Introduction to Mathematical

Philosophy

by

Bertrand Russell

Originally published byGeorge Allen & Unwin, Ltd., London. May .

Online Corrected Edition version . (February , ),based on the “second edition” (second printing) of April

, incorporating additional corrections,marked in green.

[Russell’s blurb from the original dustcover:]

This book is intended for those who have no previ-ous acquaintance with the topics of which it treats,and no more knowledge of mathematics than canbe acquired at a primary school or even at Eton. Itsets forth in elementary form the logical definitionof number, the analysis of the notion of order, themodern doctrine of the infinite, and the theory ofdescriptions and classes as symbolic fictions. Themore controversial and uncertain aspects of the sub-ject are subordinated to those which can by now beregarded as acquired scientific knowledge. Theseare explained without the use of symbols, but insuch a way as to give readers a general understand-ing of the methods and purposes of mathematicallogic, which, it is hoped, will be of interest not onlyto those who wish to proceed to a more serious studyof the subject, but also to that wider circle who feel adesire to know the bearings of this important mod-ern science.

Contents

Contents . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . viEditor’s Note . . . . . . . . . . . . . . . . . . . ix

I. The Series of Natural Numbers . . . II. Definition of Number . . . . . . . . .

III. Finitude and Mathematical Induction IV. The Definition of Order . . . . . . . V. Kinds of Relations . . . . . . . . . . .

VI. Similarity of Relations . . . . . . . . VII. Rational, Real, and Complex Numbers

VIII. Infinite Cardinal Numbers . . . . . . IX. Infinite Series and Ordinals . . . . . X. Limits and Continuity . . . . . . . .

XI. Limits and Continuity of Functions . XII. Selections and the Multiplicative Ax-

iom . . . . . . . . . . . . . . . . . . . .

iv

XIII. The Axiom of Infinity and LogicalTypes . . . . . . . . . . . . . . . . . .

XIV. Incompatibility and the Theory of De-duction . . . . . . . . . . . . . . . . .

XV. Propositional Functions . . . . . . . XVI. Descriptions . . . . . . . . . . . . . .

XVII. Classes . . . . . . . . . . . . . . . . . XVIII. Mathematics and Logic . . . . . . . . Index . . . . . . . . . . . . . . . . . . . . . . . Appendix: Changes to Online Edition . . . .

Preface

This book is intended essentially as an “Introduc- vtion,” and does not aim at giving an exhaustivediscussion of the problems with which it deals. Itseemed desirable to set forth certain results, hith-erto only available to those who have mastered log-ical symbolism, in a form offering the minimum ofdifficulty to the beginner. The utmost endeavourhas been made to avoid dogmatism on such ques-tions as are still open to serious doubt, and thisendeavour has to some extent dominated the choiceof topics considered. The beginnings of mathemati-cal logic are less definitely known than its later por-tions, but are of at least equal philosophical interest.Much of what is set forth in the following chaptersis not properly to be called “philosophy,” thoughthe matters concerned were included in philosophyso long as no satisfactory science of them existed.The nature of infinity and continuity, for example,

belonged in former days to philosophy, but belongsnow to mathematics. Mathematical philosophy, inthe strict sense, cannot, perhaps, be held to includesuch definite scientific results as have been obtainedin this region; the philosophy of mathematics willnaturally be expected to deal with questions on thefrontier of knowledge, as to which comparative cer-tainty is not yet attained. But speculation on suchquestions is hardly likely to be fruitful unless themore scientific parts of the principles of mathemat-ics are known. A book dealing with those parts may,therefore, claim to be an introduction to mathemati-cal philosophy, though it can hardly claim, exceptwhere it steps outside its province, to be actuallydealing with a part of philosophy. It does deal, |however, with a body of knowledge which, to those viwho accept it, appears to invalidate much tradi-tional philosophy, and even a good deal of what iscurrent in the present day. In this way, as well as byits bearing on still unsolved problems, mathemati-cal logic is relevant to philosophy. For this reason, aswell as on account of the intrinsic importance of thesubject, some purpose may be served by a succinctaccount of the main results of mathematical logic ina form requiring neither a knowledge of mathemat-ics nor an aptitude for mathematical symbolism.

Here, however, as elsewhere, the method is moreimportant than the results, from the point of viewof further research; and the method cannot well beexplained within the framework of such a book asthe following. It is to be hoped that some readersmay be sufficiently interested to advance to a studyof the method by which mathematical logic can bemade helpful in investigating the traditional prob-lems of philosophy. But that is a topic with whichthe following pages have not attempted to deal.

BERTRAND RUSSELL.

Editor’s Note

[The note below was written by J. H. Muirhead, viiLL.D., editor of the Library of Philosophy series inwhich Introduction to Mathematical Philosophy wasoriginally published.]

Those who, relying on the distinction between Math-ematical Philosophy and the Philosophy of Math-ematics, think that this book is out of place in thepresent Library, may be referred to what the au-thor himself says on this head in the Preface. It isnot necessary to agree with what he there suggestsas to the readjustment of the field of philosophyby the transference from it to mathematics of suchproblems as those of class, continuity, infinity, inorder to perceive the bearing of the definitions anddiscussions that follow on the work of “traditionalphilosophy.” If philosophers cannot consent to rel-egate the criticism of these categories to any of the

special sciences, it is essential, at any rate, that theyshould know the precise meaning that the science ofmathematics, in which these concepts play so large apart, assigns to them. If, on the other hand, there bemathematicians to whom these definitions and dis-cussions seem to be an elaboration and complicationof the simple, it may be well to remind them fromthe side of philosophy that here, as elsewhere, ap-parent simplicity may conceal a complexity whichit is the business of somebody, whether philosopheror mathematician, or, like the author of this volume,both in one, to unravel.

Chapter I

The Series of Natural Numbers

Mathematics is a study which, when we start from its most familiar portions, may be pursued in eitherof two opposite directions. The more familiar direc-tion is constructive, towards gradually increasingcomplexity: from integers to fractions, real num-bers, complex numbers; from addition and multipli-cation to differentiation and integration, and on tohigher mathematics. The other direction, which isless familiar, proceeds, by analysing, to greater andgreater abstractness and logical simplicity; insteadof asking what can be defined and deduced fromwhat is assumed to begin with, we ask instead whatmore general ideas and principles can be found, interms of which what was our starting-point can bedefined or deduced. It is the fact of pursuing thisopposite direction that characterises mathematical

philosophy as opposed to ordinary mathematics.But it should be understood that the distinctionis one, not in the subject matter, but in the stateof mind of the investigator. Early Greek geome-ters, passing from the empirical rules of Egyptianland-surveying to the general propositions by whichthose rules were found to be justifiable, and thenceto Euclid’s axioms and postulates, were engagedin mathematical philosophy, according to the abovedefinition; but when once the axioms and postulateshad been reached, their deductive employment, aswe find it in Euclid, belonged to mathematics in the| ordinary sense. The distinction between mathe- matics and mathematical philosophy is one whichdepends upon the interest inspiring the research,and upon the stage which the research has reached;not upon the propositions with which the researchis concerned.

We may state the same distinction in anotherway. The most obvious and easy things in mathe-matics are not those that come logically at the begin-ning; they are things that, from the point of view oflogical deduction, come somewhere in the middle.Just as the easiest bodies to see are those that areneither very near nor very far, neither very small norvery great, so the easiest conceptions to grasp are

those that are neither very complex nor very simple(using “simple” in a logical sense). And as we needtwo sorts of instruments, the telescope and the mi-croscope, for the enlargement of our visual powers,so we need two sorts of instruments for the enlarge-ment of our logical powers, one to take us forwardto the higher mathematics, the other to take us back-ward to the logical foundations of the things that weare inclined to take for granted in mathematics. Weshall find that by analysing our ordinary mathemat-ical notions we acquire fresh insight, new powers,and the means of reaching whole new mathematicalsubjects by adopting fresh lines of advance after ourbackward journey. It is the purpose of this book toexplain mathematical philosophy simply and un-technically, without enlarging upon those portionswhich are so doubtful or difficult that an elemen-tary treatment is scarcely possible. A full treatmentwill be found in Principia Mathematica; the treat-ment in the present volume is intended merely asan introduction.

To the average educated person of the presentday, the obvious starting-point of mathematics

Cambridge University Press, vol. i., ; vol. ii., ; vol.iii., . By Whitehead and Russell.

would be the series of whole numbers,

, , , , . . . etc. |

Probably only a person with some mathematical knowledge would think of beginning with insteadof with , but we will presume this degree of knowl-edge; we will take as our starting-point the series:

, , , , . . . n, n+ , . . .

and it is this series that we shall mean when wespeak of the “series of natural numbers.”

