number is wholly unnecessary. The way to find out, they may say, is to count both classes. It is such notions as this which have, until very recently, prevented the exhibition of Arithmetic as a branch of Pure Logic. For the question immediately arises: What is meant by counting? To this question we usually get only some irrelevant psychological answer, as, that counting consists in successive acts of attention. In order to count 10, I suppose that ten acts of attention are required: certainly a most useful definition of the number 10! Counting has, in fact, a good meaning, which is not psychological. But this meaning is highly complex; it is only applicable to classes which can be well-ordered, which are not known to be all classes; and it only gives the number of the class when this number is finite—a rare and exceptional case. We must not, therefore, bring in counting where the definition of numbers is in question.

The relation of similarity between classes has the three properties of
being reflexive, symmetrical, and transitive; that is to say, if `u`, `v`, `w` be
classes, `u` is similar to itself; if `u` be similar to `v`, `v` is similar to `u`; and
if `u` be similar to `v`, and `v` to `w`, then `u` is similar to `w`. These properties
all follow easily from the definition. Now these three properties of a
relation are held by Peano and common sense to indicate that when the
relation holds between two terms, those two terms have a certain common
property, and `v``i``c``e` `v``e``r``s`â. This common property we call their number*.
This is the definition of numbers by abstraction.

**110.** Now this definition by abstraction, and generally the process
employed in such definitions, suffers from an absolutely fatal formal
defect: it does not show that only one object satisfies the definition†.
Thus instead of obtaining *one* common property of similar classes, which
is *the* number of the classes in question, we obtain a *class* of such
properties, with no means of deciding how many terms this class contains.
In order to make this point clear, let us examine what is meant, in the
present instance, by a common property. What is meant is, that any
class has to a certain entity, its number, a relation which it has to nothing
else, but which all similar classes (and no other entities) have to the said
number. That is, there is a many-one relation which every class has to
its number and to nothing else. Thus, so far as the definition by
abstraction can show, any set of entities to each of which some class has
a certain many-one relation, and to one and only one of which any given
class has this relation, and which are such that all classes similar to a
given class have this relation to one and the same entity of the set,
appear as the set of numbers, and any entity of this set is *the* number of
some class. If, then, there are many such sets of entities—and it is easy
to prove that there are an infinite number of them—every class will
have many numbers, and the definition wholly fails to define *the* number
of a class. This argument is perfectly general, and shows that definition
by abstraction is never a logically valid process.

**111.** There are two ways in which we may attempt to remedy this
defect. One of these consists in defining as *the* number of a class the
whole class of entities, chosen one from each of the above sets of entities,
to which all classes similar to the given class (and no others) have some
many-one relation or other. But this method is practically useless, since
all entities, without exception, belong to every such class, so that every
class will have as its number the class of all entities of every sort and
description. The other remedy is more practicable, and applies to all
the cases in which Peano employs definition by abstraction. This
method is, to define as the number of a class the class of all classes
similar to the given class. Membership of this class of classes (considered
as a predicate) is a common property of all the similar classes and of no
others; moreover every class of the set of similar classes has to the set
a relation which it has to nothing else, and which every class has to its
own set. Thus the conditions are completely fulfilled by this class of
classes, and it has the merit of being determinate when a class is given,
and of being different for two classes which are not similar. This, then,
is an irreproachable definition of the number of a class in purely logical
terms.

To regard a number as a class of classes must appear, at first sight,
a wholly indefensible paradox. Thus Peano (*F.* 1901, §32) remarks that
“we cannot identify the number of [a class] `a` with the class of classes in
question [*i.e.* the class of classes similar to `a`], for these objects have
different properties.” He does not tell us what these properties are, and
for my part I am unable to discover them. Probably it appeared to him
immediately evident that a number is not a class of classes. But something may be said to mitigate the appearance of paradox in this view.
In the first place, such a word as *couple* or *trio* obviously does denote a
class of classes. Thus what we have to say is, for example, that “two
men” means “logical product of class of men and couple,” and “there are
two men” means “there is a class of men which is also a couple.” In the
second place, when we remember that a class-concept is not itself a collection, but a property by which a collection is defined, we see that, if we
define the number as the class-concept, not the class, a number is really
defined as a common property of a set of similar classes and of nothing
else. This view removes the appearance of paradox to a great degree.
There is, however, a philosophical difficulty in this view, and generally in
the connection of classes and predicates. It may be that there are many
predicates common to a certain collection of objects and to no others. In
this case, these predicates are all regarded by Symbolic Logic as equivalent,
and any one of them is said to be equal to any other. Thus if the
predicate were defined by the collection of objects, we should not obtain,
in general, a single predicate, but a class of predicates; for this class of
predicates we should require a new class-concept, and so on. The only
available class-concept would be “predicability of the given collection of
terms and of no others.” But in the present case, where the collection is
defined by a certain relation to one of its terms, there is some danger of
a logical error. Let `u` be a class; then the number of `u`, we said, is the
class of classes similar to `u`. But “similar to `u`” cannot be the actual
concept which constitutes the number of `u`; for, if `v` be similar to `u`,
“similar to `v`” defines the same class, although it is a different concept.
Thus we require, as the defining predicate of the class of similar classes,
some concept which does not have any special relation to one or more of
the constituent classes. In regard to every particular number that may
be mentioned, whether finite or infinite, such a predicate is, as a matter
of fact, discoverable; but when all we are told about a number is that it
is the number of some class `u`, it is natural that a special reference to `u`
should appear in the definition. This, however, is not the point at issue.
The real point is, that what is defined is the same whether we use the
predicate “similar to `u`” or “similar to `v`,” provided `u` is similar to `v`.
This shows that it is not the class-concept or defining predicate that is
defined, but the class itself whose terms are the various classes which are
similar to `u` or to `v`. It is such classes, therefore, and not predicates such
as “similar to `u`,” that must be taken to constitute numbers.

Thus, to sum up: Mathematically, a number is nothing but a class of similar classes: this definition allows the deduction of all the usual properties of numbers, whether finite or infinite, and is the only one (so far as I know) which is possible in terms of the fundamental concepts of general logic. But philosophically we may admit that every collection of similar classes has some common predicate applicable to no entities except the classes in question, and if we can find, by inspection, that there is a certain class of such common predicates, of which one and only one applies to each collection of similar classes, then we may, if we see fit, call this particular class of predicates the class of numbers. For my part, I do not know whether there is any such class of predicates, and I do know that, if there be such a class, it is wholly irrelevant to Mathematics. Wherever Mathematics derives a common property from a reflexive, symmetrical, and transitive relation, all mathematical purposes of the supposed common property are completely served when it is replaced by the class of terms having the given relation to a given term; and this is precisely the case presented by cardinal numbers. For the future, therefore, I shall adhere to the above definition, since it is at once precise and adequate to all mathematical uses.

Notes

*^{[page 111]} Cantor has shown that it is necessary to separate the study of Cardinal and Ordinal numbers, which are distinct entities, of which the former are simpler, but of which both are essential to ordinary Mathematics. On Ordinal numbers, cf. Chaps. xxix, xxxviii, *infra.*

†^{[page 111]} See Peano, *F.* 1901, p. 6ff. and Padoa, “Théorie Algébrique des Nombres Entiers,” *Congrès*, Vol. iii, p. 314ff.

*^{[page 112]} Cf. Burali-Forti, “Sur les différentes definitions du nombre réel,” *Congrès*, iii, p. 294ff.

*^{[page 114]} Cf. Peano, *F.* 1901, §32, ·0, Note.

†^{[page 114]} On the necessity of this condition, cf. Padoa, *loc. cit.*, p. 324. Padoa appears not to perceive, however, that *all* definitions define the single individual of a class: when what is defined is a class, this must be the only term of some class of classes.

Addition and Multiplication.

**112.** In most mathematical accounts of arithmetical operations we
find the error of endeavouring to give at once a definition which shall be
applicable to rationals, or even to real numbers, without dwelling at
sufficient length upon the theory of integers. For the present, integers
alone will occupy us. The definition of integers, given in the preceding
chapter, obviously does not admit of extension to fractions; and in fact
the absolute difference between integers and fractions, even between
integers and fractions whose denominator is unity, cannot possibly be too
strongly emphasized. What rational fractions are, and what real numbers
are, I shall endeavour to explain at a later stage; positive and negative
numbers also are at present excluded. The integers with which we are
now concerned are not positive, but signless. And so the addition and
multiplication to be defined in this chapter are only applicable to integers;
but they have the merit of being equally applicable to finite and infinite
integers. Indeed, for the present, I shall rigidly exclude all propositions
which involve either the finitude or the infinity of the numbers considered.

**113.** There is only one fundamental kind of addition, namely the
logical kind. All other kinds can be defined in terms of this and logical
multiplication. In the present chapter the addition of integers is to be
defined by its means. Logical addition, as was explained in Part I,
is the same as disjunction; if `p` and `q` are propositions, their logical
sum is the proposition “`p` or `q`,” and if `u` and `v` are classes, their
logical sum is the class “`u` or `v`,” *i.e.* the class to which belongs every
term which either belongs to `u` or belongs to `v`. The logical sum
of two classes `u` and `v` may be defined in terms of the logical product
of two propositions, as the class of terms belonging to every class
in which both `u` and `v` are contained*. This definition is not essentially confined to two classes, but may be extended to a class of
classes, whether finite or infinite. Thus if `k` be a class of classes, the
logical sum of the classes composing `k` (called for short the sum of `k`) is
the class of terms belonging to every class which contains every class
which is a term of `k`. It is this notion which underlies arithmetical
addition. If `k` be a class of classes no two of which have any common
terms (called for short an exclusive class of classes), then the arithmetical sum of the numbers of the various classes of `k` is the number of
terms in the logical sum of `k`. This definition is absolutely general, and
applies equally whether `k` or any of its constituent classes be finite
or infinite. In order to assure ourselves that the resulting number
depends only upon the *numbers* of the various classes belonging to `k`, and
not upon the particular class `k` that happens to be chosen, it is necessary
to prove (as is easily done) that if `k`′ be another exclusive class of classes,
similar to `k`, and every member of `k` is similar to its correlate in `k`′, and
*vice versâ*, then the number of terms in the sum of `k` is the same as the
number in the sum of `k`′. Thus, for example, suppose `k` has only two
terms, `u` and `v`, and suppose `u` and `v` have no common part. Then the
number of terms in the logical sum of `u` and `v` is the sum of the number
of terms in `u` and in `v`; and if `u`′ be similar to `u`, and `v`′ to `v`, and `u`′, `v`′
have no common part, then the sum of `u`′ and `v`′ is similar to the
sum of `u` and `v`.

**114.** With regard to this definition of a sum of numbers, it is to be
observed that it cannot be freed from reference to classes which have the
numbers in question. The number obtained by summation is essentially
the number of the logical sum of a certain class of classes or of some
similar class of similar classes. The necessity of this reference to classes
emerges when one number occurs twice or oftener in the summation. It
is to be observed that the numbers concerned have no *order* of summation,
so that we have no such proposition as the commutative law: this proposition, as introduced in Arithmetic, results only from a defective
symbolism, which causes an order among the symbols which has no
correlative order in what is symbolized. But owing to the absence of
order, if one number occurs twice in a summation, we cannot distinguish
a first and a second occurrence of the said number. If we exclude a
reference to classes which have the said number, there is no sense in the
supposition of its occurring twice: the summation of a class of numbers
can be defined, but in that case, no number can be repeated. In the
above definition of a sum, the numbers concerned are defined as the
numbers of certain classes, and therefore it is not necessary to decide
whether any number is repeated or not. But in order to define, without
reference to particular classes, a sum of numbers of which some are
repeated, it is necessary first to define multiplication.

This point may be made clearer by considering a special case, such as
1 + 1. It is plain that we cannot take the number 1 itself twice over,
for there is one number 1, and there are not two instances of it. And if
the logical addition of 1 to itself were in question, we should find that
1 and 1 is 1, according to the general principle of Symbolic Logic. Nor
can we define 1 + 1 as the arithmetical sum of a certain class of numbers.
This method can be employed as regards 1 + 2, or any sum in which no
number is repeated; but as regards 1 + 1, the only class of numbers
involved is the class whose only member is 1, and since this class has one
member, not two, we cannot define 1 + 1 by its means. Thus the full definition of 1 + 1 is as follows: 1 + 1 is the number of a class `w` which
is the logical sum of two classes `u` and `v` which have no common term
and have each only one term. The chief point to be observed is, that
logical addition of classes is the fundamental notion, while the arithmetical addition of numbers is wholly subsequent.

**115.** The general definition of multiplication is due to Mr A. N. Whitehead*. It is as follows. Let `k` be a class of classes, no two of
which have any term in common. Form what is called the multiplicative
class of `k`, *i.e.* the class each of whose terms is a class formed by choosing
one and only one term from each of the classes belonging to `k`. Then
the number of terms in the multiplicative class of `k` is the product of all
the numbers of the various classes composing `k`. This definition, like
that of addition given above, has two merits, which make it preferable
to any other hitherto suggested. In the first place, it introduces no
order among the numbers multiplied, so that there is no need of the
commutative law, which, here as in the case of addition, is concerned
rather with the symbols than with what is symbolized. In the second
place, the above definition does not require us to decide, concerning any
of the numbers involved, whether they are finite or infinite. Cantor has
given† definitions of the sum and product of *two* numbers, which do not
require a decision as to whether these numbers are finite or infinite.
These definitions can be extended to the sum and product of any *finite*
number of finite or infinite numbers; but they do not, as they stand,
allow the definition of the sum or product of an infinite number of
numbers. This grave defect is remedied in the above definitions, which
enable us to pursue Arithmetic, as it ought to be pursued, without
introducing the distinction of finite and infinite until we wish to study
it. Cantor’s definitions have also the formal defect of introducing an
order among the numbers summed or multiplied: but this is, in his
case, a mere defect in the symbols chosen, not in the ideas which he
symbolizes. Moreover it is not practically desirable, in the case of the
sum or product of *two* numbers, to avoid this formal defect, since the
resulting cumbrousness becomes intolerable.

**116.** It is easy to deduce from the above definitions the usual
connection of addition and multiplication, which may be thus stated.
If `k` be a class of `b` mutually exclusive classes, each of which contains
`a` terms, then the logical sum of `k` contains `a` × `b` terms‡. It is also
easy to obtain the definition of `a`^{b}, and to prove the associative and distributive laws, and the formal laws for powers, such as `a`^{b}`a`^{c} = `a`^{b+c} But
it is to be observed that exponentiation is not to be regarded as a new
independent operation, since it is merely an application of multiplication. It is true that exponentiation can be independently defined,
as is done by Cantor*, but there is no advantage in so doing. Moreover
exponentiation unavoidably introduces ordinal notions, since `a`^{b} is not in
general equal to `b`^{a}. For this reason we cannot define the result of an
infinite number of exponentiations. Powers, therefore, are to be regarded
simply as abbreviations for products in which all the numbers multiplied
together are equal.

From the data which we now possess, all those propositions which hold equally of finite and infinite numbers can be deduced. The next step, therefore, is to consider the distinction between the finite and the infinite.

Notes

*^{[page 117]} *F.* 1901, §2, Prop. 1·0.

*^{[page 119]} *American Journal of Mathematics*, Oct. 1902.

†^{[page 119]} *Math. Annalen*, Vol. xlvi, §3.

‡^{[page 119]} See Whitehead, *loc. cit*.

*^{[page 120]} *Loc. cit.*, §4.

Finite and Infinite.

**117.** The purpose of the present chapter is not to discuss the philosophical difficulties concerning the infinite, which are postponed to
Part V. For the present I wish merely to set forth briefly the mathematical theory of finite and infinite as it appears in the theory of
cardinal numbers. This is its most fundamental form, and must be
understood before the ordinal infinite can be adequately explained*.

Let `u` be any class, and let `u`′ be a class formed by taking away one
term `x` from `u`. Then it may or may not happen that `u` is similar to `u`′.
For example, if `u` be the class of all finite numbers, and `u`′ the class of
all finite numbers except 0, the terms of `u`′ are obtained by adding 1 to each
of the terms of `u`, and this correlates one term of `u` with one of `u`′ and *vice versâ*, no term of either being omitted or taken twice over. Thus `u`′ is
similar to `u`. But if `u` consists of all finite numbers up to `n`, where `n` is
some finite number, and `u`′ consists of all these except 0, then `u`′ is not
similar to `u`. If there is one term `x` which can be taken away from `u` to
leave a similar class `u`′, it is easily proved that if any other term `y` is
taken away instead of `x` we also get a class similar to `u`. When it is
possible to take away one term from `u` and leave a class `u`′ similar to `u`,
we say that `u` is an *infinite* class. When this is not possible, we say that
`u` is a *finite* class. From these definitions it follows that the null-class is
finite, since no term can be taken from it. It is also easy to prove that
if `u` be a finite class, the class formed by adding one term to `u` is finite;
and conversely if this class is finite, so is `u`. It follows from the definition
that the numbers of finite classes other than the null-class are altered
by subtracting 1, while those of infinite classes are unaltered by this
operation. It is easy to prove that the same holds of the addition of 1.

**118.** Among finite classes, if one is a proper part of another, the
one has a smaller number of terms than the other. (A proper part is
a part not the whole.) But among infinite classes, this no longer holds.
This distinction is, in fact, an essential part of the above definitions of
the finite and the infinite. Of two infinite classes, one may have a
greater or a smaller number of terms than the other. A class `u` is said
to be greater than a class `v`, or to have a number greater than that of `v`,
when the two are not similar, but `v` is similar to a proper part of `u`. It
is known that if `u` is similar to a proper part of `v`, and `v` to a proper
part of `u` (a case which can only arise when `u` and `v` are infinite), then `u`
is similar to `v`; hence “`u` is greater than `v`” is inconsistent with “`v` is
greater than `u`.” It is not at present known whether, of two different
infinite numbers, one must be greater and the other less. But it is known
that there is a least infinite number, *i.e.* a number which is less than any
different infinite number. This is the number of finite integers, which
will be denoted, in the present work, by `α`_{0}*. This number is capable of
several definitions in which no mention is made of the finite numbers. In
the first place it may be defined (as is implicitly done by Cantor†) by means
of the principle of mathematical induction. This definition is as follows:
`α`_{0} is the number of any class `u` which is the domain of a one-one relation
`R`, whose converse domain is contained in but not coextensive with `u`,
and which is such that, calling the term to which `x` has the relation `R`
the *successor* of `x`, if `s` be any class to which belongs a term of `u` which is
not a successor of any other term of `u`, and to which belongs the successor
of every term of `u` which belongs to `s`, then every term of `u` belongs to `s`.
Or again, we may define `α`_{0} as follows. Let `P` be a transitive and asymmetrical relation, and let any two different terms of the field of `P` have the
relation `P` or its converse. Further let any class `u` contained in the field
of `P` and having successors (*i.e.* terms to which every term of `u` has the
relation `P`) have an immediate successor, *i.e.* a term whose predecessors
either belong to `u` or precede some term of `u`; let there be one term of
the field of `P` which has no predecessors, but let every term which has
predecessors have successors and also have an immediate predecessor;
then the number of terms in the field of `P` is `α`_{0}. Other definitions may
be suggested, but as all are equivalent it is not necessary to multiply
them. The following characteristic is important: Every class whose
number is `α`_{0} can be arranged in a series having consecutive terms, a
beginning but no end, and such that the number of predecessors of any
term of the series is finite; and any series having these characteristics
has the number `α`_{0}.

It is very easy to show that every infinite class contains classes whose
number is `α`_{0}. For let `u` be such a class, and let `x`_{0} be a term of `u`.
Then `u` is similar to the class obtained by taking away `x`_{0}, which we will
call the class `u`_{1}. Thus `u`_{1} is an infinite class. From this we can take
away a term `x`_{1}, leaving an infinite class `u`_{2}, and so on. The series of
terms `x`_{1}, `x`_{2}, … is contained in `u`, and is of the type which has the
number `α`_{0}. From this point we can advance to an alternative definition
of the finite and the infinite by means of mathematical induction, which
must now be explained.

**119.** If `n` be any finite number, the number obtained by adding
1 to `n` is also finite, and is different from `n`. Thus beginning with 0
we can form a series of numbers by successive additions of 1. We
may define finite numbers, if we choose, as those numbers that can be
obtained from 0 by such steps, and that obey mathematical induction.
That is, the class of finite numbers is the class of numbers which is
contained in every class `s` to which belongs 0 and the successor of every
number belonging to `s`, where the successor of a number is the number
obtained by adding 1 to the given number. Now `α`_{0} is not such a
number, since, in virtue of propositions already proved, no such number
is similar to a part of itself. Hence also no number greater than `α`_{0}
is finite according to the new definition. But it is easy to prove that
every number less than `α`_{0} is finite with the new definition as with the
old. Hence the two definitions are equivalent. Thus we may define
finite numbers either as those that can be reached by mathematical
induction, starting from 0 and increasing by 1 at each step, or as those
of classes which are not similar to the parts of themselves obtained by
taking away single terms. These two definitions are both frequently
employed, and it is important to realize that either is a consequence
of the other. Both will occupy us much hereafter; for the present
it is only intended, without controversy, to set forth the bare outlines
of the mathematical theory of finite and infinite, leaving the details to
be filled in during the course of the work.

Notes

*^{[page 121]} On the present topic cf. Cantor, *Math. Annalen*, Vol. xlvi, §§5, 6, where most of what follows will be found.

*^{[page 122]} Cantor employs for this number the Hebrew Aleph with the suffix 0, but this notation is inconvenient.

†^{[page 122]} *Math. Annalen*, Vol. xlvi, §6.

Theory of Finite Numbers.

**120.** Having now clearly distinguished the finite from the infinite,
we can devote ourselves to the consideration of finite numbers. It is
customary, in the best treatises on the elements of Arithmetic*, not to
define number or particular finite numbers, but to begin with certain
axioms or primitive propositions, from which all the ordinary results
are shown to follow. This method makes Arithmetic into an independent study, instead of regarding it, as is done in the present
work, as merely a development, without new axioms or indefinables, of a
certain branch of general Logic. For this reason, the method in question
seems to indicate a less degree of analysis than that adopted here. I
shall nevertheless begin by an exposition of the more usual method,
and then proceed to definitions and proofs of what are usually taken
as indefinables and indemonstrables. For this purpose, I shall take
Peano’s exposition in the Formulaire†, which is, so far as I know,
the best from the point of view of accuracy and rigour. This exposition
has the inestimable merit of showing that all Arithmetic can be developed from three fundamental notions (in addition to those of general
Logic) and five fundamental propositions concerning these notions. It
proves also that, if the three notions be regarded as determined by the
five propositions, these five propositions are mutually independent. This
is shown by finding, for each set of four out of the five propositions,
an interpretation which renders the remaining proposition false. It
therefore only remains, in order to connect Peano’s theory with that
here adopted, to give a definition of the three fundamental notions and
a demonstration of the five fundamental propositions. When once this
has been accomplished, we know with certainty that everything in the
theory of finite integers follows.
Peano’s three indefinables are 0, *finite integer**, and *successor of*.
It is assumed, as part of the idea of succession (though it would,
I think, be better to state it as a separate axiom), that every number
has one and only one successor. (By *successor* is meant, of course,
immediate successor.) Peano’s primitive propositions are then the
following. (1) 0 is a number. (2) If `a` is a number, the successor of
`a` is a number. (3) If two numbers have the same successor, the two
numbers are identical. (4) 0 is not the successor of any number.
(5) If `s` be a class to which 0 belongs and also the successor of every
number belonging to `s`, then every number belongs to `s`. The last of
these propositions is the principle of mathematical induction.

**121.** The mutual independence of these five propositions has been
demonstrated by Peano and Padoa as follows†. (1) Giving the usual
meanings to 0 and *successor*, but denoting by *number* finite integers
other than 0, all the above propositions except the first are true.
(2) Giving the usual meanings to 0 and *successor*, but `d``e``n``o``t``i``n``g` by
number only finite integers less than 10, or less than any other specified
finite integer, all the above propositions are true except the second.
(3) A series which begins by an antiperiod and then becomes periodic
(for example, the digits in a decimal which becomes recurring after a
certain number of places) will satisfy all the above propositions except
the third. (4) A periodic series (such as the hours on the clock)
satisfies all except the fourth of the primitive propositions. (5) Giving
to *successor* the meaning *greater by 2*, so that the successor of 0 is 2,
and of 2 is 4, and so on, all the primitive propositions are satisfied
except the fifth, which is not satisfied if `s` be the class of even numbers
including 0. Thus no one of the five primitive propositions can be
deduced from the other four.

**122.** Peano points out (*loc. cit.*) that other classes besides that of
the finite integers satisfy the above five propositions. What he says
is as follows: “There is an infinity of systems satisfying all the primitive
propositions. They are all verified, *e.g.*, by replacing *number* and 0 by
*number other than* 0 and 1. All the systems which satisfy the primitive
propositions have a one-one correspondence with the numbers. Number
is what is obtained from all these systems by abstraction; in other
words, number is the system which has all the properties enunciated
in the primitive propositions, and those only.” This observation appears
to me lacking in logical correctness. In the first place, the question
arises: How are the various systems distinguished, which agree in satisfying the primitive propositions? How, for example, is the system
beginning with 1 distinguished from that beginning with 0? To this
question two different answers may be given. We may say that 0 and
1 are both primitive ideas, or at least that 0 is so, and that therefore
0 and 1 can be intrinsically distinguished, as yellow and blue are distinguished. But if we take this view—which, by the way, will have to
be extended to the other primitive ideas, number and succession—we
shall have to say that these three notions are what I call constants,
and that there is no need of any such process of abstraction as Peano
speaks of in the definition of number. In this method, 0, number, and
succession appear, like other indefinables, as ideas which must be simply
recognized. Their recognition yields what mathematicians call the
existence-theorem, *i.e.* it assures us that there really are numbers.
But this process leaves it doubtful whether numbers are *logical* constants
or not, and therefore makes Arithmetic, according to the definition in
Part I, Chapter i, *primâ facie* a branch of Applied Mathematics. Moreover it is evidently not the process which Peano has in mind. The
other answer to the question consists in regarding 0, number, and
succession as a class of three ideas belonging to a certain class of trios
defined by the five primitive propositions. It is very easy so to state
the matter that the five primitive propositions become transformed into
the nominal definition of a certain class of trios. There are then no
longer any indefinables or indemonstrables in our theory, which has
become a pure piece of Logic. But 0, number and succession become
variables, since they are only determined as one of the class of trios:
moreover the existence-theorem now becomes doubtful, since we cannot
know, except by the discovery of at least one actual trio of this class,
that there are any such trios at all. One *actual* trio, however, would
be a constant, and thus we require some method of giving constant
values to 0, number, and succession. What we can show is that, if there
is one such trio, there are an infinite number of them. For by striking
out the first term from any class satisfying the conditions laid down
concerning number, we always obtain a class which again satisfies the
conditions in question. But even this statement, since the meaning of
number is still in question, must be differently worded if circularity
is to be avoided. Moreover we must ask ourselves: Is any process of
abstraction from all systems satisfying the five axioms, such as Peano
contemplates, logically possible? Every term of a class is the term it
is, and satisfies some proposition which becomes false when another term
of the class is substituted. There is therefore no term of a class which
has merely the properties defining the class and no others. What
Peano’s process of abstraction really amounts to is the consideration of
the class and variable members of it, to the exclusion of constant
members. For only a variable member of the class will have only the
properties by which the class is defined. Thus Peano does not succeed
in indicating any constant meaning for 0, number, and succession, nor
in showing that any constant meaning is possible, since the existence-theorem is not proved. His only method, therefore, is to say that at
least one such constant meaning can be immediately perceived, but is
not definable. This method is not logically unsound, but it is wholly
different from the impossible abstraction which he suggests. And the
proof of the mutual independence of his five primitive propositions is
only necessary in order to show that the definition of the class of trios
determined by them is not redundant. Redundancy is not a logical
error, but merely a defect of what may be called style. My object, in
the above account of cardinal numbers, has been to prove, from general
Logic, that there is one constant meaning which satisfies the above five
propositions, and that this constant meaning should be called number,
or rather finite cardinal number. And in this way, new indefinables
and indemonstrables are wholly avoided; for when we have shown that
the class of trios in question has at least one member, and when this
member has been used to define number, we easily show that the class
of trios has an infinite number of members, and we define the class
by means of the five properties enumerated in Peano’s primitive propositions. For the comprehension of the connection between Mathematics
and Logic, this point is of very great importance, and similar points will
occur constantly throughout the present work.

**123.** In order to bring out more clearly the difference between
Peano’s procedure and mine, I shall here repeat the definition of the
class satisfying his five primitive propositions, the definition of *finite number*, and the proof, in the case of finite numbers, of his five primitive
propositions.

The class of classes satisfying his axioms is the same as the class of
classes whose cardinal number is `α`_{0}, *i.e.* the class of classes, according to
my theory, which *is* `α`_{0}. It is most simply defined as follows: `α`_{0} is the
class of classes *u* each of which is the domain of some one-one relation `R`
(the relation of a term to its successor) which is such that there is at
least one term which succeeds no other term, every term which succeeds
has a successor, and `u` is contained in any class `s` which contains a term
of `u` having no predecessors, and also contains the successor of every
term of `u` which belongs to `s`. This definition includes Peano’s five
primitive propositions and no more. Thus of every such class all the
usual propositions in the arithmetic of finite numbers can be proved:
addition, multiplication, fractions, etc. can be defined, and the whole of
analysis can be developed, in so far as complex numbers are not involved.
But in this whole development, the meaning of the entities and relations
which occur is to a certain degree indeterminate, since the entities and
the relation with which we start are variable members of a certain class.
Moreover, in this whole development, nothing shows that there are such
classes as the definition speaks of.

In the logical theory of cardinals, we start from the opposite end.
We first define a certain class of entities, and then show that this class
of entities belongs to the class `α`_{0} above defined. This is done as follows.
(1) 0 is the class of classes whose only member is the null-class. (2) A
number is the class of all classes similar to any one of themselves. (3) 1 is
the class of all classes which are not null and are such that, if `x` belongs to
the class, the class without `x` is the null-class; or such that, if `x` and `y`
belong to the class, then `x` and `y` are identical. (4) Having shown that
if two classes be similar, and a class of one term be added to each, the
sums are similar, we define that, if `n` be a number, `n` + 1 is the number
resulting from adding a unit to a class of `n` terms. (5) Finite numbers
are those belonging to every class `s` to which belongs 0, and to which
`n` + 1 belongs if `n` belongs. This completes the definition of finite
numbers. We then have, as regards the five propositions which Peano
assumes: (1) 0 is a number. (2) Meaning `n` + 1 by the successor of `n`,
if `n` be a number, then `n` + 1 is a number. (3) If `n` + 1 = `m` + 1, then
`n` = `m`. (4) If `n` be any number, `n` + 1 is different from 0. (5) If `s` be
a class, and 0 belongs to this class, and if when `n` belongs to it, `n` + 1
belongs to it, then all finite numbers belong to it. Thus all the five
essential properties are satisfied by the class of finite numbers as above
defined. Hence the class of classes `α`_{0} has members, and the class `f``i``n``i``t``e` `n``u``m``b``e``r` is one definite member of `α`_{0}. There is, therefore, from the
mathematical standpoint, no need whatever of new indefinables or
indemonstrables in the whole of Arithmetic and Analysis.

Notes

*^{[page 124]} Except Frege’s *Grundgesetze der Arithmetik* (Jena, 1893).

†^{[page 124]} *F.* 1901, Part II and *F.* 1899, §20ff. *F.* 1901 differs from earlier editions in making “number is a class” a primitive proposition. I regard this as unnecessary, since it is implied by “0 is a number.” I therefore follow the earlier editions.

*^{[page 125]} Throughout the rest of this chapter, I shall use *number* as synonymous with *finite integer*.

†^{[page 125]} *F.* 1899, p. 30.

Addition of Terms and Addition of Classes.

**124.** Having now briefly set forth the mathematical theory of
cardinal numbers, it is time to turn our attention to the philosophical
questions raised by this theory. I shall begin by a few preliminary
remarks as to the distinction between philosophy and mathematics, and
as to the function of philosophy in such a subject as the foundations of
mathematics. The following observations are not necessarily to be
regarded as applicable to other branches of philosophy, since they are
derived specially from the consideration of the problems of logic.

The distinction of philosophy and mathematics is broadly one of point of view: mathematics is constructive and deductive, philosophy is critical, and in a certain impersonal sense controversial. Wherever we have deductive reasoning, we have mathematics; but the principles of deduction, the recognition of indefinable entities, and the distinguishing between such entities, are the business of philosophy. Philosophy is, in fact, mainly a question of insight and perception. Entities which are perceived by the so-called senses, such as colours and sounds, are, for some reason, not commonly regarded as coming within the scope of philosophy, except as regards the more abstract of their relations; but it seems highly doubtful whether any such exclusion can be maintained. In any case, however, since the present work is essentially unconcerned with sensible objects, we may confine our remarks to entities which are not regarded as existing in space and time. Such entities, if we are to know anything about them, must be also in some sense perceived, and must be distinguished one from another; their relations also must be in part immediately apprehended. A certain body of indefinable entities and indemonstrable propositions must form the starting-point for any mathematical reasoning; and it is this starting-point that concerns the philosopher. When the philosopher’s work has been perfectly accomplished, its results can be wholly embodied in premisses from which deduction may proceed. Now it follows from the very nature of such inquiries that results may be disproved, but can never be proved. The disproof will consist in pointing out contradictions and inconsistencies; but the absence of these can never amount to proof. All depends, in the end, upon immediate perception; and philosophical argument, strictly speaking, consists mainly of an endeavour to cause the reader to perceive what has been perceived by the author. The argument, in short, is not of the nature of proof, but of exhortation. Thus the question of the present chapter: Is there any indefinable set of entities commonly called numbers, and different from the set of entities above defined? is an essentially philosophical question, to be settled by inspection rather than by accurate chains of reasoning.

**125.** In the present chapter, we shall examine the question whether
the above definition of cardinal numbers in any way presupposes some
more fundamental sense of number. There are several ways in which
this may be supposed to be the case. In the first place, the individuals
which compose classes seem to be each in some sense *one*, and it might
be thought that a one-one relation could not be defined without introducing the number 1. In the second place, it may very well be
questioned whether a class which has only one term can be distinguished
from that one term. And in the third place, it may be held that the
notion of *class* presupposes number in a sense different from that above
defined: it may be maintained that classes arise from the addition of
individuals, as indicated by the word *and*, and that the logical addition
of classes is subsequent to this addition of individuals. These questions
demand a new inquiry into the meaning of *one* and of *class*, and here,
I hope, we shall find ourselves aided by the theories set forth in Part I.

As regards the fact that any individual or term is in some sense *one*,
this is of course undeniable. But it does not follow that the notion of
*one* is presupposed when individuals are spoken of: it may be, on the
contrary, that the notion of term or individual is the fundamental one,
from which that of *one* is derived. This view was adopted in Part I,
and there seems no reason to reject it. And as for one-one relations,
they are defined by means of identity, without any mention of *one*, as
follows: `R` is a one-one relation if, when `x` and `x`′ have the relation `R` to
`y`, and `x` has the relation `R` to `y` and `y`′, then `x` and `x`′ are identical, and
so are *y* and `y`′. It is true that here `x`, `y`, `x`′, `y`′ are each *one* term, but
this is not (it would seem) in any way presupposed in the definition.
This disposes (pending a new inquiry into the nature of classes) of the
first of the above objections.

The next question is as to the distinction between a class containing
only one member, and the one member which it contains. If we could
identify a class with its defining predicate or class-concept, no difficulty
would arise on this point. When a certain predicate attaches to one
and only one term, it is plain that that term is not identical with the
predicate in question. But if two predicates attach to precisely the
same terms, we should say that, although the predicates are different,
the classes which they define are identical, *i.e.* there is only one class
which both define. If, for example, all featherless bipeds are men, and
all men are featherless bipeds, the classes *men* and *featherless bipeds* are
identical, though *man* differs from *featherless biped*. This shows that a
class cannot be identified with its class-concept or defining predicate.
There might seem to be nothing left except the actual terms, so that
when there is only one term, that term would have to be identical with
the class. Yet for many formal reasons this view cannot give the
meaning of the symbols which stand for classes in symbolic logic. For
example, consider the class of numbers which, added to 3, give 5. This
is a class containing no terms except the number 2. But we can say
that 2 is a member of this class, *i.e.* it has to the class that peculiar
indefinable relation which terms have to the classes they belong to.
This seems to indicate that the class is different from the one term.
The point is a prominent one in Peano’s Symbolic Logic, and is connected with his distinction between the relation of an individual to its
class and the relation of a class to another in which it is contained.
Thus the class of numbers which, added to 3, give 5, is contained in the
class of numbers, but is not a number; whereas 2 is a number, but is
not a class contained in the class of numbers. To identify the two
relations which Peano distinguishes is to cause havoc in the theory of
infinity, and to destroy the formal precision of many arguments and
definitions. It seems, in fact, indubitable that Peano’s distinction is
just, and that some way must be found of discriminating a term from
a class containing that term only.

**126.** In order to decide this point, it is necessary to pass to our
third difficulty, and reconsider the notion of *class* itself. This notion
appears to be connected with the notion of *denoting*, explained in Part I,
Chapter v. We there pointed out five ways of denoting, one of which
we called the *numerical conjunction*. This was the kind indicated by *all*.
This kind of conjunction appears to be that which is relevant in the
case of classes. For example, *man* being the class-concept, *all men* will
be the class. But it will not be *all men* quâ concept which will be the
class, but what this concept denotes, *i.e.* certain terms combined in the
particular way indicated by *all*. The way of combination is essential,
since *any man* or *some man* is plainly not the class, though either denotes
combinations of precisely the same terms. It might seem as though, if
we identify a class with the numerical conjunction of its terms, we must
deny the distinction of a term from a class whose only member is that
term. But we found in Chapter x that a class must be always an object
of a different logical type from its members, and that, in order to avoid
the proposition `x`ε`x`, this doctrine must be extended even to classes
which have only one member. How far this forbids us to identify
classes with numerical conjunctions, I do not profess to decide; in any
case, the distinction between a term and the class whose only member
it is must be made, and yet classes must be taken extensionally to the
degree involved in their being determinate when their members are
given. Such classes are called by Frege *Werthverläufe*; and cardinal
numbers are to be regarded as classes in this sense.

**127.** There is still, however, a certain difficulty, which is this: a
class *seems* to be not many terms, but to be itself a single term, even
when many terms are members of the class. This difficulty would seem
to indicate that the class cannot be identified with all its members, but
is rather to be regarded as the whole which they compose. In order,
however, to state the difficulty in an unobjectionable manner, we must
exclude unity and plurality from the statement of it, since these notions
were to be defined by means of the notion of class. And here it may be
well to clear up a point which is likely to occur to the reader. Is the
notion of *one* presupposed every time we speak of *a* term? A term,
it may be said, means *one* term, and thus no statement can be made
concerning a term without presupposing *one*. In some sense of *one*, this
proposition seems indubitable. Whatever is, is one: being and one, as
Leibniz remarks, are convertible terms*. It is difficult to be sure how
far such statements are merely grammatical. For although whatever
is, is one, yet it is equally true that whatever are, are many. But the
truth seems to be that the kind of object which is a class, *i.e.* the kind
of object denoted by *all men*, or by any concept of a class, is not *one*
except where the class has only one term, and must not be made a single
logical subject. There is, as we said in Part I, Chapter vi, in simple cases an
associated single term which is the class as a whole; but this is sometimes
absent, and is in any case not identical with the class as many. But in
this view there is not a contradiction, as in the theory that verbs and
adjectives cannot be made subjects; for assertions can be made about
classes as many, but the subject of such assertions is many, not one only
as in other assertions. “Brown and Jones are two of Miss Smith’s
suitors” is an assertion about the class “Brown and Jones,” but not
about this class considered as a single term. Thus one-ness belongs, in
this view, to a certain type of logical subjects, but classes which are not
one may yet have assertions made about them. Hence we conclude that
one-ness is implied, but not presupposed, in statements about a term,
and “a term” is to be regarded as an indefinable.

**128.** It seems necessary, however, to make a distinction as regards
the use of *one*. The sense in which every object is *one*, which is
apparently involved in speaking of *an* object is, as Frege urges†, a very
shadowy sense, since it is applicable to everything alike. But the sense
in which a class may be said to have one member is quite precise.
A class `u` has one member when `u` is not null, and “`x` and `y` are `u`’s”
implies “`x` is identical with `y`.” Here the one-ness is a property of the
class, which may therefore be called a unit-class. The `x` which is its
only member may be itself a class of many terms, and this shows that
the sense of `o``n``e` involved in `o``n``e` `t``e``r``m` or `a` `t``e``r``m` is not relevant to
Arithmetic, for many terms as such may be a single member of a class
of classes. `O``n``e`, therefore, is not to be asserted of terms, but of classes
having one member in the above-defined sense; *i.e.* “`u` is one,” or better
“`u` is a unit” means “`u` is not null, and ‘`x` and `y` are `u`’s’ implies ‘`x`
and `y` are identical’.” The member of `u`, in this case, will itself be none
or one or many if `u` is a class of classes; but if `u` is a class of terms,
the member of `u` will be neither none nor one nor many, but simply
a term.

**129.** The commonly received view, as regards finite numbers, is that
they result from counting, or, as some philosophers would prefer to
say, from synthesizing. Unfortunately, those who hold this view have
not analyzed the notion of counting: if they had done so, they would
have seen that it is very complex, and presupposes the very numbers
which it is supposed to generate.

The process of counting has, of course, a psychological aspect, but
this is quite irrelevant to the theory of Arithmetic. What I wish now
to point out is the logical process involved in the act of counting, which
is as follows. When we say one, two, three, etc., we are necessarily
considering some one-one relation which holds between the numbers used
in counting and the objects counted. What is meant by the “one, two,
three” is that the objects indicated by these numbers are their correlates
with respect to the relation which we have in mind. (This relation, by
the way, is usually extremely complex, and is apt to involve a reference
to our state of mind at the moment.) Thus we correlate a class of objects
with a class of numbers; and the class of numbers consists of all the
numbers from 1 up to some number `n`. The only immediate inference to be
drawn from this correlation is, that the number of objects is the same as
the number of numbers from 1 up to `n`. A further process is required to
show that this number of numbers is `n`, which is only true, as a matter
of fact, when `n` is finite, or, in a certain wider sense, when `n` is `α`_{0} (the
smallest of infinite numbers). Moreover the process of counting gives us
no indication as to what the numbers are, as to why they form a series,
or as to how it is to be proved (in the cases where it is true) that there
are `n` numbers from 1 up to `n`. Hence counting is irrelevant in the
foundations of Arithmetic; and with this conclusion, it may be dismissed
until we come to order and ordinal numbers.

**130.** Let us return to the notion of the numerical conjunction. It
is plain that it is of such objects as “`A` and `B`,” “`A` and `B` and `C`,”
that numbers other than one are to be asserted. We examined such
objects, in Part I, in relation to classes, with which we found them to
be identical. Now we must investigate their relation to numbers and
plurality.

The notion to be now examined is the notion of a numerical
conjunction or, more shortly, a *collection*. This is not to be identified,
to begin with, with the notion of a *class*, but is to receive a new and
independent treatment. By a collection I mean what is conveyed by
“`A` and `B`” or “`A` and `B` and `C`,” or any other enumeration of definite
terms. The collection is defined by the actual mention of the terms,
and the terms are connected by `a``n``d`. It would seem that `a``n``d` represents
a fundamental way of combining terms, and it might be urged that
just this way of combination is essential if anything is to result of which
a number other than 1 is to be asserted. Collections do not presuppose
numbers, since they result simply from the terms together with `a``n``d`:
they could only `p``r``e``s``u``p``p``o``s``e` numbers in the particular case where the
terms of the collection themselves presupposed numbers. There is a
grammatical difficulty which, since no method exists of avoiding it,
must be pointed out and allowed for. A collection, grammatically, is
one, whereas `A` and `B`, or `A` and `B` and `C`, are essentially many. The
strict meaning of `c``o``l``l``e``c``t``i``o``n` is the whole composed of many, but since a
word is needed to denote the many themselves, I choose to use the word
`c``o``l``l``e``c``t``i``o``n` in this sense, so that a collection, according to the usage here
adopted, is many and not one.