It is only at a high stage of civilisation that wecould take this series as our starting-point. It musthave required many ages to discover that a brace ofpheasants and a couple of days were both instancesof the number : the degree of abstraction involvedis far from easy. And the discovery that is a num-ber must have been difficult. As for , it is a veryrecent addition; the Greeks and Romans had nosuch digit. If we had been embarking upon mathe-matical philosophy in earlier days, we should havehad to start with something less abstract than theseries of natural numbers, which we should reachas a stage on our backward journey. When the log-ical foundations of mathematics have grown morefamiliar, we shall be able to start further back, at

what is now a late stage in our analysis. But forthe moment the natural numbers seem to representwhat is easiest and most familiar in mathematics.

But though familiar, they are not understood.Very few people are prepared with a definition ofwhat is meant by “number,” or “,” or “.” It isnot very difficult to see that, starting from , anyother of the natural numbers can be reached by re-peated additions of , but we shall have to definewhat we mean by “adding ,” and what we mean by“repeated.” These questions are by no means easy.It was believed until recently that some, at least, ofthese first notions of arithmetic must be acceptedas too simple and primitive to be defined. Since allterms that are defined are defined by means of otherterms, it is clear that human knowledge must al-ways be content to accept some terms as intelligiblewithout definition, in order | to have a starting-point for its definitions. It is not clear that there must beterms which are incapable of definition: it is possiblethat, however far back we go in defining, we alwaysmight go further still. On the other hand, it is alsopossible that, when analysis has been pushed farenough, we can reach terms that really are simple,and therefore logically incapable of the sort of defi-nition that consists in analysing. This is a question

which it is not necessary for us to decide; for ourpurposes it is sufficient to observe that, since humanpowers are finite, the definitions known to us mustalways begin somewhere, with terms undefined forthe moment, though perhaps not permanently.

All traditional pure mathematics, including an-alytical geometry, may be regarded as consistingwholly of propositions about the natural numbers.That is to say, the terms which occur can be definedby means of the natural numbers, and the propo-sitions can be deduced from the properties of thenatural numbers—with the addition, in each case,of the ideas and propositions of pure logic.

That all traditional pure mathematics can bederived from the natural numbers is a fairly re-cent discovery, though it had long been suspected.Pythagoras, who believed that not only mathemat-ics, but everything else could be deduced from num-bers, was the discoverer of the most serious obstaclein the way of what is called the “arithmetising” ofmathematics. It was Pythagoras who discovered theexistence of incommensurables, and, in particular,the incommensurability of the side of a square andthe diagonal. If the length of the side is inch, thenumber of inches in the diagonal is the square rootof , which appeared not to be a number at all. The

problem thus raised was solved only in our ownday, and was only solved completely by the help ofthe reduction of arithmetic to logic, which will beexplained in following chapters. For the present, weshall take for granted the arithmetisation of math-ematics, though this was a feat of the very greatestimportance. |

Having reduced all traditional pure mathemat- ics to the theory of the natural numbers, the nextstep in logical analysis was to reduce this theoryitself to the smallest set of premisses and undefinedterms from which it could be derived. This workwas accomplished by Peano. He showed that the en-tire theory of the natural numbers could be derivedfrom three primitive ideas and five primitive propo-sitions in addition to those of pure logic. These threeideas and five propositions thus became, as it were,hostages for the whole of traditional pure mathe-matics. If they could be defined and proved in termsof others, so could all pure mathematics. Their log-ical “weight,” if one may use such an expression,is equal to that of the whole series of sciences thathave been deduced from the theory of the naturalnumbers; the truth of this whole series is assured ifthe truth of the five primitive propositions is guar-anteed, provided, of course, that there is nothing

erroneous in the purely logical apparatus which isalso involved. The work of analysing mathematics isextraordinarily facilitated by this work of Peano’s.

The three primitive ideas in Peano’s arithmeticare:

, number, successor.

By “successor” he means the next number in thenatural order. That is to say, the successor of is ,the successor of is , and so on. By “number” hemeans, in this connection, the class of the naturalnumbers. He is not assuming that we know allthe members of this class, but only that we knowwhat we mean when we say that this or that is anumber, just as we know what we mean when wesay “Jones is a man,” though we do not know allmen individually.

The five primitive propositions which Peano as-sumes are:

() is a number.() The successor of any number is a number.() No two numbers have the same successor. |() is not the successor of any number. We shall use “number” in this sense in the present chapter.

Afterwards the word will be used in a more general sense.

() Any property which belongs to , and also tothe successor of every number which has theproperty, belongs to all numbers.

The last of these is the principle of mathematicalinduction. We shall have much to say concern-ing mathematical induction in the sequel; for thepresent, we are concerned with it only as it occursin Peano’s analysis of arithmetic.

Let us consider briefly the kind of way in whichthe theory of the natural numbers results from thesethree ideas and five propositions. To begin with, wedefine as “the successor of ,” as “the successorof ,” and so on. We can obviously go on as longas we like with these definitions, since, in virtue of(), every number that we reach will have a succes-sor, and, in virtue of (), this cannot be any of thenumbers already defined, because, if it were, twodifferent numbers would have the same successor;and in virtue of () none of the numbers we reach inthe series of successors can be . Thus the series ofsuccessors gives us an endless series of continuallynew numbers. In virtue of () all numbers comein this series, which begins with and travels onthrough successive successors: for (a) belongs tothis series, and (b) if a number n belongs to it, sodoes its successor, whence, by mathematical induc-

tion, every number belongs to the series.Suppose we wish to define the sum of two num-

bers. Taking any number m, we define m+ as m,and m+ (n+ ) as the successor of m+ n. In virtueof () this gives a definition of the sum of m andn, whatever number n may be. Similarly we candefine the product of any two numbers. The readercan easily convince himself that any ordinary ele-mentary proposition of arithmetic can be provedby means of our five premisses, and if he has anydifficulty he can find the proof in Peano.

It is time now to turn to the considerations whichmake it necessary to advance beyond the stand-point of Peano, who | represents the last perfection of the “arithmetisation” of mathematics, to that ofFrege, who first succeeded in “logicising” math-ematics, i.e. in reducing to logic the arithmeticalnotions which his predecessors had shown to besufficient for mathematics. We shall not, in thischapter, actually give Frege’s definition of numberand of particular numbers, but we shall give someof the reasons why Peano’s treatment is less finalthan it appears to be.

In the first place, Peano’s three primitive ideas—namely, “,” “number,” and “successor”—are ca-pable of an infinite number of different interpre-

tations, all of which will satisfy the five primitivepropositions. We will give some examples.

() Let “” be taken to mean , and let “num-ber” be taken to mean the numbers from on-ward in the series of natural numbers. Then all ourprimitive propositions are satisfied, even the fourth,for, though is the successor of , is not a“number” in the sense which we are now giving tothe word “number.” It is obvious that any numbermay be substituted for in this example.

() Let “” have its usual meaning, but let “num-ber” mean what we usually call “even numbers,”and let the “successor” of a number be what resultsfrom adding two to it. Then “” will stand for thenumber two, “” will stand for the number four,and so on; the series of “numbers” now will be

, two, four, six, eight . . .

All Peano’s five premisses are satisfied still.() Let “” mean the number one, let “number”

mean the set

, , , , , . . .

and let “successor” mean “half.” Then all Peano’sfive axioms will be true of this set.

It is clear that such examples might be multi-plied indefinitely. In fact, given any series

x, x, x, x, . . . xn, . . . |

which is endless, contains no repetitions, has a be- ginning, and has no terms that cannot be reachedfrom the beginning in a finite number of steps, wehave a set of terms verifying Peano’s axioms. Thisis easily seen, though the formal proof is somewhatlong. Let “” mean x, let “number” mean the wholeset of terms, and let the “successor” of xn mean xn+.Then

() “ is a number,” i.e. x is a member of theset.

() “The successor of any number is a number,”i.e. taking any term xn in the set, xn+ is also in theset.

() “No two numbers have the same successor,”i.e. if xm and xn are two different members of theset, xm+ and xn+ are different; this results from thefact that (by hypothesis) there are no repetitions inthe set.

() “ is not the successor of any number,” i.e. noterm in the set comes before x.

() This becomes: Any property which belongsto x, and belongs to xn+ provided it belongs to xn,

belongs to all the x’s.This follows from the corresponding property

for numbers.A series of the form

x, x, x, . . . xn, . . .

in which there is a first term, a successor to eachterm (so that there is no last term), no repetitions,and every term can be reached from the start in afinite number of steps, is called a progression. Pro-gressions are of great importance in the principles ofmathematics. As we have just seen, every progres-sion verifies Peano’s five axioms. It can be proved,conversely, that every series which verifies Peano’sfive axioms is a progression. Hence these five ax-ioms may be used to define the class of progres-sions: “progressions” are “those series which verifythese five axioms.” Any progression may be takenas the basis of pure mathematics: we may give thename “” to its first term, the name “number” to thewhole set of its terms, and the name “successor” tothe next in the progression. The progression neednot be composed of numbers: it may be | composed of points in space, or moments of time, or any otherterms of which there is an infinite supply. Eachdifferent progression will give rise to a different in-

terpretation of all the propositions of traditionalpure mathematics; all these possible interpretationswill be equally true.