As regards what is meant by the combination indicated by `a``n``d`, it
gives what we called before the numerical conjunction. That is `A` and
`B` is what is denoted by the concept of a class of which `A` and `B` are
the only terms, and is precisely `A` and `B` denoted in the way which is
indicated by `a``l``l`. We may say, if `u` be the class-concept corresponding
to a class of which `A` and `B` are the only terms, that “all `u`’s” is a
concept which denotes the terms `A`, `B` combined in a certain way, and
`A` and `B` are those terms combined in precisely that way. Thus `A` and
`B` appears indistinguishable from the class, though distinguishable from
the class-concept and from the concept of the class. Hence if `u` be a
class of more than one term, it seems necessary to hold that `u` is not
one, but many, since `u` is distinguished both from the class-concept and
from the whole composed of the terms of `u`*. Thus we are brought back
to the dependence of numbers upon classes; and where it is not said
that the classes in question are finite, it is practically necessary to begin
with class-concepts and the theory of denoting, not with the theory of
and which has just been given. The theory of *and* applies practically
only to finite numbers, and gives to finite numbers a position which is
different, at least psychologically, from that of infinite numbers. There
are, in short, two ways of defining particular finite classes, but there is
only one practicable way of defining particular infinite classes, namely
by intension. It is largely the habit of considering classes primarily
from the side of extension which has hitherto stood in the way of a
correct logical theory of infinity.

**131.** Addition, it should be carefully observed, is not primarily a
method of forming numbers, but of forming classes or collections. If
we add `B` to `A`, we do not obtain the number 2, but we obtain `A` and `B`,
which is a collection of two terms, or a couple. And a couple is defined
as follows: `u` is a couple if `u` has terms, and if, if `x` be a term of `u`, there
is a term of `u` different from `x`, but if `x`, `y` be different terms of `u`, and `z`
differs from `x` and from `y`, then every class to which `z` belongs differs
from `u`. In this definition, only diversity occurs, together with the
notion of a class having terms. It might no doubt be objected that we
have to take just two terms `x`, `y` in the above definition: but as a
matter of fact any finite number can be defined by induction without
introducing more than one term. For, if `n` has been defined, a class `u`
has `n` + 1 terms when, if `x` be a term of `u`, the number of terms of `u`
which differ from `x` is `n`. And the notion of the arithmetical sum `n` + 1
is obtained from that of the logical sum of a class of `n` terms and a class
of one term. When we say 1 + 1 = 2, it is not possible that we should
mean 1 and 1, since there is only one 1: if we take 1 as an individual,
1 and 1 is nonsense, while if we take it as a class, the rule of Symbolic
Logic applies, according to which 1 and 1 is 1. Thus in the corresponding
logical proposition, we have on the left-hand side terms of which 1 can
be asserted, and on the right-hand side we have a couple. That is,
1 + 1 = 2 means “one term and one term are two terms,” or, stating the
proposition in terms of variables, “if `u` has one term and `v` has one
term, and `u` differs from `v`, their logical sum has two terms.” It is to be
observed that on the left-hand side we have a numerical conjunction of
propositions, while on the right-hand side we have a proposition concerning a numerical conjunction of terms. But the true premiss, in the
above proposition, is not the conjunction of the three propositions, but
their logical product. This point, however, has little importance in the
present connection.

**132.** Thus the only point which remains is this: Does the notion
of a term presuppose the notion of 1? For we have seen that all
numbers except 0 involve in their definitions the notion of a term, and
if this in turn involves 1, the definition of 1 becomes circular, and 1 will
have to be allowed to be indefinable. This objection to our procedure
is answered by the doctrine of §128, that a term is not *one* in the sense
which is relevant to Arithmetic, or in the sense which is opposed to
*many*. The notion of *any term* is a logical indefinable, presupposed in
formal truth and in the whole theory of the variable; but this notion is
that of the variable conjunction of terms, which in no way involves the
number 1. There is therefore nothing circular in defining the number 1
by means of the notion of *a term* or of *any term*.

To sum up: Numbers are classes of classes, namely of all classes similar to a given class. Here classes have to be understood in the sense of numerical conjunctions in the case of classes having many terms; but a class may have no terms, and a class of one term is distinct from that term, so that a class is not simply the sum of its terms. Only classes have numbers; of what is commonly called one object, it is not true, at least in the sense required, to say that it is one, as appears from the fact that the object may be a class of many terms. “One object” seems to mean merely “a logical subject in some proposition.” Finite numbers are not to be regarded as generated by counting, which on the contrary presupposes them; and addition is primarily logical addition, first of propositions, then of classes, from which latter arithmetical addition is derivative. The assertion of numbers depends upon the fact that a class of many terms can be a logical subject without being arithmetically one. Thus it appeared that no philosophical argument could overthrow the mathematical theory of cardinal numbers set forth in Chapters xi to xiv.

Notes

*^{[page 132]} Ed. Gerhardt, ii, p. 300.

†^{[page 132]} *Grundlagen der Arithmetik*, Breslau, 1884, p. 40.

*^{[page 134]} A conclusive reason against identifying a class with the whole composed of its
terms is, that one of these terms may be the class itself, as in the case “class is a class,” or rather “classes are one among classes.” The logical type of the class *class* is of an infinite order, and therefore the usual objection to “`x`ε`x`” does not apply in this case.

Whole and Part.

**133.** For the comprehension of analysis, it is necessary to investigate
the notion of whole and part, a notion which has been wrapped in
obscurity—though not without certain more or less valid logical
reasons—by the writers who may be roughly called Hegelian. In the
present chapter I shall do my best to set forth a straightforward and
non-mystical theory of the subject, leaving controversy as far as possible
on one side. It may be well to point out, to begin with, that I shall
use the word *whole* as strictly correlative to *part*, so that nothing will
be called a whole unless it has parts. Simple terms, such as points,
instants, colours, or the fundamental concepts of logic, will not be called
wholes.

Terms which are not classes may be, as we saw in the preceding
chapter, of two kinds. The first kind are simple: these may be
characterized, though not defined, by the fact that the propositions
asserting the being of such terms have no presuppositions. The second
kind of terms that are not classes, on the other hand, are complex, and
in their case, their being presupposes the being of certain other terms.
Whatever is not a class is called a *unit*, and thus units are either simple
or complex. A complex unit is a *whole*; its parts are other units,
whether simple or complex, which are presupposed in it. This suggests
the possibility of defining whole and part by means of logical priority,
a suggestion which, though it must be ultimately rejected, it will be
necessary to examine at length.

**134.** Wherever we have a one-sided formal implication, it may be
urged, if the two propositional functions involved are obtainable one from
the other by the variation of a single constituent, then what is implied
is simpler than what implies it. Thus “Socrates is a man” implies
“Socrates is a mortal,” but the latter proposition does not imply the
former: also the latter proposition is simpler than the former, since
*man* is a concept of which *mortal* forms part. Again, if we take
a proposition asserting a relation of two entities `A` and `B`, this
proposition implies the being of `A` and the being of `B`, and the being of
the relation, none of which implies the proposition, and each of which is
simpler than the proposition. There will only be equal complexity—according to the theory that intension and extension vary inversely as
one another—in cases of mutual implication, such as “`A` is greater
than `B`” and “`B` is less than `A`.” Thus we might be tempted to set up
the following definition: `A` is said to be part of `B` when `B` *is* implies
`A` *is*, but `A` *is* does not imply `B` *is*. If this definition could be maintained, whole and part would not be a new indefinable, but would be
derivative from logical priority. There are, however, reasons why such
an opinion is untenable.

The first objection is, that logical priority is not a simple relation:
implication is simple, but logical priority of `A` to `B` requires not only
“`B` implies `A`,” but also “`A` does not imply `B`.” (For convenience,
I shall say that `A` implies `B` when `A` *is* implies `B` *is*.) This state of
things, it is true, is realized when `A` is part of `B`; but it seems necessary
to regard the relation of whole to part as something simple, which must
be different from any possible relation of one whole to another which is
not part of it. This would not result from the above definition. For
example, “`A` is greater and better than `B`” implies “`B` is less than `A`,”
but the converse implication does not hold: yet the latter proposition is
not part of the former*.

Another objection is derived from such cases as redness and colour.
These two concepts appear to be equally simple: there is no specification,
other and simpler than redness itself, which can be added to colour to
produce redness, in the way in which specifications will turn *mortal* into
*man*. Hence *A is red* is no more complex than *A is coloured*, although
there is here a one-sided implication. Redness, in fact, appears to be
(when taken to mean one particular shade) a simple concept, which,
although it implies colour, does not contain colour as a constituent.
The inverse relation of extension and intension, therefore, does not hold
in all cases. For these reasons, we must reject, in spite of their very
close connection, the attempt to define whole and part by means of
implication.

**135.** Having failed to define wholes by logical priority, we shall
not, I think, find it possible to define them at all. The relation of
whole and part is, it would seem, an indefinable and ultimate relation,
or rather, it is several relations, often confounded, of which one at least
is indefinable. The relation of a part to a whole must be differently
discussed according to the nature both of the whole and of the parts.
Let us begin with the simplest case, and proceed gradually to those that
are more elaborate.

(1) Whenever we have any collection of many terms, in the sense
explained in the preceding chapter, there the terms, provided there is
some non-quadratic propositional function which they all satisfy, together
form a whole. In the preceding chapter we regarded the class as formed by
all the terms, but usage seems to show no reason why the class should not
equally be regarded as the whole composed of all the terms in those cases
where there is such a whole. The first is the class as many, the second
the class as one. Each of the terms then has to the whole a certain
indefinable relation*, which is one meaning of the relation of whole and
part. The whole is, in this case, a whole of a particular kind, which
I shall call an *aggregate*: it differs from wholes of other kinds by the
fact that it is definite as soon as its constituents are known.

(2) But the above relation holds only between the aggregate and
the single terms of the collection composing the aggregate: the relation
to our aggregate of aggregates containing some but not all the terms
of our aggregate, is a different relation, though also one which would be
commonly called a relation of part to whole. For example, the relation
of the Greek nation to the human race is different from that of Socrates
to the human race; and the relation of the whole of the primes to the
whole of the numbers is different from that of 2 to the whole of the
numbers. This most vital distinction is due to Peano†. The relation
of a subordinate aggregate to one in which it is contained can be defined,
as was explained in Part I, by means of implication and the first kind of
relation of part to whole. If `u`, `v` be two aggregates, and for every
value of `x` “`x` is `u`” implies “`x` is a `v`,” then, provided the converse
implication does not hold, `u` is a proper part (in the second sense) of `v`.
This sense of whole and part, therefore, is derivative and definable.

(3) But there is another kind of whole, which may be called a *unity*.
Such a whole is always a proposition, though it need not be an *asserted*
proposition. For example, “`A` differs from `B`,” or “`A`’s difference from
`B`,” is a complex of which the parts are `A` and `B` and difference; but
this sense of whole and part is different from the previous senses, since
“`A` differs from `B`” is not an aggregate, and has no parts at all in the
first two senses of parts. It is parts in this third sense that are chiefly
considered by philosophers, while the first two senses are those usually
relevant in symbolic logic and mathematics. This third sense of *part* is
the sense which corresponds to analysis: it appears to be indefinable,
like the first sense—*i.e.*, I know no way of defining it. It must be held
that the three senses are always to be kept distinct: *i.e.*, if `A` is part
of `B` in one sense, while `B` is part of `C` in another, it must not be
inferred (in general) that `A` is part of `C` in any of the three senses. But
we may make a fourth general sense, in which anything which is part in
any sense, or part in one sense of part in another, is to be called a part.
This sense, however, has seldom, if ever, any utility in actual discussion.

**136.** The difference between the kinds of wholes is important,
and illustrates a fundamental point in Logic. I shall therefore repeat
it in other words. Any collection whatever, if defined by a non-quadratic
propositional function, though as such it is many, yet composes a whole,
whose parts are the terms of the collection or any whole composed of some
of the terms of the collection. It is highly important to realize the difference between a whole and all its parts, even in this case where the difference
is a minimum. The word *collection*, being singular, applies more strictly
to the whole than to all the parts; but convenience of expression has led
me to neglect grammar, and speak of all the terms as the collection.
The whole formed of the terms of the collection I call an *aggregate*.
Such a whole is completely specified when all its simple constituents are
specified; its parts have no direct connection *inter se*, but only the
indirect connection involved in being parts of one and the same whole.
But other wholes occur, which contain relations or what may be called
predicates, not occurring simply as terms in a collection, but as relating
or qualifying. Such wholes are always propositions. These are not
completely specified when their parts are all known. Take, as a simple
instance, the proposition “`A` differs from `B`,” where `A` and `B` are simple
terms. The simple parts of this whole are `A` and `B` and difference; but
the enumeration of these three does not specify the whole, since there
are two other wholes composed of the same parts, namely the aggregate
formed of `A` and `B` and difference, and the proposition “`B` differs
from `A`.” In the former case, although the whole was different from
all its parts, yet it was completely specified by specifying its parts; but
in the present case, not only is the whole different, but it is not even
specified by specifying its parts. We cannot explain this fact by saying
that the parts stand in certain relations which are omitted in the
analysis; for in the above case of “`A` differs from `B`,” the relation was
included in the analysis. The fact seems to be that a relation is one
thing when it relates, and another when it is merely enumerated as a
term in a collection. There are certain fundamental difficulties in this
view, which however I leave aside as irrelevant to our present purpose*.

Similar remarks apply to `A` *is*, which is a whole composed of `A` and
*Being*, but is different from the whole formed of the collection `A` and
*Being*. *A is one* raises the same point, and so does *A and B are two*.
Indeed all propositions raise this point, and we may distinguish them
among complex terms by the fact that they raise it.

Thus we see that there are two very different classes of wholes, of
which the first will be called *aggregates*, while the second will be called
*unities*. (*Unit* is a word having a quite different application, since whatever is a class which is not null, and is such that, if `x` and `y` be members
of it, `x` and `y` are identical, is a unit.) Each class of wholes consists of
terms not simply equivalent to all their parts; but in the case of unities,
the whole is not even specified by its parts. For example, the parts `A`,
greater than, `B`, may compose simply an aggregate, or either of the
propositions “`A` is greater than `B`,” “`B` is greater than `A`.” Unities
thus involve problems from which aggregates are free. As aggregates
are more specially relevant to mathematics than unities, I shall in
future generally confine myself to the former.

**137.** It is important to realize that a whole is a new single term,
distinct from each of its parts and from all of them: it is one, not many*,
and is related to the parts, but has a being distinct from theirs. The
reader may perhaps be inclined to doubt whether there is any need of
wholes other than unities; but the following reasons seem to make
aggregates logically unavoidable. (1) We speak of one collection, one
manifold, etc., and it would seem that in all these cases there really is
something that is a single term. (2) The theory of fractions, as we shall
shortly see, appears to depend partly upon aggregates. (3) We shall find
it necessary, in the theory of extensive quantity, to assume that aggregates,
even when they are infinite, have what may be called magnitude of
divisibility, and that two infinite aggregates may have the same number
of terms without having the same magnitude of divisibility: this theory,
we shall find, is indispensable in metrical geometry. For these reasons,
it would seem, the aggregate must be admitted as an entity distinct
from all its constituents, and having to each of them a certain ultimate
and indefinable relation.

**138.** I have already touched on a very important logical doctrine,
which the theory of whole and part brings into prominence—I mean the
doctrine that analysis is falsification. Whatever can be analyzed is a
whole, and we have already seen that analysis of wholes is in some
measure falsification. But it is important to realize the very narrow
limits of this doctrine. We cannot conclude that the parts of a whole
are not really its parts, nor that the parts are not presupposed in the
whole in a sense in which the whole is not presupposed in the parts, nor
yet that the logically prior is not usually simpler than the logically
subsequent. In short, though analysis gives us the truth, and nothing
but the truth, yet it can never give us the whole truth. This is the
only sense in which the doctrine is to be accepted. In any wider sense,
it becomes merely a cloak for laziness, by giving an excuse to those who
dislike the labour of analysis.

**139.** It is to be observed that what we called classes as one may
always, except where they contain one term or none, or are defined by
quadratic propositional functions, be interpreted as aggregates. The
logical product of two classes as one will be the common part (in the
second of our three senses) of the two aggregates, and their sum will
be the aggregate which is identical with or part of (again in the second
sense) any aggregate of which the two given aggregates are parts, but is
neither identical with nor part of any other aggregate*. The relation
of whole and part, in the second of our three senses, is transitive and
asymmetrical, but is distinguished from other such relations by the fact
of allowing logical addition and multiplication. It is this peculiarity
which forms the basis of the Logical Calculus as developed by writers
previous to Peano and Frege (including Schröder)†. But wherever infinite
wholes are concerned it is necessary, and in many other cases it is
practically unavoidable, to begin with a class-concept or predicate or
propositional function, and obtain the aggregate from this. Thus the
theory of whole and part is less fundamental logically than that of
predicates or class-concepts or propositional functions; and it is for
this reason that the consideration of it has been postponed to so late
a stage.

Notes

*^{[page 138]} See Part IV, Chap. xxvii.

*^{[page 139]} Which may, if we choose, be taken as Peano’s ε. The objection to this meaning for ε is that not every propositional function defines a whole of the kind required. The whole differs from the class as many by being of the same *type* as its terms.

†^{[page 139]} Cf. *e.g.* *F.* 1901, §1, Prop. 4.4, note (p. 12).

*^{[page 140]} See Part I, Chap. iv, esp. §54.

*^{[page 141]} *I.e.* it is of the same logical type as its simple parts.

*^{[page 142]} Cf. Peano, *F.* 1901, §2, Prop. 1·0 (p. 19).

†^{[page 142]} See *e.g.* his *Algebra der Logik*, Vol. i (Leipzig, 1890).

Infinite Wholes.

**140.** In the present chapter the special difficulties of infinity are
not to be considered: all these are postponed to Part V. My object
now is to consider two questions: (1) Are there any infinite wholes?
(2) If so, must an infinite whole which contains parts in the second of
our three senses be an aggregate of parts in the first sense? In order to
avoid the reference to the first, second and third senses, I propose henceforward to use the following phraseology: A part in the first sense is to
be called a *term* of the whole*; a part in the second sense is to be called
a *part* simply; and a part in the third sense will be called a *constituent*
of the whole. Thus terms and parts belong to aggregates, while constituents belong to unities. The consideration of aggregates and unities,
where infinity is concerned, must be separately conducted. I shall begin
with aggregates.

An infinite aggregate is an aggregate corresponding to an infinite
class, *i.e.* an aggregate which has an infinite number of terms. Such
aggregates are defined by the fact that they contain parts which have
as many terms as themselves. Our first question is: Are there any such
aggregates?

Infinite aggregates are often denied. Even Leibniz, favourable as
he was to the actual infinite, maintained that, where infinite classes are
concerned, it is possible to make valid statements about *any* term of the
class, but not about *all* the terms, nor yet about the whole which (as he
would say) they do *not* compose†. Kant, again, has been much criticised
for maintaining that space is an infinite given whole. Many maintain
that every aggregate must have a finite number of terms, and that
where this condition is not fulfilled there is no true whole. But I do
not believe that this view can be successfully defended. Among those
who deny that space is a given whole, not a few would admit that what
they are pleased to call a finite space may be a given whole, for instance,
the space in a room, a box, a bag, or a book-case. But such a space is
only finite in a psychological sense, *i.e.* in the sense that we can take it
in at a glance: it is not finite in the sense that it is an aggregate of a
finite number of terms, nor yet a unity of a finite number of constituents.
Thus to admit that such a space can be a whole is to admit that there
are wholes which are not finite. (This does not follow, it should be
observed, from the admission of material objects apparently occupying
finite spaces, for it is always possible to hold that such objects, though
apparently continuous, consist really of a large but finite number of
material points.) With respect to time, the same argument holds: to
say, for example, that a certain length of time elapses between sunrise
and sunset, is to admit an infinite whole, or at least a whole which is not
finite. It is customary with philosophers to deny the reality of space
and time, and to deny also that, if they were real, they would be
aggregates. I shall endeavour to show, in Part VI, that these denials
are supported by a faulty logic, and by the now resolved difficulties of
infinity. Since science and common sense join in the opposite view, it
will therefore be accepted; and thus, since no argument *à priori* can
now be adduced against infinite aggregates, we derive from space and
time an argument in their favour.

Again, the natural numbers, or the fractions between 0 and 1, or the
sum-total of all colours, are infinite, and seem to be true aggregates:
the position that, although true propositions can be made about *any*
number, yet there are no true propositions about *all* numbers, could be
supported formerly, as Leibniz supported it, by the supposed contradictions of infinity, but has become, since Cantor’s solution of these
contradictions, a wholly unnecessary paradox. And where a collection
can be defined by a non-quadratic propositional function, this must be
held, I think, to imply that there is a genuine aggregate composed
of the terms of the collection. It may be observed also that, if there
were no infinite wholes, the word *Universe* would be wholly destitute of
meaning.

**141.** We must, then, admit infinite aggregates. It remains to ask
a more difficult question, namely: Are we to admit infinite unities?
This question may also be stated in the form: Are there any
infinitely complex propositions? This question is one of great logical
importance, and we shall require much care both in stating and in
discussing it.

The first point is to be clear as to the meaning of an infinite unity.
A unity will be infinite when the aggregate of all its constituents is
infinite, but this scarcely constitutes the meaning of an infinite unity.
In order to obtain the meaning, we must introduce the notion of a
*simple* constituent. We may observe, to begin with, that a constituent
of a constituent is a constituent of the unity, *i.e.* this form of the
relation of part to whole, like the second, but unlike the first form, is
transitive. A simple constituent may now be defined as a constituent
which itself has no constituents. We may assume, in order to eliminate
the question concerning aggregates, that no constituent of our unity is
to be an aggregate, or, if there be a constituent which is an aggregate,
then this constituent is to be taken as simple. (This view of an aggregate is rendered legitimate by the fact that an aggregate is a single term,
and does not have that kind of complexity which belongs to propositions.)
With this the definition of a simple constituent is completed.

We may now define an infinite unity as follows: A unity is finite when, and only when, the aggregate of its simple constituents is finite. In all other cases a unity is said to be infinite. We have to inquire whether there are any such unities*.

If a unity is infinite, it is possible to find a constituent unity, which
again contains a constituent unity, and so on without end. If there be
any unities of this nature, two cases are *primâ facie* possible. (1) There
may be simple constituents of our unity, but these must be infinite in
number. (2) There may be no simple constituents at all, but all
constituents, without exception, may be complex; or, to take a slightly
more complicated case, it may happen that, although there are some
simple constituents, yet these and the unities composed of them do not
constitute all the constituents of the original unity. A unity of either
of these two kinds will be called infinite. The two kinds, though
distinct, may be considered together.

An infinite unity will be an infinitely complex proposition: it will
not be analyzable in any way into a finite number of constituents. It
thus differs radically from assertions about infinite aggregates. For
example, the proposition “any number has a successor” is composed of
a finite number of constituents: the number of concepts entering into it
can be enumerated, and in addition to these there is an infinite aggregate
of terms denoted in the way indicated by *any*, which counts as one
constituent. Indeed it may be said that the logical purpose which is
served by the theory of denoting is, to enable propositions of finite
complexity to deal with infinite classes of terms: this object is effected
by *all*, *any*, and *every*, and if it were not effected, every general proposition about an infinite class would have to be infinitely complex.
Now, for my part, I see no possible way of deciding whether propositions
of infinite complexity are possible or not; but this at least is clear, that
all the propositions known to us (and, it would seem, all propositions
that we *can* know) are of finite complexity. It is only by obtaining
such propositions about infinite classes that we are enabled to deal with
infinity; and it is a remarkable and fortunate fact that this method is
successful. Thus the question whether or not there are infinite unities
must be left unresolved; the only thing we can say, on this subject, is
that no such unities occur in any department of human knowledge, and
therefore none such are relevant to the foundations of mathematics.

**142.** I come now to our second question: Must an infinite whole
which contains parts be an aggregate of terms? It is often held, for
example, that spaces have parts, and can be divided *ad lib.*, but that
they have no *simple* parts, *i.e.* they are not aggregates of points. The
same view is put forward as regards periods of time. Now it is plain
that, if our definition of a part by means of terms (*i.e.* of the second
sense of part by means of the first) was correct, the present problem can
never arise, since parts only belong to aggregates. But it may be urged
that the notion of *part* ought to be taken as an indefinable, and that
therefore it may apply to other wholes than aggregates. This will
require that we should add to aggregates and unities a new kind of
whole, corresponding to the second sense of *part*. This will be a whole
which has parts in the second sense, but is not an aggregate or a unity.
Such a whole seems to be what many philosophers are fond of calling a
continuum, and space and time are often held to afford instances of such
a whole.

Now it may be admitted that, among infinite wholes, we find a
distinction which *seems* relevant, but which, I believe, is in reality
merely psychological. In some cases, we feel no doubt as to the terms,
but great doubt as to the whole, while in others, the whole seems
obvious, but the terms seem a precarious inference. The ratios between
0 and 1, for instance, are certainly indivisible entities; but the whole
aggregate of ratios between 0 and 1 seems to be of the nature of a
construction or inference. On the other hand, sensible spaces and times
seem to be obvious wholes; but the inference to indivisible points and
instants is so obscure as to be often regarded as illegitimate. This
distinction seems, however, to have no logical basis, but to be wholly
dependent on the nature of our senses. A slight familiarity with coordinate geometry suffices to make a finite length seem strictly analogous
to the stretch of fractions between 0 and 1. It must be admitted,
nevertheless, that in cases where, as with the fractions, the indivisible
parts are evident on inspection, the problem with which we are concerned does not arise. But to infer that all infinite wholes have
indivisible parts merely because this is known to be the case with some
of them, would certainly be rash. The general problem remains,
therefore, namely: Given an infinite whole, is there a universal reason
for supposing that it contains indivisible parts?

**143.** In the first place, the definition of an infinite whole must not
be held to deny that it has an assignable number of simple parts which
do not reconstitute it. For example, the stretch of fractions from 0 to 1
has three simple parts, ⅓, ½, ⅔. But these do not reconstitute the
whole, that is, the whole has other parts which are not parts of the
assigned parts or of the sum of the assigned parts. Again, if we form a
whole out of the number 1 and a line an inch long, this whole certainly
has one simple part, namely 1. Such a case as this may be excluded by
asking whether every part of our whole either is simple or contains
simple parts. In this case, if our whole be formed by adding `n` simple
terms to an infinite whole, the `n` simple terms can be taken away, and
the question can be asked concerning the infinite whole which is left.
But again, the meaning of our question seems hardly to be: Is our
infinite whole an actual aggregate of innumerable simple parts? This is
doubtless an important question, but it is subsequent to the question we
are asking, which is: Are there always simple parts at all? We may
observe that, if a finite number of simple parts be found, and taken
away from the whole, the remainder is always infinite. For if not, it
would have a finite number; and since the term of two finite numbers is
finite, the original whole would then be finite. Hence if it can be
shown that every infinite whole contains one simple part, it follows that
it contains an infinite number of them. For, taking away the one
simple part, the remainder is an infinite whole, and therefore has a new
simple part, and so on. It follows that every part of the whole either is
simple, or contains simple parts, provided that every infinite whole has
at least one simple part. But it seems as hard to prove this as to prove
that every infinite whole is an aggregate.

If an infinite whole be divided into a finite number of parts, one at least of these parts must be infinite. If this be again divided, one of its parts must be infinite, and so on. Thus no finite number of divisions will reduce all the parts to finitude. Successive divisions give an endless series of parts, and in such endless series there is (as we shall see in Parts IV and V) no manner of contradiction. Thus there is no method of proving by actual division that every infinite whole must be an aggregate. So far as this method can show, there is no more reason for simple constituents of infinite wholes than for a first moment in time or a last finite number.

But perhaps a contradiction may emerge in the present case from the connection of whole and part with logical priority. It certainly seems a greater paradox to maintain that infinite wholes do not have indivisible parts than to maintain that there is no first moment in time or furthest limit to space. This might be explained by the fact that we know many simple terms, and some infinite wholes undoubtedly composed of simple terms, whereas we know of nothing suggesting a beginning of time or space. But it may perhaps have a more solid basis in logical priority. For the simpler is always implied in the more complex, and therefore there can be no truth about the more complex unless there is truth about the simpler. Thus in the analysis of our infinite whole, we are always dealing with entities which would not be at all unless their constituents were. This makes a real difference from the time-series, for example: a moment does not logically presuppose a previous moment, and if it did it would perhaps be self-contradictory to deny a first moment, as it has been held (for the same reason) self-contradictory to deny a First Cause. It seems to follow that infinite wholes would not have Being at all, unless there were innumerable simple Beings whose Being is presupposed in that of the infinite wholes. For where the presupposition is false, the consequence is false also. Thus there seems a special reason for completing the infinite regress in the case of infinite wholes, which does not exist where other asymmetrical transitive relations are concerned. This is another instance of the peculiarity of the relation of whole and part: a relation so important and fundamental that almost all our philosophy depends upon the theory we adopt in regard to it.

The same argument may be otherwise stated by asking how our
infinite wholes are to be defined. The definition must not be infinitely
complex, since this would require an infinite unity. Now if there is any
definition which is of finite complexity, this cannot be obtained from
the parts, since these are either infinitely numerous (in the case of an
aggregate), or themselves as complex as the whole (in the case of a
whole which is not an aggregate). But any definition which is of finite
complexity will necessarily be intensional, *i.e.* it will give some characteristic of a collection of terms. There seems to be no other known method
of defining an infinite whole, or of obtaining such a whole in a way not
involving any infinite unity.

The above argument, it must be admitted, is less conclusive than could be wished, considering the great importance of the point at issue. It may, however, be urged in support of it that all the arguments on the other side depend upon the supposed difficulties of infinity, and are therefore wholly fallacious; also that the procedure of Geometry and Dynamics (as will be shown in Parts VI and VII) imperatively demands points and instants. In all applications, in short, the results of the doctrine here advocated are far simpler, less paradoxical, and more logically satisfactory, than those of the opposite view. I shall therefore assume, throughout the remainder of this work, that all the infinite wholes with which we shall have to deal are aggregates of terms.

Notes

*^{[page 143]} A part in this sense will also be sometimes called a *simple* or *indivisible* part.

†^{[page 143]} Cf. *Phil. Werke*, ed. Gerhardt, ii, p. 316; also i, p. 338, v, pp. 144–5.

*^{[page 145]} In Leibniz’s philosophy, all contingent things are infinite unities.

Ratios and Fractions.

**144.** The present chapter, in so far as it deals with relations of
integers, is essentially confined to *finite* integers: those that are infinite
have no relations strictly analogous to what are usually called ratios.
But I shall distinguish ratios, as relations between integers, from
fractions, which are relations between aggregates, or rather between
their magnitudes of divisibility; and fractions, we shall find, may
express relations which hold where both aggregates are infinite. It will
be necessary to begin with the mathematical definition of ratio, before
proceeding to more general considerations.

Ratio is commonly associated with multiplication and division, and in this way becomes indistinguishable from fractions. But multiplication and division are equally applicable to finite and infinite numbers, though in the case of infinite numbers they do not have the properties which connect them with ratio in the finite case. Hence it becomes desirable to develop a theory of ratio which shall be independent of multiplication and division.

Two finite numbers are said to be consecutive when, if `u` be a class
having one of the numbers, and one term be added to `u`, the resulting
class has the other number. To be consecutive is thus a relation which
is one-one and asymmetrical. If now a number `a` has to a number `b`
the `n`th power of this relation of consecutiveness (powers of relations
being defined by relative multiplication), then we have `a` + `n` = `b`. This
equation expresses, between `a` and `b`, a one-one relation which is determinate when `n` is given. If now the `m`th power of this relation holds
between `a`′ and `b`′, we shall have `a`′ + `m``n` = `b`′. Also we may define `m``n` as
0 + `m``n`. If now we have three numbers `a`, `b`, `c` such that `a``b` = `c`, this
equation expresses between `a` and `c` a one-one relation which is determinate when `b` is given. Let us call this relation `B`. Suppose we have
also `a`′`b`′ = `c`. Then `a` has to `a`′ a relation which is the relative product
of `B` and the converse of `B`′, where `B`′ is derived from `b`′ as `B` was derived
from `b`. This relation we define as the ratio of `a`′ to `a`. This theory
has the advantage that it applies not only to finite integers, but to
all other series of the same type, *i.e.* all series of the type which I call
progressions.

**145.** The only point which it is important, for our present purpose,
to observe as regards the above definition of ratios is, that they are
one-one relations between finite integers, which are with one exception
asymmetrical, which are such that one and only one holds between any
specified pair of finite integers, which are definable in terms of consecutiveness, and which themselves form a series having no first or last term
and having a term, and therefore an infinite number of terms, between
any two specified terms. From the fact that ratios are relations it
results that no ratios are to be identified with integers: the ratio of 2 to
1, for example, is a wholly different entity from 2. When, therefore,
we speak of the series of ratios as containing integers, the integers said
to be contained are not cardinal numbers, but relations which have a
certain one-one correspondence with cardinal numbers. The same remark
applies to positive and negative numbers. The `n`th power of the relation
of consecutiveness is the positive number +`n`, which is plainly a wholly
different concept from the cardinal number `n`. The confusion of entities
with others to which they have some important one-one relation is an
error to which mathematicians are very liable, and one which has
produced the greatest havoc in the philosophy of mathematics. We
shall find hereafter innumerable other instances of the same error, and it
is well to realize, as early as possible, that any failure in subtlety of
distinctions is sure, in this subject at least, to cause the most disastrous
consequences.

There is no difficulty in connecting the above theory of ratio with the usual theory derived from multiplication and division. But the usual theory does not show, as the present theory does, why the infinite integers do not have ratios strictly analogous to those of finite integers. The fact is, that ratio depends upon consecutiveness, and consecutiveness as above defined does not exist among infinite integers, since these are unchanged by the addition of 1.

It should be observed that what is called addition of ratios demands a new set of relations among ratios, relations which may be called positive and negative ratios, just as certain relations among integers are positive and negative integers. This subject, however, need not be further developed.

**146.** The above theory of ratio has, it must be confessed, a highly
artificial appearance, and one which makes it seem extraordinary that
ratios should occur in daily life. The fact is, it is not ratios, but
fractions, that occur, and fractions are not purely arithmetical, but are
really concerned with relations of whole and part.

Propositions asserting fractions show an important difference from
those asserting integers. We can say `A` is one, `A` and `B` are two, and
so on; but we cannot say `A` is one-third, or `A` and `B` are two-thirds.
There is always need of some second entity, to which our first has some
fractional relation. We say `A` is one-third of `C`, `A` and `B` together are
two-thirds of `C`, and so on. Fractions, in short, are either relations of
a simple part to a whole, or of two wholes to one another. But it
is not necessary that the one whole, or the simple part, should be part
of the other whole. In the case of finite wholes, the matter seems
simple: the fraction expresses the ratio of the number of parts in the
one to the number in the other. But the consideration of infinite
wholes will show us that this simple theory is inadequate to the facts.

**147.** There is no doubt that the notion of half a league, or half
a day, is a legitimate notion. It is therefore necessary to find some
sense for fractions in which they do not essentially depend upon number.
For, if a given period of twenty-four hours is to be divided into two
continuous portions, each of which is to be half of the whole period,
there is only one way of doing this: but Cantor has shown that every
possible way of dividing the period into two continuous portions divides
it into two portions having the same *number* of terms. There must be,
therefore, some other respect in which two periods of twelve hours are
equal, while a period of one hour and another of twenty-three hours
are unequal. I shall have more to say upon this subject in Part III;
for the present I will point out that what we want is of the nature of a
magnitude, and that it must be essentially a property of ordered wholes.
I shall call this property *magnitude of divisibility*. To say now that `A` is
one-half of `B` means: `B` is a whole, and if `B` be divided into two similar
parts which have both the same magnitude of divisibility as each other,
then `A` has the same magnitude of divisibility as each of these parts.
We may interpret the fraction ½ somewhat more simply, by regarding
it as a relation (analogous to ratio so long as finite wholes are concerned)
between two magnitudes of divisibility. Thus finite integral fractions
(such as `n`/1) will measure the relation of the divisibility of an aggregate
of `n` terms to the divisibility of a single term; the converse relation will
be 1/`n`. Thus here again we have a new class of entities which is in
danger of being confused with finite cardinal integers, though in reality
quite distinct. Fractions, as now interpreted, have the advantage (upon
which all metrical geometry depends) that they introduce a discrimination of greater and smaller among infinite aggregates having the same
number of terms. We shall see more and more, as the logical inadequacy
of the usual accounts of measurement is brought to light, how absolutely
essential the notion of magnitude of divisibility really is. Fractions,
then, in the sense in which they may express relations of infinite
aggregates—and this is the sense which they usually have in daily life—are really of the nature of relations between magnitudes of divisibility;
and magnitudes of divisibility are only measured by number of parts
where the aggregates concerned are finite. It may also be observed
(though this remark is anticipatory) that, whereas ratios, as above
defined, are essentially rational, fractions, in the sense here given to
them, are also capable of irrational values. But the development of
this topic must be left for Part V.

**148.** We may now sum up the results obtained in Part II. In the
first four chapters, the modern mathematical theory of cardinal integers,
as it results from the joint labours of arithmeticians and symbolic
logicians, was briefly set forth. Chapter xi explained the notion of
similar classes, and showed that the usual formal properties of integers
result from defining them as classes of similar classes. In Chapter xii,
we showed how arithmetical addition and multiplication both depend
upon logical addition, and how both may be defined in a way which
applies equally to finite and infinite numbers, and to finite, and infinite
sums and products, and which moreover introduces nowhere any idea of
order. In Chapter xiii, we gave the strict definition of an infinite class,
as one which is similar to a class resulting from taking away one of its
terms; and we showed in outline how to connect this definition with the
definition of finite numbers by mathematical induction. The special
theory of finite integers was discussed in Chapter xiv, and it was shown
how the primitive propositions, which Peano proves to be sufficient in
this subject, can all be deduced from our definition of finite cardinal
integers. This confirmed us in the opinion that Arithmetic contains no
indefinables or indemonstrables beyond those of general logic.

We then advanced, in Chapter xv, to the consideration of philosophical questions, with a view of testing critically the above mathematical
deductions. We decided to regard both *term* and *a term* as indefinable,
and to define the number 1, as well as all other numbers, by means of these
indefinables (together with certain others). We also found it necessary
to distinguish a class from its class-concept, since one class may have
several different class-concepts. We decided that a class consists of all
the terms denoted by the class-concept, denoted in a certain indefinable
manner; but it appeared that both common usage and the majority of
mathematical purposes would allow us to identify a class with the whole
formed of the terms denoted by the class-concept. The only reasons
against this view were, the necessity of distinguishing a class containing
only one term from that one term, and the fact that some classes are
members of themselves. We found also a distinction between finite and
infinite classes, that the former can, while the latter cannot, be defined
extensionally, *i.e.* by actual enumeration of their terms. We then
proceeded to discuss what may be called the addition of individuals,
*i.e.* the notion involved in “`A` and `B`”; and we found that a more or less
independent theory of *finite* integers can be based upon this notion.
But it appeared finally, in virtue of our analysis of the notion of *class*,
that this theory was really indistinguishable from the theory previously
expounded, the only difference being that it adopted an extensional
definition of classes.

Chapter xvi dealt with the relation of whole and part. We found
that there are two indefinable senses of this relation, and one definable
sense, and that there are two correspondingly different sorts of wholes,
which we called unities and aggregates respectively. We saw also that,
by extending the notion of aggregates to single terms and to the null-class, we could regard the whole of the traditional calculus of Symbolic
Logic as an algebra specially applicable to the relations of wholes and
parts in the definable sense. We considered next, in Chapter xvii, the
notion of an infinite whole. It appeared that infinite unities, even if
they be logically possible, at any rate never appear in anything accessible
to human knowledge. But infinite aggregates, we found, must be admitted; and it seemed that all infinite wholes which are not unities
must be aggregates of terms, though it is by no means necessary that the
terms should be simple. (They must, however, owing to the exclusion
of infinite unities, be assumed to be of *finite* complexity.)

In Chapter xviii, finally, we considered ratios and fractions: the former were found to be somewhat complicated relations of finite integers, while the latter were relations between the divisibilities of aggregates. These divisibilities being magnitudes, their further discussion belongs to Part III, in which the general nature of quantity is to be considered.

QUANTITY.

The Meaning of Magnitude.

**149.** Among the traditional problems of mathematical philosophy,
few are more important than the relation of quantity to number.
Opinion as to this relation has undergone many revolutions. Euclid,
as is evident from his definitions of ratio and proportion, and indeed
from his whole procedure, was not persuaded of the applicability of
numbers to spatial magnitudes. When Des Cartes and Vieta, by the
introduction of co-ordinate Geometry, made this applicability a fundamental postulate of their systems, a new method was founded, which,
however fruitful of results, involved, like most mathematical advances of
the seventeenth century, a diminution of logical precision and a loss in
subtlety of distinction. What was meant by measurement, and whether
*all* spatial magnitudes were susceptible of a numerical measure, were
questions for whose decision, until very lately, the necessary mathematical instrument was lacking; and even now much remains to be
done before a complete answer can be given. The view prevailed that
number and quantity were *the* objects of mathematical investigation,
and that the two were so similar as not to require careful separation.
Thus number was applied to quantities without any hesitation, and
conversely, where existing numbers were found inadequate to measurement, new ones were created on the sole ground that every quantity
must have a numerical measure.

All this is now happily changed. Two different lines of argument,
conducted in the main by different men, have laid the foundations both
for large generalizations, and for thorough accuracy in detail. On the
one hand, Weierstrass, Dedekind, Cantor, and their followers, have
pointed out that, if irrational numbers are to be significantly employed as
measures of quantitative fractions, they must be defined without reference
to quantity; and the same men who showed the necessity of such a
definition have supplied the want which they had created. In this way,
during the last thirty or forty years, a new subject, which has added
quite immeasurably to theoretical correctness, has been created, which
may legitimately be called Arithmetic; for, starting with integers, it
succeeds in defining whatever else it requires—rationals, limits, irrationals, continuity, and so on. It results that, for all Algebra and
Analysis, it is unnecessary to assume any material beyond the integers,
which, as we have seen, can themselves be defined in logical terms.
It is this science, far more than non-Euclidean Geometry, that is really
fatal to the Kantian theory of *à priori* intuitions as the basis of
mathematics. Continuity and irrationals were formerly the strongholds
of the school who may be called intuitionists, but these strongholds are
theirs no longer. Arithmetic has grown so as to include all that can
strictly be called pure in the traditional mathematics.

**150.** But, concurrently with this purist’s reform, an opposite advance
has been effected. New branches of mathematics, which deal neither
with number nor with quantity, have been invented; such are the
Logical Calculus, Projective Geometry, and—in its essence—the Theory
of Groups. Moreover it has appeared that measurement—if this means
the correlation, with numbers, of entities which are not numbers or
aggregates—is not a prerogative of quantities: some quantities cannot
be measured, and some things which are not quantities (for example
anharmonic ratios projectively defined) can be measured. Measurement,
in fact, as we shall see, is applicable to all series of a certain kind—a kind
which excludes some quantities and includes some things which are
not quantities. The separation between number and quantity is thus
complete: each is wholly independent of the other. Quantity, moreover,
has lost the mathematical importance which it used to possess, owing to
the fact that most theorems concerning it can be generalized so as to
become theorems concerning order. It would therefore be natural
to discuss order before quantity. As all propositions concerning order
can, however, be established independently for particular instances of
order, and as quantity will afford an illustration, requiring slightly less
effort of abstraction, of the principles to be applied to series in general;
as, further, the theory of distance, which forms a part of the theory of
order, presupposes somewhat controversial opinions as to the nature
of quantity, I shall follow the more traditional course, and consider
quantity first. My aim will be to give, in the present chapter, a theory
of quantity which does not depend upon number, and then to show the
peculiar relation to number which is possessed by two special classes of
quantities, upon which depends the measurement of quantities wherever
this is possible. The whole of this Part, however—and it is important
to realize this—is a concession to tradition; for quantity, we shall find,
is not definable in terms of logical constants, and is not properly a
notion belonging to pure mathematics at all. I shall discuss quantity
because it is traditionally supposed to occur in mathematics, and because
a thorough discussion is required for disproving this supposition; but
if the supposition did not exist, I should avoid all mention of any such
notion as quantity.

**151.** In fixing the meaning of such a term as *quantity* or *magnitude*,
one is faced with the difficulty that, however one may define the word,
one must appear to depart from usage. This difficulty arises wherever
two characteristics have been commonly supposed inseparable which,
upon closer examination, are discovered to be capable of existing apart.
In the case of magnitude, the usual meaning appears to imply (1) a
capacity for the relations of *greater* and *less*, (2) divisibility. Of these
characteristics, the first is supposed to imply the second. But as I
propose to deny the implication, I must either admit that some things
which are indivisible are magnitudes, or that some things which are
greater or less than others are not magnitudes. As one of these departures from usage is unavoidable, I shall choose the former, which
I believe to be the less serious. A magnitude, then, is to be defined as
anything which is greater or less than something else.

It might be thought that *equality* should be mentioned, along with
greater and less, in the definition of magnitude. We shall see reason
to think, however—paradoxical as such a view may appear—that what
can be greater or less than some term, can never be equal to any term
whatever, and *vice versâ*. This will require a distinction, whose necessity
will become more and more evident as we proceed, between the kind of
terms that can be equal, and the kind that can be greater or less. The
former I shall call *quantities*, the latter *magnitudes*. An actual foot-rule is a quantity: its length is a magnitude. Magnitudes are more
abstract than quantities: when two quantities are equal, they have the
*same* magnitude. The necessity of this abstraction is the first point to
be established.

**152.** Setting aside magnitudes for the moment, let us consider
quantities. A quantity is anything which is capable of quantitative
equality to something else. Quantitative equality is to be distinguished
from other kinds, such as arithmetical or logical equality. All kinds
of equality have in common the three properties of being reflexive,
symmetrical, and transitive, *i.e.* a term which has this relation at all
has this relation to itself; if `A` has the relation to `B`, `B` has it to `A`;
if `A` has it to `B`, and `B` to `C`, `A` has it to `C`*. What it is that distinguishes quantitative equality from other kinds, and whether this kind of equality is analyzable, is a further and more difficult question,
to which we must now proceed.

There are, so far as I know, three main views of quantitative
equality. There is (1) the traditional view, which denies quantity as
an independent idea, and asserts that two terms are equal when, and
only when, they have the same number of parts. (2) There is what may
be called the relative view of quantity, according to which equal, greater
and less are all direct relations between quantities. In this view we
have no need of magnitude, since sameness of magnitude is replaced
by the symmetrical and transitive relation of equality. (3) There is
the absolute theory of quantity, in which equality is not a direct relation,
but is to be analyzed into possession of a common magnitude, *i.e.* into
sameness of relation to a third term. In this case there will be a special
kind of relation of a term to its magnitude; between two magnitudes
of the same kind there will be the relation of greater and less; while
equal, greater and less will apply to quantities only in virtue of their
relation to magnitudes. The difference between the second and third
theories is exactly typical of a difference which arises in the case of many
other series, and notably in regard to space and time. The decision
is, therefore, a matter of very considerable importance.

**153.** (1) The kind of equality which consists in having the same
number of parts has been already discussed in Part II. If this be
indeed the meaning of quantitative equality, then quantity introduces
no new idea. But it may be shown, I think, that greater and less have
a wider field than whole and part, and an independent meaning. The
arguments may be enumerated as follows: (`α`) We must admit indivisible quantities; (`β`) where the number of simple parts is infinite,
there is no generalization of number which will give the recognized
results as to inequality; (`γ`) some relations must be allowed to be
quantitative, and relations are not even conceivably divisible; (`δ`) even
where there is divisibility, the axiom that the whole is greater than the
part must be allowed to be significant, and not a result of definition.

(`α`) Some quantities are indivisible. For it is generally admitted
that some psychical existents, such as pleasure and pain, are quantitative.
If now equality means sameness in the number of indivisible parts, we
shall have to regard a pleasure or a pain as consisting of a collection
of units, all perfectly simple, and not, in any significant sense, equal
*inter se*; for the equality of compound pleasures results on this hypothesis,
solely from the number of simple ones entering into their composition,
so that equality is formally inapplicable to indivisible pleasures. If, on
the other hand, we allow pleasures to be infinitely divisible, so that no
unit we can take is indivisible, then the number of units in any given
pleasure is wholly arbitrary, and if there is to be any equality of
pleasures, we shall have to admit that any two units may be significantly
called equal or unequal*. Hence we shall require for equality some
meaning other than sameness as to the number of parts. This latter
theory, however, seems unavoidable. For there is not only no reason
to regard pleasures as consisting of definite sums of indivisible units,
but further—as a candid consideration will, I think, convince anyone—two pleasures can *always* be significantly judged equal or unequal.
However small two pleasures may be, it must always be significant to
say that they are equal. But on the theory I am combating, the judgment in question would suddenly cease to be significant when both
pleasures were indivisible units. Such a view seems wholly unwarrantable, and I cannot believe that it has been consciously held by those* who have advocated the premisses from which it follows.

(`β`) Some quantities are infinitely divisible, and in these, whatever
definition we take of infinite number, equality is not coextensive with
sameness in the number of parts. In the first place, equality or
inequality must always be definite: concerning two quantities of the
same kind, one answer must be right and the other wrong, though it is
often not in our power to decide the alternative. From this it follows
that, where quantities consist of an infinite number of parts, if equality
or inequality is to be reduced to number of parts at all, it must be
reduced to number of *simple* parts; for the number of complex parts
that may be taken to make up the whole is wholly arbitrary. But
equality, for example in Geometry, is far narrower than sameness in the
number of parts. The cardinal number of parts in any two continuous
portions of space is the same, as we know from Cantor; even the ordinal
number or type is the same for any two lengths whatever. Hence if
there is to be any spatial inequality of the kind to which Geometry and
common-sense have accustomed us, we must seek some other meaning for
equality than that obtained from the number of parts. At this point
I shall be told that the meaning is very obvious: it is obtained from
superposition. Without trenching too far on discussions which belong
to a later part, I may observe (*a*) that superposition applies to matter,
not to space, (*b*) that as a criterion of equality, it presupposes that the
matter superposed is rigid, (*c*) that rigidity means constancy as regards
metrical properties. This shows that we cannot, without a vicious
circle, define spatial equality by superposition. Spatial magnitude is, in
fact, as indefinable as every other kind; and number of parts, in this case
as in all others where the number is infinite, is wholly inadequate even
as a criterion.

(`γ`) Some relations are quantities. This is suggested by the above
discussion of spatial magnitudes, where it is very natural to base equality
upon distances. Although this view, as we shall see hereafter, is not
wholly adequate, it is yet partly true. There appear to be in certain
spaces, and there certainly are in some series (for instance that of the
rational numbers), quantitative relations of distance among the various
terms. Also similarity and difference appear to be quantities. Consider
for example two shades of colour. It seems undeniable that two shades
of red are more similar to each other than either is to a shade of blue;
yet there is no common property in the one case which is not found in
the other also. *Red* is a mere collective name for a certain series of
shades, and the only reason for giving a collective name to this series
lies in the close resemblance between its terms. Hence *red* must not be
regarded as a common property in virtue of which two shades of red
resemble each other. And since relations are not even conceivably
divisible, greater and less among relations cannot depend upon number
of parts.

(`δ`) Finally, it is well to consider directly the meanings of greater
and less on the one hand, and of whole and part on the other. Euclid’s
axiom, that the whole is greater than the part, seems undeniably significant; but on the traditional view of quantity, this axiom would be
a mere tautology. This point is again connected with the question
whether superposition is to be taken as the meaning of equality, or as a
mere criterion. On the latter view, the axiom must be significant, and
we cannot identify magnitude with number of parts*.

**154.** (2) There is therefore in quantity something over and above
the ideas which we have hitherto discussed. It remains to decide between
the relative and absolute theories of magnitude.

The relative theory regards equal quantities as not possessed of any
common property over and above that of unequal quantities, but as
distinguished merely by the mutual relation of equality. There is no
such thing as a magnitude, shared by equal quantities. We must not
say: This and that are both a yard long; we must say: This and that
are equal, or are both equal to the standard yard in the Exchequer.
Inequality is also a direct relation between quantities, not between
magnitudes. There is nothing by which a set of equal quantities are
distinguished from one which is not equal to them, except the relation of
equality itself. The course of definition is, therefore, as follows: We
have first a quality or relation, say pleasure, of which there are various
instances, specialized, in the case of a quality, by temporal or spatio-temporal position, and in the case of a relation, by the terms between
which it holds. Let us, to fix ideas, consider quantities of pleasure.
Quantities of pleasure consist merely of the complexes *pleasure at such a time*, and *pleasure at such another time* (to which *place* may be added, if
it be thought that pleasures have position in space). In the analysis of
a particular pleasure, there is, according to the relational theory, no
other element to be found. But on comparing these particular pleasures,
we find that any two have one and only one of three relations, equal,
greater, and less. Why some have one relation, some another, is a
question to which it is theoretically and strictly impossible to give an
answer; for there is, *ex hypothesi*, no point of difference except temporal
or spatio-temporal position, which is obviously irrelevant. Equal quantities of pleasure do not agree in any respect in which unequal ones
differ: it merely happens that some have one relation and some another.
This state of things, it must be admitted, is curious, and it becomes
still more so when we examine the indemonstrable axioms which the
relational theory obliges us to assume. They are the following (`A`, `B`, `C`
being all quantities of one kind):

(*a*) `A` = `B`, or `A` is greater than `B`, or `A` is less than `B`.

(*b*) `A` being given, there is always a `B`, which may be identical
with `A`, such that `A` = `B`.

(*c*) If `A` = `B`, then `B` = `A`.

(*d*) If `A` = `B` and `B` = `C`, then `A` = `C`.

(*e*) If `A` is greater than `B`, then `B` is less than `A`.

(*f*) If `A` is greater than `B`, and `B` is greater than `C`, then `A` is
greater than `C`.

(*g*) If `A` is greater than `B`, and `B` = `C`, then `A` is greater than `C`.

(*h*) If `A` = `B`, and `B` is greater than `C`, then `A` is greater than `C`.

From (*b*), (*c*), and (*d*) it follows that `A` = `A`*. From (*e*) and (*f*) it
follows that, if `A` is less than `B`, and `B` is less than `C`, then `A` is less than
`C`; from (*c*), (*e*), and (*h*) it follows that, if `A` is less than `B`, and `B` = `C`,
then `A` is less than `C`; from (*c*), (*e*), and (*g*) it follows that, if `A` = `B`, and
`B` is less than `C`, then `A` is less than `C`. (In the place of (*b*) we may put
the axiom: If `A` be a quantity, then `A` = `A`.) These axioms, it will be
observed, lead to the conclusion that, in any proposition asserting
equality, excess, or defect, an equal quantity may be substituted anywhere without affecting the truth or falsehood of the proposition.
Further, the proposition `A` = `A` is an essential part of the theory. Now
the first of these facts strongly suggests that what is relevant in quantitative propositions is not the actual quantity, but some property which
it shares with other equal quantities. And this suggestion is almost
demonstrated by the second fact, `A` = `A`. For it may be laid down that
the only unanalyzable symmetrical and transitive relation which a term
can have to itself is identity, if this be indeed a relation. Hence the
relation of equality should be analyzable. Now to say that a relation is
analyzable is to say either that it consists of two or more relations
between its terms, which is plainly not the case here, or that, when it is
said to hold between two terms, there is some third term to which both
are related in ways which, when compounded, give the original relation.
Thus to assert that `A` is `B`’s grandparent is to assert that there is some
third person `C`, who is `A`’s son or daughter and `B`’s father or mother.
Hence if equality be analyzable, two equal terms must both be related to
some third term; and since a term may be equal to itself, any two equal
terms must have the *same* relation to the third term in question. But
to admit this is to admit the absolute theory of magnitude.

A direct inspection of what we mean when we say that two terms are equal or unequal will reinforce the objections to the relational theory. It seems preposterous to maintain that equal quantities have absolutely nothing in common beyond what is shared by unequal quantities. Moreover unequal quantities are not merely different: they are different in the specific manner expressed by saying that one is greater, the other less. Such a difference seems quite unintelligible unless there is some point of difference, where unequal quantities are concerned, which is absent where quantities are equal. Thus the relational theory, though apparently not absolutely self-contradictory, is complicated and paradoxical. Both the complication and the paradox, we shall find, are entirely absent in the absolute theory.

**155.** (3) In the absolute theory, there is, belonging to a set of
equal quantities, one definite concept, namely a certain magnitude.
Magnitudes are distinguished among concepts by the fact that they
have the relations of greater and less (or at least one of them) to other
terms, which are therefore also magnitudes. Two magnitudes cannot
be equal, for equality belongs to quantities, and is defined as possession
of the *same* magnitude. Every magnitude is a simple and indefinable
concept. Not any two magnitudes are one greater and the other less;
on the contrary, given any magnitude, those which are greater or less
than that magnitude form a certain definite class, within which any two
are one greater and the other less. Such a class is called a *kind* of
magnitude. A kind of magnitude may, however, be also defined in
another way, which has to be connected with the above by an axiom.
Every magnitude is a magnitude of something—pleasure, distance, area,
etc.—and has thus a certain specific relation to the something of which
it is a magnitude. This relation is very peculiar, and appears to be
incapable of further definition. All magnitudes which have this relation
to one and the same something (*e.g.* pleasure) are magnitudes of one
kind; and with this definition, it becomes an axiom to say that, of two
magnitudes of the same kind, one is greater and the other less.

**156.** An objection to the above theory may be based on the
relation of a magnitude to that whose magnitude it is. To fix our
ideas, let us consider pleasure. A magnitude of pleasure is so much
pleasure, such and such an intensity of pleasure. It seems difficult to
regard this, as the absolute theory demands, as a simple idea: there
seem to be two constituents, pleasure and intensity. Intensity need not
be intensity of pleasure, and intensity of pleasure is distinct from
abstract pleasure. But what we require for the constitution of a certain
magnitude of pleasure is, not intensity in general, but a certain specific
intensity; and a *specific* intensity cannot be indifferently of pleasure or
of something else. We cannot first settle how much we will have, and
then decide whether it is to be pleasure or mass. A specific intensity
must be of a specific kind. Thus intensity and pleasure are not independent and coordinate elements in the definition of a given amount
of pleasure. There are different kinds of intensity, and different magnitudes in each kind; but magnitudes in different kinds must be different.
Thus it seems that the common element, indicated by the term *intensity*
or *magnitude*, is not any thing intrinsic, that can be discovered by analysis
of a single term, but is merely the fact of being one term in a relation of
inequality. Magnitudes are defined by the fact that they have this
relation, and they do not, so far as the definition shows, agree in anything else. The class to which they all belong, like the married portion
of a community, is defined by mutual relations among its terms, not by
a common relation to some outside term—unless, indeed, inequality
itself were taken as such a term, which would be merely an unnecessary
complication. It is necessary to consider what may be called the
extension or field of a relation, as well as that of a class-concept: and
magnitude is the class which forms the extension of inequality. Thus
*magnitude of pleasure* is complex, because it combines magnitude and
pleasure; but a particular magnitude of pleasure is not complex, for
magnitude does not enter into its concept at all. It is only a magnitude
because it is greater or less than certain other terms; it is only a magnitude of *pleasure* because of a certain relation which it has to pleasure.
This is more easily understood where the particular magnitude has a
special name. A yard, for instance, is a magnitude, because it is greater
than a foot; it is a magnitude of length, because it is what is called
`a` length. Thus all magnitudes are simple concepts, and are classified
into kinds by their relation to some quality or relation. The quantities
which are instances of a magnitude are particularized by spatio-temporal
position or (in the case of relations which are quantities) by the terms
between which the relation holds. Quantities are not properly greater
or less, for the relations of greater and less hold between their
magnitudes, which are distinct from the quantities.

When this theory is applied in the enumeration of the necessary
axioms, we find a very notable simplification. The axioms in which
equality appears have all become demonstrable, and we require only the
following (`L`, `M`, `N` being magnitudes of one kind):

(*a*) No magnitude is greater or less than itself.

(*b*) `L` is greater than `M` or `L` is less than `M`.

(*c*) If `L` is greater than `M`, then `M` is less than `L`.

(*d*) If `L` is greater than `M` and `M` is greater than `N`, then `L` is
greater than `N`.

The difficult axiom which we formerly called (*b*) is avoided, as are the
other axioms concerning equality; and those that remain are simpler
than our former set.

**157.** The decision between the absolute and relative theories can
be made at once by appealing to a certain general principle, of very
wide application, which I propose to call the principle of Abstraction.
This principle asserts that, whenever a relation, of which there are
instances, has the two properties of being symmetrical and transitive,
then the relation in question is not primitive, but is analyzable into
sameness of relation to some other term; and that this common relation
is such that there is only one term at most to which a given term can be
so related, though many terms may be so related to a given term.
(That is, the relation is like that of son to father: a man may have
many sons, but can have only one father.)

This principle, which we have already met with in connection with
cardinals, may seem somewhat elaborate. It is, however, capable of
proof, and is merely a careful statement of a very common assumption.
It is generally held that all relations are analyzable into identity or
diversity of content. Though I entirely reject this view, I retain, so far
as symmetrical transitive relations are concerned, what is really a somewhat modified statement of the traditional doctrine. Such relations, to
adopt more usual phraseology, are always constituted by possession of
a common property. But a common property is not a very precise
conception, and will not, in most of its ordinary significations, formally
fulfil the function of analyzing the relations in question. A common
quality of two terms is usually regarded as a predicate of those terms.
But the whole doctrine of subject and predicate, as the only form of
which propositions are capable, and the whole denial of the ultimate
reality of relations, are rejected by the logic advocated in the present
work. Abandoning the word *predicate*, we may say that the most
general sense which can be given to a common property is this: A
common property of two terms is any third term to which both have
one and the same relation. In this general sense, the possession of
a common property is symmetrical, but not necessarily transitive. In
order that it may be transitive, the relation to the common property
must be such that only one term at most can be the property of any
given term*. Such is the relation of a quantity to its magnitude, or of
an event to the time at which it occurs: given one term of the relation,
namely the referent, the other is determinate, but given the other, the
one is by no means determinate. Thus it is capable of demonstration
that the possession of a common property of the type in question always
leads to a symmetrical transitive relation. What the principle of
abstraction asserts is the converse, that such relations only spring from
common properties of the above type*. It should be observed that the
relation of the terms to what I have called their common property can
never be that which is usually indicated by the relation of subject to
predicate, or of the individual to its class. For no subject (in the
received view) can have only one predicate, and no individual can belong
to only one class. The relation of the terms to their common property
is, in general, different in different cases. In the present case, the
quantity is a complex of which the magnitude forms an element: the
relation of the quantity to the magnitude is further defined by the
fact that the magnitude has to belong to a certain class, namely that of
magnitudes. It must then be taken as an axiom (as in the case of
colours) that two magnitudes of the same kind cannot coexist in one
spatio-temporal place, or subsist as relations between the same pair of
terms; and this supplies the required uniqueness of the magnitude. It
is such synthetic judgments of incompatibility that lead to negative
judgments; but this is a purely logical topic, upon which it is not
necessary to enlarge in this connection.

**158.** We may now sum up the above discussion in a brief statement
of results. There are a certain pair of indefinable relations, called
*greater* and *less*; these relations are asymmetrical and transitive, and
are inconsistent the one with the other. Each is the converse of the
other, in the sense that, whenever the one holds between `A` and `B`, the
other holds between `B` and `A`. The terms which are capable of these
relations are *magnitudes*. Every magnitude has a certain peculiar
relation to some concept, expressed by saying that it is a magnitude *of* that
concept. Two magnitudes which have this relation to the same concept
are said to be of the same kind; to be of the same kind is the necessary
and sufficient condition for the relations of greater and less. When a
magnitude can be particularized by temporal, spatial, or spatio-temporal
position, or when, being a relation, it can be particularized by taking
into a consideration a pair of terms between which it holds, then the
magnitude so particularized is called a *quantity*. Two magnitudes of
the same kind can never be particularized by exactly the same specifications. Two quantities which result from particularizing the same
magnitude are said to be *equal*.

Thus our indefinables are (1) greater and less, (2) every particular magnitude. Our indemonstrable propositions are:

(1) Every magnitude has to some term the relation which makes it of a certain kind.

(2) Any two magnitudes of the same kind are one greater and the other less.

(3) Two magnitudes of the same kind, if capable of occupying space or time, cannot both have the same spatio-temporal position; if relations, can never be both relations between the same pair of terms.

(4) No magnitude is greater than itself.

(5) If `A` is greater than `B`, `B` is less than `A`, and *vice versâ*.

(6) If `A` is greater than `B` and `B` is greater than `C`, then `A` is
greater than `C`*.

Further axioms characterize various species of magnitudes, but the above seem alone necessary to magnitude in general. None of them depend in any way upon number or measurement; hence we may be undismayed in the presence of magnitudes which cannot be divided or measured, of which, in the next chapter, we shall find an abundance of instances.

*Note to Chapter XIX.* The work of Herr Meinong on Weber’s Law,
already alluded to, is one from which I have learnt so much, and with
which I so largely agree, that it seems desirable to justify myself on
the points in which I depart from it. This work begins (§1) by a
characterization of magnitude as that which is limited towards zero.
Zero is understood as the negation of magnitude, and after a discussion,
the following statement is adopted (p. 8):

“That is or has magnitude, which allows the interpolation of terms between itself and its contradictory opposite.”

Whether this constitutes a definition, or a mere criterion, is left
doubtful (*ib.*), but in either case, it appears to me to be undesirable as
a fundamental characterization of magnitude. It derives support, as
Herr Meinong points out (p. 6*n*), from its similarity to Kant’s
“Anticipations of Perception†.” But it is, if I am not mistaken, liable
to several grave objections. In the first place, the whole theory of zero
is most difficult, and seems subsequent, rather than prior, to the theory
of other magnitudes. And to regard zero as the contradictory opposite
of other magnitudes seems erroneous. The phrase should denote the
class obtained by negation of the class “magnitudes of such and such
a kind”; but this obviously would not yield the zero of that kind of
magnitude. Whatever interpretation we give to the phrase, it would
seem to imply that we must regard zero as not a magnitude of the kind
whose zero it is. But in that case it is not less than the magnitudes of
the kind in question, and there seems no particular meaning in saying
that a lesser magnitude is *between* zero and a greater magnitude. And
in any case, the notion of *between*, as we shall see in Part IV, demands
asymmetrical relations among the terms concerned. These relations, it
would seem, are, in the case of magnitude, none other than *greater* and
*less*, which are therefore prior to the betweenness of magnitudes, and
more suitable to definition. I shall endeavour at a later stage to give
what I conceive to be the true theory of zero; and it will then appear
how difficult this subject is. It can hardly be wise, therefore, to introduce
zero in the first account of magnitude. Other objections might be urged,
as, for instance, that it is doubtful whether all kinds of magnitude have
a zero; that in discrete kinds of magnitude, zero is unimportant; and
that among distances, where the zero is simply identity, there is hardly
the same relation of zero to negation or non-existence as in the case of
qualities such as pleasure. But the main reason must be the logical
inversion involved in the introduction of *between* before any asymmetrical
relations have been specified from which it could arise. This subject
will be resumed in Chapter xxii.

Notes

*^{[page 159]} On the independence of these three properties, see Peano, *Revue de Mathématique*, VII, p. 22. The reflexive property is not strictly necessary; what is properly necessary and what is alone (at first sight at any rate) true of quantitative equality, is, that there exists at least one pair of terms having the relation in question. It follows then from the other two properties that each of these terms has to itself the relation in question.

*^{[page 160]} I shall never use the word *unequal* to mean merely *not equal*, but always to mean *greater or less*, *i.e.* not equal, though of the same kind of quantities.

*^{[page 161]} *E.g.* Mr Bradley, “What do we mean by the Intensity of Psychical States?” *Mind*, N. S. Vol. iv; see esp. p. 5.

*^{[page 162]} Compare, with the above discussion, Meinong, *Ueber die Bedeutung des Weberschen Gesetzes*, Hamburg and Leipzig, 1896; especially Chap. i, §3.

*^{[page 163]} This does not follow from (*c*) and (*d*) alone, since they do not assert that `A` is ever equal to `B`. See Peano, *loc. cit.*

*^{[page 166]} The proof of these assertions is mathematical, and depends upon the Logic of Relations; it will be found in my article “Sur la Logique des Relations,” *R. d. M.* vii, No. 2, §1, Props. 6.1, and 6.2.

*^{[page 167]} The principle is proved by showing that, if `R` be a symmetrical transitive relation, and `a` a term of the field of `R`, `a` has, to the class of terms to which it has the relation `B` taken as a whole, a many-one relation which, relationally multiplied by its converse, is equal to `R`. Thus a magnitude may, so far as formal arguments are concerned, be identified with a class of equal quantities.

*^{[page 168]} It is not necessary in (5) and (6) to add “`A`, `B`, `C` being magnitudes,” for the above relations of greater and less are what define magnitudes, and the addition would therefore be tautological.

†^{[page 168]} *Reine Vernunft*, ed. Hartenstein (1867), p. 158.

The Range of Quantity.

**159.** The questions to be discussed in the present chapter are these:
What kinds of terms are there which, by their common relation to a
number of magnitudes, constitute a class of quantities of one kind?
Have all such terms anything else in common? Is there any mark
which will ensure that a term is thus related to a set of magnitudes?
What sorts of terms are capable of degree, or intensity, or greater and
less?

The traditional view regards divisibility as a common mark of all
terms having magnitude. We have already seen that there is no
*à priori* ground for this view. We are now to examine the question
inductively, to find as many undoubted instances of quantities as possible,
and to inquire whether they all have divisibility or any other common
mark.

Any term of which a greater or less degree is possible contains under
it a collection of magnitudes of one kind. Hence the comparative form
in grammar is *primâ facie* evidence of quantity. If this evidence were
conclusive, we should have to admit that all, or almost all, qualities are
susceptible of magnitude. The praises and reproaches addressed by
poets to their mistresses would afford comparatives and superlatives
of most known adjectives. But some circumspection is required in
using evidence of this grammatical nature. There is always, I think,
*some* quantitative comparison wherever a comparative or superlative
occurs, but it is often not a comparison as regards the quality indicated
by grammar.

“

O ruddier than the cherry,

O sweeter than the berry,

O nymph more bright

Than moonshine light,”

O sweeter than the berry,

O nymph more bright

Than moonshine light,”

are lines containing three comparatives. As regards sweetness and
brightness, we have, I think, a genuine quantitative comparison; but as
regards ruddiness, this may be doubted. The comparative here—and
generally where colours are concerned—indicates, I think, not more of a
given colour, but more likeness to a standard colour. Various shades of
colour are supposed to be arranged in a series, such that the difference
of quality is greater or less according as the distance in the series is
greater or less. One of these shades is the ideal “ruddiness,” and others
are called more or less ruddy according as they are nearer to or further
from this shade in the series. The same explanation applies, I think,
to such terms as *whiter*, *blacker*, *redder*. The true quantity involved
seems to be, in all these cases, a relation, namely the relation of similarity.
The difference between two shades of colour is certainly a difference of
quality, not merely of magnitude; and when we say that one thing is
redder than another, we do not imply that the two are of the same shade.
If there were no difference of shade, we should probably say one was
*brighter* than the other, which is quite a different kind of comparison.
But though the difference of two shades is a difference of quality, yet, as
the possibility of serial arrangement shows, this difference of quality is
itself susceptible of degrees. Each shade of colour seems to be simple
and unanalyzable; but neighbouring colours in the spectrum are certainly
more similar than remote colours. It is this that gives continuity to
colours. Between two shades of colour, `A` and `B`, we should say, there
is always a third colour `C`; and this means that `C` resembles `A` or `B`
more than `B` or `A` does. But for such relations of immediate resemblance,
we should not be able to arrange colours in series. The resemblance
must be immediate, since all shades of colour are unanalyzable, as appears
from any attempt at description or definition*. Thus we have an
indubitable case of relations which have magnitude. The difference or
resemblance of two colours is a relation, and is a magnitude; for it is
greater or less than other differences or resemblances.

**160.** I have dwelt upon this case of colours, since it is one instance
of a very important class. When any number of terms can be arranged
in a series, it frequently happens that any two of them have a relation
which may, in a generalized sense, be called a *distance*. This relation
suffices to generate a serial arrangement, and is always necessarily a
magnitude. In all such cases, if the terms of the series have names, and
if these names have comparatives, the comparatives indicate, not more
of the term in question, but more likeness to that term. Thus, if we
suppose the time-series to be one in which there is distance, when an
event is said to be more recent than another, what is meant is that its
distance from the present was less than that of the other. Thus recentness
is not itself a quality of the time or of the event. What are quantitatively
compared in such cases are relations, not qualities. The case of colours
is convenient for illustration, because colours have names, and the
difference of two colours is generally admitted to be qualitative. But
the principle is of very wide application. The importance of this class
of magnitudes, and the absolute necessity of clear notions as to their
nature, will appear more and more as we proceed. The whole philosophy
of space and time, and the doctrine of so-called extensive magnitudes,
depend throughout upon a clear understanding of series and distance.

Distance must be distinguished from mere difference or unlikeness.
It holds only between terms in a series. It is intimately connected with
order, and implies that the terms between which it holds have an ultimate
and simple difference, not one capable of analysis into constituents.
It implies also that there is a more or less continuous passage, through
other terms belonging to the same series, from one of the distant terms
to the other. Mere difference *per se* appears to be the bare *minimum* of
a relation, being in fact a precondition of almost all relations. It is
always absolute, and is incapable of degrees. Moreover it holds between
any two terms whatever, and is hardly to be distinguished from the
assertion that they are two. But distance holds only between the
members of certain series, and its existence is then the source of the
series. It is a specific relation, and it has *sense*; we can distinguish
the distance of `A` from `B` from that of `B` from `A`. This last mark
alone suffices to distinguish distance from bare difference.

It might perhaps be supposed that, in a series in which there is
distance, although the distance `A``B` must be greater than or less than `A``C`,
yet the distance `B``D` need not be either greater or less than `A``C`. For
example, there is obviously more difference between the pleasure
derivable from £5 and that derivable from £100 than between that
from £5 and that from £20. But need there be either equality or
inequality between the difference for £1 and £20 and that for £5 and
£100? This question must be answered affirmatively. For `A``C` is
greater or less than `B``C`, and `B``C` is greater or less than `B``D`; hence `A``C`,
`B``C` and also `B``C`, `B``D` are magnitudes of the same kind. Hence `A``C`, `B``D`
are magnitudes of the same kind, and if not identical, one must be the
greater and the other the less. Hence, when there is distance in a series,
any two distances are quantitatively comparable.

It should be observed that all the magnitudes of one kind form a series, and that their distances, therefore, if they have distances, are again magnitudes. But it must not be supposed that these can, in general, be obtained by subtraction, or are of the same kind as the magnitudes whose differences they express. Subtraction depends, as a rule, upon divisibility, and is therefore in general inapplicable to indivisible quantities. The point is important, and will be treated in detail in the following chapter.

Thus nearness and distance are relations which have magnitude. Are there any other relations having magnitude? This may, I think, be doubted*. At least I am unaware of any other such relation, though I know no way of disproving their existence.

**161.** There is a difficult class of terms, usually regarded as magnitudes, apparently implying relations, though certainly not always
relational. These are differential coefficients, such as velocity and
acceleration. They must be borne in mind in all attempts to generalize
about magnitude, but owing to their complexity they require a special
discussion. This will be given in Part V; and we shall then find that
differential coefficients are never magnitudes, but only real numbers, or
segments in some series.

**162.** All the magnitudes dealt with hitherto have been, strictly
speaking, indivisible. Thus the question arises: Are there any divisible
magnitudes? Here I think a distinction must be made. A magnitude
is essentially one, not many. Thus no magnitude is correctly expressed
as a number of terms. But may not the quantity which has magnitude
be a sum of parts, and the magnitude a magnitude of divisibility? If so,
every whole consisting of parts will be a single term possessed of the property of divisibility. The more parts it consists of, the greater is its
divisibility. On this supposition, divisibility is a magnitude, of which we
may have a greater or less degree; and the degree of divisibility corresponds
exactly, in finite wholes, to the number of parts. But though the whole
which has divisibility is of course divisible, yet its divisibility, which alone
is strictly a magnitude, is not properly speaking divisible. The divisibility
does not itself consist of parts, but only of the property of having parts.
It is necessary, in order to obtain divisibility, to take the whole strictly
as *one*, and to regard divisibility as its adjective. Thus although, in
this case, we have numerical measurement, and all the mathematical
consequences of division, yet, philosophically speaking, our magnitude is
still indivisible.

There are difficulties, however, in the way of admitting divisibility as
a kind of magnitude. It seems to be not a property of the whole, but
merely a relation to the parts. It is difficult to decide this point, but a
good deal may be said, I think, in support of divisibility as a simple
quality. The whole has a certain relation, which for convenience we may
call that of inclusion, to all its parts. This relation is the same whether
there be many parts or few; what distinguishes a whole of many parts is
that it has many such relations of inclusion. But it seems reasonable to
suppose that a whole of many parts differs from a whole of few parts in
some intrinsic respect. In fact, wholes may be arranged in a series
according as they have more or fewer parts, and the serial arrangement
implies, as we have already seen, some series of properties differing more
or less from each other, and agreeing when two wholes have the same
finite number of parts, but distinct from number of parts in finite
wholes. These properties can be none other than greater or less degrees
of divisibility. Thus magnitude of divisibility would *appear* to be a
simple property of a whole, distinct from the number of parts included
in the whole, but correlated with it, provided this number be finite. If
this view can be maintained, divisibility may be allowed to remain as a
numerically measurable, but not divisible, class of magnitudes. In this
class we should have to place lengths, areas and volumes, but not
distances. At a later stage, however, we shall find that the divisibility
of infinite wholes, in the sense in which this is not measured by cardinal
numbers, must be derived through relations in a way analogous to that
in which distance is derived, and must be really a property of relations*.

Thus it would appear, in any case, that all magnitudes are indivisible. This is one common mark which they all possess, and so far as I know, it is the only one to be added to those enumerated in Chapter xix. Concerning the range of quantity, there seems to be no further general proposition. Very many simple non-relational terms have magnitude, the principal exceptions being colours, points, instants and numbers.

**163.** Finally, it is important to remember that, on the theory
adopted in Chapter xix, a given magnitude of a given kind is a simple
concept, having to the kind a relation analogous to that of inclusion in
a class. When the kind is a kind of existents, such as pleasure, what
actually exists is never the kind, but various particular magnitudes of
the kind. Pleasure, abstractly taken, does not exist, but various amounts
of it exist. This degree of abstraction is essential to the theory of
quantity: there must be entities which differ from each other in nothing
except magnitude. The grounds for the theory adopted may perhaps
appear more clearly from a further examination of this case.

Let us start with Bentham’s famous proposition: “Quantity of pleasure being equal, pushpin is as good as poetry.” Here the qualitative difference of the pleasures is the very point of the judgment; but in order to be able to say that the quantities of pleasure are equal, we must be able to abstract the qualitative differences, and leave a certain magnitude of pleasure. If this abstraction is legitimate, the qualitative difference must be not truly a difference of quality, but only a difference of relation to other terms, as, in the present case, a difference in the causal relation. For it is not the whole pleasurable states that are compared, but only—as the form of the judgment aptly illustrates—their quality of pleasure. If we suppose the magnitude of pleasure to be not a separate entity, a difficulty will arise. For the mere element of pleasure must be identical in the two cases, whereas we require a possible difference of magnitude. Hence we can neither hold that only the whole concrete state exists, and any part of it is an abstraction, nor that what exists is abstract pleasure, not magnitude of pleasure. Nor can we say: We abstract, from the whole states, the two elements magnitude and pleasure. For then we should not get a quantitative comparison of the pleasures. The two states would agree in being pleasures, and in being magnitudes. But this would not give us a magnitude of pleasure; and it would give a magnitude to the states as a whole, which is not admissible. Hence we cannot abstract magnitude in general from the states, since as wholes they have no magnitude. And we have seen that we must not abstract bare pleasure, if we are to have any possibility of different magnitudes. Thus what we have to abstract is a magnitude of pleasure as a whole. This must not be analyzed into magnitude and pleasure, but must be abstracted as a whole. And the magnitude of pleasure must exist as a part of the whole pleasurable states, for it is only where there is no difference save at most one of magnitude that quantitative comparison is possible. Thus the discussion of this particular case fully confirms the theory that every magnitude is unanalyzable, and has only the relation analogous to inclusion in a class to that abstract quality or relation of which it is a magnitude.

Having seen that all magnitudes are indivisible, we have next to consider the extent to which numbers can be used to express magnitudes, and the nature and limits of measurement.

Notes

*^{[page 171]} On the subject of the resemblances of colours, see Meinong, “Abstrahiren und Vergleichen,” *Zeitschrift f. Psych. u. Phys. d. Sinnesorgane*, Vol. xxiv, p. 72ff. I am not sure that I agree with the whole of Meinong’s argument, but his general conclusion, “dass die Umfangscollective des Aehnlichen Allgemeinheiten darstellen, an denen die Abstraction wenigstens unmittelbar keinen Antheil hat” (p. 78), appears to me to be a correct and important logical principle.

*^{[page 173]} Meinong, *Ueber die Bedeutung des Weber’schen Gesetzes*, Hamburg and Leipzig, 1896, p. 23.

Numbers As Expressing Magnitudes: Measurement.

**164.** It is one of the assumptions of educated common-sense that
two magnitudes of the same kind must be numerically comparable.
People are apt to say that they are thirty per cent, healthier or happier
than they were, without any suspicion that such phrases are destitute of
meaning. The purpose of the present chapter is to explain what is
meant by measurement, what are the classes of magnitudes to which it
applies, and how it is applied to those classes.

Measurement of magnitudes is, in its most general sense, any method
by which a unique and reciprocal correspondence is established between
all or some of the magnitudes of a kind and all or some of the numbers,
integral, rational, or real, as the case may be. (It might be thought
that complex numbers ought to be included; but what can *only* be
measured by complex numbers is in fact always an aggregrate of magnitudes of different kinds, not a single magnitude.) In this general sense,
measurement demands some one-one relation between the numbers and
magnitudes in question—a relation which may be direct or indirect,
important or trivial, according to circumstances. Measurement in this
sense can be applied to very many classes of magnitudes; to two great
classes, distances and divisibilities, it applies, as we shall see, in a more
important and intimate sense.

Concerning measurement in the most general sense, there is very
little to be said. Since the numbers form a series, and since every kind
of magnitude also forms a series, it will be desirable that the order of
the magnitudes measured should correspond to that of the numbers, *i.e.* that all relations of *between* should be the same for magnitudes and their
measures. Wherever there is a zero, it is well that this should be
measured by the number zero. These and other conditions, which a
measure should fulfil if possible, may be laid down; but they are of
practical rather than theoretical importance.

**165.** There are two general metaphysical opinions, either of which,
if accepted, shows that *all* magnitudes are theoretically capable of
measurement in the above sense. The first of these is the theory that
all events either are, or are correlated with, events in the dynamical
causal series. In regard to the so-called secondary qualities, this view
has been so far acted upon by physical science that it has provided most
of the so-called intensive quantities that appear in space with spatial,
and thence numerical, measures. And with regard to mental quantities
the theory in question is that of psychophysical parallelism. Here the
motion which is correlated with any psychical quantity always theoretically affords a means of measuring that quantity. The other metaphysical
opinion, which leads to universal measurability, is one suggested by
Kant’s “Anticipations of Perception*,” namely that, among intensive
magnitudes, an increase is always accompanied by an increase of reality.
Reality, in this connection, seems synonymous with existence; hence
the doctrine may be stated thus: Existence is a kind of intensive
magnitude, of which, where a greater magnitude exists, there is always
more than where a less magnitude exists. (That this is exactly Kant’s
doctrine seems improbable; but it is at least a tenable view.) In this
case, since two instances of the same magnitude (*i.e.* two equal quantities)
must have more existence than one, it follows that, if a single magnitude
of the same kind can be found having the same amount of existence as
the two equal quantities together, then that magnitude may be called
double that of each of the equal quantities. In this way all intensive
magnitudes become theoretically capable of measurement. That this
method has any practical importance it would be absurd to maintain;
but it may contribute to the appearance of meaning belonging to *twice as happy*. It gives a sense, for example, in which we may say that a
child derives as much pleasure from one chocolate as from two acid
drops; and on the basis of such judgments the hedonistic Calculus
could theoretically be built.

There is one other general observation of some importance. If it be maintained that all series of magnitudes are either continuous in Cantor’s sense, or are similar to series which can be chosen out of continuous series, then it is theoretically possible to correlate any kind of magnitudes with all or some of the real numbers, so that the zeros correspond, and the greater magnitudes correspond to the greater numbers. But if any series of magnitudes, without being continuous, contains continuous series, then such a series of magnitudes will be strictly and theoretically incapable of measurement by the real numbers†.

**166.** Leaving now these somewhat vague generalities, let us examine
the more usual and concrete sense of measurement. What we require is
some sense in which we may say that one magnitude is double of another.
In the above instances this sense was derived by correlation with spatio-temporal magnitudes, or with existence. This presupposed that in these
cases a meaning had been found for the phrase. Hence measurement
demands that, in some cases, there should be an intrinsic meaning to the
proposition “this magnitude is double of that.” (In what sense the
meaning is intrinsic will appear as we proceed.) Now so long as
quantities are regarded as inherently divisible, there is a perfectly
obvious meaning to such a proposition: a magnitude `A` is double of `B`
when it is the magnitude of two quantities together, each of these
having the magnitude `B`. (It should be observed that to divide a
magnitude into two equal parts must always be impossible, since there
are no such things as equal magnitudes.) Such an interpretation will
still apply to magnitudes of divisibility; but since we have admitted
other magnitudes, a different interpretation (if any) must be found for
these. Let us first examine the case of divisibilities, and then proceed
to the other cases where measurement is intrinsically possible.