In Peano’s system there is nothing to enable us todistinguish between these different interpretationsof his primitive ideas. It is assumed that we knowwhat is meant by “,” and that we shall not supposethat this symbol means or Cleopatra’s Needleor any of the other things that it might mean.

This point, that “” and “number” and “suc-cessor” cannot be defined by means of Peano’s fiveaxioms, but must be independently understood, isimportant. We want our numbers not merely to ver-ify mathematical formulæ, but to apply in the rightway to common objects. We want to have ten fingersand two eyes and one nose. A system in which “”meant , and “” meant , and so on, might beall right for pure mathematics, but would not suitdaily life. We want “” and “number” and “succes-sor” to have meanings which will give us the rightallowance of fingers and eyes and noses. We havealready some knowledge (though not sufficientlyarticulate or analytic) of what we mean by “” and“” and so on, and our use of numbers in arithmeticmust conform to this knowledge. We cannot securethat this shall be the case by Peano’s method; all

that we can do, if we adopt his method, is to say “weknow what we mean by ‘’ and ‘number’ and ‘suc-cessor,’ though we cannot explain what we mean interms of other simpler concepts.” It is quite legiti-mate to say this when we must, and at some pointwe all must; but it is the object of mathematical phi-losophy to put off saying it as long as possible. Bythe logical theory of arithmetic we are able to put itoff for a very long time.

It might be suggested that, instead of settingup “” and “number” and “successor” as terms ofwhich we know the meaning although we cannotdefine them, we might let them | stand for any three terms that verify Peano’s five axioms. They will thenno longer be terms which have a meaning that isdefinite though undefined: they will be “variables,”terms concerning which we make certain hypothe-ses, namely, those stated in the five axioms, butwhich are otherwise undetermined. If we adopt thisplan, our theorems will not be proved concerningan ascertained set of terms called “the natural num-bers,” but concerning all sets of terms having certainproperties. Such a procedure is not fallacious; in-deed for certain purposes it represents a valuablegeneralisation. But from two points of view it failsto give an adequate basis for arithmetic. In the first

place, it does not enable us to know whether thereare any sets of terms verifying Peano’s axioms; itdoes not even give the faintest suggestion of anyway of discovering whether there are such sets. Inthe second place, as already observed, we want ournumbers to be such as can be used for countingcommon objects, and this requires that our num-bers should have a definite meaning, not merely thatthey should have certain formal properties. Thisdefinite meaning is defined by the logical theory ofarithmetic.

Chapter II

Definition of Number

The question “What is a number?” is one which has been often asked, but has only been correctlyanswered in our own time. The answer was givenby Frege in , in his Grundlagen der Arithmetik.

Although this book is quite short, not difficult, andof the very highest importance, it attracted almostno attention, and the definition of number which itcontains remained practically unknown until it wasrediscovered by the present author in .

In seeking a definition of number, the first thingto be clear about is what we may call the grammar ofour inquiry. Many philosophers, when attemptingto define number, are really setting to work to defineplurality, which is quite a different thing. Number isThe same answer is given more fully and with more devel-

opment in his Grundgesetze der Arithmetik, vol. i., .

what is characteristic of numbers, as man is what ischaracteristic of men. A plurality is not an instanceof number, but of some particular number. A trio ofmen, for example, is an instance of the number ,and the number is an instance of number; but thetrio is not an instance of number. This point mayseem elementary and scarcely worth mentioning;yet it has proved too subtle for the philosophers,with few exceptions.

A particular number is not identical with anycollection of terms having that number: the number is not identical with | the trio consisting of Brown, Jones, and Robinson. The number is somethingwhich all trios have in common, and which dis-tinguishes them from other collections. A numberis something that characterises certain collections,namely, those that have that number.

Instead of speaking of a “collection,” we shall asa rule speak of a “class,” or sometimes a “set.” Otherwords used in mathematics for the same thing are“aggregate” and “manifold.” We shall have much tosay later on about classes. For the present, we willsay as little as possible. But there are some remarksthat must be made immediately.

A class or collection may be defined in two waysthat at first sight seem quite distinct. We may enu-

merate its members, as when we say, “The collectionI mean is Brown, Jones, and Robinson.” Or we maymention a defining property, as when we speak of“mankind” or “the inhabitants of London.” The def-inition which enumerates is called a definition by“extension,” and the one which mentions a definingproperty is called a definition by “intension.” Ofthese two kinds of definition, the one by intension islogically more fundamental. This is shown by twoconsiderations: () that the extensional definitioncan always be reduced to an intensional one; () thatthe intensional one often cannot even theoreticallybe reduced to the extensional one. Each of thesepoints needs a word of explanation.

() Brown, Jones, and Robinson all of them pos-sess a certain property which is possessed by noth-ing else in the whole universe, namely, the propertyof being either Brown or Jones or Robinson. Thisproperty can be used to give a definition by inten-sion of the class consisting of Brown and Jones andRobinson. Consider such a formula as “x is Brownor x is Jones or x is Robinson.” This formula willbe true for just three x’s, namely, Brown and Jonesand Robinson. In this respect it resembles a cubicequation with its three roots. It may be taken asassigning a property common to the members of the

class consisting of these three |men, and peculiar to them. A similar treatment can obviously be appliedto any other class given in extension.

() It is obvious that in practice we can oftenknow a great deal about a class without being ableto enumerate its members. No one man could actu-ally enumerate all men, or even all the inhabitantsof London, yet a great deal is known about each ofthese classes. This is enough to show that definitionby extension is not necessary to knowledge about aclass. But when we come to consider infinite classes,we find that enumeration is not even theoreticallypossible for beings who only live for a finite time.We cannot enumerate all the natural numbers: theyare , , , , and so on. At some point we mustcontent ourselves with “and so on.” We cannot enu-merate all fractions or all irrational numbers, or allof any other infinite collection. Thus our knowledgein regard to all such collections can only be derivedfrom a definition by intension.

These remarks are relevant, when we are seekingthe definition of number, in three different ways.In the first place, numbers themselves form an infi-nite collection, and cannot therefore be defined byenumeration. In the second place, the collectionshaving a given number of terms themselves presum-

ably form an infinite collection: it is to be presumed,for example, that there are an infinite collection oftrios in the world, for if this were not the case thetotal number of things in the world would be fi-nite, which, though possible, seems unlikely. In thethird place, we wish to define “number” in such away that infinite numbers may be possible; thus wemust be able to speak of the number of terms inan infinite collection, and such a collection must bedefined by intension, i.e. by a property common toall its members and peculiar to them.

For many purposes, a class and a defining char-acteristic of it are practically interchangeable. Thevital difference between the two consists in the factthat there is only one class having a given set ofmembers, whereas there are always many differentcharacteristics by which a given class may be de-fined. Men | may be defined as featherless bipeds, or as rational animals, or (more correctly) by thetraits by which Swift delineates the Yahoos. It is thisfact that a defining characteristic is never uniquewhich makes classes useful; otherwise we could becontent with the properties common and peculiarto their members. Any one of these properties can

As will be explained later, classes may be regarded as logicalfictions, manufactured out of defining characteristics. But for

be used in place of the class whenever uniquenessis not important.

Returning now to the definition of number, itis clear that number is a way of bringing togethercertain collections, namely, those that have a givennumber of terms. We can suppose all couples inone bundle, all trios in another, and so on. In thisway we obtain various bundles of collections, eachbundle consisting of all the collections that havea certain number of terms. Each bundle is a classwhose members are collections, i.e. classes; thuseach is a class of classes. The bundle consistingof all couples, for example, is a class of classes:each couple is a class with two members, and thewhole bundle of couples is a class with an infinitenumber of members, each of which is a class of twomembers.

How shall we decide whether two collections areto belong to the same bundle? The answer that sug-gests itself is: “Find out how many members eachhas, and put them in the same bundle if they havethe same number of members.” But this presup-poses that we have defined numbers, and that weknow how to discover how many terms a collection

the present it will simplify our exposition to treat classes as ifthey were real.

has. We are so used to the operation of countingthat such a presupposition might easily pass unno-ticed. In fact, however, counting, though familiar,is logically a very complex operation; moreover itis only available, as a means of discovering howmany terms a collection has, when the collectionis finite. Our definition of number must not as-sume in advance that all numbers are finite; and wecannot in any case, without a vicious circle, | use counting to define numbers, because numbers areused in counting. We need, therefore, some othermethod of deciding when two collections have thesame number of terms.

In actual fact, it is simpler logically to find outwhether two collections have the same number ofterms than it is to define what that number is. Anillustration will make this clear. If there were nopolygamy or polyandry anywhere in the world, itis clear that the number of husbands living at anymoment would be exactly the same as the number ofwives. We do not need a census to assure us of this,nor do we need to know what is the actual numberof husbands and of wives. We know the numbermust be the same in both collections, because eachhusband has one wife and each wife has one hus-band. The relation of husband and wife is what is

called “one-one.”A relation is said to be “one-one” when, if x

has the relation in question to y, no other term x′

has the same relation to y, and x does not have thesame relation to any term y′ other than y. Whenonly the first of these two conditions is fulfilled,the relation is called “one-many”; when only thesecond is fulfilled, it is called “many-one.” It shouldbe observed that the number is not used in thesedefinitions.

In Christian countries, the relation of husbandto wife is one-one; in Mahometan countries it isone-many; in Tibet it is many-one. The relation offather to son is one-many; that of son to father ismany-one, but that of eldest son to father is one-one. If n is any number, the relation of n to n + is one-one; so is the relation of n to n or to n.When we are considering only positive numbers,the relation of n to n is one-one; but when negativenumbers are admitted, it becomes two-one, sincen and −n have the same square. These instancesshould suffice to make clear the notions of one-one,one-many, and many-one relations, which play agreat part in the principles of mathematics, not onlyin relation to the definition of numbers, but in manyother connections.

Two classes are said to be “similar” when thereis a one-one | relation which correlates the terms of the one class each with one term of the otherclass, in the same manner in which the relation ofmarriage correlates husbands with wives. A fewpreliminary definitions will help us to state thisdefinition more precisely. The class of those termsthat have a given relation to something or other iscalled the domain of that relation: thus fathers arethe domain of the relation of father to child, hus-bands are the domain of the relation of husband towife, wives are the domain of the relation of wife tohusband, and husbands and wives together are thedomain of the relation of marriage. The relation ofwife to husband is called the converse of the relationof husband to wife. Similarly less is the converseof greater, later is the converse of earlier, and so on.Generally, the converse of a given relation is thatrelation which holds between y and x whenever thegiven relation holds between x and y. The conversedomain of a relation is the domain of its converse:thus the class of wives is the converse domain of therelation of husband to wife. We may now state ourdefinition of similarity as follows:—

One class is said to be “similar” to another whenthere is a one-one relation of which the one class is the

domain, while the other is the converse domain.It is easy to prove () that every class is similar to

itself, () that if a class α is similar to a class β, thenβ is similar to α, () that if α is similar to β and β toγ , then α is similar to γ . A relation is said to be re-flexive when it possesses the first of these properties,symmetrical when it possesses the second, and tran-sitive when it possesses the third. It is obvious thata relation which is symmetrical and transitive mustbe reflexive throughout its domain. Relations whichpossess these properties are an important kind, andit is worth while to note that similarity is one of thiskind of relations.

It is obvious to common sense that two finiteclasses have the same number of terms if they aresimilar, but not otherwise. The act of counting con-sists in establishing a one-one correlation | between the set of objects counted and the natural numbers(excluding ) that are used up in the process. Ac-cordingly common sense concludes that there areas many objects in the set to be counted as there arenumbers up to the last number used in the count-ing. And we also know that, so long as we confineourselves to finite numbers, there are just n num-bers from up to n. Hence it follows that the lastnumber used in counting a collection is the number

of terms in the collection, provided the collectionis finite. But this result, besides being only appli-cable to finite collections, depends upon and as-sumes the fact that two classes which are similarhave the same number of terms; for what we dowhen we count (say) objects is to show that theset of these objects is similar to the set of numbers to . The notion of similarity is logically presup-posed in the operation of counting, and is logicallysimpler though less familiar. In counting, it is nec-essary to take the objects counted in a certain order,as first, second, third, etc., but order is not of theessence of number: it is an irrelevant addition, anunnecessary complication from the logical point ofview. The notion of similarity does not demand anorder: for example, we saw that the number of hus-bands is the same as the number of wives, withouthaving to establish an order of precedence amongthem. The notion of similarity also does not requirethat the classes which are similar should be finite.Take, for example, the natural numbers (excluding) on the one hand, and the fractions which have for their numerator on the other hand: it is obviousthat we can correlate with /, with /, and soon, thus proving that the two classes are similar.

We may thus use the notion of “similarity” to de-

cide when two collections are to belong to the samebundle, in the sense in which we were asking thisquestion earlier in this chapter. We want to makeone bundle containing the class that has no mem-bers: this will be for the number . Then we want abundle of all the classes that have one member: thiswill be for the number . Then, for the number ,we want a bundle consisting | of all couples; then one of all trios; and so on. Given any collection, wecan define the bundle it is to belong to as being theclass of all those collections that are “similar” to it.It is very easy to see that if (for example) a collectionhas three members, the class of all those collectionsthat are similar to it will be the class of trios. Andwhatever number of terms a collection may have,those collections that are “similar” to it will havethe same number of terms. We may take this as adefinition of “having the same number of terms.” Itis obvious that it gives results conformable to usageso long as we confine ourselves to finite collections.

So far we have not suggested anything in theslightest degree paradoxical. But when we come tothe actual definition of numbers we cannot avoidwhat must at first sight seem a paradox, though thisimpression will soon wear off. We naturally thinkthat the class of couples (for example) is something

different from the number . But there is no doubtabout the class of couples: it is indubitable and notdifficult to define, whereas the number , in anyother sense, is a metaphysical entity about whichwe can never feel sure that it exists or that we havetracked it down. It is therefore more prudent to con-tent ourselves with the class of couples, which weare sure of, than to hunt for a problematical number which must always remain elusive. Accordinglywe set up the following definition:—

The number of a class is the class of all those classesthat are similar to it.

Thus the number of a couple will be the classof all couples. In fact, the class of all couples willbe the number , according to our definition. Atthe expense of a little oddity, this definition securesdefiniteness and indubitableness; and it is not diffi-cult to prove that numbers so defined have all theproperties that we expect numbers to have.

We may now go on to define numbers in generalas any one of the bundles into which similarity col-lects classes. A number will be a set of classes suchas that any two are similar to each | other, and none outside the set are similar to any inside the set. Inother words, a number (in general) is any collectionwhich is the number of one of its members; or, more

simply still:A number is anything which is the number of some

class.Such a definition has a verbal appearance of be-

ing circular, but in fact it is not. We define “thenumber of a given class” without using the notionof number in general; therefore we may define num-ber in general in terms of “the number of a givenclass” without committing any logical error.

Definitions of this sort are in fact very common.The class of fathers, for example, would have to bedefined by first defining what it is to be the fatherof somebody; then the class of fathers will be allthose who are somebody’s father. Similarly if wewant to define square numbers (say), we must firstdefine what we mean by saying that one number isthe square of another, and then define square num-bers as those that are the squares of other numbers.This kind of procedure is very common, and it isimportant to realise that it is legitimate and evenoften necessary.

We have now given a definition of numbers whichwill serve for finite collections. It remains to beseen how it will serve for infinite collections. Butfirst we must decide what we mean by “finite” and

“infinite,” which cannot be done within the limits ofthe present chapter.

Chapter III

Finitude and Mathematical Induction

The series of natural numbers, as we saw in Chapter I., can all be defined if we know what we meanby the three terms “,” “number,” and “successor.”But we may go a step farther: we can define all thenatural numbers if we know what we mean by “”and “successor.” It will help us to understand thedifference between finite and infinite to see howthis can be done, and why the method by which itis done cannot be extended beyond the finite. Wewill not yet consider how “” and “successor” are tobe defined: we will for the moment assume that weknow what these terms mean, and show how thenceall other natural numbers can be obtained.

It is easy to see that we can reach any assignednumber, say ,. We first define “” as “the suc-cessor of ,” then we define “” as “the successor

of ,” and so on. In the case of an assigned num-ber, such as ,, the proof that we can reach itby proceeding step by step in this fashion may bemade, if we have the patience, by actual experiment:we can go on until we actually arrive at ,. Butalthough the method of experiment is available foreach particular natural number, it is not availablefor proving the general proposition that all suchnumbers can be reached in this way, i.e. by proceed-ing from step by step from each number to itssuccessor. Is there any other way by which this canbe proved?

Let us consider the question the other way round.What are the numbers that can be reached, giventhe terms “” and | “successor”? Is there any way by which we can define the whole class of such num-bers? We reach , as the successor of ; , as thesuccessor of ; , as the successor of ; and so on.It is this “and so on” that we wish to replace bysomething less vague and indefinite. We might betempted to say that “and so on” means that the pro-cess of proceeding to the successor may be repeatedany finite number of times; but the problem uponwhich we are engaged is the problem of defining“finite number,” and therefore we must not use thisnotion in our definition. Our definition must not

assume that we know what a finite number is.The key to our problem lies in mathematical in-

duction. It will be remembered that, in Chapter I.,this was the fifth of the five primitive propositionswhich we laid down about the natural numbers. Itstated that any property which belongs to , and tothe successor of any number which has the property,belongs to all the natural numbers. This was thenpresented as a principle, but we shall now adoptit as a definition. It is not difficult to see that theterms obeying it are the same as the numbers thatcan be reached from by successive steps from nextto next, but as the point is important we will setforth the matter in some detail.