**167.** The divisibility of a finite whole is immediately and inherently
correlated with the number of simple parts in the whole. In this case,
although the magnitudes are even now incapable of addition of the sort
required, the quantities can be added in the manner explained in Part II.
The addition of two magnitudes of divisibility yields merely two magnitudes, not a new magnitude. But the addition of two quantities of
divisibility, *i.e.* two wholes, does yield a new single whole, provided the
addition is of the kind which results from logical addition by regarding
classes as the wholes formed by their terms. Thus there is a good
meaning in saying that one magnitude of divisibility is double of
another, when it applies to a whole containing twice as many parts.
But in the case of infinite wholes, the matter is by no means so simple.
Here the number of simple parts (in the only senses of infinite number
hitherto discovered) may be equal without equality in the magnitude of
divisibility. We require here a method which does not go back to
simple parts. In actual space, we have immediate judgments of equality
as regards two infinite wholes. When we have such judgments, we can
regard the sum of `n` equal wholes as `n` times each of them; for addition
of wholes does not demand their finitude. In this way numerical comparison of some pairs of wholes becomes possible. By the usual well-known methods, by continual subdivision and the method of limits, this
is extended to all pairs of wholes which are such that immediate comparisons are possible. Without these immediate comparisons, which
are necessary both logically and psychologically*, nothing can be
accomplished: we are always reduced in the last resort to the immediate
judgment that our foot-rule has not greatly changed its size during
measurement, and this judgment is prior to the results of physical
science as to the extent to which bodies do actually change their sizes.
But where immediate comparison is psychologically impossible, we may
theoretically substitute a logical variety of measurement, which, however,
gives a property not of the divisible whole, but of some relation or class
of relations more or less analogous to those that hold between points in
space.

That divisibility, in the sense required for areas and volumes, is not a property of a whole, results from the fact (which will be established in Part VI) that between the points of a space there are always relations which generate a different space. Thus two sets of points which, with regard to one set of relations, form equal areas, form unequal areas with respect to another set, or even form one an area and the other a line or a volume. If divisibility in the relevant sense were an intrinsic property of wholes, this would be impossible. But this subject cannot be fully discussed until we come to Metrical Geometry.

Where our magnitudes are divisibilities, not only do numbers measure
them, but the difference of two measuring numbers, with certain limitations, measures the magnitude of the difference (in the sense of dissimilarity) between the divisibilities. If one of the magnitudes be
fixed, its difference from the other increases as the difference of the
measuring numbers increases; for this difference depends upon the
difference in the number of parts. But I do not think it can be shown
generally that, if `A`, `B`, `C`, `D` be the numbers measuring four magnitudes,
and `A` − `B` = `C` − `D`, then the differences of the magnitudes are equal.
It would seem, for instance, that the difference between one inch and
two inches is greater than that between 1001 inches and 1002 inches.
This remark has no importance in the present case, since differences of
divisibility are never required; but in the case of distances it has a
curious connection with non-Euclidean Geometry. But it is theoretically
important to observe that, if divisibility be indeed a magnitude—as the
equality of areas and volumes seems to require—then there is strictly no
ground for saying that the divisibility of a sum of two units is twice as
great as that of one unit. Indeed this proposition cannot be strictly
taken, for no magnitude *is* a sum of parts, and no magnitude therefore is
double of another. We can only mean that the sum of two units contains twice as many parts, which is an arithmetical, not a quantitative,
judgment, and is adequate only in the case where the number of parts is
finite, since in other cases the double of a number is in general equal to
it. Thus even the measurement of divisibility by numbers contains
an element of convention; and this element, we shall find, is still more
prominent in the case of distances.

**168.** In the above case we still had addition in one of its two
fundamental senses, *i.e.* the combination of wholes to form a new whole.
But in other cases of magnitude we do not have any such addition.
The sum of two pleasures is not a new pleasure, but is merely two
pleasures. The sum of two distances is also not properly one distance.
But in this case we have an extension of the idea of addition. Some
such extension must always be possible where measurement is to be
effected in the more natural and restricted sense which we are now
discussing. I shall first explain this generalized addition in abstract
terms, and then illustrate its application to distances.

It sometimes happens that two quantities, which are not capable of
addition proper, have a relation, which has itself a one-one relation to
a quantity of the same kind as those between which it holds. Supposing
`a`, `b`, `c` to be such quantities, we have, in the case supposed, some proposition `a``B``c`, where `B` is a relation which uniquely determines and is
uniquely determined by some quantity `b` of the same kind as that to
which `a` and `c` belong. Thus for example two ratios have a relation,
which we may call their difference, which is itself wholly determined by
another ratio, namely the difference, in the arithmetical sense, of the
two given ratios. If `α`, `β`, `γ` be terms in a series in which there is
distance, the distances `α``β`, `α``γ` have a relation which is measured by
(though not identical with) the distance `β``γ`. In all such cases, by an
extension of addition, we may put `a` + `b` = `c` in place of `a``B``c`. Wherever
a set of quantities have relations of this kind, if further `a``B``c` implies `b``A``c`,
so that `a` + `b` = `b` + `a`, we shall be able to proceed as if we had ordinary
addition, and shall be able in consequence to introduce numerical
measurement.

The conception of distance will be discussed fully in Part IV, in
connection with order: for the present I am concerned only to show
how distances come to be measurable. The word will be used to cover
a far more general conception than that of distance in space. I shall
mean by a kind of distance a set of quantitative asymmetrical relations of
which one and only one holds between any pair of terms of a given
class; which are such that, if there is a relation of the kind between `a`
and `b`, and also between `b` and `c`, then there is one of the kind between
`a` and `c`, the relation between `a` and `c` being the relative product of
those between `a` and `b`, `b` and `c`; this product is to be commutative,
*i.e.* independent of the order of its factors; and finally, if the distance
`a``b` be greater than the distance `a``c`, then, `d` being any other member of
the class, `d``b` is greater than `d``c`. Although distances are thus relations,
and therefore indivisible and incapable of addition proper, there is a
simple and natural convention by which such distances become numerically measurable.

The convention is this. Let it be agreed that, when the distances
`a`_{0}`a`_{1}, `a`_{1}`a`_{2} … `a`_{n−1}`a`` _{n}` are all equal and in the same sense, then

**169.** The importance of the numerical measurement of distance, at
least as applied to space and time, depends partly upon a further fact,
by which it is brought into relation with the numerical measurement of
divisibility. In all series there are terms intermediate between any two
whose distance is not the minimum. These terms are determinate when
the two distant terms are specified. The intermediate terms may be
called the *stretch* from `a`_{0} to `a`` _{n}`†. The whole composed of these terms
is a quantity, and has a divisibility measured by the number of terms,
provided their number is finite. If the series is such that the distances
of consecutive terms are all equal, then, if there are

**170.** The above analysis explains a curious problem which must
have troubled most people who have endeavoured to philosophize about
Geometry. Starting from one-dimensional magnitudes connected with
the straight line, most theories may be divided into two classes, those
appropriate to areas and volumes, and those appropriate to angles
between lines or planes. Areas and volumes are radically different
from angles, and are generally neglected in philosophies which hold
to relational views of space or start from projective Geometry. The
reason of this is plain enough. On the straight line, if, as is usually
supposed, there is such a relation as distance, we have two philosophically distinct but practically conjoined magnitudes, namely the distance,
and the divisibility of the stretch. The former is similar to angles; the
latter, to areas and volumes. Angles may also be regarded as distances
between terms in a series, namely between lines through a point or
planes through a line. Areas and volumes, on the contrary, are sums,
or magnitudes of divisibility. Owing to the confusion of the two kinds
of magnitude connected with the line, either angles, or else areas and
volumes, are usually incompatible with the philosophy invented to
suit the line. By the above analysis, this incompatibility is at once
explained and overcome*.

**171.** We thus see how two great classes of magnitudes—divisibilities
and distances—are rendered amenable to measure. These two classes
practically cover what are usually called extensive magnitudes, and it
will be convenient to continue to allow the name to them. I shall
extend this name to cover all distances and divisibilities, whether they
have any relation to space and time or not. But the word *extensive*
must not be supposed to indicate, as it usually does, that the magnitudes
so designated are divisible. We have already seen that no magnitude is
divisible. *Quantities* are only divisible into other quantities in the one
case of wholes which are quantities of divisibility. Quantities which are
distances, though I shall call them extensive, are not divisible into
smaller distances; but they allow the important kind of addition explained above, which I shall call in future *relational* addition†.

All other magnitudes and quantities may be properly called *intensive*.
Concerning these, unless by some causal relation, or by means of some
more or less roundabout relation such as those explained at the beginning
of the present chapter, numerical measurement is impossible. Those
mathematicians who are accustomed to an exclusive emphasis on numbers,
will think that not much can be said with definiteness concerning magnitudes incapable of measurement. This, however, is by no means the
case. The immediate judgments of equality, upon which (as we saw)
all measurements depend, are still possible where measurement fails, as
are also the immediate judgments of greater and less. Doubt only
arises where the difference is small; and all that measurement does,
in this respect, is to make the margin of doubt smaller—an achievement
which is purely psychological, and of no philosophical importance.
Quantities not susceptible of numerical measurement can thus be arranged in a scale of greater and smaller magnitudes, and this is the
only strictly quantitative achievement of even numerical measurement.
We can know that one magnitude is greater than another, and that
a third is intermediate between them; also, since the differences of
magnitudes are always magnitudes, there is always (theoretically, at
least) an answer to the question whether the difference of one pair
of magnitudes is greater than, less than, or the same as the difference of
another pair of the same kind. And such propositions, though to the
mathematician they may appear approximate, are just as precise and
definite as the propositions of Arithmetic. Without numerical measurement, therefore, the quantitative relations of magnitudes have all the
definiteness of which they are capable—nothing is added, from the
theoretical standpoint, by the assignment of correlated numbers. The
whole subject of the measurement of quantities is, in fact, one of more
practical than theoretical importance. What is theoretically important
in it is merged in the wider question of the correlation of series, which
will occupy us much hereafter. The chief reason why I have treated
the subject thus at length is derived from its traditional importance, but
for which it might have been far more summarily treated.

Notes

*^{[page 177]} *Reine Vernunft*, ed. Hart. (1867), p. 160. The wording of the first edition illustrates better than that of the second the doctrine to which I allude. See *e.g.* Erdmann’s edition, p. 161.

†^{[page 177]} See Part V, Chap. xxxiii ff.

*^{[page 178]} Cf. Meinong, *op. cit.*, pp. 63–4.

*^{[page 181]} See Part IV, Chap. xxxi. This axiom asserts that a magnitude can be divided into `n` equal parts, and forms part of Du Bois Raymond’s definition of linear magnitude. See his *Allgemeine Functionentheorie* (Tübingen, 1882), Chap. i, §16; also Bettazzi, *Teoria delle Grandezze* (Pisa, 1890), p. 44. The axiom of Archimedes asserts that, given any two magnitudes of a kind, some finite multiple of the lesser exceeds the greater.

†^{[page 181]} Called *Strecke* by Meinong, *op. cit.*, *e.g.* p. 22.

*^{[page 182]} In Part VI, we shall find reason to deny distance in most spaces. But there is still a distinction between stretches, consisting of the terms of some series, and such quantities as areas and volumes, where the terms do not, in any simple sense, form a one-dimensional series.

†^{[page 182]} Not to be confounded with the *relative* addition of the Algebra of Relatives. It is connected rather with relative multiplication.

Zero.

**172.** The present chapter is concerned, not with any form of the
numerical zero, nor yet with the infinitesimal, but with the pure zero
of magnitude. This is the zero which Kant has in mind, in his refutation of Mendelssohn’s proof of the immortality of the soul*. Kant
points out that an intensive magnitude, while remaining of the same
kind, can become zero; and that, though zero is a definite magnitude,
no quantity whose magnitude is zero can exist. This kind of zero, we
shall find, is a fundamental quantitative notion, and is one of the points
in which the theory of quantity presents features peculiar to itself. The
quantitative zero has a certain connection both with the number 0 and
with the null-class in Logic, but it is not (I think) definable in terms of
either. What is less universally realized is its complete independence
of the infinitesimal. The latter notion will not be discussed until the
following chapter.

The meaning of zero, in any kind of quantity, is a question of much
difficulty, upon which the greatest care must be bestowed, if contradictions are to be avoided. Zero seems to be definable by some general
characteristic, without reference to any special peculiarity of the kind of
quantity to which it belongs. To find such a definition, however, is far
from easy. Zero *seems* to be a radically distinct conception according as
the magnitudes concerned are discrete or continuous. To prove that
this is not the case, let us examine various suggested definitions.

**173.** (1) Herr Meinong (*op. cit.*, p. 8) regards zero as the contradictory opposite of each magnitude of its kind. The phrase
“contradictory opposite” is one which is not free from ambiguity.
The opposite of a class, in symbolic logic, is the class containing all
individuals not belonging to the first class; and hence the opposite
of an individual should be all other individuals. But this meaning is
evidently inappropriate: zero is not everything except one magnitude
of its kind, nor yet everything except the class of magnitudes of its
kind. It can hardly be regarded as true to say that a pain is a zero
pleasure. On the other hand, a zero pleasure is said to be *no pleasure*,
and this is evidently what Herr Meinong means. But although we
shall find this view to be correct, the meaning of the phrase is very
difficult to seize. It does not mean something other than pleasure,
as when our friends assure us that it is no pleasure to tell us our faults.
It seems to mean what is neither pleasure, nor yet anything else. But
this would be merely a cumbrous way of saying *nothing*, and the
reference to pleasure might be wholly dropped. This gives a zero
which is the same for all kinds of magnitude, and if this be the true
meaning of zero, then zero is not one among the magnitudes of a kind,
nor yet a term in the series formed by magnitudes of a kind. For
though it is often true that there is nothing smaller than all the
magnitudes of a kind, yet it is always false that nothing itself is
smaller than all of them. This zero, therefore, has no special reference
to any particular kind of magnitude, and is incapable of fulfilling the
functions which Herr Meinong demands of it*. The phrase, however,
as we shall see, is capable of an interpretation which avoids this difficulty.
But let us first examine some other suggested meanings of the word.

**174.** (2) Zero may be defined as the least magnitude of its kind.
Where a kind of magnitude is discrete, and generally when it has what
Professor Bettazzi calls a *limiting* magnitude of the kind†, such a
definition is insufficient. For in such a case, the limiting magnitude
seems to be really the least of its kind. And in any case, the definition
gives rather a characteristic than a true definition, which must be sought
in some more purely logical notion, for zero cannot fail to be in some
sense a denial of all other magnitudes of the kind. The phrase that
zero is the smallest of magnitudes is like the phrase Which De Morgan
commends for its rhetoric: “Achilles was the strongest of all his enemies.”
Thus it would be obviously false to say that 0 is the least of the positive
integers, or that the interval between `A` and `A` is the least interval
between any two letters of the alphabet. On the other hand, where a
kind of magnitude is continuous, and has no limiting magnitude, although
we have apparently a gradual and unlimited approach to zero, yet now a
new objection arises. Magnitudes of this kind are essentially such as
have no minimum. Hence we cannot without express contradiction take
zero as their minimum. We may, however, avoid this contradiction by
saying that there is always a magnitude less than any other, but not
zero, unless that other be zero. This emendation avoids any formal
contradiction, and is only inadequate because it gives rather a mark of
zero than its true meaning. Whatever else is a magnitude of the kind
in question might have been diminished; and we wish to know what it
is that makes zero obviously incapable of any further diminution. This
the suggested definition does not tell us, and therefore, though it gives a
characteristic which often belongs to no other magnitude of the kind, it
cannot be considered philosophically sufficient. Moreover, where there
are negative magnitudes, it precludes us from regarding these as less
than zero.

**175.** (3) Where our magnitudes are differences or distances, zero
has, at first sight, an obvious meaning, namely identity. But here again,
the zero so defined seems to have no relation to one kind of distances
rather than another: a zero distance in time would seem to be the same
as a zero distance in space. This can, however, be avoided, by substituting,
for identity simply, identity with some member of the class of terms
between which the distances in question hold. By this device, the zero
of any class of relations which are magnitudes is made perfectly definite
and free from contradiction; moreover we have both zero quantities and
zero magnitudes, for if `A` and `B` be terms of the class which has distances,
identity with `A` and identity with `B` are distinct zero quantities*. This
case, therefore, is thoroughly clear. And yet the definition must be
rejected: for it is plain that zero has some general logical meaning, if
only this could be clearly stated, which is the same for all classes of
quantities; and that a zero distance is not actually the same concept as
identity.

**176.** (4) In any class of magnitudes which is continuous, in the
sense of having a term between any two, and which also has no limiting
magnitude, we can introduce zero in the manner in which real numbers are
obtained from rationals. Any collection of magnitudes defines a class of
magnitudes less than all of them. This class of magnitudes can be made as
small as we please, and can actually be made to be the null-class, *i.e.* to
contain no members at all. (This is effected, for instance, if our collection
consists of all magnitudes of the kind.) The classes so defined form a
series, closely related to the series of original magnitudes, and in this
new series the null-class is definitely the first term. Thus taking the
classes as quantities, the null-class is a zero quantity. There is no class
containing a finite number of members, so that there is not, as in
Arithmetic, a discrete approach to the null-class; on the contrary, the
approach is (in several senses of the word) continuous. This method of
defining zero, which is identical with that by which the real number zero
is introduced, is important, and will be discussed in Part V. But for the
present we may observe that it again makes zero the same for all kinds
of magnitude, and makes it not one among the magnitudes whose zero
it is.

**177.** (5) We are compelled, in this question, to face the problem
as to the nature of negation. “No pleasure” is obviously a different
concept from “no pain,” even when these terms are taken strictly as
mere denials of pleasure and pain respectively. It would seem that “no
pleasure” has the same relation to pleasure as the various magnitudes of
pleasure have, though it has also, of course, the special relation of
negation. If this be allowed, we see that, if a kind of magnitudes be
defined by that of which they are magnitudes, then *no pleasure* is one
among the various magnitudes of pleasure. If, then, we are to hold to
our axiom, that all pairs of magnitudes of one kind have relations of
inequality, we shall be compelled to admit that zero is less than all other
magnitudes of its kind. It seems, indeed, to be rendered evident that
this must be admitted, by the fact that zero is obviously *not greater*
than all other magnitudes of its kind. This shows that zero has a
connection with *less* which it does not have with *greater*. And if we
adopt this theory, we shall no longer accept the clear and simple account
of zero distances given above, but we shall hold that a zero distance is
strictly and merely *no distance*, and is only *correlated* with identity.

Thus it would seem that Herr Meinong’s theory, with which we
began, is substantially correct; it requires emendation, on the above
view, only in this, that a zero magnitude is the denial of the defining
concept of a kind of magnitudes, not the denial of any one particular
magnitude, or of all of them. We shall have to hold that any concept
which defines a kind of magnitudes defines also, by its negation, a
particular magnitude of the kind, which is called the zero of that kind,
and is less than all other members of the kind. And we now reap the
benefit of the absolute distinction which we made between the defining
concept of a kind of magnitude, and the various magnitudes of the kind.
The relation which we allowed between a particular magnitude and that
of which it is a magnitude was not identified with the class-relation, but
was held to be *sui generis*; there is thus no contradiction, as there
would be in most theories, in supposing this relation to hold between *no pleasure* and *pleasure*, or between *no distance* and *distance*.

**178.** But finally, it must be observed that *no pleasure*, the zero
magnitude, is not obtained by the logical denial of pleasure, and is not
the same as the logical notion of *not pleasure*. On the contrary, *no pleasure* is essentially a quantitative concept, having a curious and
intimate relation to logical denial, just as 0 has a very intimate relation
to the null-class. The relation is this, that there is no *quantity* whose
*magnitude* is zero, so that the class of zero quantities is the null-class*.
The zero of any kind of magnitude is incapable of that relation to
existence or to particulars, of which the other magnitudes are capable.
But this is a synthetic proposition, to be accepted only on account of its
self-evidence. The zero magnitude of any kind, like the other magnitudes,
is properly speaking indefinable, but is capable of specification by means
of its peculiar relation to the logical zero.

Notes

*^{[page 184]} *Kritik der Reinen Vernunft*, ed. Hartenstein, p. 281ff.

*^{[page 185]} See note to Chap. xix, *supra*.

†^{[page 185]} *Teoria delle Grandezze*, Pisa, 1890, p. 24.

*^{[page 186]} On this point, however, see §55 above.

*^{[page 187]} This must be applied in correction of what was formerly said about zero distances.

Infinity, the Infinitesimal, and Continuity.

**179.** Almost all mathematical ideas present one great difficulty:
the difficulty of infinity. This is usually regarded by philosophers as
an antinomy, and as showing that the propositions of mathematics are
not metaphysically true. From this received opinion I am compelled to
dissent. Although all apparent antinomies, except such as are quite
easily disposed of, and such as belong to the fundamentals of logic, are,
in my opinion, reducible to the one difficulty of infinite number, yet this
difficulty itself appears to be soluble by a correct philosophy of *any*, and
to have been generated very largely by confusions due to the ambiguity
in the meaning of finite integers. The problem in general will be
discussed in Part V; the purpose of the present chapter is merely to
show that quantity, which has been regarded as the true home of infinity,
the infinitesimal, and continuity, must give place, in this respect, to
order; while the statement of the difficulties which arise in regard to
quantity can be made in a form which is at once ordinal and arithmetical,
but involves no reference to the special peculiarities of quantity.

**180.** The three problems of infinity, the infinitesimal, and continuity, as they occur in connection with quantity, are closely related.
None of them can be fully discussed at this stage, since all depend
essentially upon order, while the infinitesimal depends also upon number.
The question of infinite quantity, though traditionally considered more
formidable than that of zero, is in reality far less so, and might be
briefly disposed of, but for the great devotion commonly shown by
philosophers to a proposition which I shall call the axiom of finitude.
Of some kinds of magnitude (for example ratios, or distances in space
and time), it appears to be true that there is a magnitude greater than
any given magnitude. That is, any magnitude being mentioned, another
can be found which is greater than it. The deduction of infinity from
this fact is, when correctly performed, a mere fiction to facilitate compression in the statement of results obtained by the method of limits.
Any class `u` of magnitudes of our kind being defined, three cases may
arise: (1) There may be a class of terms greater than any of our class `u`,
and this new class of terms may have a smallest member; (2) there may
be such a class, but it may have no smallest member; (3) there may be
no magnitudes which are greater than any term of our class `u`. Supposing our kind of magnitudes to be one in which there is no greatest
magnitude, case (2) will always arise where the class `u` contains a finite
number of terms. On the other hand, if our series be what is called
*condensed in itself*, case (2) will never arise when `u` is an infinite class,
and has no greatest term; and if our series is not condensed in itself,
but does have a term between any two, another which has this property
can always be obtained from it*. Thus all infinite series which have
no greatest term will have limits, except in case (3). To avoid circumlocution, case (3) is defined as that in which the limit is infinite.
But this is a mere device, and it is generally admitted by mathematicians to be such. Apart from special circumstances, there is no
reason, merely because a kind of magnitudes has no maximum, to
admit that there is an infinite magnitude of the kind, or that there
are many such. When magnitudes of a kind having no maximum
are capable of numerical measurement, they very often obey the axiom
of Archimedes, in virtue of which the ratio of any two magnitudes of
the kind is finite. Thus, so far, there might appear to be no problem
connected with infinity.

But at this point the philosopher is apt to step in, and to declare that, by all true philosophic principles, every well-defined series of terms must have a last term. If he insists upon creating this last term, and calling it infinity, he easily deduces intolerable contradictions, from which he infers the inadequacy of mathematics to obtain absolute truth. For my part, however, I see no reason for the philosopher’s axiom. To show, if possible, that it is not a necessary philosophic principle, let us undertake its analysis, and see what it really involves.

The problem of infinity, as it has now emerged, is not properly a quantitative problem, but rather one concerning order. It is only because our magnitudes form a series having no last term that the problem arises: the fact that the series is composed of magnitudes is wholly irrelevant. With this remark I might leave the subject to a later stage. But it will be worth while now to elicit, if not to examine, the philosopher’s axiom of finitude.

**181.** It will be well, in the first place, to show how the problem
concerning infinity is the same as that concerning continuity and the
infinitesimal. For this purpose, we shall find it convenient to ignore the
absolute zero, and to mean, when we speak of any kind of magnitudes,
all the magnitudes of the kind except zero. This is a mere change of
diction, without which intolerable repetitions would be necessary. Now
there certainly are some kinds of magnitude where the three following
axioms hold:

(1) If `A` and `B` be any two magnitudes of the kind, and `A` is
greater than `B`, there is always a third magnitude `C` such that `A`
is greater than `C` and `C` greater than `B`. (This I shall call, for the
present, the axiom of continuity.)

(2) There is always a magnitude less than any given magnitude `B`.

(3) There is always a magnitude greater than any given magnitude `A`.

From these it follows:—

(1) That no two magnitudes of the kind are consecutive.

(2) That there is no least magnitude.

(3) That there is no greatest magnitude.

The above propositions are certainly true of `s``o``m``e` kinds of magnitude; whether they are true of `a``l``l` kinds remains to be examined. The
following three propositions, which directly contradict the previous three,
must be always true, if the philosopher’s axiom of finitude is to be
accepted:

(*a*) There are consecutive magnitudes, *i.e.* magnitudes such that
no other magnitude of the same kind is greater than the less and less
than the greater of the two given magnitudes.

(*b*) There is a magnitude smaller than any other of the same kind.

(*c*) There is a magnitude greater than any other of the same
kind*.

As these three propositions directly contradict the previous three, it would seem that both sets cannot be true. We have to examine the grounds for both, and let one set of alternatives fall.

**182.** Let us begin with the propositions (*a*), (*b*), (*c*), and examine
the nature of their grounds.

(*a*) A definite magnitude `A` being given, all the magnitudes greater
than `A` form a series, whose differences from `A` are magnitudes of a new
kind. If there be a magnitude `B` consecutive to `A`, its difference from `A`
will be the least magnitude of its kind, provided equal stretches correspond to equal distances in the series. And conversely, if there be
a smallest difference between two magnitudes. `A`, `B`, then these two
magnitudes must always be consecutive; for if not, any intermediate
magnitude would have a smaller difference from `A` than `B` has. Thus
if (*b*) is universally true, (*a*) must also be true; and conversely, if (*a*) is
true, and if the series of magnitudes be such that equal stretches correspond to equal distances, then (*b*) is true of the distances between the
magnitudes considered. We might rest content with the reduction of
(*a*) to (*b*), and proceed to the proof of (*b*); but it seems worth while
to offer a direct proof, such as presumably the finitist philosopher has in
his mind.

Between `A` and `B` there is a certain number of magnitudes, unless `A`
and `B` are consecutive. The intermediate magnitudes all have order, so
that in passing from `A` to `B` all the intermediate magnitudes would
be met with. In such an enumeration, there must be *some* magnitude
which comes next after any magnitude `C`; or, to put the matter otherwise, since the enumeration has to begin, it must begin somewhere, and
the term with which it begins must be the magnitude next to `A`. If
this were not the case, there would be no definite series; for if all the
terms have an order, some of them must be consecutive.

In the above argument, what is important is its dependence upon
number. The whole argument turns upon the principle by which infinite
number is shown to be self-contradictory, namely: *A given collection of many terms must contain some finite number of terms.* We say: All
the magnitudes between `A` and `B` form a given collection. If there
are no such magnitudes, `A` and `B` are consecutive, and the question
is decided. If there are such magnitudes, there must be a finite
number of them, say `n`. Since they form a series, there is a definite
way of assigning to them the ordinal numbers from 1 to `n`. The `m`th
and (`m` + 1)th are then consecutive.

If the axiom in italics be denied, the whole argument collapses; and
this, we shall find, is also the case as regards (*b*) and (*c*).

(*b*) The proof here is precisely similar to the proof of (*a*). If there
are no magnitudes less than `A`, then `A` is the least of its kind, and the
question is decided. If there are any, they form a definite collection,
and therefore (by our axiom) have a finite number, say `n`. Since they
form a series, ordinal numbers may be assigned to them growing higher
as the magnitudes become more distant from `A`. Thus the `n`th magnitude is the smallest of its kind.

(*c*) The proof here is obtained as in (*b*), by considering the collection
of magnitudes greater than `A`. Thus everything depends upon our
axiom, without which no case can be made out against continuity, or
against the absence of a greatest and least magnitude.

As regards the axiom itself, it will be seen that it has no particular
reference to quantity, and at first sight it might seem to have no
reference to order. But the word *finite*, which occurs in it, requires
definition; and this definition, in the form suited to the present discussion, has, we shall find, an essential reference to order.

**183.** Of all the philosophers who have inveighed against infinite
number, I doubt whether there is one who has known the difference
between finite and infinite numbers. The difference is simply this.
Finite numbers obey the law of mathematical induction; infinite
numbers do not. That is to say, given any number `n`, if `n` belongs
to every class `s` to which 0 belongs, and to which belongs also the
number next after any number which is an `s`, then `n` is finite; if not,
not. It is in this *alone*, and in its consequences, that finite and infinite
numbers differ*.

The principle may be otherwise stated thus: If every proposition
which holds concerning 0, and also holds concerning the immediate
successor of every number of which it holds, holds concerning the number
`n`, then `n` is finite; if not, not. This is the precise sense of what may be
popularly expressed by saying that every finite number can be reached
from 0 by successive steps, or by successive additions of 1. This is the
principle which the philosopher must be held to lay down as obviously
applicable to all numbers, though he will have to admit that the more
precisely his principle is stated, the less obvious it becomes.

**184.** It may be worth while to show exactly how mathematical
induction enters into the above proofs. Let us take the proof of (*a*),
and suppose there are `n` magnitudes between `A` and `B`. Then to begin
with, we supposed these magnitudes capable of enumeration, *i.e.* of an
order in which there are consecutive terms and a first term, and a term
immediately preceding any term except the first. This property presupposes mathematical induction, and was in fact the very property in
dispute. Hence we must not presuppose the possibility of enumeration,
which would be a *petitio principii*. But to come to the kernel of the
argument: we supposed that, in any series, there must be a definite way
of assigning ordinal numbers to the terms. This property belongs to
a series of one term, and belongs to every series having `m` + 1 terms,
if it belongs to every series having `m` terms. Hence, by mathematical
induction, it belongs to all series having a finite number of terms. But
if it be allowed that the number of terms may not be finite, the whole
argument collapses.

As regards (*b*) and (*c*), the argument is similar. Every series having
a finite number of terms can be shown by mathematical induction to
have a first and last term; but no way exists of proving this concerning
other series, or of proving that all series are finite. Mathematical
induction, in short, like the axiom of parallels, is useful and convenient
in its proper place; but to suppose it always true is to yield to the
tyranny of mere prejudice. The philosopher’s finitist arguments, therefore, rest on a principle of which he is ignorant, which there is no reason
to affirm, and every reason to deny. With this conclusion, the apparent
antinomies may be considered solved.

**185.** It remains to consider what kinds of magnitude satisfy the
propositions (1), (2), (3). There is no general principle from which
these can be proved or disproved, but there are certainly cases where
they are true, and others where they are false. It is generally held by
philosophers that numbers are essentially discrete, while magnitudes are
essentially continuous. This we shall find to be not the case. Real
numbers possess the most complete continuity known, while many kinds
of magnitude possess no continuity at all. The word *continuity* has
many meanings, but in mathematics it has only two—one old, the other
new. For present purposes the old meaning will suffice. I therefore
set up, for the present, the following definition:

*Continuity* applies to series (and only to series) whenever these are
such that there is a term between any two given terms*. Whatever is
not a series, or a compound of series, or whatever is a series not fulfilling
the above condition, is discontinuous.

Thus the series of rational numbers is continuous, for the arithmetic mean of two of them is always a third rational number between the two. The letters of the alphabet are not continuous.

We have seen that any two terms in a series have a distance, or a
stretch which has magnitude. Since there are certainly discrete series
(*e.g.* the alphabet), there are certainly discrete magnitudes, namely, the
distances or the stretches of terms in discrete series. The distance
between the letters `A` and `C` is greater than that between the letters
`A` and `B`, but there is no magnitude which is greater than one of these
and less than the other. In this case, there is also a greatest possible
and a least possible distance, so that all three propositions (1), (2), (3)
fail. It must not be supposed, however, that the three propositions
have any necessary connection. In the case of the integers, for example,
there are consecutive distances, and there is a least possible distance,
namely, that between consecutive integers, but there is no greatest
possible distance. Thus (3) is true, while (1) and (2) are false. In
the case of the series of notes, or of colours of the rainbow, the series
has a beginning and end, so that there is a greatest distance; but there
is no least distance, and there is a term between any two. Thus (1)
and (2) are true, while (3) is false. Or again, if we take the series
composed of zero and the fractions having one for numerator, there is a
greatest distance, but no least distance, though the series is discrete.
Thus (2) is true, while (1) and (3) are false. And other combinations
might be obtained from other series.

Thus the three propositions (1), (2), (3), have no necessary connection, and all of them, or any selection, may be false as applied to any given kind of magnitude. We cannot hope, therefore, to prove their truth from the nature of magnitude. If they are ever to be true, this must be proved independently, or discovered by mere inspection in each particular case. That they are sometimes true, appears from a consideration of the distances between terms of the number-continuum or of the rational numbers. Either of these series is continuous in the above sense, and has no first or last term (when zero is excluded). Hence its distances or stretches fulfil all three conditions. The same might be inferred from space and time, but I do not wish to anticipate what is to be said of these. Quantities of divisibility do not fulfil these conditions when the wholes which are divisible consist of a finite number of indivisible parts. But where the number of parts is infinite in a whole class of differing magnitudes, all three conditions are satisfied, as appears from the properties of the number-continuum.

We thus see that the problems of infinity and continuity have no essential connection with quantity, but are due, where magnitudes present them at all, to characteristics depending upon number and order. Hence the discussion of these problems can only be undertaken after the pure theory of order has been set forth*. To do this will be the aim of the following Part.

**186.** We may now sum up the results obtained in Part III. In
Chapter xix we determined to define a magnitude as whatever is either
greater or less than something else. We found that magnitude has no
necessary connection with divisibility, and that greater and less are indefinable. Every magnitude, we saw, has a certain relation—analogous to,
but not identical with, that of inclusion in a class—to a certain quality
or relation; and this fact is expressed by saying that the magnitude
in question is a magnitude *of* that quality or relation. We defined a
*quantity* as a particular contained under a magnitude, *i.e.* as the complex
consisting of a magnitude with a certain spatio-temporal position, or with
a pair of terms between which it is a relation. We decided, by means of
a general principle concerning transitive symmetrical relations, that
it is impossible to content ourselves with quantities, and deny the
further abstraction involved in magnitudes; that equality is not a direct
relation between quantities, but consists in being particularizations of
the same magnitude. Thus equal quantities are instances of the same
magnitude. Similarly greater and less are not direct relations between
quantities, but between magnitudes: quantities are only greater and
less in virtue of being instances of greater and less magnitudes. Any
two magnitudes which are of the same quality or relation are one
greater, the other less; and greater and less are asymmetrical transitive
relations.

Among the terms which have magnitude are not only many qualities,
but also asymmetrical relations by which certain kinds of series are
constituted. These may be called *distances*. When there are distances
in a series, any two terms of the series have a distance, which is the same
as, greater than, or less than, the distance of any two other terms in the
series. Another peculiar class of magnitudes discussed in Chapter xx is
constituted by the degrees of divisibility of different wholes. This, we
found, is the only case in which quantities are divisible, while there is no
instance of divisible magnitudes.

Numerical measurement, which was discussed in Chapter xxi, required,
owing to the decision that most quantities and all magnitudes are indivisible, a somewhat unusual treatment. The problem lies, we found,
in establishing a one-one relation between numbers and the magnitudes
of the kind to be measured. On certain metaphysical hypotheses (which
were neither accepted nor rejected), this was found to be always theoretically possible as regards existents actual or possible, though often
not practically feasible or important. In regard to two classes of
magnitudes, namely divisibilities and distances, measurement was found
to proceed from a very natural convention, which defines what is
meant by saying (what can never have the simple sense which it has in
connection with finite wholes and parts) that one such magnitude is
double of, or `n` times, another. The relation of distance to stretch
was discussed, and it was found that, apart from a special axiom to
that effect, there was no *à priori* reason for regarding equal distances as
corresponding to equal stretches.

In Chapter xxii we discussed the definition of zero. The problem of zero was found to have no connection with that of the infinitesimal, being in fact closely related to the purely logical problem as to the nature of negation. We decided that, just as there are the distinct logical and arithmetical negations, so there is a third fundamental kind, the quantitative negation; but that this is negation of that quality or relation of which the magnitudes are, not of magnitude of that quality or relation. Hence we were able to regard zero as one among the magnitudes contained in a kind of magnitude, and to distinguish the zeroes of different kinds. We showed also that quantitative negation is connected with logical negation by the fact that there cannot be any quantities whose magnitude is zero.

In the present Chapter the problems of continuity, the infinite, and the infinitesimal, were shown to belong, not specially to the theory of quantity, but to those of number and order. It was shown that, though there are kinds of magnitude in which there is no greatest and no least magnitude, this fact does not require us to admit infinite or infinitesimal magnitudes; and that there is no contradiction in supposing a kind of magnitudes to form a series in which there is a term between any two, and in which, consequently, there is no term consecutive to a given term. The supposed contradiction was shown to result from an undue use of mathematical induction—a principle, the full discussion of which presupposes the philosophy of order.

Notes

*^{[page 189]} This will be further explained in Part V, Chap. xxxvi.

*^{[page 190]} Those Hegelians who search for a chance of an antinomy may proceed to the definition of zero and infinity by means of the above propositions. When (2) and (*b*) both hold, they may say, the magnitude satisfying (*b*) is called zero; when (3) and (*c*) both hold, the magnitude satisfying (*c*) is called infinity. We have seen, however, that zero is to be otherwise defined, and has to be excluded before (2) becomes true; while infinity is not a magnitude of the kind in question at all, but merely a piece of mathematical shorthand. (Not infinity in general, that is, but infinite magnitude in the cases we are discussing.)

*^{[page 192]} It must, however, be mentioned that one of these consequences gives a logical difference between finite and infinite numbers, which may be taken as an independent definition. This has been already explained in Part II, Chap. xiii, and will be further discussed in Part V.

*^{[page 193]} The objection to this definition (as we shall see in Part V) is, that it does not give the usual properties of the existence of limits to convergent series which are commonly associated with continuity. Series of the above kind will be called *compact*, except in the present discussion.

*^{[page 194]} Cf. Couturat, “Sur la Definition du Continu,” *Revue de Métaphysique et de Morale*, 1900.

ORDER.

The Genesis of Series.

**187.** The notion of order or series is one with which, in connection
with distance, and with the order of magnitude, we have already
had to deal. The discussion of continuity in the last chapter of
Part III showed us that this is properly an ordinal notion, and
prepared us for the fundamental importance of order. It is now high
time to examine this concept on its own account. The importance of
order, from a purely mathematical standpoint, has been immeasurably
increased by many modern developments. Dedekind, Cantor, and Peano
have shown how to base all Arithmetic and Analysis upon series of a
certain kind—*i.e.* upon those properties of finite numbers in virtue
of which they form what I shall call a *progression*. Irrationals are
defined (as we shall see) entirely by the help of order; and a new
class of transfinite ordinals is introduced, by which the most important
and interesting results are obtained. In Geometry, von Staudt’s quadrilateral construction and Pieri’s work on Projective Geometry have shown
how to give points, lines, and planes an order independent of metrical
considerations and of quantity; while descriptive Geometry proves that
a very large part of Geometry demands only the possibility of serial
arrangement. Moreover the whole philosophy of space and time depends
upon the view we take of order. Thus a discussion of order, which
is lacking in the current philosophies, has become essential to any
understanding of the foundations of mathematics.

**188.** The notion of order is more complex than any hitherto
analyzed. Two terms cannot have an order, and even three cannot
have a cyclic order. Owing to this complexity, the logical analysis
of order presents considerable difficulties. I shall therefore approach
the problem gradually, considering, in this chapter, the circumstances
under which order arises, and reserving for the second chapter the
discussion as to what order really is. This analysis will raise several
fundamental points in general logic, which will demand considerable
discussion of an almost purely philosophical nature. From this I shall
pass to more mathematical topics, such as the types of series and
the ordinal definition of numbers, thus gradually preparing the way for
the discussion of infinity and continuity in the following Part.

There are two different ways in which order may arise, though we
shall find in the end that the second way is reducible to the first. In
the first, what may be called the ordinal element consists of three terms
`a`, `b`, `c`, one of which (`b` say) is *between* the other two. This happens
whenever there is a relation of `a` to `b` and of `b` to `c`, which is not a
relation of `b` to `a`, of `c` to `b`, or of `c` to `a`. This is the definition, or
better perhaps, the necessary and sufficient condition, of the proposition
“`b` is between `a` and `c`.” But there are other cases of order where, at
first sight, the above conditions are not satisfied, and where *between*
is not obviously applicable. These are cases where we have four terms
`a`, `b`, `c`, `d`, as the ordinal element, of which we can say that `a` and `c` are
separated by `b` and `d`. This relation is more complicated, but the
following seems to characterize it: `a` and `c` are separated from `b` and `d`,
when there is an asymmetrical relation which holds between `a` and `b`,
`b` and `c`, `c` and `d`, or between `a` and `d`, `d` and `c`, `c` and `b`, or between
`c` and `d`, `d` and `a`, `a` and `b`; while if we have the first case, the same
relation must hold either between `d` and `a`, or else between both `a`
and `c`, and `a` and `d`; with similar assumptions for the other two cases*.
(No further special assumption is required as to the relation between
`a` and `c` or between `b` and `d`; it is the absence of such an assumption
which prevents our reducing this case to the former in a simple manner.)
There are cases—notably where our series is closed—in which it *seems*
formally impossible to reduce this second case to the first, though this
appearance, as we shall see, is in part deceptive. We have to show,
in the present chapter, the principal ways in which series arise from
collections of such ordinal elements.

Although two terms alone cannot have an order, we must not
assume that order is possible except where there are relations between
two terms. In all series, we shall find, there are asymmetrical relations
between two terms. But an asymmetrical relation of which there is
only one instance does not constitute order. We require at least two
instances for *between*, and at least three for separation of pairs. Thus
although order is a relation between three or four terms, it is only
possible where there are other relations which hold between pairs of
terms. These relations may be of various kinds, giving different ways
of generating series. I shall now enumerate the principal ways with
which I am acquainted.

**189.** (1) The simplest method of generating a series is as follows.
Let there be a collection of terms, finite or infinite, such that every
term (with the possible exception of a single one) has to one and only
one other term of the collection a certain asymmetrical relation (which
must of course be intransitive), and that every term (with again one
possible exception, which must not be the same as the term formerly
excepted) has also to one and only one other term of the collection
the relation which is the converse of the former one*. Further, let
it be assumed that, if `a` has the first relation to `b`, and `b` to `c`, then `c`
does not have the first relation to `a`. Then every term of the collection
except the two peculiar terms has one relation to a second term, and
the converse relation to a third, while these terms themselves do not
have to each other either of the relations in question. Consequently,
by the definition of *between*, our first term is between our second and
third terms. The term to which a given term has one of the two
relations in question is called *next after* the given term; the term to
which the given term has the converse relation is called *next before*
the given term. Two terms between which the relations in question
hold are called *consecutive*. The exceptional terms (when they exist)
are not between any pair of terms; they are called the two ends of
the series, or one is called the beginning and the other the end. The
existence of the one does not imply that of the other—for example
the natural numbers have a beginning but no end—and neither need
exist—for example, the positive and negative integers together have
neither†.

The above method may perhaps become clear by a formal exhibition.
Let `R` be one of our relations, and let its converse be denoted by ˘`R`‡.
Then if `e` be any term of our set, there are two terms `d`, `f`, such that
`e`˘`R``d`, `e``R``f`, *i.e.* such that `d``R``e`, `e``R``f`. Since each term only has the
relation `R` to one other, we cannot have `d``R``f`; and it was one of
the initial assumptions that we were not to have `f``R``d`. Hence `e` is
between `d` and `f`§. If `a` be a term which has only the relation `R`, then
obviously `a` is not between any pair of terms. We may extend the
notion of *between* by defining that, if `c` be between `b` and `d`, and `d`
between `c` and `e`, then `c` or `d` will be said to be also between `b` and `e`.
In this way, unless we either reach an end or come back to the term
with which we started, we can find any number of terms between which
and `b` the term `c` will lie. But if the total number of terms be not
less than seven, we cannot show in this way that of *any* three terms
one must be between the other two, since the collection may consist
of two distinct series, of which, if the collection is finite, one at least
must be closed, in order to avoid more than two ends.