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

A property is said to be “hereditary” in the na-tural-number series if, whenever it belongs to anumber n, it also belongs to n + , the successorof n. Similarly a class is said to be “hereditary” if,whenever n is a member of the class, so is n+. It iseasy to see, though we are not yet supposed to know,that to say a property is hereditary is equivalent tosaying that it belongs to all the natural numbers notless than some one of them, e.g. it must belong toall that are not less than , or all that are not less

than , or it may be that it belongs to all that arenot less than , i.e. to all without exception.

A property is said to be “inductive” when it is ahereditary | property which belongs to . Similarly a class is “inductive” when it is a hereditary class ofwhich is a member.

Given a hereditary class of which is a member,it follows that is a member of it, because a hered-itary class contains the successors of its members,and is the successor of . Similarly, given a hered-itary class of which is a member, it follows that is a member of it; and so on. Thus we can prove bya step-by-step procedure that any assigned naturalnumber, say ,, is a member of every inductiveclass.

We will define the “posterity” of a given naturalnumber with respect to the relation “immediatepredecessor” (which is the converse of “successor”)as all those terms that belong to every hereditaryclass to which the given number belongs. It is againeasy to see that the posterity of a natural numberconsists of itself and all greater natural numbers;but this also we do not yet officially know.

By the above definitions, the posterity of willconsist of those terms which belong to every induc-tive class.

It is now not difficult to make it obvious thatthe posterity of is the same set as those terms thatcan be reached from by successive steps from nextto next. For, in the first place, belongs to boththese sets (in the sense in which we have definedour terms); in the second place, if n belongs to bothsets, so does n+ . It is to be observed that we aredealing here with the kind of matter that does notadmit of precise proof, namely, the comparison of arelatively vague idea with a relatively precise one.The notion of “those terms that can be reached from by successive steps from next to next” is vague,though it seems as if it conveyed a definite meaning;on the other hand, “the posterity of ” is precise andexplicit just where the other idea is hazy. It may betaken as giving what we meant to mean when wespoke of the terms that can be reached from bysuccessive steps.

We now lay down the following definition:—The “natural numbers” are the posterity of with

respect to the | relation “immediate predecessor” (which is the converse of “successor”).

We have thus arrived at a definition of one ofPeano’s three primitive ideas in terms of the othertwo. As a result of this definition, two of his prim-itive propositions—namely, the one asserting that

is a number and the one asserting mathematicalinduction—become unnecessary, since they resultfrom the definition. The one asserting that the suc-cessor of a natural number is a natural number isonly needed in the weakened form “every naturalnumber has a successor.”

We can, of course, easily define “” and “succes-sor” by means of the definition of number in generalwhich we arrived at in Chapter II. The number isthe number of terms in a class which has no mem-bers, i.e. in the class which is called the “null-class.”By the general definition of number, the number ofterms in the null-class is the set of all classes similarto the null-class, i.e. (as is easily proved) the set con-sisting of the null-class all alone, i.e. the class whoseonly member is the null-class. (This is not identi-cal with the null-class: it has one member, namely,the null-class, whereas the null-class itself has nomembers. A class which has one member is neveridentical with that one member, as we shall explainwhen we come to the theory of classes.) Thus wehave the following purely logical definition:— is the class whose only member is the null-class.It remains to define “successor.” Given any num-

ber n, let α be a class which has n members, and letx be a term which is not a member of α. Then the

class consisting of α with x added on will have n+members. Thus we have the following definition:—

The successor of the number of terms in the classα is the number of terms in the class consisting of αtogether with x, where x is any term not belonging tothe class.

Certain niceties are required to make this def-inition perfect, but they need not concern us. Itwill be remembered that we | have already given (in Chapter II.) a logical definition of the number ofterms in a class, namely, we defined it as the set ofall classes that are similar to the given class.

We have thus reduced Peano’s three primitiveideas to ideas of logic: we have given definitions ofthem which make them definite, no longer capableof an infinity of different meanings, as they werewhen they were only determinate to the extent ofobeying Peano’s five axioms. We have removed themfrom the fundamental apparatus of terms that mustbe merely apprehended, and have thus increasedthe deductive articulation of mathematics.

As regards the five primitive propositions, wehave already succeeded in making two of themdemonstrable by our definition of “natural num-ber.” How stands it with the remaining three? It is

See Principia Mathematica, vol. ii. ∗.

very easy to prove that is not the successor of anynumber, and that the successor of any number is anumber. But there is a difficulty about the remain-ing primitive proposition, namely, “no two numbershave the same successor.” The difficulty does notarise unless the total number of individuals in theuniverse is finite; for given two numbers m and n,neither of which is the total number of individualsin the universe, it is easy to prove that we cannothave m + = n + unless we have m = n. But letus suppose that the total number of individuals inthe universe were (say) ; then there would be noclass of individuals, and the number wouldbe the null-class. So would the number . Thus weshould have = ; therefore the successor of would be the same as the successor of , although would not be the same as . Thus we shouldhave two different numbers with the same succes-sor. This failure of the third axiom cannot arise,however, if the number of individuals in the worldis not finite. We shall return to this topic at a laterstage.

Assuming that the number of individuals in theuniverse is not finite, we have now succeeded notonly in defining Peano’s | three primitive ideas, but

See Chapter XIII.

in seeing how to prove his five primitive proposi-tions, by means of primitive ideas and propositionsbelonging to logic. It follows that all pure mathe-matics, in so far as it is deducible from the theoryof the natural numbers, is only a prolongation oflogic. The extension of this result to those modernbranches of mathematics which are not deduciblefrom the theory of the natural numbers offers no dif-ficulty of principle, as we have shown elsewhere.

The process of mathematical induction, by meansof which we defined the natural numbers, is capableof generalisation. We defined the natural numbersas the “posterity” of with respect to the relation ofa number to its immediate successor. If we call thisrelation N, any number m will have this relation tom+ . A property is “hereditary with respect to N,”or simply “N-hereditary,” if, whenever the propertybelongs to a number m, it also belongs to m+ , i.e.to the number to which m has the relation N. And anumber n will be said to belong to the “posterity”of m with respect to the relation N if n has everyN-hereditary property belonging to m. These defi-nitions can all be applied to any other relation just

For geometry, in so far as it is not purely analytical, seePrinciples of Mathematics, part vi.; for rational dynamics, ibid.,part vii.

as well as to N. Thus if R is any relation whatever,we can lay down the following definitions:—

A property is called “R-hereditary” when, if itbelongs to a term x, and x has the relation R to y,then it belongs to y.

A class is R-hereditary when its defining prop-erty is R-hereditary.

A term x is said to be an “R-ancestor” of the termy if y has every R-hereditary property that x has,provided x is a term which has the relation R tosomething or to which something has the relationR. (This is only to exclude trivial cases.) |

The “R-posterity” of x is all the terms of which x is an R-ancestor.

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

It will be observed that if we take for R the rela-tion “parent,” “ancestor” and “posterity” will havethe usual meanings, except that a person will be in-cluded among his own ancestors and posterity. It is,

These definitions, and the generalised theory of induction,are due to Frege, and were published so long ago as in hisBegriffsschrift. In spite of the great value of this work, I was,I believe, the first person who ever read it—more than twentyyears after its publication.

of course, obvious at once that “ancestor” must becapable of definition in terms of “parent,” but untilFrege developed his generalised theory of induc-tion, no one could have defined “ancestor” preciselyin terms of “parent.” A brief consideration of thispoint will serve to show the importance of the the-ory. A person confronted for the first time with theproblem of defining “ancestor” in terms of “parent”would naturally say that A is an ancestor of Z if,between A and Z, there are a certain number of peo-ple, B, C, . . . , of whom B is a child of A, each is aparent of the next, until the last, who is a parent ofZ. But this definition is not adequate unless we addthat the number of intermediate terms is to be finite.Take, for example, such a series as the following:—

−, − , − , −

, . . .

, , , .

Here we have first a series of negative fractions withno end, and then a series of positive fractions withno beginning. Shall we say that, in this series, −/is an ancestor of /? It will be so according to thebeginner’s definition suggested above, but it willnot be so according to any definition which will givethe kind of idea that we wish to define. For thispurpose, it is essential that the number of interme-diaries should be finite. But, as we saw, “finite” is

to be defined by means of mathematical induction,and it is simpler to define the ancestral relation gen-erally at once than to define it first only for the caseof the relation of n to n + , and then extend it toother cases. Here, as constantly elsewhere, gener-ality from the first, though it may | require more thought at the start, will be found in the long run toeconomise thought and increase logical power.

The use of mathematical induction in demon-strations was, in the past, something of a mystery.There seemed no reasonable doubt that it was avalid method of proof, but no one quite knew whyit was valid. Some believed it to be really a case ofinduction, in the sense in which that word is usedin logic. Poincare considered it to be a principle ofthe utmost importance, by means of which an infi-nite number of syllogisms could be condensed intoone argument. We now know that all such viewsare mistaken, and that mathematical induction isa definition, not a principle. There are some num-bers to which it can be applied, and there are others(as we shall see in Chapter VIII.) to which it can-not be applied. We define the “natural numbers” asthose to which proofs by mathematical inductioncan be applied, i.e. as those that possess all induc-

Science and Method, chap. iv.

tive properties. It follows that such proofs can beapplied to the natural numbers, not in virtue of anymysterious intuition or axiom or principle, but asa purely verbal proposition. If “quadrupeds” aredefined as animals having four legs, it will followthat animals that have four legs are quadrupeds;and the case of numbers that obey mathematicalinduction is exactly similar.

We shall use the phrase “inductive numbers” tomean the same set as we have hitherto spoken ofas the “natural numbers.” The phrase “inductivenumbers” is preferable as affording a reminder thatthe definition of this set of numbers is obtained frommathematical induction.

Mathematical induction affords, more than any-thing else, the essential characteristic by which thefinite is distinguished from the infinite. The prin-ciple of mathematical induction might be statedpopularly in some such form as “what can be in-ferred from next to next can be inferred from first tolast.” This is true when the number of intermediatesteps between first and last is finite, not otherwise.Anyone who has ever | watched a goods train begin- ning to move will have noticed how the impulse iscommunicated with a jerk from each truck to thenext, until at last even the hindmost truck is in mo-

tion. When the train is very long, it is a very longtime before the last truck moves. If the train wereinfinitely long, there would be an infinite succes-sion of jerks, and the time would never come whenthe whole train would be in motion. Nevertheless,if there were a series of trucks no longer than theseries of inductive numbers (which, as we shall see,is an instance of the smallest of infinites), everytruck would begin to move sooner or later if theengine persevered, though there would always beother trucks further back which had not yet begunto move. This image will help to elucidate the ar-gument from next to next, and its connection withfinitude. When we come to infinite numbers, wherearguments from mathematical induction will be nolonger valid, the properties of such numbers willhelp to make clear, by contrast, the almost uncon-scious use that is made of mathematical inductionwhere finite numbers are concerned.

Chapter IV

The Definition of Order

We have now carried our analysis of the series of natural numbers to the point where we have ob-tained logical definitions of the members of thisseries, of the whole class of its members, and of therelation of a number to its immediate successor. Wemust now consider the serial character of the naturalnumbers in the order , , , , . . . We ordinarilythink of the numbers as in this order, and it is anessential part of the work of analysing our data toseek a definition of “order” or “series” in logicalterms.

The notion of order is one which has enormousimportance in mathematics. Not only the integers,but also rational fractions and all real numbershave an order of magnitude, and this is essentialto most of their mathematical properties. The order

of points on a line is essential to geometry; so is theslightly more complicated order of lines through apoint in a plane, or of planes through a line. Di-mensions, in geometry, are a development of order.The conception of a limit, which underlies all highermathematics, is a serial conception. There are partsof mathematics which do not depend upon the no-tion of order, but they are very few in comparisonwith the parts in which this notion is involved.

In seeking a definition of order, the first thingto realise is that no set of terms has just one orderto the exclusion of others. A set of terms has allthe orders of which it is capable. Sometimes oneorder is so much more familiar and natural to our| thoughts that we are inclined to regard it as the order of that set of terms; but this is a mistake. Thenatural numbers—or the “inductive” numbers, aswe shall also call them—occur to us most readilyin order of magnitude; but they are capable of aninfinite number of other arrangements. We might,for example, consider first all the odd numbers andthen all the even numbers; or first , then all theeven numbers, then all the odd multiples of , thenall the multiples of but not of or , then allthe multiples of but not of or or , and soon through the whole series of primes. When we

say that we “arrange” the numbers in these variousorders, that is an inaccurate expression: what wereally do is to turn our attention to certain relationsbetween the natural numbers, which themselvesgenerate such-and-such an arrangement. We can nomore “arrange” the natural numbers than we can thestarry heavens; but just as we may notice among thefixed stars either their order of brightness or theirdistribution in the sky, so there are various relationsamong numbers which may be observed, and whichgive rise to various different orders among numbers,all equally legitimate. And what is true of numbersis equally true of points on a line or of the momentsof time: one order is more familiar, but others areequally valid. We might, for example, take first, ona line, all the points that have integral co-ordinates,then all those that have non-integral rational co-ordinates, then all those that have algebraic non-rational co-ordinates, and so on, through any set ofcomplications we please. The resulting order willbe one which the points of the line certainly have,whether we choose to notice it or not; the only thingthat is arbitrary about the various orders of a setof terms is our attention, for the terms themselveshave always all the orders of which they are capable.

One important result of this consideration is that

we must not look for the definition of order in thenature of the set of terms to be ordered, since one setof terms has many orders. The order lies, not in theclass of terms, but in a relation among | the members of the class, in respect of which some appear asearlier and some as later. The fact that a class mayhave many orders is due to the fact that there can bemany relations holding among the members of onesingle class. What properties must a relation havein order to give rise to an order?

The essential characteristics of a relation whichis to give rise to order may be discovered by con-sidering that in respect of such a relation we mustbe able to say, of any two terms in the class whichis to be ordered, that one “precedes” and the other“follows.” Now, in order that we may be able to usethese words in the way in which we should natu-rally understand them, we require that the orderingrelation should have three properties:—

() If x precedes y, y must not also precede x.This is an obvious characteristic of the kind of re-lations that lead to series. If x is less than y, y isnot also less than x. If x is earlier in time than y, yis not also earlier than x. If x is to the left of y, yis not to the left of x. On the other hand, relationswhich do not give rise to series often do not have

this property. If x is a brother or sister of y, y is abrother or sister of x. If x is of the same height asy, y is of the same height as x. If x is of a differentheight from y, y is of a different height from x. Inall these cases, when the relation holds between xand y, it also holds between y and x. But with serialrelations such a thing cannot happen. A relationhaving this first property is called asymmetrical.

() If x precedes y and y precedes z, x must pre-cede z. This may be illustrated by the same instancesas before: less, earlier, left of. But as instances of re-lations which do not have this property only twoof our previous three instances will serve. If x isbrother or sister of y, and y of z, x may not be brotheror sister of z, since x and z may be the same person.The same applies to difference of height, but notto sameness of height, which has our second prop-erty but not our first. The relation “father,” on theother hand, has our first property but not | our sec- ond. A relation having our second property is calledtransitive.

() Given any two terms of the class which isto be ordered, there must be one which precedesand the other which follows. For example, of anytwo integers, or fractions, or real numbers, one issmaller and the other greater; but of any two com-

plex numbers this is not true. Of any two momentsin time, one must be earlier than the other; but ofevents, which may be simultaneous, this cannot besaid. Of two points on a line, one must be to the leftof the other. A relation having this third property iscalled connected.

When a relation possesses these three proper-ties, it is of the sort to give rise to an order amongthe terms between which it holds; and whereveran order exists, some relation having these threeproperties can be found generating it.

Before illustrating this thesis, we will introducea few definitions.

() A relation is said to be an aliorelative, orto be contained in or imply diversity, if no term hasthis relation to itself. Thus, for example, “greater,”“different in size,” “brother,” “husband,” “father”are aliorelatives; but “equal,” “born of the sameparents,” “dear friend” are not.

() The square of a relation is that relation whichholds between two terms x and z when there is anintermediate term y such that the given relationholds between x and y and between y and z. Thus“paternal grandfather” is the square of “father,”“greater by ” is the square of “greater by ,” and

This term is due to C. S. Peirce.

so on.() The domain of a relation consists of all those

terms that have the relation to something or other,and the converse domain consists of all those termsto which something or other has the relation. Thesewords have been already defined, but are recalledhere for the sake of the following definition:—

() The field of a relation consists of its domainand converse domain together. |

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

It will be seen that an asymmetrical relation isthe same thing as a relation whose square is analiorelative. It often happens that a relation is analiorelative without being asymmetrical, though anasymmetrical relation is always an aliorelative. Forexample, “spouse” is an aliorelative, but is symmet-rical, since if x is the spouse of y, y is the spouse ofx. But among transitive relations, all aliorelativesare asymmetrical as well as vice versa.

From the definitions it will be seen that a tran-sitive relation is one which is implied by its square,or, as we also say, “contains” its square. Thus “an-cestor” is transitive, because an ancestor’s ancestoris an ancestor; but “father” is not transitive, becausea father’s father is not a father. A transitive aliorel-

ative is one which contains its square and is con-tained in diversity; or, what comes to the same thing,one whose square implies both it and diversity—because, when a relation is transitive, asymmetry isequivalent to being an aliorelative.

A relation is connected when, given any two dif-ferent terms of its field, the relation holds betweenthe first and the second or between the second andthe first (not excluding the possibility that both mayhappen, though both cannot happen if the relationis asymmetrical).

It will be seen that the relation “ancestor,” forexample, is an aliorelative and transitive, but notconnected; it is because it is not connected that itdoes not suffice to arrange the human race in a se-ries.

The relation “less than or equal to,” among num-bers, is transitive and connected, but not asymmet-rical or an aliorelative.

The relation “greater or less” among numbers isan aliorelative and is connected, but is not transi-tive, for if x is greater or less than y, and y is greateror less than z, it may happen that x and z are thesame number.