This remark shows that, if the above method is to give a single
series, to which any term of our collection is to belong, we need a
further condition, which may be expressed by saying that the collection
must be *connected*. We shall find means hereafter of expressing this
condition without reference to number, but for the present we may
content ourselves by saying that our collection is connected when, given
any two of its terms, there is a certain finite number (not necessarily
unique) of steps from one term to the next, by which we can pass
from one of our two terms to the other. When this condition is
fulfilled, we are assured that, of any three terms of our collection, one
must be between the other two.

Assuming now that our collection is connected, and therefore forms
a single series, four cases may arise: (`a`) our series may have two ends,
(`b`) it may have one end, (`c`) it may have no end and be open, (*d*) it may
have no end and be closed. Concerning (`a`), it is to be observed that
our series must be finite. For, taking the two ends, since the collection
is connected, there is some finite number `n` of steps which will take
us from one end to the other, and hence `n` + 1 is the number of terms
of the series. Every term except the two ends is between them, and
neither of them is between any other pair of terms. In case (`b`), on
the other hand, our collection must be infinite, and this would hold
even if it were not connected. For suppose the end which exists to
have the relation `R`, but not ˘`R`. Then every other term of the collection
has both relations, and can never have both to the same term, since `R`
is asymmetrical. Hence the term to which (say) `e` has the relation `R` is
not that to which it had the relation ˘`R`, but is either some new term,
or one of `e`’s predecessors. Now it cannot be the end-term `a`, since
`a` does not have the relation ˘`R` to any term. Nor can it be any term
which can be reached by successive steps from `a` without passing
through `e`, for if it were, this term would have two predecessors,
contrary to the hypothesis that `R` is a one-one relation. Hence, if
`k` be any term which can be reached by successive steps from `a`,
`k` has a successor which is not `a` or any of the terms between `a`
and `k`; and hence the collection is infinite, whether it be connected
or not. In case (`c`), the collection must again be infinite. For here,
by hypothesis, the series is open—*i.e.*, starting from any term `e`, no
number of steps in either direction brings us back to `e`. And there
cannot be a finite limit to the number of possible steps, since, if there
were, the series would have an end. Here again, it is not necessary to
suppose the series connected. In case (`d`), on the contrary, we must
assume connection. By saying that the series is closed, we mean that
there exists some number `n` of steps by which, starting from a certain
term `a`, we shall be brought back to `a`. In this case, `n` is the number
of terms, and it makes no difference with which term we start. In this
case, *between* is not definite except where three terms are consecutive,
and the series contains more than three terms. Otherwise, we need the
more complicated relation of separation.

**190.** (2) The above method, as we have seen, will give either open
or closed series, but only such as have consecutive terms. The second
method, which is now to be discussed, will give series in which there
are no consecutive terms, but will not give closed series*. In this
method we have a transitive asymmetrical relation `P`, and a collection
of terms any two of which are such that either `x``P``y` or `y``P``x`. When
these conditions are satisfied our terms necessarily form a single series.
Since the relation is asymmetrical, we can distinguish `x``P``y` from `y``P``x`,
and the two cannot both subsist†. Since `P` is transitive, `x``P``y` and `y``P``z`
involve `x``P``z`. It follows that ˘`P` is also asymmetrical and transitive‡.
Thus with respect to any term `x` of our collection, all other terms of
the collection fall into two classes, those for which `x``P``y`, and those for
which `z``P``x`. Calling these two classes ˘`π``x` and `π``x` respectively, we see
that, owing to the transitiveness of `P`, if `y` belongs to the class ˘`π``x`,
˘`π``y` is contained in ˘`π``x`; and if `z` belongs to the class `π``x`, `π``z` is contained
in `π``x`. Taking now two terms `x`, `y`, for which `x``P``y`, all other terms fall
into three classes: (1) Those belonging to `π``x`, and therefore to `π``y`;
(2) those belonging to ˘`π``y`, and therefore to ˘`π``x`; (3) those belonging to
˘`π``x` but not to ˘`π``y`. If `z` be of the first class, we have `z``P``x`, `z``P``y`; if `v` be
of the second, `x``P``v` and `y``P``v`; if `w` be of the third, `x``P``w` and `w``P``y`. The
case `y``P``u` and `u``P``x` is excluded: for `x``P``y`, `y``P``u` imply `x``P``u`, which is inconsistent with `u``P``x`. Thus we have, in the three cases, (1) `x` is between
`z` and `y`; (2) `y` is between `x` and `v`; (3) `w` is between `x` and `y`. Hence
any three terms of our collection are such that one is between the other
two, and the whole collection forms a single series. If the class (3)
contains no terms, `x` and `y` are said to be consecutive; but many relations `P` can be assigned, for which there are always terms in the class (3).
If for example `P` be `b``e``f``o``r``e`, and our collection be the moments in a
certain interval, or in all time, there is a moment between any two of
our collection. Similarly in the case of the magnitudes which, in the
last chapter of Part III, we called continuous. There is nothing in
the present method, as there was in the first, to show that there *must*
be consecutive terms, unless the total number of terms in our collection
be finite. On the other hand, the present method will not allow closed
series; for owing to the transitiveness of the relation `P`, if the series
were closed, and `x` were any one of its terms, we should have `x``P``x`, which
is impossible because `P` is asymmetrical. Thus in a closed series, the
generating relation can never be transitive*. As in the former method,
the series may have two ends, or one, or none. In the first case only,
it may be finite; but even in this case it *may* be infinite, and in the
other two cases it must be so.

**191.** (3) A series may be generated by means of distances, as was
already partially explained in Part III, and as we shall see more fully
hereafter. In this case, starting with a certain term `x`, we are to have
relations, which are magnitudes, between `x` and a number of other terms
`y`, `z` …. According as these relations are greater or less, we can order
the corresponding terms. If there are no similar relations between the
remaining terms `y`, `z`, …, we require nothing further. But if these
have relations which are magnitudes of the same kind, certain axioms
are necessary to insure that the order may be independent of the
particular term from which we start. Denoting by `x``z` the distance of
`x` and `z`, if `x``z` is less than `x``w`, we must have `y``z` less than `y``w`. A consequence, which did not follow when `x` was the only term that had
a distance, is that the distances must be asymmetrical relations, and
those which have one sense must be considered less than zero. For
“`x``z` is less than `x``w`” must involve “`w``z` is less than `w``w`,” *i.e.* `w``z` is less
than 0. In this way the present case is practically reduced to the
second; for every pair of terms `x`, `y` will be such that `x``y` is less than 0
or else `x``y` is greater than 0; and we may put in the first case `y``P``x`,
in the second `x``P``y`. But we require one further axiom in order that
the arrangement may be thus effected unambiguously. If `x``z` = `y``w`, and
`z``w`′= `x``y`, `w` and `w`′ must be the same point. With this further axiom,
the reduction to case (2) becomes complete.

**192.** (4) Cases of triangular relations are capable of giving rise to
order. Let there be a relation `R` which holds between `y` and (`x`, `z`),
between `z` and (`y`, `u`), between `u` and (`z`, `w`), and so on. *Between* is itself
such a relation, and this might therefore seem the most direct and
natural way of generating order. We should say, in such a case, that `y`
is between `x` and `z`, when the relation `R` holds between `y` and the couple
`x`, `z`. We should need assumptions concerning `R` which should show
that, if `y` is between `x` and `z`, and `z` between `y` and `w`, then `y` and `z` are
each between `x` and `w`. That is, if we have `y``R`(`x`, `z`), `z``R`(`y`, `w`), we must
have `y``R`(`x`, `w`) and `z``R`(`x`, `w`). This is a kind of three-term transitiveness.
Also if `y` be between `x` and `w`, and `z` between `y` and `w`, then `z` must be
between `x` and `w`, and `y` between `x` and `z`: that is, if `y``R`(`x`, `w`) and
`z``R`(`y`, `w`), then `z``R`(`x`, `w`) and `y``R`(`x`, `z`). Also `y``R`(`x`, `z`) must be equivalent to `y``R`(`z`, `x`)*. With these assumptions, an unambiguous order
will be generated among any number of terms such that any triad has
the relation `R`. Whether such a state of things can ever be incapable of
further analysis, is a question which I leave for the next chapter.

**193.** (5) We have found hitherto no way of generating closed
continuous series. There are, however, instances of such series, *e.g.*
angles, the elliptic straight line, the complex numbers with a given
modulus. It is therefore necessary to have some theory which allows of
their possibility. In the case where our terms are asymmetrical relations,
as straight lines are, or are correlated uniquely and reciprocally with
such relations, the following theory will effect this object. In other
cases, the sixth method (below) seems adequate to the end in view.

Let `x`, `y`, `z` … be a set of asymmetrical relations, and let `R` be an
asymmetrical relation which holds between any two `x`, `y` or `y`, `x` except
when `y` is the converse relation to `x`. Also let `R` be such that, if it holds
between `x` and `y`, it holds between `y` and the converse of `x`; and if `x` be
any term of the collection, let all the terms to which `x` has either of the
relations `R`, ˘`R` be terms of the collection. All these conditions are
satisfied by angles, and whenever they are satisfied, the resulting series is
closed. For `x``R``y` implies `y``R`˘`x`, and hence ˘`x``R`˘`y`, and thence ˘`y``R``x`; so
that by means of relations `R` it is possible to travel from `x` back to `x`.
Also there is nothing in the definition to show that our series cannot be
continuous. Since it is closed, we cannot apply universally the notion of
*between*; but the notion of separation can be always applied. The
reason why it is necessary to suppose that our terms either are, or are
correlated with, asymmetrical relations, is, that such series often have
antipodes, *opposite* terms as they may be called; and that the notion of
*opposite* seems to be essentially bound up with that of the converse of an
asymmetrical relation.

**194.** (6) In the same way in which, in (4), we showed how to
construct a series by relations of *between*, we can construct a series
directly by four-term relations of separation. For this purpose, as
before, certain axioms are necessary. The following five axioms have
been shown by Vailati† to be sufficient, and by Padoa to possess ordered
independence, *i.e.* to be such that none can be deduced from its predecessors‡. Denoting “`a` and `b` separate `c` from `d`” by `a``b`‖`c``d`, we must
have:

(`α`) `a``b`‖`c``d` is equivalent to `c``d`‖`a``b`;

(`β`) `a``b`‖`c``d` is equivalent to `a``b`‖`d``c`;

(`γ`) `a``b`‖`c``d` excludes `a``c`‖`b``d`;

(`δ`) For any four terms of our collection, we must have `a``b`‖`c``d`, or
`a``c`‖`b``d`, or `a``d`‖`b``c`;

(`ε`) If `a``b`‖`c``d`, and `a``c`‖`b``e`, then `a``c`‖`d``e`.

By means of these five assumptions, our terms `a`, `b`, `c`, `d`, `e` … acquire
an unambiguous order, in which we start from a relation between two
pairs of terms, which is undefined except to the extent to which the
above assumptions define it. The further consideration of this case, as
generally of the relation of separation, I postpone to a later stage.

The above six methods of generating series are the principal ones with which I am acquainted, and all other methods, so far as I know, are reducible to one of these six. The last alone gives a method of generating closed continuous series whose terms neither are, nor are correlated with, asymmetrical relations*. This last method should therefore be applied in projective and elliptic Geometry, where the correlation of the points on a line with the lines through a point appears to be logically subsequent to the order of the points on a line. But before we can decide whether these six methods (especially the fourth and sixth) are irreducible and independent, we must discuss (what has not hitherto been analyzed) the meaning of order, and the logical constituents (if any) of which this meaning is compounded. This will be done in the following chapter.

Notes

*^{[page 200]} This gives a sufficient but not a necessary condition for the separation of couples.

*^{[page 201]} The converse of a relation is the relation which must hold between `y` and `x` when the given relation holds between `x` and `y`.

†^{[page 201]} The above is the only method of generating series given by Bolzano, “Paradoxien des Unendlichen,” §7.

‡^{[page 201]} This is the notation adopted by Professor Schröder.

§^{[page 201]} The denial of `d``R``f` is only necessary to this special method, but the denial of `f``R``d` is essential to the definition of *between*.

*^{[page 203]} The following method is the only one given by Vivanti in the *Formulaire de Mathématiques*, (1895), vi, §2, No. 7; also by Gilman, “On the properties of a one-dimensional manifold,” *Mind*, N.S. Vol. i. We shall find that it is general in a sense in which none of our other methods are so.

†^{[page 203]} I use the term *asymmetrical* as the contrary, rather than the contradictory, of *symmetrical*. If `x``P``y` and the relation is symmetrical, we have always `y``P``x`; if asymmetrical, we never have `y``P``x`. Some relations—*e.g.* logical implication—are neither symmetrical nor asymmetrical. Instead of assuming `P` to be asymmetrical, we may make the equivalent assumption that it is what Professor Peirce calls an
*aliorelative*, *i.e.* a relation which no term has to itself. (This assumption is not equivalent to asymmetry in general, but only when combined with transitiveness.)

‡^{[page 203]} `P` may be read *precedes*, and ˘`P` may be read *follows*, provided no temporal or spatial ideas are allowed to intrude themselves.

*^{[page 204]} For more precise statements, see Chap. xxviii.

*^{[page 205]} See Peano, *I Principii di Geometria*, Turin, 1889, Axioms viii, ix, x, xi.

†^{[page 205]} *Rivista di Matematica*, v, pp. 76, 183.

‡^{[page 205]} *Ibid.*, p. 185.

The Meaning of Order.

**195.** We have now seen under what circumstances there is an order
among a set of terms, and by this means we have acquired a certain
inductive familiarity with the nature of order. But we have not yet
faced the question: What *is* order? This is a difficult question, and
one upon which, so far as I know, nothing at all has been written. All
the authors with whom I am acquainted are content to exhibit the
genesis of order; and since most of them give only one of the six
methods enumerated in Chapter xxiv, it is easy for them to confound the
genesis of order with its nature. This confusion is rendered evident to
us by the multiplicity of the above methods; for it is evident that we
mean by *order* something perfectly definite, which, being generated
equally in all our six cases, is clearly distinct from each and all of the
ways in which it may be generated, unless one of these ways should turn
out to be fundamental, and the others to be reducible to it. To elicit
this common element in all series, and to broach the logical discussions
connected with it, is the purpose of the present chapter. This discussion
is of purely philosophical interest, and might be wholly omitted in a
mathematical treatment of the subject.

In order to approach the subject gradually, let us separate the
discussion of *between* from that of separation of couples. When we have
decided upon the nature of each of these separately, it will be time to
combine them, and examine what it is that both have in common.
I shall begin with *between*, as being the simpler of the two.

**196.** *Between* may be characterized (as in Chapter xxiv) as a relation
of one term `y` to two others `x` and `z`, which holds whenever `x` has to `y`, and
`y` has to `z`, some relation which `y` does not have to `x`, nor `z` to `y`, nor `z` to `x`*.
These conditions are undoubtedly *sufficient* for betweenness, but it may
be questioned whether they are *necessary*. Several possible opinions
must be distinguished in this respect. (1) We may hold that the above
conditions give the very *meaning* of between, that they constitute an
actual analysis of it, and not merely a set of conditions insuring its
presence. (2) We may hold that *between* is not a relation of the terms
`x`, `y`, `z` at all, but a relation of the relation of `y` to `x` to that of `y` to `z`,
namely the relation of difference of sense. (3) We may hold that
*between* is an indefinable notion, like *greater* and *less*; that the above
conditions allow us to infer that `y` is between `x` and `z`, but that there
may be other circumstances under which this occurs, and even that it
may occur without involving any relation except diversity among the
pairs (`x`, `y`), (`y`, `z`), (`x`, `z`). In order to decide between these theories, it
will be well to develop each in turn.

**197.** (1) In this theory, we define “`y` is between `x` and `z`” to mean:
“There is a relation `R` such that `x``R``y`, `y``R``z` but not `y``R``x`, `z``R``y`”; and it
remains a question whether we are to add “not `z``R``x`.” We will suppose
to begin with that this addition is not made. The following propositions
will be generally admitted to be self-evident: (`α`) If `y` be between `x` and `z`,
and `z` between `y` and `w`, then `y` is between `x` and `w`; (`β`) if `y` be between
`x` and `z`, and `w` between `x` and `y`, then `y` is between `w` and `z`. For brevity,
let us express “`y` is between `x` and `z`” by the symbol `x``y``z`. Then our two
propositions are: (`α`) `x``y``z` and `y``z``w` imply `x``y``w`; (`β`) `x``y``z` and `x``w``y` imply
`w``y``z`. We must add that the relation of *between* is symmetrical so far as
the extremes are concerned: *i.e.* `x``y``z` implies `z``y``x`. This condition follows
directly from our definition. With regard to the axioms (`α`) and (`β`), it
is to be observed that *between*, on our present view, is always relative to
some relation `R`, and that the axioms are only assumed to hold when it
is the same relation `R` that is in question in both the premisses. Let us
see whether these axioms are consequences of our definition. For this
purpose, let us write `R` for not-`R`.

`x``y``z` means `x``R``y`, `y``R``z`, `y``R``x`, `z``R``y`.

`y``z``w` means `y``R``z`, `z``R``w`, `z``R``y`, `w``R``z`.

Thus `y``z``w` only adds to `x``y``z` the two conditions `z``R``w`, `w``R``z`. If `R` is
transitive, these conditions insure `x``y``w`; if not, not. Now we have seen
that some series are generated by one-one relations `R`, which are not
transitive. In these cases, however, denoting by `R`^{2} the relation between
`x` and `z` implied by `x``R``y`, `y``R``z`, and so on for higher powers, we can
substitute a transitive relation `R`′ for `R`, where `R`′ means “some positive
power of `R`.” In this way, if `x``y``z` holds for a relation which is some
definite power of `R`, then `x``y``z` holds for `R`′, provided only that no positive
power of `R` is equivalent to ˘`R`. For, in this latter event, we should
have `y``R`′`x` whenever `x``R`′`y`, and `R`′ could not be substituted for `R` in the
explanation of `x``y``z`. Now this condition, that the converse of `R` is not
to be a positive power of `R`, is equivalent to the condition that our
series is not to be closed. For if ˘`R` = `R`^{n}, then `R`˘`R` = `R`^{n+1}; but since `R`
is a one-one relation, `R`˘`R` implies the relation of identity. Thus `n` + 1
steps bring us back from `x` to `x`, and our series is a closed series of
`n` + 1 terms. Now we have agreed already that *between* is not properly
applicable to closed series. Hence this condition, that ˘`R` is not to
be a power of `R`, imposes only such restrictions upon our axiom (`α`) as
we should expect it to be subject to.

With regard to (`β`), we have

`x``y``z` = `x``R``y` . `y``R``z` . `y``R``x` . `z``R``y`.

`x``w``y` = `x``R``w` . `w``R``y` . `w``R``x` . `y``R``w`.

The case contemplated by this axiom is only possible if `R` be not
a one-one relation, since we have `x``R``y` and `x``R``w`. The deduction `w``y``z`
is here an immediate consequence of the definition, without the need of
any further conditions.

It remains to examine whether we can dispense with the condition
`z``R``x` in the definition of *between*. If we suppose `R` to be a one-one
relation, and `z``R``x` to be satisfied, we shall have

`x``y``z` = `x``R``y` . `y``R``z` . `z``R``y` . `y``R``x`,

and we have further by hypothesis `z``R``x`, and since `R` is one-one, and
`x``R``y`, we have `x``R``z`. Hence, in virtue of the definition, we have `y``z``x`;
and similarly we shall obtain `z``x``y`. If we now adhere to our axiom (`α`),
we shall have `x``z``x`, which is impossible; for it is certainly part of the
meaning of *between* that the three terms in the relation should be
different, and it is impossible that a term should be between `x` and `x`.
Thus we must either insert our condition `z``R``x`, or we must set up the
new condition in the definition, that `x` and `z` are to be different. (It
should be observed that our definition implies that `x` is different from `y`
and `y` from `z`; for if not, `x``R``y` would involve `y``R``x`, and `y``R``z` would
involve `z``R``y`.) It would seem preferable to insert the condition that `x`
and `z` are to be different: for this is in any case necessary, and is not
implied by `z``R``x`. This condition must then be added to our axiom (`α`);
`x``y``z` and `y``z``w` are to imply `x``y``w`, unless `x` and `w` are identical. In axiom
(`β`), this addition is not necessary, since it is implied in the premisses.
Thus the condition `z``R``x` is not necessary, if we are willing to admit that
`x``y``z` is compatible with `y``z``x`—an admission which such cases as the
angles of a triangle render possible. Or we may insert, in place of
`z``R``x`, the condition which we found necessary before to the universal
validity of our axiom (`α`), namely that no power of `R` is to be equivalent
to the converse of `R`: for if we have both `x``y``z` and `y``z``x`, we shall have (so
far at least as `x`, `y`, `z` are concerned) `R`^{2} = `R`, *i.e.* if `x``R``y` and `y``R``z`, then
`z``R``x`. This last course seems to be the best. Hence in all cases where
our first instance of *between* is defined by a one-one relation `R`, we shall
substitute the relation `R`′, which means “some positive power of `R`.”
The relation `R`′ is then transitive, and the condition that no positive
power of `R` is to be equivalent to ˘`R` is equivalent to the condition that
`R`′ is to be asymmetrical. Hence, finally, the whole matter is simplified
into the following:

To say that `y` is between `x` and `z` is equivalent to saying that there
is some transitive asymmetrical relation which relates both `x` and `y`, and
`y` and `z`.

This short and simple statement, as the above lengthy argument
shows, contains neither more nor less than our original definition, together with the emendations which we gradually found to be necessary.
The question remains, however: Is this the *meaning* of *between*?

**198.** A negative instance can be at once established if we allow the
phrase: `R` is a relation *between* `x` and `y`. The phrase, as the reader will
have observed, has been with difficulty excluded from the definitions of
*between*, which its introduction would have rendered at least verbally
circular. The phrase may have none but a linguistic importance, or
again it may point to a real insufficiency in the above definition. Let
us examine the relation of a relation `R` to its terms `x` and `y`. In the
first place, there certainly is such a relation. To be a term which has
the relation `R` to some other term is certainly to have a relation to `R`,
a relation which we may express as “belonging to the domain of `R`.”
Thus if `x``R``y`, `x` will belong to the domain of `R`, and `y` to that of ˘`R`.
If we express this relation between `x` and `R`, or between `y` and ˘`R`, by `E`,
we shall have `x``E``R`, `y``E`˘`R`. If further we express the relation of `R` to ˘`R`
by `I`, we shall have ˘`R``I``R` and `R``I`˘`R`. Thus we have `x``E``R`, `y``E``I``R`. Now
`E``I` is by no means the converse of `E`, and thus the above definition of
*between*, if for this reason only, does not apply; also neither `E` nor `E``I`
is transitive. Thus our definition of *between* is wholly inapplicable to
such a case. Now it may well be doubted whether *between*, in this case,
has at all the same meaning as in other cases. Certainly we do not in
this way obtain series: `x` and `y` are not, in the same sense as `R`, between
`R` and other terms. Moreover, if we admit relations of a term to itself,
we shall have to admit that such relations are *between* a term and
itself, which we agreed to be impossible. Hence we may be tempted
to regard the use of *between* in this case as due to the linguistic accident
that the relation is usually mentioned between the subject and the
object, as in “`A` is the father of `B`.” On the other hand, it may be
urged that a relation does have a very peculiar relation to the pair of
terms which it relates, and that *between* should denote a relation of one
term to two others. To the objection concerning relations of a term
to itself, it may be answered that such relations, in any system, constitute a grave logical difficulty; that they would, if possible, be denied
philosophic validity; and that even where the relation asserted is
identity, there must be *two* identical terms, which are therefore not
quite identical. As this raises a fundamental difficulty, which we cannot
discuss here, it will be prudent to allow the answer to pass*. And it
may be further urged that use of the same word in two connections
points always to some analogy, the extent of which should be carefully
indicated by those who deny that the meaning is the same in both
cases; and that the analogy here is certainly profounder than the mere
order of words in a sentence, which is, in any case, far more variable
in this respect than the phrase that a relation is between its terms.
To these remarks, however, it may be retorted that the objector has
himself indicated the precise extent of the analogy: the relation of a
relation to its terms is a relation of one term to two others, just as
*between* is, and this is what makes the two cases similar. This last
retort is, I think, valid, and we may allow that the relation of a relation
to its terms, though involving a most important logical problem, is
not the same as the relation of *between* by which order is to be constituted.

Nevertheless, the above definition of *between*, though we shall be
ultimately forced to accept it, seems, at first sight, scarcely adequate
from a philosophical point of view. The reference to *some* asymmetrical
relation is vague, and seems to require to be replaced by some phrase
in which no such undefined relation appears, but only the terms and
the betweenness. This brings us to the second of the above opinions
concerning *between*.

**199.** (2) *Between*, it may be said, is not a relation of three terms
at all, but a relation of two relations, namely difference of sense. Now
if we take this view, the first point to be observed is, that we require
the two opposite relations, not merely in general, but as particularized
by belonging to one and the same term. This distinction is already
familiar from the case of magnitudes and quantities. *Before* and *after*
in the abstract do not constitute *between*: it is only when one and the
same term is both before and after that *between* arises: this term is
then between what it is before and what it is after. Hence there is
a difficulty in the reduction of *between* to difference of sense. The particularized relation is a logically puzzling entity, which in Part I (§55)
we found it necessary to deny; and it is not quite easy to distinguish
a relation of two relations, particularized as belonging to the same term,
from a relation of the term in question to two others. At the same
time, great advantages are secured by this reduction. We get rid of
the necessity for a triangular relation, to which many philosophers may
object, and we assign a common element to all cases of *between*, namely
difference of sense, *i.e.* the difference between an asymmetrical relation
and its converse.

**200.** The question whether there can be an ultimate triangular
relation is one whose actual solution is both difficult and unimportant,
but whose precise statement is of very great importance. Philosophers
seem usually to assume—though not, so far as I know, explicitly—that
relations never have more than two terms; and even such relations they
reduce, by force or guile, to predications. Mathematicians, on the other
hand, almost invariably speak of relations of many terms. We cannot,
however, settle the question by a simple appeal to mathematical instances,
for it remains a question whether these are, or are not, susceptible of
analysis. Suppose, for example, that the projective plane has been
defined as a relation of three points: the philosopher may always say
that it should have been defined as a relation of a point and a line,
or of two intersecting lines—a change which makes little or no mathematical difference. Let us see what is the precise meaning of the question.
There are among terms two radically different kinds, whose difference
constitutes the truth underlying the doctrine of substance and attribute.
There are terms which can never occur except as terms; such are points,
instants, colours, sounds, bits of matter, and generally terms of the kind
of which existents consist. There are, on the other hand, terms which
can occur otherwise than as terms; such are being, adjectives generally,
and relations. Such terms we agreed to call concepts*. It is the presence
of concepts not occurring as terms which distinguishes propositions from
mere concepts; in every proposition there is at least one more concept
than there are terms. The traditional view—which may be called the
subject-predicate theory—holds that in every proposition there is one
term, the subject, and one concept which is not a term, the predicate.
This view, for many reasons, must be abandoned†. The smallest
departure from the traditional opinion lies in holding that, where
propositions are not reducible to the subject-predicate form, there are
always two terms only, and one concept which is not a term. (The
two terms may, of course, be complex, and may each contain concepts
which are not terms.) This gives the opinion that relations are always
between only two terms; for a relation may be defined as any concept
which occurs in a proposition containing more than one term. But
there seems no *à priori* reason for limiting relations to two terms,
and there are instances which lead to an opposite view. In the first
place, when the concept of a number is asserted of a collection, if the
collection has `n` terms, there are `n` terms, and only one concept (namely
`n`) which is not a term. In the second place, such relations as those
of an existent to the place and time of its existence are only reducible
by a very cumbrous method to relations of two terms‡. If, however,
the reduction be held essential, it seems to be always formally possible,
by compounding part of the proposition into one complex term, and
then asserting a relation between this part and the remainder, which
can be similarly reduced to one term. There may be cases where this
is not possible, but I do not know of them. The question whether such
a formal reduction is to be always undertaken is not, however, so far
as I have been able to discover, one of any great practical or theoretical
importance.

**201.** There is thus no valid *à priori* reason in favour of analyzing
*between* into a relation of two relations, if a triangular relation seems
otherwise preferable. The other reason in favour of the analysis of
*between* is more considerable. So long as *between* is a triangular relation
of the terms, it must be taken either as indefinable, or as involving a
reference to *some* transitive asymmetrical relation. But if we make
*between* consist essentially in the opposition of two relations belonging
to one term, there seems to be no longer any undue indeterminateness.
Against this view we may urge, however, that no reason now appears
why the relations in question should have to be transitive, and that—what is more important—the very meaning of *between* involves the
terms, for it is they, and not their relations, that have order. And
if it were only the relations that were relevant, it would not be necessary,
as in fact it is, to particularize them by the mention of the terms
between which they hold. Thus on the whole, the opinion that *between*
is not a triangular relation must be abandoned.

**202.** (3) We come now to the view that *between* is an ultimate
and indefinable relation. In favour of this view it might be urged that,
in all our ways of generating open series, we could see that cases of
*between* did arise, and that we could apply a test to suggested definitions.
This seems to show that the suggested definitions were merely conditions
which imply relations of *between*, and were not true definitions of this
relation. The question: Do such and such conditions insure that `y`
shall be between `x` and `z`? is always one which we can answer, without
having to appeal (at least consciously) to any previous definition. And
the unanalyzable nature of *between* may be supported by the fact that
the relation is symmetrical with respect to the two extremes, which was
not the case with the relations of pairs from which *between* was inferred.
There is, however, a very grave difficulty in the way of such a view, and
that is, that sets of terms have many different orders, so that in one we
may have `y` between `x` and `z`, while in another we have `x` between
`y` and `z`*. This seems to show that *between* essentially involves reference
to the relations from which it is inferred. If not, we shall at least have
to admit that these relations are relevant to the genesis of series; for
series require imperatively that there should be at most one relevant
relation of *between* among three terms. Hence we must, apparently,
allow that *between* is not the sole source of series, but must always be
supplemented by the mention of some transitive asymmetrical relation
with respect to which the betweenness arises. The most that can be
said is, that this transitive asymmetrical relation of two terms may
itself be logically subsequent to, and derived from, some relation of
three terms, such as those considered in Chapter xxiv, in the fourth way
of generating series. When such relations fulfil the axioms which were
then mentioned, they lead of themselves to relations between pairs of
terms. For we may say that `b` precedes `c` when `a``c``d` implies `b``c``d`, and
that `b` follows `c` when `a``b``d` implies `c``b``d`, where `a` and `d` are fixed terms.
Though such relations are merely derivative, it is in virtue of them
that *between* occurs in such cases. Hence we seem finally compelled to
leave the reference to an asymmetrical relation in our definition. We
shall therefore say:

A term `y` is between two terms `x` and `z` with reference to a transitive
asymmetrical relation `R` when `x``R``y` and `y``R``z`. In no other case can `y`
be said properly to be between `x` and `z`; and this definition gives not
merely a criterion, but the very *meaning* of betweenness.

**203.** We have next to consider the meaning of separation of
couples. This is a more complicated relation than *between*, and was
but little considered until elliptic Geometry brought it into prominence.
It has been shown by Vailati* that this relation, like *between*, always
involves a transitive asymmetrical relation of two terms; but this relation of a pair of terms is itself relative to three other fixed terms of the
set, as, in the case of *between*, it was relative to two fixed terms. It is
further sufficiently evident that wherever there is a transitive asymmetrical relation, which relates every pair of terms in a collection of not
less than four terms, there there are pairs of couples having the relation
of separation. Thus we shall find it possible to express separation, as
well as *between*, by means of transitive asymmetrical relations and their
terms. But let us first examine directly the meaning of separation.

We may denote the fact that `a` and `c` are separated by `b` and `d` by
the symbol `a``b``c``d`. If, then, `a`, `b`, `c`, `d`, `e` be any five terms of the set we
require the following properties to hold of the relation of separation (of
which, it will be observed, only the last involves five
terms):

1. `a``b``c``d` = `b``a``d``c`.

2. `a``b``c``d` = `a``d``c``b`.

3. `a``b``c``d` excludes `a``c``b``d`.

4. We must have `a``b``c``d` or `a``c``d``b` or `a``d``b``c`.

5. `a``b``c``d` and `a``c``d``e` together imply `a``b``d``e`†.

These properties may be illustrated by the consideration of five points on a circle, as in the accompanying figure. Whatever relation of two pairs of terms possesses these properties we shall call a relation of separation between the pairs. It will be seen that the relation is symmetrical, but not in general transitive.

**204.** Wherever we have a transitive asymmetrical relation `R` between any two terms of a set of not less than four terms, the relation of
separation necessarily arises. For in any series, if four terms have the
order `a``b``c``d`, then `a` and `c` are separated by `b` and `d`; and every transitive
asymmetrical relation, as we have seen, provided there are at least two
consecutive instances of it, gives rise to a series. Thus in this case,
separation is a mere extension of *between*: if `R` be asymmetrical and
transitive, and `a``R``b`, `b``R``c`, `c``R``d`, then `a` and `c` are separated by `b` and `d`.
The existence of such a relation is therefore a sufficient condition of
separation.

It is also a necessary condition. For, suppose a relation of separation
to exist, and let `a`, `b`, `c`, `d`, `e` be five terms of the set to which the relation
applies. Then, considering `a`, `b`, `c` as fixed, and `d` and `e` as variable,
twelve cases may arise. In virtue of the five fundamental properties, we
may introduce the symbol `a``b``c``d``e` to denote that, striking out any one
of these five letters, the remaining four have the relation of separation
which is indicated by the resulting symbol. Thus by the fifth property,
`a``b``c``d` and `a``c``d``e` imply `a``b``c``d``e`*. Thus the twelve cases arise from permuting
`d` and `e`, while keeping `a`, `b`, `c` fixed. (It should be observed that it
makes no difference whether a letter appears at the end or the beginning:
*i.e.* `a``b``c``d``e` is the same case as `e``a``b``c``d`. We may therefore decide not to put
either `d` or `e` before `a`.) Of these twelve cases, six will have `d` before `e`,
and six will have `e` before `d`. In the first six cases, we say that, with
respect to the sense `a``b``c`, `d` precedes `e`; in the other six cases, we say that
`e` precedes `d`. In order to deal with limiting cases, we shall say further
that `a` precedes every other term, and that `b` precedes `c`†. We shall then
find that the relation of preceding is asymmetrical and transitive, and
that every pair of terms of our set is such that one precedes and the
other follows. In this way our relation of separation is reduced, formally
at least, to the combination of “`a` precedes `b`,” “`b` precedes `c`,” and “`c`
precedes `d`.”

The above reduction is for many reasons highly interesting. In the
first place, it shows the distinction between open and closed series to be
somewhat superficial. For although our series may initially be of the
sort which is called closed, it becomes, by the introduction of the above
transitive relation, an open series, having `a` for its beginning, but having
possibly no last term, and not in any sense returning to `a`. Again it is
of the highest importance in Geometry, since it shows how order may
arise on the elliptic straight line, by purely projective considerations,
in a manner which is far more satisfactory than that obtained from
von Staudt’s construction*. And finally, it is of great importance as
unifying the two sources of order, *between* and separation; since it
shows that transitive asymmetrical relations are always present where
either occurs, and that either implies the other. For, by the relation of
preceding, we can say that one term is between two others, although we
started solely from separation of pairs.

**205.** At the same time, the above reduction (and also, it would
seem, the corresponding reduction in the case of *between*) cannot be
allowed to be more than formal. That is, the three terms `a`, `b`, `c` by
relation to which our transitive asymmetrical relation was defined, are
essential to the definition, and cannot be omitted. The reduction shows
no reason for supposing that there is any transitive asymmetrical relation
independent of *all* other terms than those related, though it is arbitrary
what other terms we choose. And the fact that the term `a`, which is
not essentially peculiar, appears as the beginning of the series, illustrates
this fact. Where there are transitive asymmetrical relations independent
of all outside reference, our series cannot have an arbitrary beginning,
though it may have none at all. Thus the four-term relation of separation remains logically prior to the resulting two-term relation, and
cannot be analyzed into the latter.

**206.** But when we have said that the reduction is formal, we have
not said that it is irrelevant to the genesis of order. On the contrary,
it is just because such a reduction is possible that the four-term relation
leads to order. The resulting asymmetrical transitive relation is in
reality a relation of five terms; but when three of these are kept fixed,
it becomes asymmetrical and transitive as regards the other two. Thus
although *between* applies to such series, and although the essence of
order consists, here as elsewhere, in the fact that one term has, to two
others, converse relations which are asymmetrical and transitive, yet
such an order can only arise in a collection containing at least five terms,
because five terms are needed for the characteristic relation. And it
should be observed that *all* series, when thus explained, are open series,
in the sense that there is some relation between pairs of terms, no power
of which is equal to its converse, or to identity.

**207.** Thus finally, to sum up this long and complicated discussion:
The six methods of generating series enumerated in Chapter xxiv are all
genuinely distinct; but the second is the only one which is fundamental,
and the other five agree in this, that they are all reducible to the second.
Moreover, it is solely in virtue of their reducibility to the second that
they give rise to order. The minimum ordinal proposition, which can
always be made wherever there is an order at all, is of the form: “`y` is
between `x` and `z`”; and this proposition means; “There is some
asymmetrical transitive relation which holds between `x` and `y` and
between `y` and `z`.” This very simple conclusion might have been guessed
from the beginning; but it was only by discussing all the apparently
exceptional cases that the conclusion could be solidly established.

Notes

*^{[page 207]} The condition that `z` does not have to `x` the relation in question is comparatively inessential, being only required in order that, if `y` be between `x` and `z`, we may not have `x` between `y` and `z`, or `z` between `x` and `y`. If we are willing to allow that in such cases, for example, as the angles of a triangle, each is between the other two, we may drop the condition in question altogether. The other four conditions, on the contrary, seem more essential.

*^{[page 211]} Cf. §95.

*^{[page 212]} See Part I, Chap. iv.

†^{[page 212]} See *The Philosophy of Leibniz*, by the present author, Cambridge, 1900; Chapter ii, §10.

‡^{[page 212]} See Part VII, Chap. liv.

*^{[page 213]} This case is illustrated by the rational numbers, which may be taken in order of magnitude, or in one of the orders (*e.g.* the logical order) in which they are denumerable. The logical order is the order 1, 2, 1/2, 3, 1/3, 2/3, 4, …….

*^{[page 214]} *Rivista di Matematica*, v, pp. 75–78. See also Pieri, *I Principii delta Geometria di Posizione*, Turin, 1898, §7.

†^{[page 214]} These five properties are taken from Vailati, *loc. cit.* and *ib.* p. 183.

*^{[page 215]} The argument is somewhat tedious, and I therefore omit it. It will be found in Vailati, *loc. cit.*

†^{[page 215]} Pieri, *op. cit.* p. 32.

*^{[page 216]} The advantages of this method are evident from Pieri’s work quoted above, where many things which seemed incapable of projective proof are rigidly deduced from projective premisses. See Part VI, Chap. xlv.

Asymmetrical Relations.

**208.** We have now seen that all order depends upon transitive
asymmetrical relations. As such relations are of a kind which traditional
logic is unwilling to admit, and as the refusal to admit them is one of
the main sources of the contradictions which the Critical Philosophy has
found in mathematics, it will be desirable, before proceeding further, to
make an excursion into pure logic, and to set forth the grounds which
make the admission of such relations necessary. At a later stage (in
Part VI, Chap. li), I shall endeavour to answer the general objections
of philosophers to relations; for the present, I am concerned only with
asymmetrical relations.

Relations may be divided into four classes, according as they do
or do not possess either of two attributes, transitiveness* and symmetry.
Relations such that `x``R``y` always implies `y``R``x` are called *symmetrical*;
relations such that `x``R``y`, `y``R``z` together always imply `x``R``z` are called
*transitive*. Relations which do not possess the first property I shall
call *not symmetrical*; relations which do possess the opposite property,
*i.e.* for which `x``R``y` always excludes `y``R``x`, I shall call *asymmetrical*.
Relations which do not possess the second property I shall call *not transitive*; those which possess the property that `x``R``y`, `y``R``z` always
exclude `x``R``z` I shall call *intransitive*. All these cases may be illustrated from human relationships. The relation *brother or sister* is
symmetrical, and is transitive if we allow that a man may be his
own brother, and a woman her own sister. The relation *brother* is not
symmetrical, but is transitive. *Half-brother or half-sister* is symmetrical
but not transitive. *Spouse* is symmetrical but intransitive; *descendant*
is asymmetrical but transitive. *Half-brother* is not symmetrical and not
transitive; if third marriages were forbidden, it would be intransitive.
*Son-in-law* is asymmetrical and not transitive; if second marriages were
forbidden, it would be intransitive. *Brother-in-law* is not symmetrical
and not transitive. Finally, `f``a``t``h``e``r` is both asymmetrical and intransitive.
Of not-transitive but not intransitive relations there is, so far as I know,
only one *important* instance, namely diversity; of not-symmetrical but not
asymmetrical relations there seems to be similarly only one important
instance, namely *implication*. In other cases, of the kind that usually
occur, relations are either transitive or intransitive, and either symmetrical or asymmetrical.

**209.** Relations which are both symmetrical and transitive are formally
of the nature of equality. Any term of the field of such a relation has the
relation in question to itself, though it may not have the relation to any
other term. For denoting the relation by the sign of equality, if `a` be
of the field of the relation, there is some term `b` such that `a` = `b`. If
`a` and `b` be identical, then `a` = `a`. But if not, then, since the relation
is symmetrical, `b` = `a`; since it is transitive, and we have `a` = `b`, `b` = `a`,
it follows that `a` = `a`. The property of a relation which insures that
it holds between a term and itself is called by Peano *reflexiveness*, and
he has shown, contrary to what was previously believed, that this
property cannot be inferred from symmetry and transitiveness. For
neither of these properties asserts that there is a `b` such that `a` = `b`, but
only what follows in case there is such a `b`; and if there is no such `b`,
then the proof of `a` = `a` fails*. This property of reflexiveness, however,
introduces some difficulty. There is only one relation of which it is true
without limitation, and that is identity. In all other cases, it holds
only of the terms of a certain class. Quantitative equality, for example,
is only reflexive as applied to quantities; of other terms, it is absurd
to assert that they have quantitative equality with themselves. Logical
equality, again, is only reflexive for classes, or propositions, or relations.
Simultaneity is only reflexive for events, and so on. Thus, with any
given symmetrical transitive relation, other than identity, we can only
assert reflexiveness within a certain class: and of this class, apart from
the principle of abstraction (already mentioned in Part III, Chap. xix,
and shortly to be discussed at length), there need be no definition
except as the extension of the transitive symmetrical relation in question.
And when the class is so defined, reflexiveness within that class, as we
have seen, follows from transitiveness and symmetry.

**210.** By introducing what I have called the principle of abstraction†,
a somewhat better account of reflexiveness becomes possible. Peano has
defined‡ a process which he calls definition by abstraction, of which, as
he shows, frequent use is made in Mathematics. This process is as
follows: when there is any relation which is transitive, symmetrical and
(within its field) reflexive, then, if this relation holds between `u` and `v`,
we define a new entity `φ`(`u`), which is to be identical with `φ`(`v`). Thus
our relation is analyzed into sameness of relation to the new term
`φ`(`u`) or `φ`(`v`). Now the legitimacy of this process, as set forth by
Peano, requires an axiom, namely the axiom that, if there is any
instance of the relation in question, then there is such an entity as
`φ`(`u`) or `φ`(`v`). This axiom is my principle of abstraction, which,
precisely stated, is as follows: “Every transitive symmetrical relation,
of which there is at least one instance, is analyzable into joint possession
of a new relation to a new term, the new relation being such that no
term can have this relation to more than one term, but that its converse
does not have this property.” This principle amounts, in common
language, to the assertion that transitive symmetrical relations arise
from a common property, with the addition that this property stands,
to the terms which have it, in a relation in which nothing else stands
to those terms. It gives the precise statement of the principle, often
applied by philosophers, that symmetrical transitive relations always
spring from identity of content. Identity of content is, however, an
extremely vague phrase, to which the above proposition gives, in the
present case, a precise signification, but one which in no way answers
the purpose of the phrase, which is, apparently, the reduction of relations
to adjectives of the related terms.

It is now possible to give a clearer account of the reflexive property.
Let `R` be our symmetrical relation, and let `S` be the asymmetrical
relation which two terms having the relation `R` must have to some
third term. Then the proposition `x``R``y` is equivalent to this: “There
is some term `a` such that `x``S``a` and `y``S``a`.” Hence it follows that, if `x`
belongs to what we have called the domain of `S`, *i.e.* if there is any
term `a` such that `x``S``a`, then `x``R``x`; for `x``R``x` is merely `x``S``a` and `x``S``a`. It
does not of course follow that there is any other term `y` such that `x``R``y`,
and thus Peano’s objections to the usual proof of reflexiveness are valid.
But by means of the analysis of symmetrical transitive relations, we
obtain the proof of the reflexive property, together with the exact
limitation to which it is subject.

**211.** We can now see the reason for excluding from our accounts
of the methods of generating series a seventh method, which some
readers may have expected to find. This is the method in which
position is merely relative—a method which, in Chap. xix, §154,
we rejected as regards quantity. As the whole philosophy of space
and time is bound up with the question as to the legitimacy of this
method, which is in fact the question as to absolute and relative
position, it may be well to give an account of it here, and to show
how the principle of abstraction leads to the absolute theory of position.

If we consider such a series as that of events, and if we refuse to
allow absolute time, we shall have to admit three fundamental relations
among events, namely, simultaneity, priority, and posteriority. Such a
theory may be formally stated as follows: Let there be a class of terms,
such that any two, `x` and `y`, have either an asymmetrical transitive
relation `P`, or the converse relation ˘`P`, or a symmetrical transitive
relation `R`. Also let `x``R``y`, `y``P``z` imply `x``P``z`, and let `x``P``y`, `y``R``z` imply `x``P``z`.
Then all the terms can be arranged in a series, in which, however, there
may be many terms which have the same place in the series. This
place, according to the relational theory of position, is nothing but
the transitive symmetrical relation `R` to a number of other terms. But
it follows from the principle of abstraction that there is some relation `S`,
such that, if `x``R``y`, there is some one entity `t` for which `x``S``t`, `y``S``t`. We
shall then find that the different entities `t`, corresponding to different
groups of our original terms, also form a series, but one in which
any two different terms have an asymmetrical relation (formally, the
product ˘`S``R``S`). These terms `t` will then be the absolute positions of
our `x`’s and `y`’s, and our supposed seventh method of generating series
is reduced to the fundamental second method. Thus there will be no
series having only relative position, but in all series it is the positions
themselves that constitute the series*.

**212.** We are now in a position to meet the philosophic dislike of
relations. The whole account of order given above, and the present
argument concerning abstraction, will be necessarily objected to by
those philosophers—and they are, I fear, the major part—who hold
that no relations can possess absolute and metaphysical validity. It
is not my intention here to enter upon the general question, but merely
to exhibit the objections to any analysis of asymmetrical relations.

It is a common opinion—often held unconsciously, and employed
in argument, even by those who do not explicitly advocate it—that
all propositions, ultimately, consist of a subject and a predicate. When
this opinion is confronted by a relational proposition, it has two ways
of dealing with it, of which the one may be called monadistic,
the other monistic. Given, say, the proposition `a``R``b`, where `R` is some
relation, the monadistic view will analyse this into two propositions,
which we may call `a``r`_{1} and `b``r`_{2}, which give to `a` and `b` respectively
adjectives supposed to be together equivalent to `R`. The monistic
view, on the contrary, regards the relation as a property of the whole
composed of `a` and `b`, and as thus equivalent to a proposition which
we may denote by (`a``b`)`r`. Of these views, the first is represented by
Leibniz and (on the whole) by Lotze, the second by Spinoza and
Mr Bradley. Let us examine these views successively, as applied to
asymmetrical relations; and for the sake of definiteness, let us take
the relations of greater and less.

**213.** The monadistic view is stated with admirable lucidity by
Leibniz in the following passage*:

“The ratio or proportion between two lines `L` and `M` may be
conceived three several ways; as a ratio of the greater `L` to the
lesser `M`; as a ratio of the lesser `M` to the greater `L`; and lastly, as
something abstracted from both, that is, as the ratio between `L` and `M`,
without considering which is the antecedent, or which the consequent;
which the subject, and which the object, …. In the first way of considering
them, `L` the greater, in the second `M` the lesser, is the subject of that
accident which philosophers call *relation*. But which of them will be
the subject, in the third way of considering them? It cannot be said
that both of them, `L` and `M` together, are the subject of such an
accident; for if so, we should have an accident in two subjects, with
one leg in one, and the other in the other; which is contrary to the
notion of accidents. Therefore we must say that this relation, in this
third way of considering it, is indeed *out of* the subjects; but being
neither a substance nor an accident, it must be a mere ideal thing,
the consideration of which is nevertheless useful.”

**214.** The third of the above ways of considering the relation of
greater and less is, roughly speaking, that which the monists advocate,
holding, as they do, that the whole composed of `L` and `M` is one subject,
so that their way of considering ratio does not compel us, as Leibniz
supposed, to place it among bipeds. For the present our concern is only
with the first two ways. In the first way of considering the matter, we
have “`L` is (greater than `M`),” the words in brackets being considered
as an adjective of `L`. But when we examine this adjective it is at once
evident that it is complex: it consists, at least, of the parts *greater*
and `M`, and both these parts are essential. To say that `L` is greater
does not at all convey our meaning, and it is highly probable that `M` is
also greater. The supposed adjective of `L` involves some reference to `M`;
but what can be meant by a reference the theory leaves unintelligible.
An adjective involving a reference to `M` is plainly an adjective which is
relative to `M`, and this is merely a cumbrous way of describing a relation.
Or, to put the matter otherwise, if `L` has an adjective corresponding
to the fact that it is greater than `M`, this adjective is logically subsequent to, and is merely derived from, the direct relation of `L` to `M`.
Apart from `M`, nothing appears in the analysis of `L` to differentiate it
from `M`; and yet, on the theory of relations in question, `L` should differ
intrinsically from `M`. Thus we should be forced, in all cases of asymmetrical relations, to admit a specific difference between the related
terms, although no analysis of either singly will reveal any relevant
property which it possesses and the other lacks. For the monadistic
theory of relations, this constitutes a contradiction; and it is a contradiction which condemns the theory from which it springs*.

Let us examine further the application of the monadistic theory to
quantitative relations. The proposition “`A` is greater than `B`” is to be
analyzable into two propositions, one giving an adjective to `A`, the
other giving one to `B`. The advocate of the opinion in question will
probably hold that `A` and `B` are quantities, not magnitudes, and will
say that the adjectives required are the magnitudes of `A` and `B`. But
then he will have to admit a relation between the magnitudes, which
will be as asymmetrical as the relation which the magnitudes were to
explain. Hence the magnitudes will need new adjectives, and so on
*ad infinitum*; and the infinite process will have to be completed before
any *meaning* can be assigned to our original proposition. This kind
of infinite process is undoubtedly objectionable, since its sole object
is to explain the meaning of a certain proposition, and yet none of its
steps bring it any nearer to that meaning†. Thus we cannot take
the magnitudes of `A` and `B` as the required adjectives. But further,
if we take any adjectives whatever except such as have each a reference
to the other term, we shall not be able, even formally, to give any
account of the relation, without assuming just such a relation between
the adjectives. For the mere fact that the adjectives are different will
yield only a symmetrical relation. Thus if our two terms have different
colours we find that `A` has to `B` the relation of differing in colour,
a relation which no amount of careful handling will render asymmetrical.
Or if we were to recur to magnitudes, we could merely say that `A` and
`B` differ in magnitude, which gives us no indication as to which is
the greater. Thus the adjectives of `A` and `B` must be, as in Leibniz’s
analysis, adjectives having a reference each to the other term. The
adjective of `A` must be “greater than `B`,” and that of `B` must be “less
than `A`.” Thus `A` and `B` differ, since they have different adjectives—`B` is not greater than `B`, and `A` is not less than `A`—but the adjectives
are extrinsic, in the sense that `A`’s adjective has reference to `B`, and
`B`’s to `A`. Hence the attempted analysis of the relation fails, and we
are forced to admit what the theory was designed to avoid, a so-called
“external” relation, *i.e.* one implying no complexity in either of the
related terms.

The same result may be proved of asymmetrical relations generally,
since it depends solely upon the fact that both identity and diversity
are symmetrical. Let `a` and `b` have an asymmetrical relation `R`, so
that `a``R``b` and `b`˘`R``a`. Let the supposed adjectives (which, as we have
seen, must each have a reference to the other term) be denoted by `β`
and `α` respectively. Thus our terms become `a``β` and `b``α`. `α` involves
a reference to `a`, and `β` to `b`; and `α` and `β` differ, since the relation
is asymmetrical. But `a` and `b` have no intrinsic differences corresponding
to the relation `R`, and prior to it; or, if they have, the points of
difference must themselves have a relation analogous to `R`, so that
nothing is gained. Either `α` or `β` expresses a difference between `a`
and `b`, but one which, since either `α` or `β` involves reference to a term
other than that whose adjective it is, so far from being prior to `R`,
is in fact the relation `R` itself. And since `α` and `β` both presuppose `R`,
the difference between `α` and `β` cannot be used to supply an intrinsic
difference between `a` and `b`. Thus we have again a difference without
a prior point of difference. This shows that some asymmetrical relations must be ultimate, and that at least one such ultimate asymmetrical
relation must be a component in any asymmetrical relation that may be
suggested.

It is easy to criticize the monadistic theory from a general standpoint, by developing the contradictions which spring from the relations of the terms to the adjectives into which our first relation has been analyzed. These considerations, which have no special connection with asymmetry, belong to general philosophy, and have been urged by advocates of the monistic theory. Thus Mr Bradley says of the monadistic theory*: “We, in brief, are led by a principle of fission which conducts us to no end. Every quality in relation has, in consequence, a diversity within its own nature, and this diversity cannot immediately be asserted of the quality. Hence the quality must exchange its unity for an internal relation. But, thus set free, the diverse aspects, because each something in relation, must each be something also beyond. This diversity is fatal to the internal unity of each; and it demands a new relation, and so on without limit.” It remains to be seen whether the monistic theory, in avoiding this difficulty, does not become subject to others quite as serious.

**215.** The monistic theory holds that every relational proposition
`a``R``b` is to be resolved into a proposition concerning the whole which
`a` and `b` compose—a proposition which we may denote by (`a``b`)`r`. This
view, like the other, may be examined with special reference to asymmetrical relations, or from the standpoint of general philosophy. We
are told, by those who advocate this opinion, that the whole contains
diversity within itself, that it synthesizes differences, and that it performs
other similar feats. For my part, I am unable to attach any precise
significance to these phrases. But let us do our best.

The proposition “`a` is greater than `b`,” we are told, does not really
say anything about either `a` or `b`, but about the two together. Denoting
the whole which they compose by (`a``b`) it says, we will suppose, “(`a``b`)
contains diversity of magnitude.” Now to this statement—neglecting
for the present all general arguments—there is a special objection in
the case of asymmetry, (`a``b`) is symmetrical with regard to `a` and `b`,
and thus the property of the whole will be exactly the same in the case
where `a` is greater than `b` as in the case where `b` is greater than `a`.
Leibniz, who did not accept the monistic theory, and had therefore
no reason to render it plausible, clearly perceived this fact, as appears
from the above quotation. For, in his third way of regarding ratio,
we do not consider which is the antecedent, which the consequent;
and it is indeed sufficiently evident that, in the whole (`a``b`) as such,
there is neither antecedent nor consequent. In order to distinguish
a whole (`a``b`) from a whole (`b``a`), as we must do if we are to explain
asymmetry, we shall be forced back from the whole to the parts and
their relation. For (`a``b`) and (`b``a`) consist of precisely the same parts,
and differ in no respect whatever save the sense of the relation between
`a` and `b`. “`a` is greater than `b`” and “`b` is greater than `a`” are propositions containing precisely the same constituents, and giving rise therefore
to precisely the same whole; their difference lies solely in the fact that
*greater* is, in the first case, a relation of `a` to `b`, in the second, a relation
of `b` to `a`. Thus the distinction of sense, *i.e.* the distinction between an
asymmetrical relation and its converse, is one which the monistic theory
of relations is wholly unable to explain.

Arguments of a more general nature might be multiplied almost
indefinitely, but the following argument seems peculiarly relevant. The
relation of whole and part is itself an asymmetrical relation, and the
whole—as monists are peculiarly fond of telling us—is distinct from all
its parts, both severally and collectively. Hence when we say “`a` is
part of `b`,” we really mean, if the monistic theory be correct, to assert
something of the whole composed of `a` and `b`, which is not to be
confounded with `b`. If the proposition concerning this new whole be not
one of whole and part there will be no true judgments of whole and
part, and it will therefore be false to say that a relation between the
parts is really an adjective of the whole. If the new proposition is one
of whole and part, it will require a new one for its meaning, and so on.
If, as a desperate measure, the monist asserts that the whole composed
of `a` and `b` is not distinct from `b`, he is compelled to admit that a whole
is the sum (in the sense of Symbolic Logic) of its parts, which, besides
being an abandonment of his whole position, renders it inevitable that
the whole should be symmetrical as regards its parts—a view which we
have already seen to be fatal. And hence we find monists driven to
the view that the only true whole, the Absolute, has no parts at all,
and that no propositions in regard to it or anything else are quite
true—a view which, in the mere statement, unavoidably contradicts
itself. And surely an opinion which holds all propositions to be in the
end self-contradictory is sufficiently condemned by the fact that, if it
be accepted, it also must be self-contradictory.

**216.** We have now seen that asymmetrical relations are unintelligible on both the usual theories of relation*. Hence, since such
relations are involved in Number, Quantity, Order, Space, Time, and
Motion, we can hardly hope for a satisfactory philosophy of Mathematics
so long as we adhere to the view that no relation can be “purely
external.” As soon, however, as we adopt a different theory, the logical
puzzles, which have hitherto obstructed philosophers, are seen to be
artificial. Among the terms commonly regarded as relational, those
that are symmetrical and transitive—such as equality and simultaneity—*are* capable of reduction to what has been vaguely called identity of
content, but this in turn must be analyzed into sameness of relation
to some other term. For the so-called properties of a term are, in fact,
only other terms to which it stands in some relation; and a common
property of two terms is a term to which both stand in the same
relation.

The present long digression into the realm of logic is necessitated by the fundamental importance of order, and by the total impossibility of explaining order without abandoning the most cherished and widespread of philosophic dogmas. Everything depends, where order is concerned, upon asymmetry and difference of sense, but these two concepts are unintelligible to the traditional logic. In the next chapter we shall have to examine the connection of difference of sense with what appears in Mathematics as difference of sign. In this examination, though some pure logic will still be requisite, we shall approach again to mathematical topics; and these will occupy us wholly throughout the succeeding chapters of this Part.

Notes

*^{[page 218]} This term appears to have been first used in the present sense by De Morgan; see *Camb. Phil. Trans.* ix, p. 104; x, p. 346. The term is now in general use.

*^{[page 219]} See *e.g.* *Revue de Mathématiques*, T. vii, p. 22; *Notations de Logique Mathématique*, Turin, 1894, p. 45, *F*. 1901, p. 193.

†^{[page 219]} An axiom virtually identical with this principle, but not stated with the necessary precision, and not demonstrated, will be found in De Morgan, *Camb. Phil. Trans.* Vol. x, p. 345.

‡^{[page 219]} *Notations de Logique Mathématique*, p. 45.

*^{[page 221]} A formal treatment of relative position is given by Schröder, *Sur une extension
de l’idée d’ordre*, *Congrès*, Vol. iii, p. 235.

*^{[page 222]} *Phil. Werke*, Gerhardt’s ed., Vol. vii, p. 401.

*^{[page 223]} See a paper on “The Relations of Number and Quantity,” *Mind*, N.S. No. 23. This paper was written while I still adhered to the monadistic theory of relations: the contradiction in question, therefore, was regarded as inevitable. The following passage from Kant raises the same point: “Die rechte Hand ist der linken ähnlich und gleich, und wenn man blos auf eine derselben allein sieht, auf die Proportion der Lage der Theile unter einander und auf die Grösse des Ganzen, so muss eine vollständige Beschreibung der einen in allen Stücken auch von der andern gelten.” (*Von dem ersten Grunde des Unterschiedes der Gegenden im Raume*, ed. Hart. Vol. ii, p. 389.)

†^{[page 223]} Where an infinite process of this kind is required we are necessarily dealing with a proposition which is an infinite unity, in the sense of Part II, Chap. xvii.

*^{[page 224]} *Appearance and Reality*, 1st edition, p. 31.

*^{[page 226]} The grounds of these theories will be examined from a more general point of view in Part VI, Chap. li.

Difference of Sense and Difference of Sign.

**217.** We have now seen that order depends upon asymmetrical
relations, and that these always have two senses, as before and after,
greater and less, east and west, etc. The difference of sense is closely
connected (though not identical) with the mathematical difference of
sign. It is a notion of fundamental importance in Mathematics, and
is, so far as I can see, not explicable in terms of any other notions.
The first philosopher who realized its importance would seem to be Kant.
In the *Versuch den Begriff der negativen Grösse in die Weltweisheit einzuführen* (1763), we find him aware of the difference between logical
opposition and the opposition of positive and negative. In the discussion
*Von dem ersten Grunde des Unterschiedes der Gegenden im Raume* (1768),
we find a full realization of the importance of asymmetry in spatial
relations, and a proof, based on this fact, that space cannot be wholly
relational*. But it seems doubtful whether he realized the connection of
this asymmetry with difference of sign. In 1763 he certainly was not
aware of the connection, since he regarded pain as a negative amount of
pleasure, and supposed that a great pleasure and a small pain can be
added to give a less pleasure†—a view which seems both logically and
psychologically false. In the *Prolegomena* (§13), as is well known,
he made the asymmetry of spatial relations a ground for regarding space
as a mere form of intuition, perceiving, as appears from the discussion
of 1768, that space could not consist, as Leibniz supposed, of mere
relations among objects, and being unable, owing to his adherence to
the logical objection to relations discussed in the preceding chapter,
to free from contradiction the notion of absolute space with asymmetrical relations between its points. Although I cannot regard this
later and more distinctively Kantian theory as an advance upon that
of 1768, yet credit is undoubtedly due to Kant for having first called
attention to the logical importance of asymmetrical relations.

**218.** By difference of sense I mean, in the present discussion at least,
the difference between an asymmetrical relation and its converse. It is a
fundamental logical fact that, given any relation `R`, and any two terms
`a`, `b`, there are two propositions to be formed of these elements, the one
relating `a` to `b` (which I call `a``R``b`), the other (`b``R``a`) relating `b` to `a`. These
two propositions are always different, though sometimes (as in the case
of diversity) either implies the other. In other cases, such as logical implication, the one does not imply either the other or its negation; while
in a third set of cases, the one implies the negation of the other. It is
only in cases of the third kind that I shall speak of difference of sense.
In these cases, `a``R``b` excludes `b``R``a`. But here another fundamental logical
fact becomes relevant. In all cases where `a``R``b` does not imply `b``R``a` there
is another relation, related to `R`, which must hold between `b` and `a`. That
is, there is a relation ˘`R` such that `a``R``b` implies `b`˘`R``a`; and further, `b`˘`R``a`
implies `a``R``b`. The relation of `R` to ˘`R` is difference of sense. This
relation is one-one, symmetrical, and intransitive. Its existence is the
source of series, of the distinction of signs, and indeed of the greater
part of mathematics.

**219.** A question of considerable importance to logic, and especially
to the theory of inference, may be raised with regard to difference of sense.
Are `a``R``b` and `b`˘`R``a` really different propositions, or do they only differ
linguistically? It may be held that there is only one relation `R`, and
that all necessary distinctions can be obtained from that between `a``R``b`
and `b``R``a`. It may be said that, owing to the exigencies of speech and
writing, we are compelled to mention either `a` or `b` first, and that this gives
a seeming difference between “`a` is greater than `b`” and “`b` is less than
`a`”; but that, in reality, these two propositions are identical. But if
we take this view we shall find it hard to explain the indubitable
distinction between *greater* and *less*. These two words have certainly
each a meaning, even when no terms are mentioned as related by them.
And they certainly have different meanings, and are certainly relations.
Hence if we are to hold that “`a` is greater than `b`” and “`b` is less than `a`”
are the same proposition, we shall have to maintain that both *greater*
and *less* enter into each of these propositions, which seems obviously
false; or else we shall have to hold that what really occurs is neither
of the two, but that third abstract relation mentioned by Leibniz in the
passage quoted above. In this case the difference between *greater* and
*less* would be one essentially involving a reference to the terms `a` and `b`.
But this view cannot be maintained without circularity; for neither the
greater nor the less is inherently the antecedent, and we can only say
that, when the greater is the antecedent, the relation is *greater*; when
the less, the relation is *less*. Hence, it would seem, we must admit that
`R` and ˘`R` are distinct relations. We cannot escape this conclusion by
the analysis into adjectives attempted in the last chapter. We there
analyzed `a``R``b` into `a``β` and `b``α`. But, corresponding to every `b`, there will
be two adjectives, `β` and ˘`β`, and corresponding to every `a` there will also
be two, `α` and ˘`α`. Thus if `R` be *greater*, `α` will be “greater than `A`”
and ˘`α` “less than `A`,” or *vice versâ*. But the difference between `α` and ˘`α`
presupposes that between greater and less, between `R` and ˘`R`, and therefore
cannot explain it. Hence `R` and ˘`R` must be distinct, and “`a``R``b` implies
`b`˘`R``a`” must be a genuine inference.

I come now to the connection between difference of sense and difference of sign. We shall find that the latter is derivative from the former, being a difference which only exists between terms which either are, or are correlated with, asymmetrical relations. But in certain cases we shall find some complications of detail which will demand discussion.

The difference of signs belongs, traditionally, only to numbers and magnitudes, and is intimately associated with addition. It may be allowed that the notation cannot be usefully employed where there is no addition, and even that, where distinction of sign is possible, addition in some sense is in general also possible. But we shall find that the difference of sign has no very intimate connection with addition and subtraction. To make this clear, we must, in the first place, clearly realize that numbers and magnitudes which have no sign are radically different from such as are positive. Confusion on this point is quite fatal to any just theory of signs.

**220.** Taking first finite numbers, the positive and negative numbers
arise as follows*. Denoting by `R` the relation between two integers in
virtue of which the second is next after the first, the proposition `m``R``n`
is equivalent to what is usually expressed by `m` + 1 = `n`. But the present
theory will apply to progressions generally and does not depend upon
the logical theory of cardinals developed in Part II. In the proposition
`m``R``n`, the integers `m` and `n` are considered, as when they result from the
logical definition, to be wholly destitute of sign. If now `m``R``n` and `n``R``p`,
we put `m``R`^{2}`p`; and so on for higher powers. Every power of `R` is an
asymmetrical relation, and its converse is easily shown to be the same
power of ˘`R` as it is itself of `R`. Thus `m``R`^{a}`q` is equivalent to `q`˘`R`^{a}`m`.
These are the two propositions which are commonly written `m` + `a` = `q`
and `q` − `a` = `m`. Thus the relations `R`^{a}, ˘`R`^{a} are the true positive and
negative integers; and these, though associated with `a`, are both wholly
distinct from it. Thus in this case the connection with difference of
sense is obvious and straightforward.

**221.** As regards magnitudes, several cases must be distinguished.
We have (1) magnitudes which are not either relations or, stretches,
(2) stretches, (3) magnitudes which are relations.

(1) Magnitudes of this class are themselves neither positive nor negative. But two such magnitudes, as explained in Part III, determine either a distance or a stretch, and these are always positive or negative. These are moreover always capable of addition. But since our original magnitudes are neither relations nor stretches, the new magnitudes thus obtained are of a different kind from the original set. Thus the difference of two pleasures, or the collection of pleasures intermediate between two pleasures, is not a pleasure, but in the one case a relation, in the other a class.

(2) Magnitudes of divisibility in general have no sign, but when
they are magnitudes of stretches they acquire sign by correlation.
A stretch is distinguished from other collections by the fact that it
consists of all the terms of a series intermediate between two given
terms. By combining the stretch with one sense of the asymmetrical
relation which must exist between its end-terms, the stretch itself
acquires sense, and becomes asymmetrical. That is, we can distinguish
(1) the collection of terms between `a` and `b` without regard to order, (2)
the terms from `a` to `b`, (3) the terms from `b` to `a`. Here (2) and (3) are
complex, being compounded of (1) and one sense of the constitutive
relation. Of these two, one must be called positive, the other negative.
Where our series consists of magnitudes, usage and the connection with
addition have decided that, if `a` is less than `b`, (2) is positive and (3) is
negative. But where, as in Geometry, our series is not composed of
magnitudes, it becomes wholly arbitrary which is to be positive and
which negative. In either case, we have the same relation to addition,
which is as follows. Any pair of collections can be added to form a new
collection, but not any pair of stretches can be added to form a new
stretch. For this to be possible the end of one stretch must be consecutive to the beginning of the other. In this way, the stretches `a``b`, `b``c`
can be added to form the stretch `a``c`. If `a``b`, `b``c` have the same sense, `a``c` is
greater than either; if they have different senses, `a``c` is less than one
of them. In this second case the addition of `a``b` and `b``c` is regarded
as the subtraction of `a``b` and `c``b`, `b``c` and `c``b` being negative and positive
respectively. If our stretches are numerically measurable, addition or
subtraction of their measures will give the measure of the result of
adding or subtracting the stretches, where these are such as to allow
addition or subtraction. But the whole opposition of positive and
negative, as is evident, depends upon the fundamental fact that our
series is generated by an asymmetrical relation.

(3) Magnitudes which are relations may be either symmetrical or
asymmetrical relations. In the former case, if `a` be a term of the field
of one of them, the other terms of the various fields, if certain conditions
are fulfilled*, may be arranged in series according as their relations to a
are greater or smaller. This arrangement may be different when we choose
some term other than `a`; for the present, therefore, we shall suppose `a` to
be chosen once for all. When the terms have been arranged in a series,
it may happen that some or all places in the series are occupied by more
than one term; but in any case the assemblage of terms between `a` and
some other term `m` is definite, and leads to a stretch with two senses.
We may then combine the magnitude of the relation of `a` to `m` with one
or other of these two senses, and so obtain an asymmetrical relation of
`a` to `m`, which, like the original relation, will have magnitude. Thus the
case of symmetrical relations may be reduced to that of asymmetrical
relations. These latter lead to signs, and to addition and subtraction,
in exactly the same way as stretches with sense; the only difference being
that the addition and subtraction are now of the kind which, in Part III,
we called relational. Thus in all cases of magnitudes having sign, the
difference between the two senses of an asymmetrical relation is the
source of the difference of sign.

The case which we discussed in connection with stretches is of fundamental importance in Geometry. We have here a magnitude without sign, an asymmetrical relation without magnitude, and some intimate connection between the two. The combination of both then gives a magnitude which has sign. All geometrical magnitudes having sign arise in this way. But there is a curious complication in the case of volumes. Volumes are, in the first instance, signless quantities; but in analytical Geometry they always appear as positive or negative. Here the asymmetrical relations (for there are two) appear as terms, between which there is a symmetrical relation, but one which yet has an opposite of a kind very similar to the converse of an asymmetrical relation. This relation, as an exceptional case, must be here briefly discussed.

**222.** The descriptive straight line is a serial relation in virtue of
which the points of the line form a series*. Either sense of the descriptive
straight line may be called a ray, the sense being indicated by an
arrow. Any two non-coplanar rays have one or other of two relations,
which may be called right and left-handedness respectively†. This
relation is symmetrical but not transitive, and is the essence of the usual
distinction of right and left. Thus the relation of the upward vertical
to a line from north to east is right-handed, and to a line from south to
east is left-handed. But though the relation is symmetrical, it is
changed into its opposite by changing either of the terms of the relation
into its converse. That is, denoting right-handedness by `R`, left-handedness by `L` (which is not ˘`R`), if `A` and `B` be two rays which are mutually
right-handed, we shall have

`A``R``B`, ˘`A``L``B`, `A``L`˘`B`, ˘`A``R`˘`B`, `B``R``A`, ˘`B``L``A`, `B``L`˘`A`, ˘`B``R`˘`A`.

That is, every pair of non-coplanar straight lines gives rise to eight such
relations, of which four are right-handed, and four left-handed. The
difference between `L` and `R`, though not, as it stands, a difference of
sense, is, nevertheless, the difference of positive and negative, and is the
reason why the volumes of tetrahedra, as given by determinants, always
have signs. But there is no difficulty in following the plain man’s
reduction of right and left to asymmetrical relations. The plain man
takes one of the rays (say `A`) as fixed—when he is sober, he takes `A` to
be the upward vertical—and then regards right and left as properties of
the single ray `B`, or, what comes to the same thing, as relations of any
two points which determine `B`. In this way, right and left become
asymmetrical relations, and even have a limited degree of transitiveness, of
the kind explained in the fifth way of generating series (in Chapter xxiv).
It is to be observed that what is fixed must be a ray, not a mere straight
line. For example, two planes which are not mutually perpendicular
are not one right and the other left with regard to their line of intersection, but only with regard to either of the rays belonging to this
line.* But when this is borne in mind, and when we consider, not
semi-planes, but complete planes, through the ray in question, right and
left become asymmetrical and each other’s converses. Thus the signs
associated with right and left, like all other signs, depend upon the
asymmetry of relations. This conclusion, therefore, may now be allowed
to be general.

**223.** Difference of sense is, of course, more general than difference of
sign, since it exists in cases with which mathematics (at least at present)
is unable to deal. And difference of sign seems scarcely applicable to
relations which are not transitive, or are not intimately connected with
some transitive relation. It would be absurd, for example, to regard the
relation of an event to the time of its occurrence, or of a quantity to its
magnitude, as conferring a difference of sign. These relations are what
Professor Schröder calls *erschöpft*†, *i.e.* if they hold between `a` and `b`,
they can never hold between `b` and some third term. Mathematically,
their square is null. These relations, then, do not give rise to difference
of sign.

All magnitudes with sign, so the above account has led us to believe, are either relations or compound concepts into which relations enter. But what are we to say of the usual instances of opposites: good and evil, pleasure and pain, beauty and ugliness, desire and aversion? The last pair are very complex, and if I were to attempt an analysis of them, I should emit some universally condemned opinions. With regard to the others, they seem to me to have an opposition of a very different kind from that of two mutually converse asymmetrical relations, and analogous rather to the opposition of red and blue, or of two different magnitudes of the same kind. From these oppositions, which are constituted by what may be called synthetic incompatibility*, the oppositions above mentioned differ only in the fact that there are only two incompatible terms, instead of a whole series. The incompatibility consists in the fact that two terms which are thus incompatible cannot coexist in the same spatio-temporal place, or cannot be predicates of the same existent, or, more generally, cannot both enter into true propositions of a certain form, which differ only in the fact that one contains one of the incompatibles while the other contains the other. This kind of incompatibility (which usually belongs, with respect to some class of propositions, to the terms of a given series) is a most important notion in general logic, but is by no means to be identified with the difference between mutually converse relations. This latter is, in fact, a special case of such incompatibility; but it is the special case only that gives rise to the difference of sign. All difference of sign—so we may conclude our argument—is primarily derived from transitive asymmetrical relations, from which it may be extended by correlation to terms variously related to such relations†; but such extensions are always subsequent to the original opposition derived from difference of sense.

Notes

*^{[page 227]} See especially ed. Hart, Vol. II, pp. 386, 391.

†^{[page 227]} Ed. Hart, Vol. II, p. 83.

*^{[page 229]} I give the theory briefly here, as it will be dealt with more fully and generally in the chapter on Progressions, §233.

*^{[page 230]} Cf. §245.

*^{[page 231]} See Part VI.

†^{[page 231]} The two cases are illustrated in the figure. The difference is the same as that between the two sorts of coordinate axes.

*^{[page 232]} This requires that the passage from the one plane to the other should be made *viâ* one of the acute angles made by their intersection.

†^{[page 232]} *Algebra der Logik*, Vol. III, p. 328. Professor Peirce calls such relations *non-repeating* (reference in Schröder, *ib.*).

*^{[page 233]} See *The Philosophy of Leibniz*, by the present author (Cambridge 1900), pp. 19, 20.

†^{[page 233]} Thus in mathematical Economics, pleasure and pain may be taken as positive and negative without logical error, by the theory (whose psychological correctness we need not examine) that a man must be paid to endure pain, and must pay to obtain pleasure. The opposition of pleasure and pain is thus correlated with that of money paid and money received, which is an opposition of positive and negative in the sense of elementary Arithmetic.

On the Difference Between Open and Closed Series.

**224.** We have now come to the end of the purely logical discussions
concerned with order, and can turn our attention with a free mind to
the more mathematical aspects of the subject. As the solution of the
most ancient and respectable contradictions in the notion of infinity
depends mainly upon a correct philosophy of order, it has been necessary
to go into philosophical questions at some length—not so much because
they *are* relevant, as because most philosophers think them so. But we
shall reap our reward throughout the remainder of this work.

The question to be discussed in this chapter is this: Can we ultimately distinguish open from closed series, and if so, in what does
the distinction consist? We have seen that, mathematically, *all* series
are open, in the sense that all are generated by an asymmetrical transitive relation. But philosophically, we must distinguish the different
ways in which this relation may arise, and especially we must not
confound the case where this relation involves no reference to other
terms with that where such terms are essential. And practically, it is
plain that there is some difference between open and closed series—between, for instance, a straight line and a circle, or a pedigree and a
mutual admiration society. But it is not quite easy to express the
difference precisely.

**225.** Where the number of terms in the series is finite, and
the series is generated in the first of the ways explained in
Chapter xxiv, the method of obtaining a transitive relation out of the
intransitive relation with which we start is radically different according
as the series is open or closed. If `R` be the generating relation, and `n` be
the number of terms in our series, two cases may arise. Denoting the
relation of any term to the next but one by `R`^{2}, and so on for higher
powers, the relation `R`^{n} can have only one of two values, zero and
identity. (It is assumed that `R` is a one-one relation.) For starting
with the first term, if there be one, `R`^{n−1} brings us to the last term; and
thus `R`^{n} gives no new term, and there is no instance of the relation
`R`^{n}. On the other hand, it may happen that, starting with any term,
`R`^{n} brings us back to that term again. These two are the only possible
alternatives. In the first case, we call the series open; in the second, we
call it closed. In the first case, the series has a definite beginning and
end; in the second case, like the angles of a polygon, it has no peculiar
terms. In the first case, our transitive asymmetrical relation is the
disjunctive relation “a power of `R` not greater than the (`n` − 1)th.”
By substituting this relation, which we may call `R`′, for `R`, our series
becomes of the second of the six types. But in the second case no such
simple reduction to the second type is possible. For now, the relation
of any two terms `a` and `m` of our series may be just as well taken to be
a power of ˘`R` as a power of `R`, and the question which of any three terms
is between the other two becomes wholly arbitrary. We might now introduce, first the relation of separation of four terms, and then the resulting
five-term relation explained in Chapter xxv. We should then regard
three of the terms in the five-term relation as fixed, and find that the
resulting relation of the other two is transitive and asymmetrical. But
here the first term of our series is wholly arbitrary, which was not the
case before; and the generating relation is, in reality, one of five terms,
not one of two. There is, however, in the case contemplated, a simpler
method. This may be illustrated as follows: In an open series, any two
terms `a` and `m` define two senses in which the series may be described,
the one in which `a` comes before `m`, and the other in which `m` comes
before `a`. We can then say of any two other terms `c` and `g` that the
sense of the order from `c` to `g` is the same as that of the order from
`a` to `m`, or different, as the case may be. In this way, considering
`a` and `m` fixed, and `c` and `g` variable, we get a transitive asymmetrical
relation between `c` and `g`, obtained from a transitive symmetrical relation
of the pair `c`, `g` to the pair `a`, `m` (or `m`, `a`, as the case may be). But this
transitive symmetrical relation can, by the principle of abstraction, be
analyzed into possession of a common property, which is, in this case,
the fact that `a`, `m` and `c`, `g` have the generating relation with the same
sense. Thus the four-term relation is, in this case, not essential. But in a
closed series, `a` and `m` do not define a sense of the series, even when we
are told that `a` is to precede `m`: we can start from `a` and get to `m` in
either direction. But if now we take a third term `d`, and decide that we
are to start from `a` and reach `m` taking `d` on the way, then a sense of the
series is defined. The stretch `a``d``m` includes one portion of the series, but
not the other. Thus we may go from England to New Zealand either
by the east or by the west; but if we are to take India on the way, we
must go by the east. If now we consider any other term, say `k`, this
will have some definite position in the series which starts with `a` and
reaches `m` by way of `d`. In this series, `k` will come either between `a` and
`d`, or between `d` and `m`, or after `m`. Thus the three-term relation of
`a`, `d`, `m` seems in this case sufficient to generate a perfectly definite series.
Vailati’s five-term relation will then consist in this, that with regard to
the order `a``d``m`, `k` comes before (or after) any other term `l` of the collection.
But it is not necessary to call in this relation in the present case, since the
three-term relation suffices. This three-term relation may be formally
defined as follows. There is between any two terms of our collection a
relation which is a power of `R` less than the `n`th. Let the relation between
`a` and `d` be `R`^{x}, that between `a` and `m` `R`^{y}. Then if `x` is less than `y`, we
assign one sense to `a``d``m`; if `x` is greater than `y`, we assign the other.
There will be also between `a` and `d` the ˘`R`^{n−x}, and between `a` and
`m` the relation ˘`R`^{n−y}. If `x` is less than `y`, then `n` − `x` is greater than `n` − `y`;
hence the asymmetry of the two cases corresponds to that of `R` and ˘`R`.
The terms of the series are simply ordered by correlation with their
numbers `x` and `y`, those with smaller numbers preceding those with
larger ones. Thus there is here no need of the five-term relation, everything being effected by the three-term relation, which is itself reduced to
an asymmetrical transitive relation of two numbers. But the closed
series is still distinguished from the open one by the fact that its first
term is arbitrary.

**226.** A very similar discussion will apply to the case where our
series is generated by relations of three terms. To keep the analogy
with the one-one relation of the above case, we will make the following
assumptions. Let there be a relation `B` of one term to two others, and
let the one, term be called the mean, the two others the extremes. Let
the mean be uniquely determined when the extremes are given, and let
one extreme be uniquely determined by the mean and the other extreme.
Further let each term that occurs as mean occur also as extreme, and
each term that occurs as extreme (with at most two exceptions) occur
also as mean. Finally, if there be a relation in which `c` is mean, and `b`
and `d` are extremes, let there be always (except when `b` or `d` is one of the
two possible exceptional terms) a relation in which `b` is the mean and `c`
one of the extremes, and another in which `d` is the mean and `c` one of the
extremes. Then `b` and `c` will occur together in only two relations. This
fact constitutes a relation between `b` and `c`, and only one other term
besides `b` will have this new relation to `c`. By means of this relation, if
there are two exceptional terms, or if, our collection being infinite, there
is only one, we can construct an open series. If our two-term relation be
asymmetrical, this is sufficiently evident; but the same result can be
proved if our two-term relation is symmetrical. For there will be at
either end, say `a`, an asymmetrical relation of `a` to the only term which is
the mean between `a` and some other term. This relation multiplied by
the `n`th power of our two-term relation, where `n` + 1 is any integer less
than the number of terms in our collection, will give a relation which
holds between `a` and a number (not exceeding `n` + 1) of terms of our
collection, of which terms one and only one is such that no number less
than `n` gives a relation of `a` to this term. Thus we obtain a correlation
of our terms with the natural numbers, which generates an open
series with `a` for one of its ends. If, on the other hand, our collection has no exceptional terms, but is finite, then we shall obtain
a closed series. Let our two-term relation be `P`, and first suppose it
symmetrical. (It will be symmetrical if our original three-term relation
was symmetrical with regard to the extremes.) Then every term `c` of
our collection will have the relation `P` to two others, which will have
to each other the relation `P`^{2}. Of all the relations of the form `P`^{m}
which hold between two given terms, there will be one in which `m` is
least: this may be called the principal relation of our two terms. Let
the number of terms of the collection be `n`. Then every term of our
collection will have to every other a principal relation `P`^{x}, where `x` is
some integer not greater than `n`/2. Given any two terms `c` and `g` of the
collection, provided we do not have `c``P`^{n/2}`g` (a case which will not arise
if `n` be odd), let us have `c``P`^{x}`g`, where `x` is less than `n`/2. This assumption
defines a sense of the series, which may be shown as follows. If `c``P`^{y}`k`,
where `y` is also less than `n`/2, three cases may arise, assuming `y` is greater
than `x`. We may have `g``P`^{y−z}`k`, or, if `x` + `y` is less than `n`/2, we may
have `g``P`^{x+y}`k`, or, if `x` + `y` is greater than `n`/2, we may have `g``P``n`2^{−x−y}`k`.
(We choose always the principal relation.) These three cases are illustrated in the accompanying figure. We shall say, in these three cases,
that, with regard to the sense `c``g`, (1) `k` comes after `c` and `g`, (2) and (3)
`k` comes before `c` and `g`. If `y` is less than `x`, and `k``P`^{x−y}`g`, we shall say
that `k` is between `c` and `g` in the sense `c``g`. If `n` is odd, this covers all
possible cases. But if `n` is even, we have to consider the term `c`′, which
is such that `c``P`^{n/2}`c`′. This term is, in a certain sense, antipodal to `c`; we
may define it as the first term in the series when the above method of
definition is adopted. If `n` is odd, the first term will be that term of
class (3) for which `c``P`^{(n−1)/2}`k`. Thus the series acquires a definite order,
but one in which, as in all closed series, the first term is arbitrary.

**227.** The only remaining case is that where we start from four-term
relations, and the generating relation has, strictly speaking, five terms.
This is the case of projective Geometry. Here the series is necessarily
closed; that is, in choosing our three fixed terms for the five-term
relation, there is never any restriction upon our choice; and any one of
these three may be defined to be the first.

**228.** Thus, to sum up: Every series being generated by a transitive
asymmetrical relation between any two terms of the series, a series is
open when it has either no beginning, or a beginning which is not
arbitrary; it is closed when it has an arbitrary beginning. Now if `R`
be the constitutive relation, the beginning of the series is a term having
the relation `R` but not the relation ˘`R`. Whenever `R` is genuinely a
two-term relation, the beginning, if it exists, must be perfectly definite.
It is only when `R` involves some other term (which may be considered
fixed) besides the two with regard to which it is transitive and asymmetrical (which are to be regarded as variable), that the beginning can
be arbitrary. Hence in all cases of closed series, though there may be
an asymmetrical one-one relation if the series is discrete, the *transitive*
asymmetrical relation must be one involving one or more fixed terms
in addition to the two variable terms with regard to which it generates
the series. Thus although, mathematically, every closed series can be
rendered open, and every open series closed, yet there is, in regard
to the nature of the generating relation, a genuine distinction between
them—a distinction, however, which is of philosophical rather than
mathematical importance.