Thus the three properties of being () an aliorel-ative, () | transitive, and () connected, are mutu-

ally independent, since a relation may have any twowithout having the third.

We now lay down the following definition:—A relation is serial when it is an aliorelative, tran-

sitive, and connected; or, what is equivalent, whenit is asymmetrical, transitive, and connected.

A series is the same thing as a serial relation.It might have been thought that a series should

be the field of a serial relation, not the serial relationitself. But this would be an error. For example,

, , ; , , ; , , ; , , ; , , ; , ,

are six different series which all have the same field.If the field were the series, there could only be oneseries with a given field. What distinguishes theabove six series is simply the different ordering re-lations in the six cases. Given the ordering relation,the field and the order are both determinate. Thusthe ordering relation may be taken to be the series,but the field cannot be so taken.

Given any serial relation, say P, we shall say that,in respect of this relation, x “precedes” y if x hasthe relation P to y, which we shall write “xPy” forshort. The three characteristics which P must havein order to be serial are:

() We must never have xPx, i.e. no term mustprecede itself.

() P must imply P, i.e. if x precedes y and yprecedes z, x must precede z.

() If x and y are two different terms in the fieldof P, we shall have xPy or yPx, i.e. one of thetwo must precede the other.

The reader can easily convince himself that, wherethese three properties are found in an ordering rela-tion, the characteristics we expect of series will alsobe found, and vice versa. We are therefore justifiedin taking the above as a definition of order | or se- ries. And it will be observed that the definition iseffected in purely logical terms.

Although a transitive asymmetrical connectedrelation always exists wherever there is a series, itis not always the relation which would most nat-urally be regarded as generating the series. Thenatural-number series may serve as an illustration.The relation we assumed in considering the naturalnumbers was the relation of immediate succession,i.e. the relation between consecutive integers. Thisrelation is asymmetrical, but not transitive or con-nected. We can, however, derive from it, by themethod of mathematical induction, the “ancestral”relation which we considered in the preceding chap-

ter. This relation will be the same as “less than orequal to” among inductive integers. For purposes ofgenerating the series of natural numbers, we wantthe relation “less than,” excluding “equal to.” Thisis the relation of m to n when m is an ancestor of nbut not identical with n, or (what comes to the samething) when the successor of m is an ancestor of nin the sense in which a number is its own ances-tor. That is to say, we shall lay down the followingdefinition:—

An inductive number m is said to be less than an-other number n when n possesses every hereditaryproperty possessed by the successor of m.

It is easy to see, and not difficult to prove, thatthe relation “less than,” so defined, is asymmetri-cal, transitive, and connected, and has the inductivenumbers for its field. Thus by means of this rela-tion the inductive numbers acquire an order in thesense in which we defined the term “order,” andthis order is the so-called “natural” order, or orderof magnitude.

The generation of series by means of relationsmore or less resembling that of n to n + is verycommon. The series of the Kings of England, forexample, is generated by relations of each to his suc-cessor. This is probably the easiest way, where it is

applicable, of conceiving the generation of a series.In this method we pass on from each term to thenext, as long as there | is a next, or back to the one before, as long as there is one before. This methodalways requires the generalised form of mathemati-cal induction in order to enable us to define “earlier”and “later” in a series so generated. On the analogyof “proper fractions,” let us give the name “properposterity of x with respect to R” to the class of thoseterms that belong to the R-posterity of some termto which x has the relation R, in the sense which wegave before to “posterity,” which includes a termin its own posterity. Reverting to the fundamentaldefinitions, we find that the “proper posterity” maybe defined as follows:—

The “proper posterity” of x with respect to Rconsists of all terms that possess every R-hereditaryproperty possessed by every term to which x has therelation R.

It is to be observed that this definition has to beso framed as to be applicable not only when there isonly one term to which x has the relation R, but alsoin cases (as e.g. that of father and child) where theremay be many terms to which x has the relation R.We define further:

A term x is a “proper ancestor” of y with respect

to R if y belongs to the proper posterity of x withrespect to R.

We shall speak for short of “R-posterity” and“R-ancestors” when these terms seem more conve-nient.

Reverting now to the generation of series by therelation R between consecutive terms, we see that,if this method is to be possible, the relation “properR-ancestor” must be an aliorelative, transitive, andconnected. Under what circumstances will this oc-cur? It will always be transitive: no matter whatsort of relation R may be, “R-ancestor” and “properR-ancestor” are always both transitive. But it isonly under certain circumstances that it will bean aliorelative or connected. Consider, for exam-ple, the relation to one’s left-hand neighbour at around dinner-table at which there are twelve peo-ple. If we call this relation R, the proper R-posterityof a person consists of all who can be reached bygoing round the table from right to left. This in-cludes everybody at the table, including the personhimself, since | twelve steps bring us back to our starting-point. Thus in such a case, though the rela-tion “proper R-ancestor” is connected, and thoughR itself is an aliorelative, we do not get a series be-cause “proper R-ancestor” is not an aliorelative. It

is for this reason that we cannot say that one personcomes before another with respect to the relation“right of” or to its ancestral derivative.

The above was an instance in which the ances-tral relation was connected but not contained indiversity. An instance where it is contained in diver-sity but not connected is derived from the ordinarysense of the word “ancestor.” If x is a proper ances-tor of y, x and y cannot be the same person; but itis not true that of any two persons one must be anancestor of the other.

The question of the circumstances under whichseries can be generated by ancestral relations de-rived from relations of consecutiveness is often im-portant. Some of the most important cases are thefollowing: Let R be a many-one relation, and let usconfine our attention to the posterity of some term x.When so confined, the relation “proper R-ancestor”must be connected; therefore all that remains to en-sure its being serial is that it shall be contained indiversity. This is a generalisation of the instance ofthe dinner-table. Another generalisation consists intaking R to be a one-one relation, and including theancestry of x as well as the posterity. Here again,the one condition required to secure the generationof a series is that the relation “proper R-ancestor”

shall be contained in diversity.The generation of order by means of relations

of consecutiveness, though important in its ownsphere, is less general than the method which uses atransitive relation to define the order. It often hap-pens in a series that there are an infinite number ofintermediate terms between any two that may beselected, however near together these may be. Take,for instance, fractions in order of magnitude. Be-tween any two fractions there are others—for exam-ple, the arithmetic mean of the two. Consequentlythere is no such thing as a pair of consecutive frac-tions. If we depended | upon consecutiveness for defining order, we should not be able to define theorder of magnitude among fractions. But in fact therelations of greater and less among fractions do notdemand generation from relations of consecutive-ness, and the relations of greater and less amongfractions have the three characteristics which weneed for defining serial relations. In all such casesthe order must be defined by means of a transitiverelation, since only such a relation is able to leapover an infinite number of intermediate terms. Themethod of consecutiveness, like that of counting fordiscovering the number of a collection, is appropri-ate to the finite; it may even be extended to certain

infinite series, namely, those in which, though thetotal number of terms is infinite, the number ofterms between any two is always finite; but it mustnot be regarded as general. Not only so, but caremust be taken to eradicate from the imagination allhabits of thought resulting from supposing it gen-eral. If this is not done, series in which there areno consecutive terms will remain difficult and puz-zling. And such series are of vital importance forthe understanding of continuity, space, time, andmotion.

There are many ways in which series may begenerated, but all depend upon the finding or con-struction of an asymmetrical transitive connectedrelation. Some of these ways have considerable im-portance. We may take as illustrative the generationof series by means of a three-term relation whichwe may call “between.” This method is very usefulin geometry, and may serve as an introduction torelations having more than two terms; it is best in-troduced in connection with elementary geometry.

Given any three points on a straight line in or-dinary space, there must be one of them which isbetween the other two. This will not be the casewith the points on a circle or any other closed curve,because, given any three points on a circle, we can

travel from any one to any other without passingthrough the third. In fact, the notion “between” ischaracteristic of open series—or series in the strictsense—as opposed to what may be called | “cyclic” series, where, as with people at the dinner-table,a sufficient journey brings us back to our starting-point. This notion of “between” may be chosen asthe fundamental notion of ordinary geometry; butfor the present we will only consider its applicationto a single straight line and to the ordering of thepoints on a straight line. Taking any two points a,b, the line (ab) consists of three parts (besides a andb themselves):

() Points between a and b.() Points x such that a is between x and b.() Points y such that b is between y and a.

Thus the line (ab) can be defined in terms of therelation “between.”

In order that this relation “between” may ar-range the points of the line in an order from leftto right, we need certain assumptions, namely, thefollowing:—

Cf. Rivista di Matematica, iv. pp. ff.; Principles of Mathemat-ics, p. (§).

() If anything is between a and b, a and b arenot identical.

() Anything between a and b is also between band a.

() Anything between a and b is not identicalwith a (nor, consequently, with b, in virtue of ()).

() If x is between a and b, anything between aand x is also between a and b.

() If x is between a and b, and b is between xand y, then b is between a and y.