Progressions and Ordinal Numbers.

**229.** It is now time to consider the simplest type of infinite series,
namely that to which the natural numbers themselves belong. I shall
postpone to the next Part all the supposed difficulties arising out of
the infinity of such series, and concern myself here only to give the
elementary theory of them in a form not presupposing numbers*.

The series now to be considered are those which can be correlated, term for term, with the natural numbers, without requiring any change in the order of the terms. But since the natural numbers are a particular case of such series, and since the whole of Arithmetic and Analysis can be developed out of any one such series, without any appeal to number, it is better to give a definition of progressions which involves no appeal to number.

A progression is a discrete series having consecutive terms, and a
beginning but no end, and being also *connected*. The meaning of
connection was explained in Chapter xxiv by means of number, but this
explanation cannot be given now. Speaking popularly, when a series
is not connected it falls into two or more parts, each being a series
for itself. Thus numbers and instants together form a series which
is not connected, and so do two parallel straight lines. Whenever
a series is originally given by means of a transitive asymmetrical relation, we can express connection by the condition that any two terms
of our series are to have the generating relation. But progressions
are series of the kind that may be generated in the first of our six
ways, namely, by an asymmetrical one-one relation. In order to pass
from this to a transitive relation, we before employed numbers, defining
the transitive relation as any power of the one-one relation. This
definition will not serve now, since numbers are to be excluded. It
is one of the triumphs of modern mathematics to have adapted an
ancient principle to the needs of this case.

The definition which we want is to be obtained from mathematical induction. This principle, which used to be regarded as a mere subterfuge for eliciting results of which no other proof was forthcoming, has gradually grown in importance as the foundations of mathematics have been more closely investigated. It is now seen to be the principle upon which depend, so far as ordinals are concerned, the commutative law and one form of the distributive law*. This principle, which gives the widest possible extension to the finite, is the distinguishing mark of progressions. It may be stated as follows:

Given any class of terms `s`, to which belongs the first term of any
progression, and to which belongs the term of the progression next after
any term of the progression belonging to `s`, then every term of the
progression belongs to `s`.

We may state the same principle in another form. Let `φ`(`x`) be
a propositional function, which is a determinate proposition as soon
as `x` is given. Then `φ`(`x`) is a function of `x`, and will in general be
true or false according to the value of `x`. If `x` be a member of a
progression, let seq `x` denote the term next after `x`. Let `φ`(`x`) be true
when `x` is the first term of a certain progression, and let `φ`(seq `x`)
be true whenever `φ`(`x`) is true, where `x` is any term of the progression.
It then follows, by the principle of mathematical induction, that `φ`(`x`)
is always true if `x` be any term of the progression in question.

The complete definition of a progression is as follows. Let `R` be
any asymmetrical one-one relation, and `u` a class such that every term
of `u` has the relation of `R` to some term also belonging to the class `u`.
Let there be at least one term of the class `u` which does not have
the relation ˘`R` to any term of `u`. Let `s` be any class to which belongs
at least one of the terms of `u` which do not have the relation ˘`R` to any
term of `u`, and to which belongs also every term of `u` which has the
relation ˘`R` to some term belonging to both `u` and `s`; and let `u` be such
as to be wholly contained in any class `s` satisfying the above conditions.
Then `u`, considered as ordered by the relation `R`, is a progression†.

**230.** Of such progressions, everything relevant to finite Arithmetic
can be proved. In the first place, we show that there can only be
one term of `u` which does not have the relation ˘`R` to any term of `u`.
We then define the term to which `x` has the relation `R` as the successor
of `x` (`x` being a `u`), which may be written seq `x`. The definitions and
properties of addition, subtraction, multiplication, division, positive and
negative terms, and rational fractions are easily given; and it is easily
shown that between any two rational fractions there is always a third.
From this point it is easy to advance to irrationals and the real
numbers*.

Apart from the principle of mathematical induction, what is chiefly
interesting about this process is, that it shows that only the serial or
ordinal properties of finite numbers are used by ordinary mathematics,
what may be called the logical properties being wholly irrelevant. By
the logical properties of numbers, I mean their definition by means of
purely logical ideas. This process, which has been explained in Part II,
may be here briefly recapitulated. We show, to begin with, that a one-one correlation can be effected between any two null classes, or between
any two classes `u`, `v` which are such that, if `x` is a `u`, and `x`′ differs from
`x`, then `x`′ cannot be a `u`, with a like condition for `v`. The possibility
of such one-one correlation we call similarity of the two classes `u`, `v`.
Similarity, being symmetrical and transitive, must be analyzable (by the
principle of abstraction) into possession of a common property. This
we define as the *number* of either of the classes. When the two classes
`u`, `v` have the above-defined property, we say their number is *one*; and
so on for higher numbers; the general definition of finite numbers
demanding mathematical induction, or the non-similarity of whole and
part, but being always given in purely logical terms.

It is numbers so defined that are used in daily life, and that are
essential to any assertion of numbers. It is the fact that numbers have
these logical properties that makes them important. But it is not
these properties that ordinary mathematics employs, and numbers might
be bereft of them without any injury to the truth of Arithmetic and
Analysis. What is relevant to mathematics is solely the fact that
finite numbers form a progression. This is the reason why mathematicians—*e.g.* Helmholtz, Dedekind, and Kronecker—have maintained
that ordinal numbers are prior to cardinals; for it is solely the ordinal
properties of number that are relevant. But the conclusion that ordinals are prior to cardinals seems to have resulted from a confusion.
Ordinals and cardinals alike form a progression, and have exactly the
same ordinal properties. Of either, all Arithmetic can be proved
without any appeal to the other, the propositions being symbolically
identical, but different in meaning. In order to prove that ordinals
are prior to cardinals, it would be necessary to show that the cardinals
can only be defined in terms of the ordinals. But this is false, for the
logical definition of the cardinals is wholly independent of the ordinals†.
There seems, in fact, to be nothing to choose, as regards logical priority,
between ordinals and cardinals, except that the existence of the ordinals
is inferred from the series of cardinals. The ordinals, as we shall see
in the next paragraph, can be defined without any appeal to the
cardinals; but when defined, they are seen to imply the cardinals.
Similarly, the cardinals can be defined without any appeal to the
ordinals; but they essentially form a progression, and all progressions,
as I shall now show, necessarily imply the ordinals.

**231.** The correct analysis of ordinals has been prevented hitherto by
the prevailing prejudice against relations. People speak of a series as
consisting of certain terms *taken* in a certain order, and in this idea
there is commonly a psychological element. All sets of terms have,
apart from psychological considerations, all orders of which they are
capable; that is, there are serial relations, whose fields are a given set of
terms, which arrange those terms in any possible order. In some cases,
one or more serial relations are specially prominent, either on account of
their simplicity, or of their importance. Thus the order of magnitude
among numbers, or of before and after among instants, seems emphatically the natural order, and any other seems to be artificially introduced
by our arbitrary choice. But this is a sheer error. Omnipotence itself
cannot give terms an order which they do not possess already: all that
is psychological is the *consideration* of such and such an order. Thus
when it is said that we can arrange a set of terms in any order we please,
what is really meant is, that we can consider any of the serial relations
whose field is the given set, and that these serial relations will give
between them any combinations of before and after that are compatible
with transitiveness and connection. From this it results that an order
is not, properly speaking, a property of a given set of terms, but of a serial
relation whose field is the given set. Given the relation, its field is given
with it; but given the field, the relation is by no means given. The
notion of a set of terms in a given order is the notion of a set of terms
considered as the field of a given serial relation; but the consideration
of the terms is superfluous, and that of the relation alone is quite
sufficient.

We may, then, regard an ordinal number as a common property of
sets of serial relations which generate ordinally similar series. Such
relations have what I shall call *likeness*, *i.e.* if `P`, `Q` be two such relations,
their fields can be so correlated term for term that two terms of which
the first has to the second the relation `P` will always be correlated with
two terms of which the first has to the second the relation `Q`, and
*vice versâ*. As in the case of cardinal numbers*, so here, we may, in
virtue of the principle of abstraction, define the ordinal number of
a given finite serial relation as the class of like relations. It is easy to
show that the generating relations of progressions are all alike; the
class of such relations will be the ordinal number of the finite integers
in order of magnitude. When a class is finite, all series that can be
formed of its terms are ordinally similar, and are ordinally different from
series having a different cardinal number of terms. Hence there is a
one-one correlation of finite ordinals and cardinals, for which, as we
shall see in Part V, there is no analogy in respect of infinite numbers. We
may therefore define the ordinal number `n` as the class of serial relations
whose *domains* have `n` terms, where `n` is a finite cardinal. It is necessary,
unless 1 is to be excluded, to take domains instead of fields here, for no
relation which implies diversity can have one term in its field, though it
may have none. This has a practical inconvenience, owing to the fact
that `n` + 1 must be obtained by adding *one* term to the field; but the
point involved is one for conventions as to notation, and is quite
destitute of philosophical importance.

**232.** The above definition of ordinal numbers is direct and simple,
but does not yield the notion of “`n`th,” which would usually be regarded
as *the* ordinal number. This notion is far more complex: a term is not
intrinsically the `n`th, and does not become so by the mere specification
of `n` − 1 other terms. A term is the `n`th in respect of a certain serial
relation, when, in respect of that relation, the term in question has `n` − 1
predecessors. This is the definition of “`n`th,” showing that this notion
is relative, not merely to predecessors, but also to a specified serial
relation. By induction, the various finite ordinals can be defined
without mentioning the cardinals. A finite serial relation is one which
is not like (in the above sense) any relation implying it but not equivalent
to it; and a finite ordinal is one consisting of finite serial relations. If
`n` be a finite ordinal, `n` + 1 is an ordinal such that, if the last term* of
a series of the type `n` + 1 be cut off, the remainder, in the same order, is
of the type `n`. In more technical language, a serial relation of the type
`n` + 1 is one which, when confined to its domain instead of its field,
becomes of the type `n`. This gives by induction a definition of every
particular finite ordinal, in which cardinals are never mentioned. Thus
we cannot say that ordinals presuppose cardinals, though they are more
complex, since they presuppose both serial and one-one relations, whereas
cardinals only presuppose one-one relations.

Of the ordinal number of the finite ordinals in order of magnitude, several equivalent definitions may be given. One of the simplest is, that this number belongs to any serial relation, which is such that any class contained in its field and not null has a first term, while every term of the series has an immediate successor, and every term except the first has an immediate predecessor. Here, again, cardinal numbers are in no way presupposed.

Throughout the above discussions our serial relations are taken to be transitive, not one-one. The one-one relations are easily derived from the transitive ones, while the converse derivation is somewhat complicated. Moreover the one-one relations are only adequate to define finite series, and thus their use cannot be extended to the study of infinite series unless they are taken as derivative from the transitive ones.

**233.** A few words concerning positive and negative ordinals seem to
be here in place. If the first `n` terms of a progression be taken away
(`n` being any finite number), the remainder still form a progression.
With regard to the new progression, negative ordinals may be assigned
to the terms that have been abstracted; but for this purpose it is
convenient to regard the beginning of the smaller progression as the
0th term. In order to have a series giving *any* positive or negative
ordinal, we need what may be called a double progression. This is a
series such that, choosing any term `x` out of it, two progressions start
from `x`, the one generated by a serial relation `R`, the other by ˘`R`. To
`x` we shall then assign the ordinal 0, and to the other terms we shall
assign positive or negative ordinals according as they belong to the one
or the other of the two progressions starting from `x`. The positive and
negative ordinals themselves form such a double progression. They
express essentially a relation to the arbitrarily chosen origin of the two
progressions, and +`n` and −`n` express mutually converse relations.
Thus they have all the properties which we recognize in Chapter xxvii
as characterizing terms which have signs.

Notes

*^{[page 239]} The present chapter closely follows Peano’s Arithmetic. See *Formulaire de Mathématiques*, Vol. 11, §2. I have given a mathematical treatment of the subject in *RdM*, Vols. VII and VIII. The subject is due, in the main, to Dedekind and Georg Cantor.

*^{[page 240]} Namely (`α` + `β`)`γ` = `α``γ` + `β``γ`. The other form, `α`(`β` + `γ`) = `α``β` + `α``γ`, holds also
for infinite ordinal numbers, and is thus independent of mathematical induction.

†^{[page 240]} It should be observed that a discrete open series generated by a transitive relation can always be reduced, as we saw in the preceding chapter, to one generated by an asymmetrical one-one relation, provided only that the series is finite or a progression.

*^{[page 241]} See my article on the Logic of Relations, *RdM*, VII.

†^{[page 241]} Professor Peano, who has a rare immunity from error, has recognized this fact. See *Formulaire*, 1898, 210, note (p. 39).

*^{[page 242]} Cf. §111.

*^{[page 243]} The last term of a series (if it exists) is the term belonging to the converse domain but not to the domain of the generating relation, *i.e.* the term which is after but not before other terms.

Dedekind’s Theory of Number.

**234.** The theory of progressions and of ordinal numbers, with which
we have been occupied in the last chapter, is due in the main to two
men—Dedekind and Cantor. Cantor’s contributions, being specially
concerned with infinity, need not be considered at present; and
Dedekind’s theory of irrationals is also to be postponed. It is his theory
of integers of which I wish now to give an account-—the theory, that is
to say, which is contained in his “*Was sind und was sollen die Zahlen*?”*
In reviewing this work, I shall not adhere strictly to Dedekind’s
phraseology. He appears to have been, at the time of writing, unacquainted with symbolic logic; and although he invented as much of
this subject as was relevant to his purpose, he naturally adopted phrases
which were not usual, and were not always so convenient as their conventional equivalents.

The fundamental ideas of the pamphlet in question are these†:
(1) the representation (*Abbildung*) of a system (21); (2) the notion of a
chain (37); (3) the chain of an element (44); (4) the generalized form
of mathematical induction (59); (5) the definition of a singly infinite
system (71). From these five notions Dedekind deduces numbers and
ordinary Arithmetic. Let us first explain the notions, and then examine
the deduction.

**235.** (1) A *representation* of a class `u` is any law by which, to every
term of `u`, say `x`, corresponds some one and only one term `φ`(`x`). No
assumption is made, to begin with, as to whether `φ`(`x`) belongs to the
class `u`, or as to whether `φ`(`x`) may be the same as `φ`(`y`) when `x` and `y`
are different terms of `u`. The definition thus amounts to this:

A *representation* of a class `u` is a many-one relation, whose domain
contains `u`, by which terms, which may or may not also belong to `u`, are
correlated one with each of the terms of `u`*. The representation is
similar when, if `x` differs from `y`, both being `u`’s, then `φ`(`x`) differs from
`φ`(`y`); that is, when the relation in question is one-one. He shows that
similarity between classes is reflexive, symmetrical and transitive, and
remarks (34) that classes can be classified by similarity to a given class—a suggestion of an idea which is fundamental in Cantor’s work.

**236.** (2) If there exists a relation, whether one-one or many-one,
which correlates with a class `u` only terms belonging to that class, then
this relation is said to constitute a representation of `u` in itself (36),
and with respect to this relation `u` is called a chain (37). That is to
say, any class `u` is, with respect to any many-one relation, a chain, if `u` is
contained in the domain of the relation, and the correlate of a `u` is
always itself a `u`. The collection of correlates of a class is called the
image (*Bild*) of the class. Thus a chain is a class whose image is
part or the whole of itself. For the benefit of the non-mathematical
reader, it may be not superfluous to remark that a chain with regard to
a one-one relation, provided it has any term not belonging to the image
of the chain, cannot be finite, for such a chain must contain the same
number of terms as a proper part of itself†.

**237.** (3) If `a` be any term or collection of terms, there may be,
with respect to a given many-one relation, many chains in which `a` is
contained. The common part of all these chains, which is denoted by `a`_{0},
is what Dedekind calls the *chain of* `a` (44). For example, if `a` be the
number `n`, or any set of numbers of which `n` is the least, the chain of `a`
with regard to the relation “less by 1” will be all numbers not less
than `n`.

**238.** (4) Dedekind now proceeds (59) to a theorem which is
a generalized form of mathematical induction. This theorem is as
follows: Let `a` be any term or set of terms contained in a class `s`, and let
the image of the common part of `s` and the chain of `a` be also contained
in `s`; then it follows that the chain of `a` is contained in `s`. This somewhat complicated theorem may become clearer by being put in other
language. Let us call the relation by which the chain is generated (or
rather the converse of this relation) succession, so that the correlate or
image of a term will be its successor. Let `a` be a term which has a
successor, or a collection of such terms. A chain in general (with regard
to succession) will be any set of terms such that the successor of any
one of them also belongs to the set. The chain of `a` will be the common
part of all the chains containing `a`. Then the data of the theorem
inform us that `a` is contained in `s`, and, if any term of the chain of `a` be
an `s`, so is its successor; and the conclusion is, that every term in the
chain of `a` is an `s`. This theorem, as is evident, is very similar to
mathematical induction, from which it differs, first by the fact that `a`
need not be a single term, secondly by the fact that the constitutive
relation need not be one-one, but may be many-one. It is a most
remarkable fact that Dedekind’s previous assumptions suffice to demonstrate this theorem.

**239.** (5) I come next to the definition of a singly infinite system
or class (71). This is defined as a class which can be represented in
itself by means of a one-one relation, and which is further such as to be
the chain, with regard to this one-one relation, of a single term of the
class not contained in the image of the class. Calling the class `N`, and
the one-one relation `R`, there are, as Dedekind remarks, four points in
this definition. (1) The image of `N` is contained in `N`; that is, every
term to which an `N` has the relation `R` is an `N`. (2) `N` is the chain of
one of its terms. (3) This one term is such that no `N` has the relation
`R` to it, *i.e.* it is not the image of any other term of `N`. (4) The
relation `R` is one-one, in other words, the representation is similar. The
abstract system, defined simply as possessing these properties, is defined
by Dedekind as the ordinal numbers (73). It is evident that his singly
infinite system is the same as what we called a *progression*, and he
proceeds to deduce the various properties of progressions, in particular
mathematical induction (80), which follows from the above generalized
form. One number `m` is said to be less than another `n`, when the chain
of `n` is contained in the image of the chain of `m` (89); and it is shown
(88, 90) that of two different numbers, one must be the less. From this
point everything proceeds simply.

**240.** The only further point that seems important for our present
purpose is the definition of cardinals. It is shown (132) that all singly
infinite systems are similar to each other and to the ordinals, and that
conversely (133) any system which is similar to a singly infinite system
is singly infinite. When a system is finite, it is similar to some system
`Z`` _{n}`, where

**241.** Of the merits of the above deduction it is not necessary for
me to speak, for they are universally acknowledged. But some points
call for discussion. In the first case, Dedekind *proves* mathematical
induction, while Peano regards it as an axiom. This gives Dedekind
an apparent superiority, which must be examined. In the second place,
there is no reason, merely because the numbers which Dedekind obtains
*have* an order, to hold that they *are* ordinal numbers; in the third
place, his definition of cardinals is unnecessarily complicated, and the
dependence of cardinals upon order is only apparent. I shall take these
points in turn.

As regards the proof of mathematical induction, it is to be observed
that it makes the practically equivalent assumption that numbers form
the chain of one of them. Either can be deduced from the other, and
the choice as to which is to be an axiom, which a theorem, is mainly
a matter of taste. On the whole, though the consideration of chains
is most ingenious, it is somewhat difficult, and has the disadvantage
that theorems concerning the finite class of numbers not greater than `n`
as a rule have to be deduced from corresponding theorems concerning
the infinite class of numbers greater than `n`. For these reasons, and
not because of any logical superiority, it seems simpler to begin with
mathematical induction. And it should be observed that, in Peano’s
method, it is only when theorems are to be proved concerning *any*
number that mathematical induction is required. The elementary
Arithmetic of our childhood, which discusses only particular numbers,
is wholly independent of mathematical induction; though to prove that
this is so for *every* particular number would itself require mathematical
induction. In Dedekind’s method, on the other hand, propositions
concerning particular numbers, like general propositions, demand the
consideration of chains. Thus there is, in Peano’s method, a distinct
advantage of simplicity, and a clearer separation between the particular
and the general propositions of Arithmetic. But from a purely logical
point of view, the two methods seem equally sound; and it is to be
remembered that, with the logical theory of cardinals, both Peano’s and
Dedekind’s axioms become demonstrable*.

**242.** On the second point, there is some deficiency of clearness in
what Dedekind says. His words are (73): “If in the contemplation
of a singly infinite system `N`, ordered by a representation `φ`, we disregard
entirely the peculiar nature of the elements, retaining only the possibility
of distinguishing them, and considering only the relations in which they
are placed by the ordering representation `φ`, then these elements are
called *natural numbers* or *ordinal numbers* or simply *numbers*.” Now
it is impossible that this account should be quite correct. For it implies
that the terms of all progressions other than the ordinals are complex,
and that the ordinals are elements in all such terms, obtainable by
abstraction. But this is plainly not the case. A progression can be
formed of points or instants, or of transfinite ordinals, or of cardinals,
in which, as we shall shortly see, the ordinals are not elements. Moreover it is impossible that the ordinals should be, as Dedekind suggests,
nothing but the terms of such relations as constitute a progression.
If they are to be anything at all, they must be intrinsically something;
they must differ from other entities as points from instants, or colours
from sounds. What Dedekind intended to indicate was probably a
definition by means of the principle of abstraction, such as we attempted
to give in the preceding chapter. But a definition so made always
indicates some class of entities having (or being) a genuine nature of
their own, and not logically dependent upon the manner in which they
have been defined. The entities defined should be visible, at least to
the mind’s eye; what the principle asserts is that, under certain conditions, there are such entities, if only we knew where to look for them.
But whether, when we have found them, they will be ordinals or
cardinals, or even something quite different, is not to be decided
off-hand. And in any case, Dedekind does not show us what it is
that all progressions have in common, nor give any reason for supposing
it to be the ordinal numbers, except that all progressions obey the same
laws as ordinals do, which would prove equally that *any* assigned
progression is what all progressions have in common.

**243.** This brings us to the third point, namely the definition of
cardinals by means of ordinals. Dedekind remarks in his preface (p. ix)
that many will not recognize their old friends the natural numbers in
the shadowy shapes which he introduces to them. In this, it seems
to me, the supposed persons are in the right—in other words, I am one
among them. What Dedekind presents to us is not the numbers,
but any progression: what he says is true of all progressions alike,
and his demonstrations nowhere—not even where he comes to cardinals—involve any property distinguishing numbers from other progressions.
No evidence is brought forward to show that numbers are prior to
other progressions. We are told, indeed, that they are what all progressions have in common; but no reason is given for thinking that
progressions have anything in common beyond the properties assigned
in the definition, which do not themselves constitute a new progression.
The fact is that all depends upon one-one relations, which Dedekind
has been using throughout without perceiving that they alone suffice
for the definition of cardinals. The relation of similarity between
classes, which he employs consciously, combined with the principle of
abstraction, which he implicitly assumes, suffice for the definition of
cardinals; for the definition of ordinals these do not suffice; we
require, as we saw in the preceding chapter, the relation of likeness
between well-ordered serial relations. The definition of particular
finite ordinals is effected explicitly in terms of the corresponding
cardinals: if `n` be a finite cardinal number, the ordinal number `n` is
the class of serial relations which have `n` terms in their domain
(or in their field, if we prefer this definition). In order to define
the notion of “`n`th,” we need, besides the ordinal number `n`, the
notion of powers of a relation, *i.e.* of the relative product of a relation multiplied into itself a finite number of times. Thus if `R` be any
one-one serial relation, generating a finite series or a progression, the first
term of the field of `R` (which field we will call `r`) is the term belonging
to the domain, but not to the converse domain, *i.e.*, having the relation
`R` but not the relation ˘`R`. If `r` has `n` or more terms, where `n` is a finite
number, the `n`th term of `r` is the term to which the first term has the
relation `R`^{n−1} or, again, it is the term having the relation ˘`R`^{n−1} but not
the relation ˘`R`^{n}. Through the notion of powers of a relation, the
introduction of cardinals is here unavoidable; and as powers are defined
by mathematical induction, the notion of `n`th, according to the above
definition, cannot be extended beyond finite numbers. We can however
extend the notion by the following definition: If `P` be a transitive
aliorelative generating a well-ordered series `p`, the `n`th term of `p` is the
term `x` such that, if `P`′ be the relation `P` limited to `x` and its predecessors, then `P`′ has the ordinal number `n`. Here the dependence
upon cardinals results from the fact that the ordinal `n` can, in general,
only be defined by means of the cardinal `n`.

It is important to observe that no set of terms has inherently one
order rather than another, and that no term is the `n`th of a set except
in relation to a particular generating relation whose field is the set or
part of the set. For example, since in any progression, any finite
number of consecutive terms including the first may be taken away,
and the remainder will still form a progression, the ordinal number
of a term in a progression may be diminished to any smaller number
we choose. Thus the ordinal number of a term is relative to the series
to which it belongs. This may be reduced to a relation to the first
term of the series; and lest a vicious circle should be suspected, it may
be explained that the *first* term can always be defined non-numerically.
It is, in Dedekind’s singly infinite system, the only term not contained
in the image of the system; and generally, in any series, it is the only
term which has the constitutive relation with one sense, but not with
the other*. Thus the relation expressed by `n`th is not only a relation
to `n`, but also to the first term of the series; and *first* itself depends
upon the terms included in the series, and upon the relation by which
they are ordered, so that what was first may cease to be so, and what
was not first may become so. Thus the first term of a series must be
assigned, as is done in Dedekind’s view of a progression as the chain
of its first term. Hence `n`th expresses a four-cornered relation, between
the term which is `n`th, an assigned term (the first), a generating serial
relation, and the cardinal number `n`. Thus it is plain that ordinals,
either as classes of like serial relations, or as notions like “`n`th,” are
more complex than cardinals; that the logical theory of cardinals is
wholly independent of the general theory of progressions, requiring
independent development in order to show that the cardinals form a
progression; and that Dedekind’s ordinals are not essentially either
ordinals or cardinals, but the members of any progression whatever.
I have dwelt on this point, as it is important, and my opinion is at
variance with that of most of the best authorities. If Dedekind’s view
were correct, it would have been a logical error to begin, as this work
does, with the theory of cardinal numbers rather than with order.
For my part, I do not hold it an absolute error to begin with order,
since the properties of progressions, and even most of the properties of
series in general, seem to be largely independent of number. But
the properties of number must be capable of proof without appeal to
the general properties of progressions, since cardinal numbers can be
independently defined, and must be seen to form a progression before
theorems concerning progressions can be applied to them. Hence the
question, whether to begin with order or with numbers, resolves itself
into one of convenience and simplicity; and from this point of view,
the cardinal numbers seem naturally to precede the very difficult considerations as to series which have occupied us in the present Part.

Notes

*^{[page 245]} 2nd ed. Brunswick, 1893 (1st ed. 1887). The principal contents of this book, expressed by the Algebra of Relations, will be found in my article in *RdM*, VII, 2, 3.

†^{[page 245]} The numbers in brackets refer, not to pages, but to the small sections into which the work is divided.

*^{[page 246]} A many-one relation is one in which, as in the relation of a quantity to its magnitude, the right-hand term, *to* which the relation is, is uniquely determined when the left-hand term is given. Whether the converse holds is left undecided. Thus a one-one relation is a particular case of a many-one relation.

†^{[page 246]} A *proper part* (Echter Theil) is a phrase analogous to “proper fraction”; it means a part not the whole.

*^{[page 248]} Cf. Chap. xiii.

*^{[page 250]} Though when the series has two ends, we have to make an arbitrary selection as to which we will call first, which last. The obviously non-numerical nature of *last* illustrates that of its correlative, *first*.

Distance.

**244.** The notion of distance is one which is often supposed essential
to series*, but which seldom receives precise definition. An emphasis on
distance characterizes, generally speaking, those who believe in relative
position. Thus Leibniz, in the course of his controversy with Clarke,
remarks:

“As for the objection, that space and time are quantities, or rather things endowed with quantity, and that situation and order are not so: I answer, that order also has its quantity; there is that in it which goes before, and that which follows; there is distance or interval. Relative things have their quantity, as well as absolute ones. For instance, ratios or proportions in mathematics have their quantity, and are measured by logarithms; and yet they are relations. And therefore, though time and space consist in relations, yet they have their quantity†.”

In this passage, the remark: “There is that which goes before, and
that which follows; there is distance or interval,” if considered as an
inference, is a *non sequitur*; the mere fact of order does *not* prove that
there is distance or interval. It proves, as we have seen, that there
are stretches, that these are capable of a special form of addition
closely analogous to what I have called relational addition, that they
have sign, and that (theoretically at least) stretches which fulfil the
axioms of Archimedes and of linearity are always capable of numerical
measurement. But the idea, as Meinong rightly points out, is entirely
distinct from that of stretch. Whether any particular series does or
does not contain distances, will be, in most compact series (*i.e.* such as
have a term between any two), a question not to be decided by argument.
In discrete series there must be distance; in others, there may be—unless, indeed, they are series obtained from progressions as the
rationals or the real numbers are obtained from the integers, in which
case there must be distance. But we shall find that stretches are mathematically sufficient, and that distances are complicated and unimportant.

**245.** The definition of distance, to begin with, is no easy matter.
What has been done hitherto towards this end is chiefly due to non-Euclidean Geometry*; something also has been done towards settling the
definition by Meinong†. But in both these cases, there is more concern for
numerical measurement of distance than for its actual definition. Nevertheless, distance is by no means indefinable. Let us endeavour to generalize the notion as much as possible. In the first place, distance need not
be asymmetrical; but the other properties of distance always allow us to
render it so, and we may therefore take it to be so. Secondly, a distance
need not be a quantity or a magnitude; although it is usually taken to
be such, we shall find the taking it so to be irrelevant to its other
properties, and in particular to its numerical measurement. Thirdly,
when distance is taken asymmetrically, there must be only one term to
which a given term has a given distance, and the converse relation to the
given distance must be a distance of the same kind. (It will be observed
that we must first define a *kind* of distance, and proceed thence to the
general definition of distance.) Thus every distance is a one-one
relation; and in respect to such relations it is convenient to respect the
converse of a relation as its −1th power. Further the relative product
of two distances of a kind must be a distance of the same kind. When
the two distances are mutually converse, their product will be identity,
which is thus one among distances (their zero, in fact), and must be the
only one which is not asymmetrical. Again the product of two distances
of a kind must be commutative‡. If the distances of a kind be magnitudes, they must form a kind of magnitude—*i.e.* any two must be equal
or unequal. If they are not magnitudes, they must still form a series
generated in the second of our six ways, *i.e.* every pair of different
distances must have a certain asymmetrical relation, the same for all
pairs except as regards sense. And finally, if `Q` be this relation, and
`R`_{1}`Q``R`_{2} (`R`_{1}, `R`_{2}, being distances of the kind), then if `R`_{3} be any other
distance of the kind, we must have `R`_{1}`R`_{3}`Q``R`_{2}`R`_{3}. All these properties,
so far as I can discover, are independent; and we ought to add a
property of the field, namely this: any two terms, each of which belongs
to the field of some distance of the kind (not necessarily the same for
both), have a relation which is a distance of the kind. Having now
defined a kind of distance, a distance is any relation belonging to some
kind of distance; and thus the work of definition seems completed.

The notion of distance, it will be seen, is enormously complex. The
properties of distances are analogous to those of stretches with sign, but
are far less capable of mutual deduction. The properties of stretches
corresponding to many of the above properties of distances are capable
of proof. The difference is largely due to the fact that stretches can be
added in the elementary logical (not arithmetical) way, whereas distances
require what I have called *relational* addition, which is much the same as
relative multiplication.

**246.** The numerical measurement of distances has already been partially explained in Part III. It requires, as we saw, for its full application,
two further postulates, which, however, do not belong to the definition of
distances, but to certain kinds of distances only. These are, the postulate of Archimedes: given any two distances of a kind, there exists
a finite integer `n` such that the `n`th power of the first distance is greater
than the second distance; and Du Bois Reymond’s postulate of linearity:
Any distance has an `n`th root, where `n` is any integer (or any prime,
whence the result follows for any integer). When these two postulates
are satisfied, we can find a meaning for `R`^{x}, where `R` is a distance of the
kind other than identity, and `x` is any real number*. Moreover, any distance of the kind is of the form `R`^{x}, for some value of `x`. And `x` is, of course, the numerical measure of the distance.

In the case of series generated in the first of our six ways, the various
powers of the generating relation `R` give the distances of terms. These
various powers, as the reader can see for himself, verify all the above
characteristics of distances. In the case of series generated from progressions as rationals or real numbers from integers, there are always
distances; thus in the case of the rationals themselves, which are one-one relations, their differences, which are again rationals, measure or
indicate relations between them, and these relations are of the nature of
distances. And we shall see, in Part V, that these distances have some
importance in connection with limits. For numerical measurement in
some form is essential to certain theorems about limits, and the numerical measurement of distances is apt to be more practically feasible than
that of stretches.

**247.** On the general question, however, whether series unconnected
with number—for instance spatial and temporal series—are such as to
contain distances, it is difficult to speak positively. Some things may
be said against this view. In the first place, there must be stretches, and
these must be magnitudes. It then becomes a sheer assumption—which
must be set up as an axiom—that equal stretches correspond to equal
distances. This may, of course, be denied, and we might even seek an
interpretation of non-Euclidean Geometry in the denial. We might
regard the usual coordinates as expressing stretches, and the logarithms
of their anharmonic ratios as expressing distances; hyperbolic Geometry,
at least, might thus find a somewhat curious interpretation. Herr
Meinong, who regards all series as containing distances, maintains an
analogous principle with regard to distance and stretch in general. The
distance, he thinks, increases only as the logarithm of the stretch. It
may be observed that, where the distance itself is a rational number
(which is possible, since rationals are one-one relations), the opposite
theory can be made formally convenient by the following fact. The
square of a distance, as we saw generally, is said to be twice as great as
the distance whose square it is. We might, where the distance is a
rational, say instead that the *stretch* is twice as great, but that the
*distance* is truly the square of the former distance. For where the
distance is already numerical, the usual interpretation of numerical
measurement conflicts with the notation `R`^{2}. Thus we shall be compelled to regard the stretch as proportional to the logarithm of the
distance. But since, outside the theory of progressions, it is usually
doubtful whether there are distances, and since, in almost all other
series, stretches seem adequate for all the results that are obtainable, the
retention of distance adds a complication for which, as a rule, no
necessity appears. It is therefore generally better, at least in a philosophy of mathematics, to eschew distances except in the theory of
progressions, and to measure them, in that theory, merely by the
indices of the powers of the generating relation. There is no logical
reason, so far as I know, to suppose that there are distances elsewhere,
except in a finite space of two dimensions and in a projective space; and
if there are, they are not mathematically important. We shall see in
Part VI how the theory of space and time may be developed without presupposing distance; the distances which appear in projective Geometry are
derivative relations, not required in defining the properties of our space;
and in Part V we shall see how few are the functions of distance with
regard to series in general. And as against distance it may be remarked
that, if every series must contain distances, an endless regress becomes
unavoidable, since every kind of distance is itself a series. This is not,
I think, a logical objection, since the regress is of the logically permissible kind; but it shows that great complications are introduced by
regarding distances as essential to every series. On the whole, then, it
seems doubtful whether distances in general exist; and if they do, their
existence seems unimportant and a source of very great complications.

**248.** We have now completed our review of order, in so far as is
possible without introducing the difficulties of continuity and infinity.
We have seen that all order involves asymmetrical transitive relations, and
that every series as such is open. But closed series, we found, could be
distinguished by the mode of their generation, and by the fact that,
though they always have a first term, this term may always be selected
arbitrarily. We saw that asymmetrical relations must be sometimes
unanalyzable, and that when analyzable, other asymmetrical relations
must appear in the analysis. The difference of sign, we found, depends
always upon the difference between an asymmetrical relation and its
converse. In discussing the particular type of series which we called
progressions, we saw how all Arithmetic applies to every such series, and
how finite ordinals may be defined by means of them. But though we
found this theory to be to a certain extent independent of the cardinals,
we saw no reason to agree with Dedekind in regarding cardinals as
logically subsequent to ordinals. Finally, we agreed that distance is
a notion which is not essential to series, and of little importance outside
Arithmetic. With this equipment, we shall be able, I hope, to dispose
of all the difficulties which philosophers have usually found in infinity
and continuity. If this can be accomplished, one of the greatest of
philosophical problems will have been solved. To this problem Part V is to be devoted.

Notes

*^{[page 252]} *E.g.* by Meinong, *op. cit.* §17.

†^{[page 252]} *Phil. Werke*, Gerhardt’s ed. Vol. vii, p. 404.

*^{[page 253]} See *e.g.* Whitehead, *Universal Algebra*, Cambridge, 1898, Book vi, Chap. i.

†^{[page 253]} *Op. cit.* Section iv.

‡^{[page 253]} This is an independent property; consider for instance the difference between “maternal grandfather” and “paternal grandmother.”

*^{[page 254]} The powers of distances are here understood in the sense resulting from relative multiplication; thus if `a` and `b` have the same distance as `b` and `c`, this distance is the square root of the distance of `a` and `c`. The postulate of linearity, whose expression in ordinary language is: “every linear quantity can be divided into `n` equal parts, where `n` is any integer,” will he found in Du Bois Reymond’s *Allgemeine Functionentheorie* (Tübingen, 1882), p. 46.

INFINITY AND CONTINUITY.

The Correlation of Series.

**249.** We come now to what has been generally considered the
fundamental problem of mathematical philosophy—I mean, the problem
of infinity and continuity. This problem has undergone, through the
labours of Weierstrass and Cantor, a complete transformation. Since
the time of Newton and Leibniz, the nature of infinity and continuity
had been sought in discussions of the so-called Infinitesimal Calculus.
But it has been shown that this Calculus is not, as a matter of fact,
in any way concerned with the infinitesimal, and that a large and most
important branch of mathematics is logically prior to it. The problem
of continuity, moreover, has been to a great extent separated from that
of infinity. It was formerly supposed—and herein lay the real strength
of Kant’s mathematical philosophy—that continuity had an essential
reference to space and time, and that the Calculus (as the word *fluxion*
suggests) in some way presupposed motion or at least change. In this
view, the philosophy of space and time was prior to that of continuity,
the Transcendental Aesthetic preceded the Transcendental Dialectic, and
the antinomies (at least the mathematical ones) were essentially spatio-temporal. All this has been changed by modern mathematics. What
is called the arithmetization of mathematics has shown that all the
problems presented, in this respect, by space and time, are already
present in pure arithmetic. The theory of infinity has two forms,
cardinal and ordinal, of which the former springs from the logical
theory of number; the theory of continuity is purely ordinal. In the
theory of continuity and the ordinal theory of infinity, the problems
that arise are not specially concerned with numbers, but with all series
of certain types which occur in arithmetic and geometry alike. What
makes the problems in question peculiarly easy to deal with in the case
of numbers is, that the series of rationals, which is what I shall call a
*compact* series, arises from a progression, namely that of the integers, and
that this fact enables us to give a proper name to every term of the
series of rationals—a point in which this series differs from others of the
same type. But theorems of the kind which will occupy us in most of
the following chapters, though obtained in arithmetic, have a far wider
application, since they are purely ordinal, and involve none of the
logical properties of numbers. That is to say, the idea which the
Germans call *Anzahl*, the idea of the number of terms in some class,
is irrelevant, save only in the theory of transfinite cardinals—an
important but very distinct part of Cantor’s contributions to the theory
of infinity. We shall find it possible to give a general definition of
continuity, in which no appeal is made to the mass of unanalyzed
prejudice which Kantians call “intuition”; and in Part VI we shall
find that no other continuity is involved in space and time. And we
shall find that, by a strict adherence to the doctrine of limits, it is
possible to dispense entirely with the infinitesimal, even in the definition
of continuity and the foundations of the Calculus.

**250.** It is a singular fact that, in proportion as the infinitesimal
has been extruded from mathematics, the infinite has been allowed
a freer development. From Cantor’s work it appears that there are
two respects in which infinite numbers differ from those that are finite.
The first, which applies to both cardinals and ordinals, is, that they do
not obey mathematical induction—or rather, they do not form part of
a series of numbers beginning with 1 or 0, proceeding in order of
magnitude, containing all numbers intermediate in magnitude between
any two of its terms, and obeying mathematical induction. The
second, which applies only to cardinals, is, that a whole of an infinite
number of terms always contains a part consisting of the same
number of terms. The first respect constitutes the true definition
of an infinite series, or rather of what we may call an infinite
term in a series: it gives the essence of the ordinal infinite. The
second gives the definition of an infinite collection, and will doubtless
be pronounced by the philosopher to be plainly self-contradictory. But
if he will condescend to attempt to exhibit the contradiction, he will
find that it can only be proved by admitting mathematical induction,
so that he has merely established a connection with the ordinal infinite.
Thus he will be compelled to maintain that the denial of mathematical
induction is self-contradictory; and as he has probably reflected little,
if at all, on this subject, he will do well to examine the matter before
pronouncing judgment. And when it is admitted that mathematical
induction may be denied without contradiction, the supposed antinomies
of infinity and continuity one and all disappear. This I shall endeavour
to prove in detail in the following chapters.

**251.** Throughout this Part we shall often have occasion for a
notion which has hitherto been scarcely mentioned, namely the correlation of series. In the preceding Part we examined the nature of
isolated series, but we scarcely considered the relations between different
series. These relations, however, are of an importance which philosophers have wholly overlooked, and mathematicians have but lately
realized. It has long been known how much could be done in Geometry
by means of homography, which is an example of correlation; and it
has been shown by Cantor how important it is to know whether a series
is denumerable, and how similar two series capable of correlation are.
But it is not usually pointed out that a dependent variable and its
independent variable are, in most mathematical cases, merely correlated
series, nor has the general idea of correlation been adequately dealt
with. In the present work only the philosophical aspects of the subject
are relevant.

Two *series* `s`, `s`′ are said to be correlated when there is a one-one
relation `R` coupling every term of `s` with a term of `s`′, and *vice versâ*, and
when, if `x`, `y` be terms of `s`, and `x` precedes `y`, then their correlates `x`′, `y`′ in
`s`′ are such that `x`′ precedes `y`′. Two *classes* or collections are correlated
whenever there is a one-one relation between the terms of the one and
the terms of the other, none being left over. Thus two series may be
correlated as classes without being correlated as series; for correlation
as classes involves only the same cardinal number, whereas correlation
as series involves also the same ordinal type—a distinction whose
importance will be explained hereafter. In order to distinguish these
cases, it will be well to speak of the correlation of classes as correlation
simply, and of the correlation of series as ordinal correlation. Thus
whenever correlation is mentioned without an adjective, it is to be
understood as being not necessarily ordinal. Correlated classes will be
called *similar*; correlated series will be called *ordinally similar*; and
their generating relations will be said to have the relation of
*likeness*.

Correlation is a method by which, when one series is given, others
may be generated. If there be any series whose generating relation
is `P`, and any one-one relation which holds between any term `x` of the
series and some term which we may call `x`_{R}, then the class of terms
`x`_{R}, will form a series of the same type as the class of terms `x`. For
suppose `y` to be any other term of our original series, and assume `x``P``y`.
Then we have `x`_{R}˘`R``x`, `x``P``y`, and `y``R``y`_{R}. Hence `x`_{R}˘`R``P``R``y`_{R}. Now it may
be shown* that, if `P` be transitive and asymmetrical, so is ˘`R``P``R`; hence
the correlates of terms of the `P`-series form a series whose generating
relation is ˘`R``P``R`. Between these two series there is ordinal correlation,
and the series have complete ordinal similarity. In this way a new
series, similar to the original one, is generated by any one-one relation
whose field includes the original series. It can also be shown that,
conversely, if `P`, `P`′ be the generating relations of two similar series,
there is a one-one relation `R`, whose domain is the field of `P`, which
is such that `P`′ = ˘`R``P``R`.

**252.** We can now understand a distinction of great importance,
namely that between self-sufficient or independent series, and series by
correlation. In the case just explained there is perfect mathematical
symmetry between the original series and the series by correlation; for, if
we denote by `Q` the relation ˘`R``P``R`, we shall find `P` = `R``Q`˘`R`. Thus we may
take either the `Q`-series or the `P`-series as the original, and regard the
other as derivative. But if it should happen that `R`, instead of being
one-one, is many-one, the terms of the field of `Q`, which we will call `q`,
will have an order in which there is repetition, the same term occurring
in different positions corresponding to its different correlates in the field
of `P`, which we will call `p`. This is the ordinary case of mathematical
functions which are not linear. It is owing to preoccupation with such
series that most mathematicians fail to realize the impossibility, in an
independent series, of any recurrence of the same term. In every
sentence of print, for example, the letters acquire an order by correlation
with the points of space, and the same letter will be repeated in different
positions. Here the series of letters is essentially derivative, for we
cannot order the points of space by relation to the letters: this would
give us several points in the same position, instead of one letter in several
positions. In fact, if `P` be a serial relation, and `R` be a many-one relation
whose domain is the field of `P`, and `Q` = ˘`R``P``R`, then `Q` has all the characteristics of a serial relation except that of implying diversity; but `R``Q`˘`R` is
not equivalent to `P`, and thus there is a lack of symmetry. It is for
this reason that inverse functions in mathematics, such as sin^{−1} `x`, are
genuinely distinct from direct functions, and require some device or
convention before they become unambiguous. Series obtained from
a many-one correlation as `q` was obtained above will be called series
by correlation. They are not genuine series, and it is highly important
to eliminate them from discussions of fundamental points.

**253.** The notion of *likeness* corresponds, among relations, to similarity
among classes. It is defined as follows: Two relations `P`, `Q` are like
when there is a one-one relation `S` such that the domain of `S` is the field
of `P`, and `Q` = ˘`S``P``S`. This notion is not confined to serial relations, but
may be extended to all relations. We may define the *relation-number*
of a relation `P` as the class of all relations that are like `P`; and we can
proceed to a very general subject which may be called relation-arithmetic.
Concerning relation-numbers we can prove those of the formal laws of
addition and multiplication that hold for transfinite ordinals, and thus
obtain an extension of a part of ordinal arithmetic to relations in
general. By means of likeness we can define a finite relation as one
which is not like any proper part of itself—a proper part of a relation
being a relation which implies it but is not equivalent to it. In this
way we can completely emancipate ourselves from cardinal arithmetic.
Moreover the properties of likeness are in themselves interesting and
important. One curious property is that, if `S` be one-one and have the
field of `P` for its domain, the above equation `Q` = ˘`S``P``S` is equivalent to
`S``Q` = `P``S` or to `Q`˘`S` = ˘`S``P`*.

**254.** Since the correlation of series constitutes most of the mathematical examples of functions, and since function is a notion which is
not often clearly explained, it will be well at this point to say something
concerning the nature of this notion. In its most general form, functionality does not differ from relation. For the present purpose it will be
well to recall two technical terms, which were defined in Part I. If `x`
has a certain relation to `y`, I shall call `x` the *referent*, and `y` the *relatum*,
with regard to the relation in question. If now `x` be defined as belonging
to some class contained in the domain of the relation, then the relation
defines `y` as a function of `x`. That is to say, an independent variable
is constituted by a collection of terms, each of which can be referent
in regard to a certain relation. Then each of these terms has one or
more relata, and any one of these is a certain function of its referent,
the function being defined by the relation. Thus *father* defines a
function, provided the independent variable be a class contained in that
of male animals who have or will have propagated their kind; and
if `A` be the father of `B`, `B` is said to be a function of `A`. What is
essential is an independent variable, *i.e.* any term of some class, and
a relation whose extension includes the variable. Then the referent
is the independent variable, and its function is any one of the corresponding relata.

But this most general idea of a function is of little use in mathematics.
There are two principal ways of particularizing the function: first, we
may confine the relations to be considered to such as are one-one or
many-one, *i.e.* such as give to every referent a unique relatum; secondly,
we may confine the independent variable to series. The second particularization is very important, and is specially relevant to our present
topics. But as it almost wholly excludes functions from Symbolic
Logic, where series have little importance, we may as well postpone it for
a moment while we consider the first particularization alone.

The idea of function is so important, and has been so often considered with exclusive reference to numbers, that it is well to fill our
minds with instances of non-numerical functions. Thus a very important
class of functions are propositions containing a variable† . Let there be
some proposition in which the phrase “any `a`” occurs, where `a` is some
class. Then in place of “any `a`” we may put `x`, where `x` is an undefined
member of the class `a`—in other words, any `a`. The proposition then
becomes a function of `x`, which is unique when `x` is given. This proposition will, in general, be true for some values of `x` and false for others.
The values for which the function is true form what might be called,
by analogy with Analytic Geometry, a logical curve. This general
view may, in fact, be made to include that of Analytic Geometry. The
equation of a plane curve, for example, is a propositional function which
is a function of two variables `x` and `y`, and the curve is the assemblage of
points which give to the variables values that make the proposition true.
A proposition containing the word *any* is the assertion that a certain
propositional function is true for all values of the variable for which it is
significant. Thus “any man is mortal” asserts that “`x` is a man implies
`x` is a mortal” is true for all values of `x` for which it is significant, which
may be called the admissible values. Propositional functions, such as
“`x` is a number,” have the peculiarity that they look like propositions,
and *seem* capable of implying other propositional functions, while yet
they are neither true nor false. The fact is, they are propositions for all
admissible values of the variable, but not while the variable remains a
variable, whose value is not assigned; and although they may, for every
admissible value of the variable, imply the corresponding value of some
other propositional function, yet while the variable remains as a variable
they can imply nothing. The question concerning the nature of a
propositional function as opposed to a proposition, and generally of a
function as opposed to its values, is a difficult one, which can only be
solved by an analysis of the nature of the variable. It is important,
however, to observe that propositional functions, as was shown in
Chapter vii, are more fundamental than other functions, or even than
relations. For most purposes, it is convenient to identify the function
and the relation, *i.e.*, if `y` = `f`(`x`) is equivalent to `x``R``y`, where `R` is a
relation, it is convenient to speak of `R` as the function, and this will be
done in what follows; the reader, however, should remember that the
idea of functionality is more fundamental than that of relation. But
the investigation of these points has been already undertaken in Part I,
and enough has been said to illustrate how a proposition may be a
function of a variable.

Other instances of non-numerical functions are afforded by dictionaries. The French for a word is a function of the English, and *vice versâ*, and both are functions of the term which both designate. The
press-mark of a book in a library catalogue is a function of the book,
and a number in a cipher is a function of the word for which it stands.
In all these cases there is a relation by which the relatum becomes unique
(or, in the case of languages, generally unique) when the referent
is given; but the terms of the independent variable do not form a
series, except in the purely external order resulting from the alphabet.

**255.** Let us now introduce the second specification, that our
independent variable is to be a series. The dependent variable is then
a series by correlation, and may be also an independent series. For
example, the positions occupied by a material point at a series of instants
form a series by correlation with the instants, of which they are a
function; but in virtue of the continuity of motion, they also form,
as a rule, a geometrical series independent of all reference to time.
Thus motion affords an admirable example of the correlation of series.
At the same time it illustrates a most important mark by which, when it
is present, we can tell that a series is not independent. When the
time is known, the position of a material particle is uniquely determined;
but when the position is given, there may be several moments, or even an
infinite number of them, corresponding to the given position. (There
will be an infinite number of such moments if, as is commonly said, the
particle has been at rest in the position in question. *Rest* is a loose and
ambiguous expression, but I defer its consideration to Part VII.) Thus
the relation of the time to the position is not strictly one-one, but may
be many-one. This was a case considered in our general account of
correlation, as giving rise to dependent series. We inferred, it will
be remembered, that two correlated independent series are mathematically on the same level, because if `P`, `Q` be their generating relations, and
`R` the correlating relation, we infer `P` = `R``Q`˘`R` from `Q` = ˘`R``P``R`. But
this inference fails as soon as `R` is not strictly one-one, since then we no
longer have `R`˘`R` contained in 1’, where 1’ means identity. For example,
my father’s son need not be myself, though my son’s father must be.
This illustrates the fact that, if `R` be a many-one relation, `R`˘`R` and ˘`R``R`
must be carefully distinguished: the latter is contained in identity, but
not the former. Hence whenever `R` is a many-one relation, it may be
used to form a series by correlation, but the series so formed cannot be
independent. This is an important point, which is absolutely fatal to
the relational theory of time*. For the present let us return to the
case of motion. When a particle describes a closed curve, or one
which has double points, or when the particle is sometimes at rest
during a finite time, then the series of points which it occupies
is essentially a series by correlation, not an independent series. But,
as I remarked above, a curve is not only obtainable by motion,
but is also a purely geometrical figure, which can be defined without
reference to any supposed material point. When, however, a curve is
so defined, it must not contain points of rest: the path of a material
point which sometimes moves, but is sometimes at rest for a finite time,
is different when considered kinematically and when considered geometrically; for geometrically the point in which there is rest is one, whereas
kinematically it corresponds to many terms in the series.

The above discussion of motion illustrates, in a non-numerical
instance, a case which normally occurs among the functions of pure
mathematics. These functions (when they are functions of a real
variable) usually fulfil the following conditions: Both the independent
and the dependent variable are classes of numbers, and the defining
relation of the function is many-one*. This case covers rational
functions, circular and elliptic functions of a real variable, and the
great majority of the direct functions of pure mathematics. In all such
cases, the independent variable is a series of numbers, which may be
restricted in any way we please—to positive numbers, rationals, integers,
primes, or any other class. The dependent variable consists also of
numbers, but the order of these numbers is determined by their relation
to the corresponding term of the independent variable, not by that of
the numbers forming the dependent variable themselves. In a large
class of functions the two orders happen to coincide; in others, again,
where there are maxima and minima at finite intervals, the two orders
coincide throughout a finite stretch, then they become exactly opposite
throughout another finite stretch, and so on. If `x` be the independent
variable, `y` the dependent variable, and the constitutive relation be
many-one, the same number `y` will, in general, be a function of, *i.e.*
correspond to, several numbers `x`. Hence the `y`-series is essentially by
correlation, and cannot be taken as an independent series. If, then, we
wish to consider the inverse function, which is defined by the converse
relation, we need certain devices if we are still to have correlation of
series. One of these, which seems the most important, consists in
dividing the values of `x` corresponding to the same value of `y` into
classes, so that (what may happen) we can distinguish (say) `n` different
`x`’s, each of which has a distinct one-one relation to `y`, and is therefore
simply reversible. This is the usual course, for example, in distinguishing positive and negative square roots. It is possible wherever the
generating relation of our original function is formally capable of
exhibition as a disjunction of one-one relations. It is plain that the
disjunctive relation formed of `n` one-one relations, each of which contains
in its domain a certain class `u`, will, throughout the class `u`, be an
`n`-one relation. Thus it may happen that the independent variable
can be divided into `n` classes, within each of which the defining relation
is one-one, *i.e.* within each of which there is only one `x` having the
defining relation to a given `y`. In such cases, which are usual in pure
mathematics, our many-one relation can be made into a disjunction of
one-one relations, each of which separately is reversible. In the case of
complex functions, this is, *mutatis mutandis*, the method of Riemann
surfaces. But it must be clearly remembered that, where our function
is not naturally one-one, the `y` which appears as dependent variable is
ordinally distinct from the `y` which appears as independent variable in
the inverse function.

The above remarks, which will receive illustration as we proceed,
have shown, I hope, how intimately the correlation of series is associated
with the usual mathematical employment of functions. Many other
cases of the importance of correlation will meet us as we proceed. It
may be observed that every denumerable class is related by a one-valued
function to the finite integers, and *vice versâ*. As ordered by correlation
with the integers, such a class becomes a series having the type of order
which Cantor calls `ω`. The fundamental importance of correlation to
Cantor’s theory of transfinite numbers will appear when we come to the
definition of the transfinite ordinals.

**256.** In connection with functions, it seems desirable to say something concerning the necessity of a formula for definition. A function
was originally, after it had ceased to be merely a power, essentially
something that could be expressed by a formula. It was usual to start
with some expression containing a variable `x`, and to say nothing to
begin with as to what `x` was to be, beyond a usually tacit assumption
that `x` was some kind of number. Any further limitations upon `x` were
derived, if at all, from the formula itself; and it was mainly the desire
to remove such limitations which led to the various generalizations of
number. This algebraical generalization* has now been superseded by
a more ordinal treatment, in which all classes of numbers are defined by
means of the integers, and formulae are not relevant to the process.
Nevertheless, for the use of functions, where both the independent and
the dependent variables are infinite classes, the formula has a certain
importance. Let us see what is its definition.

A formula, in its most general sense, is a proposition, or more
properly a propositional function, containing one or more variables,
a variable being any term of some defined class, or even any term
without restriction. The kind of formula which is relevant in connection
with functions of a single variable is a formula containing two variables.
If both variables are defined, say one as belonging to the class *u*,
the other as belonging to the class *v*, the formula is true or false. It is
true if every *u* has to every *v* the relation expressed by the formula;
otherwise it is false. But if one of the variables, say *x*, be defined as
belonging to the class *u*, while the other, *y*, is only defined by the
formula, then the formula may be regarded as defining *y* as a function
of *x*. Let us call the formula `P`_{xy}. If in the class *u* there are terms *x*
such that there is no term *y* which makes `P`_{xy} a true proposition, then
the formula, as regards those terms, is impossible. We must therefore
assume that *u* is a class every term of which will, for a suitable value
of `y`, make the proposition `P`_{xy} true. If, then, for every term `x` of `u`,
there are some entities `y`, which make `P`_{xy} true, and others which do not
do so, then `P`_{xy} correlates to every `x` a certain class of terms `y`. In
this way `y` is defined as a function of `x`.

But the usual meaning of *formula* in mathematics involves another
element, which may also be expressed by the word *law*. It is difficult to
say precisely what this element is, but it seems to consist in a certain
degree of intensional simplicity of the proposition `P`_{xy}. In the case of
two languages, for example, it would be said that there is no formula
connecting them, except in such cases as Grimm’s law. Apart from the
dictionary, the relation which correlates words in different languages is
sameness of meaning; but this gives no method by which, given a word
in one language, we can infer the corresponding word in the other.
What is absent is the possibility of calculation. A formula, on the
other hand (say `y` = 2`x`), gives the means, when we know `x`, of discovering `y`. In the case of languages, only enumeration of all pairs
will define the dependent variable. In the case of an algebraical
formula, the independent variable and the relation enable us to know
all about the dependent variable. If functions are to extend to infinite
classes, this state of things is essential, for enumeration has become
impossible. It is therefore essential to the correlation of infinite classes,
and to the study of functions of infinite classes, that the formula `P`_{xy}
should be one in which, given `x`, the class of terms `y` satisfying the
formula should be one which we can discover. I am unable to give
a logical account of this condition, and I suspect it of being purely
psychological. Its practical importance is great, but its theoretical
importance seems highly doubtful.

There is, however, a logical condition connected with the above,
though perhaps not quite identical with it. Given any two terms,
there is some relation which holds between those two terms and
no others. It follows that, given any two classes of terms `u`, `v`,
there is a disjunctive relation which any one term of `u` has to at
least one term of `v`, and which no term not belonging to `u` has
to any term. By this method, when two classes are both finite,
we can carry out a correlation (which may be one-one, many-one, or
one-many) which correlates terms of these classes and no others. In
this way any set of terms is theoretically a function of any other; and
it is only thus, for example, that diplomatic ciphers are made up. But
if the number of terms in the class constituting the independent variable
be infinite, we cannot in this way practically define a function, unless
the disjunctive relation consists of relations developed one from the
other by a law, in which case the formula is merely transferred to the
relation. This amounts to saying that the defining relation of a function
must not be infinitely complex, or, if it be so, must be itself a function
defined by some relation of finite complexity. This condition, though
it is itself logical, has again, I think, only psychological necessity, in
virtue of which we can only master the infinite by means of a law of
order. The discussion of this point, however, would involve a discussion
of the relation of infinity to order—a question which will be resumed
later, but which we are not yet in a position to treat intelligently. In
any case, we may say that a formula containing two variables and
defining a function must, if it is to be practically useful, give a relation
between the two variables by which, when one of them is given, all the
corresponding values of the other can be found; and this seems to
constitute the mathematical essence of all formulae.

**257.** There remains an entirely distinct logical notion of much
importance in connection with limits, namely the notion of a complete series. If `R` be the defining relation of a series, the series
is complete when there is a term `x` belonging to the series, such
that every other term which has to `x` either the relation `R` or the
relation ˘`R` belongs to the series. It is *connected* (as was explained in
Part IV) when no other terms belong to the series. Thus a complete
series consists of those terms, and only those terms, which have the
generating relation or its converse to some one term, together with that
one term. Since the generating relation is transitive, a series which
fulfils this condition for one of its terms fulfils it for all of them.
A series which is connected but not complete will be called incomplete
or partial. Instances of complete series are the cardinal integers, the
positive and negative integers and zero, the rational numbers, the
moments of time, or the points on a straight line. Any selection from
such a series is incomplete with respect to the generating relations of the
above complete series. Thus the positive numbers are an incomplete
series, and so are the rationals between 0 and 1. When a series is
complete, no term can come before or after any term of the series
without belonging to the series; when the series is incomplete, this is
no longer the case. A series may be complete with respect to one
generating relation, but not with respect to another. Thus the finite
integers are a complete series when the series is defined by powers of
the relation of consecutiveness, as in the discussion of progressions in
Part IV; but when they are ordered by correlation with whole and part,
they form only part of the series of finite and transfinite integers, as we
shall see hereafter. A complete series may be regarded as the extension
of a term with respect to a given relation, together with this term itself;
and owing to this fact it has, as we shall find, some important differences
from ordinally similar incomplete series. But it can be shown, by the
Logic of Relations, that any incomplete series can be rendered complete
by a change in the generating relation, and *vice versâ*. The distinction
between complete and incomplete series is, therefore, essentially relative
to a given generating relation.

Notes

*^{[page 261]} See my article in *RdM*, Vol. viii, No. 2.

*^{[page 263]} On this subject see my article in *RdM*, Vol. viii, especially Nos. 2, 6.

†^{[page 263]} These are what in Part I we called propositional functions.

*^{[page 265]} See my article “Is position in Time and Space absolute or relative?” *Mind*, July 1901.

*^{[page 266]} I omit for the present complex variables, which, by introducing dimensions, lead to complications of an entirely distinct kind.

*^{[page 267]} Of which an excellent account will be found in Couturat, *De l’Infini Mathématique*, Paris, 1896, Part I, Book II.

Real Numbers.

**258.** The philosopher may be surprised, after all that has already
been said concerning numbers, to find that he is only now to learn about
*real* numbers; and his surprise will be turned to horror when he learns
that *real* is opposed to *rational*. But he will be relieved to learn that
real numbers are really not numbers at all, but something quite different.

The series of real numbers, as ordinarily defined, consists of the whole assemblage of rational and irrational numbers, the irrationals being defined as the limits of such series of rationals as have neither a rational nor an infinite limit. This definition, however, introduces grave difficulties, which will be considered in the next chapter. For my part I see no reason whatever to suppose that there are any irrational numbers in the above sense; and if there are any, it seems certain that they cannot be greater or less than rational numbers. When mathematicians have effected a generalization of number they are apt to be unduly modest about it—they think that the difference between the generalized and the original notions is less than it really is. We have already seen that the finite cardinals are not to be identified with the positive integers, nor yet with the ratios of the natural numbers to 1, both of which express relations, which the natural numbers do not. In like manner there is a real number associated with every rational number, but distinct from it. A real number, so I shall contend, is nothing but a certain class of rational numbers. Thus the class of rationals less than ½ is a real number, associated with, but obviously not identical with, the rational number ½. This theory is not, so far as I know, explicitly advocated by any other author, though Peano suggests it, and Cantor comes very near to it*. My grounds in favour of this opinion are, first, that such classes of rationals have all the mathematical properties commonly assigned to real numbers, secondly, that the opposite theory presents logical difficulties which appear to me insuperable. The second point will be discussed in the next chapter; for the present I shall merely expound my own view, and endeavour to show that real numbers, so understood, have all the requisite characteristics. It will be observed that the following theory is independent of the doctrine of limits, which will only be introduced in the next chapter.

**259.** The rational numbers in order of magnitude form a series in
which there is a term between any two. Such series, which in Part III
we provisionally called continuous, must now receive another name, since
we shall have to reserve the word *continuous* for the sense which Cantor
has given to it. I propose to call such series *compact**. The rational
numbers, then, form a compact series. It is to be observed that, in a
compact series, there are an infinite number of terms between any two,
there are no consecutive terms, and the stretch between any two terms
(whether these be included or not) is again a compact series. If now we
consider any one rational number†, say `r`, we can define, by relation to `r`,
four infinite classes of rationals: (1) those less than `r`, (2) those not
greater than `r`, (3) those greater than `r`, (4) those not less than `r`.
(2) and (4) differ from (1) and (3) respectively solely by the fact that
the former contain `r`, while the latter do not. But this fact leads to
curious differences of properties. (2) has a last term, while (1) has
none; (1) is identical with the class of rational numbers less than a
variable term of (1), while (2) does not have this characteristic. Similar
remarks apply to (3) and (4), but these two classes have less importance
in the present case than in (1) and (2). Classes of rationals having
the properties of (1) are called *segments*. A segment of rationals may
be defined as a class of rationals which is not null, nor yet coextensive
with the rationals themselves (*i.e.* which contains some but not all
rationals), and which is identical with the class of rationals less than a
(variable) term of itself, *i.e.* with the class of rationals `x` such that there
is a rational `y` of the said class such that `x` is less than `y`‡. Now we shall
find that segments are obtained by the above method, not only from
single rationals, but also from finite or infinite classes of rationals, with
the proviso, for infinite classes, that there must be some rational greater
than any member of the class. This is very simply done as follows.

Let `u` be any finite or infinite class of rationals. Then four classes
may be defined by relation to `u`§, namely (1) those less than every `u`,
(2) those less than a variable `u`, (3) those greater than every `u`, (4) those
greater than a variable `u`, *i.e.* those such that for each a term of `u` can be
found which is smaller than it. If `u` be a finite class, it must have a maximum
and a minimum term; in this case the former alone is relevant to (2)
and (3), the latter alone to (1) and (4). Thus this case is reduced to
the former, in which we had only a single rational. I shall therefore
assume in future that `u` is an infinite class, and further, to prevent
reduction to our former case, I shall assume, in considering (2) and (3),
that `u` has no maximum, that is, that every term of `u` is less than some
other term of `u`; and in considering (1) and (4), I shall assume that `u`
has no minimum. For the present I confine myself to (2) and (3), and
I assume, in addition to the absence of a maximum, the existence of
rationals greater than any `u`, that is, the existence of the class (3).
Under these circumstances, the class (2) will be a segment. For (2)
consists of all rationals which are less than a variable `u`; hence, in the
first place, since `u` has no maximum, (2) contains the whole of `u`. In the
second place, since every term of (2) is less than some `u`, which in turn
belongs to (2), every term of (2) is less than some other term of (2);
and every term less than some term of (2) is *a fortiori* less than some `u`,
and is therefore a term of (2). Hence (2) is identical with the class of
terms less than some term of (2), and is therefore a segment.

Thus we have the following conclusion: If `u` be a single rational, or
a class of rationals all of which are less than some fixed rational, then
the rationals less than `u`, if `u` be a single term, or less than a variable
term of `u`, if `u` be a class of terms, always form a segment of rationals.
My contention is, that a segment of rationals *is* a real number.

**260.** So far, the method employed has been one which may be
employed in any compact series. In what follows, some of the theorems
will depend upon the fact that the rationals are a denumerable series.
I leave for the present the disentangling of the theorems dependent
upon this fact, and proceed to the properties of segments of rationals.

Some segments, as we have seen, consist of the rationals less than
some given rational. Some, it will be found, though not so defined, are
nevertheless capable of being so defined. For example, the rationals
less than a variable term of the series ·9, ·99, ·999, *etc*., are the same as
the rationals less than 1. But other segments, which correspond to
what are usually called irrationals, are incapable of any such definition.
How this fact has led to irrationals we shall see in the next chapter.
For the present I merely wish to point out the well-known fact that
segments are not capable of a one-one correlation with rationals. There
are classes of rationals defined as being composed of all terms less than
a *variable* term of an infinite class of rationals, which are not definable
as all the rationals less than some one definite rational*. Moreover
there are more segments than rationals, and hence the series of segments
has continuity of a higher order than the rationals. Segments form a
series in virtue of the relation of whole and part, or of logical inclusion
(excluding identity). Any two segments are such that one of them
is wholly contained in the other, and in virtue of this fact they form
a series. It can be easily shown that they form a compact series.
What is more remarkable is this: if we apply the above process to the
series of segments, forming segments of segments by reference to
classes of segments, we find that every segment of segments can be
defined as all segments contained in a certain definite segment. Thus
the segment of segments defined by a class of segments is always
identical with the segment of segments defined by some one segment.
Also every segment defines a segment of segments which can be defined
by an infinite class of segments. These two properties render the
series of segments *perfect*, in Cantor’s language; but the explanation of
this term must be left till we come to the doctrine of limits.

We might have defined our segments as all rationals greater than
some term of a class *u* of rationals. If we had done this, and inserted
the conditions that `u` was to have no minimum, and that there were to
be rationals less than every `u`, we should have obtained what may be
called upper segments, as distinguished from the former kind, which
may be called lower segments. We should then have found that, corresponding to every upper segment, there is a lower segment which contains
all rationals not contained in the upper segment, with the occasional
exception of a single rational. There will be one rational not belonging
to either the upper or the lower segment, when the upper segment
can be defined as all rationals greater than a single rational. In this
case, the corresponding lower segment will consist of all rationals less
than this single rational, which will itself belong to neither segment.
Since there is a rational between any two, the class of rationals not
greater than a given rational cannot ever be identical with the class of
rationals less than some other; and a class of rationals having a
maximum can never be a segment. Hence it is impossible, in the case
in question, to find a lower segment containing all the rationals not
belonging to the given upper segment. But when the upper segment
cannot be defined by a single rational, it will always be possible
to find a lower segment containing *all* rationals not belonging to the
upper segment.

Zero and infinity may be introduced as limiting cases of segments,
but in the case of zero the segment must be of the kind which we
called (1) above, not of the kind (2) hitherto discussed. It is easy to
construct a class of rationals such that some term of the class will be less
than any given rational. In this case, the class (1) will contain no terms,
and will be the null-class. This is the real number zero, which, however,
is not a segment, since a segment was defined as a class which is not null.
In order to introduce zero as a class of the kind which we called (2), we
should have to start with a null class of rationals. No rational is less
than a term of a null class of rationals, and thus the class (2), in such a
case, is null. Similarly the real number infinity may be introduced.
This is identical with the whole class of rationals. If we have any
class `u` of rationals such that no rational is greater than all `u`’s, then
every rational is contained in the class of rationals less than some
`u`. Or again, if we have a class of rationals of which a term is less than
any assigned rational, the resulting class (4) (of terms greater than
some `u`) will contain every rational, and will thus be the real number
infinity. Thus both zero and infinity may be introduced as extreme
terms among the real numbers, but neither is a segment according to the
definition.

**261.** A given segment may be defined by many different classes of
rationals. Two such classes `u` and `v` may be regarded as having the
segment as a common property. Two infinite classes `u` and `v` will define
the same lower segment if, given any `u`, there is a `v` greater than it, and
given any `v`, there is a `u` greater than it. If each class has no maximum,
this is also a *necessary* condition. The classes `u` and `v` are then what
Cantor calls coherent (*zusammengehörig*). It can be shown, without
considering segments, that the relation of being coherent is symmetrical
and transitive*, whence we should infer, by the principle of abstraction,
that both have to some third term a common relation which neither has
to any other term. This third term, as we see from the preceding
discussion, may be taken to be the segment which both define. We
may extend the word *coherent* to two classes `u` and `v`, of which one
defines an upper segment, the other a lower segment, which between
them include all rationals with at most one exception. Similar remarks,
*mutatis mutandis*, will still apply in this case.

We have now seen that the usual properties of real numbers belong to segments of rationals. There is therefore no mathematical reason for distinguishing such segments from real numbers. It remains to set forth, first the nature of a limit, then the current theories of irrationals, and then the objections which make the above theory seem preferable.

*Note.* The above theory is virtually contained in Professor Peano’s
article already referred to (“Sui Numeri Irrazionali,” *Rivista di Matematica*, vi, pp. 126–140), and it was from this article, as well as from the
*Formulaire de Mathématiques*, that I was led to adopt the theory. In
this article, separate definitions of real numbers (§2, No. 5) and of
segments (§8, ·0) are given, which makes it seem as though the two
were distinguished. But after the definition of segments, we find the
remark (p. 133): “Segments so defined differ only in nomenclature from
real numbers.” Professor Peano proceeds first to give purely technical
reasons for distinguishing the two by the notation, namely that the
addition, subtraction, *etc.* of real numbers is to be differently conducted
from analogous operations which are to be performed on segments.
Hence it would appear that the whole of the view I have advocated is
contained in this article. At the same time, there is some lack of
clearness, since it appears from the definition of real numbers that they
are regarded as the limits of classes of rationals, whereas a segment is
in no sense a limit of a class of rationals. Also it is nowhere suggested—indeed, from the definition of real numbers the opposite is to be
inferred—that no real number can be a rational, and no rational can be
a real number. And this appears where he points out (p. 134) that 1
differs from the class of proper fractions (which is no longer the case as
regards the real number 1, when this is distinguished both from the
integer 1 and from the rational number 1 : 1), or that we say 1 is less
than √2, (in which case, I should say, 1 must be interpreted as the class
of proper fractions, and the assertion must be taken to mean: the
proper fractions are some, but not all, of the rationals whose square
is less than 2). And again he says (*ib.*): “The real number, although
determined by, and determining, a segment `u`, is commonly regarded as
the extremity, or end, or upper limit, of the segment”; whereas there is
no reason to suppose that segments not having a rational limit have a
limit at all. Thus although he confesses (*ib.*) that a complete theory
of irrationals *can* be constructed by means of segments, he does not
seem to perceive the reasons (which will be given in the next chapter)
why this *must* be done—reasons which, in fact, are rather philosophical
than mathematical.

Notes

*^{[page 270]} Cf. Cantor, *Math. Annalen*, Vol. xlvi, §10; Peano,

*^{[page 271]} Such series are called by Cantor *überall dicht*.

†^{[page 271]} I shall for simplicity confine myself entirely to rationals without sign. The
extension to such as are positive or negative presents no difficulty whatever.

‡^{[page 271]} See *Formulaire de Mathématiques*, Vol. ii. Part iii, §61 (Turin, 1899).

§^{[page 271]} Eight classes *may* be defined, but four are all that we need.

*^{[page 272]} Cf. Part I, chap. v, p. 60.

*^{[page 274]} Cf. Cantor, *Math. Annalen*, xlvi, and *Rivista di Matematica*, v, pp. 158, 159.

Limits and Irrational Numbers.

**262.** The mathematical treatment of continuity rests wholly upon
the doctrine of limits. It has been thought by some mathematicians
and some philosophers that this doctrine had been superseded by the
Infinitesimal Calculus, and that this has shown true infinitesimals
to be presupposed in limits*. But modern mathematics has shown,
conclusively as it seems to me, that such a view is erroneous. The
method of limits has more and more emerged as fundamental. In this
Chapter, I shall first set forth the general definition of a limit, and
then examine its application to the creation of irrationals.

A compact series we defined as one in which there is a term between
any two. But in such a series it is always possible to find two *classes* of
terms which have no term between them, and it is always possible to
reduce *one* of these classes to a single term. For example, if `P` be the
generating relation and `x` any term of the series, then the class of terms
having to `x` the relation `P` is one between which and `x` there is no term†.
The class of terms so defined is one of the two segments determined
by `x`; the idea of a segment is one which demands only a series in
general, not necessarily a numerical series. In this case, if the series be
compact, `x` is said to be the *limit* of the class; when there is such a
term as `x`, the segment is said to be terminated, and thus every
terminated segment in a compact series has its defining term as a limit.
But this does not constitute a definition of a limit. To obtain the
general definition of a limit, consider any class `u` contained in the series
generated by `P`. Then the class `u` will in general, with respect to any
term `x` not belonging to it, be divisible into two classes, that whose
terms have to `x` the relation `P` (which I shall call the class of terms preceding `x`), and that whose terms have to `x` the relation ˘`P` (which I shall
call the class of terms following `x`). If `x` be itself a term of `u`, we
consider all the terms of `u` other than `x`, and these are still divisible into
the above two classes, which we may call `π`_{w}`x` and ˘`π`_{w}`x` respectively.
If, now, `π`_{w}`x` be such that, if `y` be any term preceding `x`, there is a term
of `π`_{w}`x` following `y`, *i.e.* between `x` and `y`, then `x` is a limit of `π`_{w}`x`. Similarly
if ˘`π`_{w}`x` be such that, if `z` be any term after `x`, there is a term of
˘`π`_{w}`x` between `x` and `z`, then `x` is a limit of ˘`π`_{w}`x`. We now define that `x` is
a limit of `u` if it is a limit of either `π`_{w}`x` or ˘`π`_{w}`x`. It is to be observed that
`u` may have many limits, and that all the limits together form a new
class contained in the series generated by `P`. This is the class (or rather
this, by the help of certain further assumptions, becomes the class)
which Cantor designates as the first derivative of the class `u`.

**263.** Before proceeding further, it may be well to make some
general remarks of an elementary character on the subject of limits.
In the first place, limits belong usually to classes contained in compact
series—classes which may, as an extreme case, be identical with the
compact series in question. In the second place, a limit may or may
not belong to the class `u` of which it is a limit, but it always belongs to
some series in which `u` is contained, and if it is a term of `u`, it is still a
limit of the class consisting of all terms of `u` except itself. In the
third place, no class can have a limit unless it contains an infinite
number of terms. For, to revert to our former division, if `u` be finite, `π`_{w}`x` and ˘`π`_{w}`x`
will both be finite. Hence each of them will have a term
nearest to `x`, and between this term and `x` no term of `u` will lie. Hence
`x` is not a limit of `u`; and since `x` is any term of the series, `u` will have
no limits at all. It is common to add a theorem that every infinite
class, provided its terms are all contained between two specified terms
of the series generated by `P`, must have at least one limit; but this
theorem, we shall find, demands an interpretation in terms of segments,
and is not true as it stands. In the fourth place, if `u` be co-extensive
with the whole compact series generated by `P`, then every term of this
series is a limit of `u`. There can be no other terms that are limits
in the same sense, since limits have only been defined in relation to this
compact series. To obtain other limits, we should have to regard the
series generated by `P` as forming part of some other compact series—a
case which, as we shall see, may arise. In any case, if `u` be any compact
series, every term of `u` is a limit of `u`; whether `u` has also other limits,
depends upon further circumstances. A limit may be defined generally
as a term which immediately follows (or precedes) some class of terms
belonging to an infinite series, without immediately following (or
preceding, as the case may be) any one term of the series. In this way,
we shall find, limits may be defined generally in all infinite series which
are not progressions—as, for instance, in the series of finite and transfinite integers.

**264.** We may now proceed to the various arithmetical theories of
irrationals, all of which depend upon limits. We shall find that, in the
exact form in which they have been given by their inventors, they all
involve an axiom, for which there are no arguments, either of philosophical necessity or of mathematical convenience; to which there are
grave logical objections; and of which the theory of real numbers given
in the preceding Chapter is wholly independent.

Arithmetical theories of irrationals could not be treated in Part II, since they depend essentially upon the notion of order. It is only by means of them that numbers become continuous in the sense now usual among mathematicians; and we shall find in Part VI that no other sense of continuity is required for space and time. It is very important to realize the logical reasons for which an arithmetical theory of irrationals is imperatively necessary. In the past, the definition of irrationals was commonly effected by geometrical considerations. This procedure was, however, highly illogical; for if the application of numbers to space is to yield anything but tautologies, the numbers applied must be independently defined; and if none but a geometrical definition were possible, there would be, properly speaking, no such arithmetical entities as the definition pretended to define. The algebraical definition, in which irrationals were introduced as the roots of algebraic equations having no rational roots, was liable to similar objections, since it remained to be shown that such equations have roots; moreover this method will only yield the so-called algebraic numbers, which are an infinitesimal proportion of the real numbers, and do not have continuity in Cantor’s sense, or in the sense required by Geometry. And in any case, if it is possible, without any further assumption, to pass from Arithmetic to Analysis, from rationals to irrationals, it is a logical advance to show how this can be done. The generalizations of number—with the exception of the introduction of imaginaries, which must be independently effected—are all necessary consequences of the admission that the natural numbers form a progression. In every progression the terms have two kinds of relations, the one constituting the general analogue of positive and negative integers, the other that of rational numbers. The rational numbers form a denumerable compact series; and segments of a denumerable compact series, as we saw in the preceding Chapter, form a series which is continuous in the strictest sense. Thus all follows from the assumption of a progression. But in the present Chapter we have to examine irrationals as based on limits; and in this sense, we shall find that they do not follow without a new assumption.

There are several somewhat similar theories of irrational numbers. I will begin with that of Dedekind*.

**265.** Although rational numbers are such that, between any two,
there is always a third, yet there are many ways of dividing *all* rational
numbers into two classes, such that all numbers of one class come after
all numbers of the other class, and no rational number lies between the
two classes, while yet the first class has no first term and the second has
no last term. For example, all rational numbers, without exception,
may be classified according as their squares are greater or less than 2.
All the terms of both classes may be arranged in a single series, in which
there exists a definite section, before which comes one of the classes,
and after which comes the other. Continuity *seems* to demand that
some term should correspond to this section. A number which lies
between the two classes must be a new number, since all the old numbers
are classified. This new number, which is thus defined by its position in
a series, is an *irrational* number. When these numbers are introduced,
not only is there always a number between any two numbers, but there
is a number between any two classes of which one comes wholly after the
other, and the first has no minimum, while the second has no maximum.
Thus we can extend to numbers the axiom by which Dedekind defines
the continuity of the straight line (*op. cit.* p. 11):—

“If all the points of a line can be divided into two classes such that every point of one class is to the left of every point of the other class, then there exists one and only one point which brings about this division of all points into two classes, this section of the line into two parts.”

**266.** This axiom of Dedekind’s is, however, rather loosely worded, and
requires an emendation suggested by the derivation of irrational numbers.
If *all* the points of a line are divided into two classes, no point is left
over to represent the section. If *all* be meant to exclude the point representing the section, the axiom no longer characterizes continuous series,
but applies equally to all series, *e.g.* the series of integers. The axiom
must be held to apply, as regards the division, not to all the points of the
line, but to all the points forming some compact series, and distributed
throughout the line, but consisting only of a portion of the points
of the line. When this emendation is made, the axiom becomes admissible. If, from among the terms of a series, some can be chosen
out to form a compact series which is distributed throughout the
previous series; and if this new series can always be divided in
Dedekind’s manner into two portions, between which lies no term of
the new series, but one and only one term of the original series, then
the original series is continuous in Dedekind’s sense of the word. The
emendation, however, destroys entirely the self-evidence upon which
alone Dedekind relies (p. 11) for the proof of his axiom as applied
to the straight line.

Another somewhat less complicated emendation may be made, which
gives, I think, what Dedekind *meant* to state in his axiom. A series,
we may say, is continuous in Dedekind’s sense when, and only when,
if *all* the terms of the series, without exception, be divided into two
classes, such that the whole of the first class precedes the whole of
the second, then, however the division be effected, either the first class
has a last term, or the second class has a first term, but never both.
This term, which comes at one end of one of the two classes, may then
be used, in Dedekind’s manner, to define the section. In discrete series,
such as that of finite integers, there is both a last term of the first
class and a first term of the second class*; while in compact series
such as the rationals, where there is not continuity, it sometimes
happens (though not for every possible division) that the first class
has no last term and the last class has no first term. Both these cases
are excluded by the above axiom. But I cannot see any vestige of
self-evidence in such an axiom, either as applied to numbers or as applied
to space.

**267.** Leaving aside, for the moment, the general problem of continuity, let us return to Dedekind’s definition of irrational numbers.
The first question that arises is this: What right have we to assume
the existence of such numbers? What reason have we for supposing
that there must be a position between two classes of which one is wholly
to the right of the other, and of which one has no minimum and the
other no maximum? This is not true of series in general, since many
series are discrete. It is not demanded by the nature of order. And,
as we have seen, continuity in a certain sense is possible without it.
Why then should we postulate such a number at all? It must be
remembered that the algebraical and geometrical problems, which irrationals are intended to solve, must not here be brought into the
account. The existence of irrationals has, in the past, been inferred
from such problems. The equation `x`^{2} − 2 = 0 must have a root, it was
argued, because, as `x` grows from 0 to 2, `x`^{2} − 2 increases, and is first
negative and then positive; if `x` changes continuously, so does `x`^{2} − 2;
hence `x`^{2} − 2 must assume the value in passing from negative to positive.
Or again, it was argued that the diagonal of unit square has evidently a
precise and definite length `x`, and that this length is such that `x`^{2} − 2 = 0.
But such arguments were powerless to show that `x` is truly a number.
They might equally well be regarded as showing the inadequacy of
numbers to Algebra and Geometry. The present theory is designed
to prove the arithmetical existence of irrationals. In its design, it is
preferable to the previous theories; but the execution seems to fall short
of the design.

Let us examine in detail the definition of √2 by Dedekind’s method.
It is a singular fact that, although a rational number lies between any
two single rational numbers, two classes of rational numbers may be
defined so that no rational number lies between them, though all of
one class are higher than all of the other. It is evident that one at
least of these classes must consist of an infinite number of terms. For
if not, we could pick out the two of opposite kinds which were nearest
together, and insert a new number between them. This one would be
between the two classes, contrary to the hypothesis. But when one of
the classes is infinite, we may arrange all or some of the terms in a series
of terms continually approaching the other class, without reaching it,
and without having a last term. Let us, for the moment, suppose our
infinite class to be denumerable. We then obtain a denumerable series
of numbers `a`` _{n}`, all belonging to the one class, but continually approaching
the other class. Let

`x` = √2 + 1, `x`^{2} − 2`x` − 1 = 0.

Thus `x` = 2 + 1/`x` = 2 + 12 + 1`x`, and
`x` − 1 = 1 + 12 + 12 + 1`x` = etc.

The successive convergents to the continued fraction 1 + 12 + 12 + 12 + … are such that all the odd convergents are less than all the even convergents, while the odd convergents continually grow, and the even ones continually diminish. Moreover the difference between the odd and the next even convergent continually diminishes. Thus both series, if they have a limit, have the same limit, and this limit is defined as √2.

But the existence of a limit, in this case, is evidently a sheer assumption. In the beginning of this Chapter, we saw that the existence
of a limit demands a larger series of which the limit forms part. To
create the limit by means of the series whose limit is to be found would
therefore be a logical error. It is essential that the distance from the
limit should diminish indefinitely. But here, it is only the distance of
consecutive terms which is known to diminish indefinitely. Moreover
all the `a`’s are less than `b`` _{n}`. Hence they continually differ less and less
from

**268.** The theory of Weierstrass concerning irrationals is somewhat
similar to that of Dedekind. In Weierstrass’s theory, we have a series
of terms `a`_{1}