() If x and y are between a and b, then either xand y are identical, or x is between a and y, or x isbetween y and b.

() If b is between a and x and also between a andy, then either x and y are identical, or x is between band y, or y is between b and x.

These seven properties are obviously verifiedin the case of points on a straight line in ordinaryspace. Any three-term relation which verifies themgives rise to series, as may be seen from the follow-ing definitions. For the sake of definiteness, let usassume | that a is to the left of b. Then the points of the line (ab) are () those between which and b, alies—these we will call to the left of a; () a itself; ()those between a and b; () b itself; () those betweenwhich and a lies b—these we will call to the right of

b. We may now define generally that of two pointsx, y, on the line (ab), we shall say that x is “to theleft of” y in any of the following cases:—

() When x and y are both to the left of a, and y isbetween x and a;

() When x is to the left of a, and y is a or b orbetween a and b or to the right of b;

() When x is a, and y is between a and b or is b oris to the right of b;

() When x and y are both between a and b, and yis between x and b;

() When x is between a and b, and y is b or to theright of b;

() When x is b and y is to the right of b;() When x and y are both to the right of b and x

is between b and y.

It will be found that, from the seven propertieswhich we have assigned to the relation “between,”it can be deduced that the relation “to the left of,”as above defined, is a serial relation as we definedthat term. It is important to notice that nothing inthe definitions or the argument depends upon ourmeaning by “between” the actual relation of thatname which occurs in empirical space: any three-term relation having the above seven purely formal

properties will serve the purpose of the argumentequally well.

Cyclic order, such as that of the points on a cir-cle, cannot be generated by means of three-termrelations of “between.” We need a relation of fourterms, which may be called “separation of couples.”The point may be illustrated by considering a jour-ney round the world. One may go from England toNew Zealand by way of Suez or by way of San Fran-cisco; we cannot | say definitely that either of these two places is “between” England and New Zealand.But if a man chooses that route to go round theworld, whichever way round he goes, his times inEngland and New Zealand are separated from eachother by his times in Suez and San Francisco, andconversely. Generalising, if we take any four pointson a circle, we can separate them into two couples,say a and b and x and y, such that, in order to getfrom a to b one must pass through either x or y, andin order to get from x to y one must pass througheither a or b. Under these circumstances we say thatthe couple (a, b) are “separated” by the couple (x, y).Out of this relation a cyclic order can be generated,in a way resembling that in which we generated anopen order from “between,” but somewhat more

complicated.

The purpose of the latter half of this chapterhas been to suggest the subject which one may call“generation of serial relations.” When such relationshave been defined, the generation of them fromother relations possessing only some of the proper-ties required for series becomes very important, es-pecially in the philosophy of geometry and physics.But we cannot, within the limits of the present vol-ume, do more than make the reader aware that sucha subject exists.

Cf. Principles of Mathematics, p. (§), and referencesthere given.

Chapter V

Kinds of Relations

A great part of the philosophy of mathematics is concerned with relations, and many different kindsof relations have different kinds of uses. It oftenhappens that a property which belongs to all rela-tions is only important as regards relations of cer-tain sorts; in these cases the reader will not see thebearing of the proposition asserting such a prop-erty unless he has in mind the sorts of relations forwhich it is useful. For reasons of this description, aswell as from the intrinsic interest of the subject, itis well to have in our minds a rough list of the moremathematically serviceable varieties of relations.

We dealt in the preceding chapter with a su-premely important class, namely, serial relations.Each of the three properties which we combined indefining series—namely, asymmetry, transitiveness,

and connexity—has its own importance. We willbegin by saying something on each of these three.

Asymmetry, i.e. the property of being incompati-ble with the converse, is a characteristic of the verygreatest interest and importance. In order to de-velop its functions, we will consider various exam-ples. The relation husband is asymmetrical, and sois the relation wife; i.e. if a is husband of b, b cannotbe husband of a, and similarly in the case of wife.On the other hand, the relation “spouse” is sym-metrical: if a is spouse of b, then b is spouse of a.Suppose now we are given the relation spouse, andwe wish to derive the relation husband. Husband isthe same as male spouse or spouse of a female; thusthe relation husband can | be derived from spouse ei- ther by limiting the domain to males or by limitingthe converse domain to females. We see from thisinstance that, when a symmetrical relation is given,it is sometimes possible, without the help of anyfurther relation, to separate it into two asymmetri-cal relations. But the cases where this is possible arerare and exceptional: they are cases where there aretwo mutually exclusive classes, say α and β, suchthat whenever the relation holds between two terms,one of the terms is a member of α and the other isa member of β—as, in the case of spouse, one term

of the relation belongs to the class of males and oneto the class of females. In such a case, the relationwith its domain confined to α will be asymmetrical,and so will the relation with its domain confinedto β. But such cases are not of the sort that occurwhen we are dealing with series of more than twoterms; for in a series, all terms, except the first andlast (if these exist), belong both to the domain andto the converse domain of the generating relation,so that a relation like husband, where the domainand converse domain do not overlap, is excluded.

The question how to construct relations havingsome useful property by means of operations uponrelations which only have rudiments of the propertyis one of considerable importance. Transitivenessand connexity are easily constructed in many caseswhere the originally given relation does not possessthem: for example, if R is any relation whatever, theancestral relation derived from R by generalised in-duction is transitive; and if R is a many-one relation,the ancestral relation will be connected if confinedto the posterity of a given term. But asymmetry is amuch more difficult property to secure by construc-tion. The method by which we derived husband fromspouse is, as we have seen, not available in the mostimportant cases, such as greater, before, to the right

of, where domain and converse domain overlap. Inall these cases, we can of course obtain a symmet-rical relation by adding together the given relationand its converse, but we cannot pass back from thissymmetrical relation to the original asymmetricalrelation except by the help of some asymmetrical |relation. Take, for example, the relation greater: the relation greater or less—i.e. unequal—is symmetri-cal, but there is nothing in this relation to show thatit is the sum of two asymmetrical relations. Takesuch a relation as “differing in shape.” This is notthe sum of an asymmetrical relation and its con-verse, since shapes do not form a single series; butthere is nothing to show that it differs from “differ-ing in magnitude” if we did not already know thatmagnitudes have relations of greater and less. Thisillustrates the fundamental character of asymmetryas a property of relations.

From the point of view of the classification of re-lations, being asymmetrical is a much more impor-tant characteristic than implying diversity. Asym-metrical relations imply diversity, but the converseis not the case. “Unequal,” for example, impliesdiversity, but is symmetrical. Broadly speaking, wemay say that, if we wished as far as possible to dis-pense with relational propositions and replace them

by such as ascribed predicates to subjects, we couldsucceed in this so long as we confined ourselves tosymmetrical relations: those that do not imply diver-sity, if they are transitive, may be regarded as assert-ing a common predicate, while those that do implydiversity may be regarded as asserting incompati-ble predicates. For example, consider the relationof similarity between classes, by means of which wedefined numbers. This relation is symmetrical andtransitive and does not imply diversity. It would bepossible, though less simple than the procedure weadopted, to regard the number of a collection as apredicate of the collection: then two similar classeswill be two that have the same numerical predicate,while two that are not similar will be two that havedifferent numerical predicates. Such a method of re-placing relations by predicates is formally possible(though often very inconvenient) so long as the rela-tions concerned are symmetrical; but it is formallyimpossible when the relations are asymmetrical, be-cause both sameness and difference of predicatesare symmetrical. Asymmetrical relations are, wemay | say, the most characteristically relational of relations, and the most important to the philosopherwho wishes to study the ultimate logical nature ofrelations.

Another class of relations that is of the greatestuse is the class of one-many relations, i.e. relationswhich at most one term can have to a given term.Such are father, mother, husband (except in Tibet),square of, sine of, and so on. But parent, square root,and so on, are not one-many. It is possible, formally,to replace all relations by one-many relations bymeans of a device. Take (say) the relation less amongthe inductive numbers. Given any number n greaterthan , there will not be only one number having therelation less to n, but we can form the whole class ofnumbers that are less than n. This is one class, andits relation to n is not shared by any other class. Wemay call the class of numbers that are less than nthe “proper ancestry” of n, in the sense in which wespoke of ancestry and posterity in connection withmathematical induction. Then “proper ancestry”is a one-many relation (one-many will always beused so as to include one-one), since each numberdetermines a single class of numbers as constitutingits proper ancestry. Thus the relation less than canbe replaced by being a member of the proper ancestryof. In this way a one-many relation in which theone is a class, together with membership of thisclass, can always formally replace a relation whichis not one-many. Peano, who for some reason always

instinctively conceives of a relation as one-many,deals in this way with those that are naturally notso. Reduction to one-many relations by this method,however, though possible as a matter of form, doesnot represent a technical simplification, and thereis every reason to think that it does not represent aphilosophical analysis, if only because classes mustbe regarded as “logical fictions.” We shall thereforecontinue to regard one-many relations as a specialkind of relations.

One-many relations are involved in all phrasesof the form “the so-and-so of such-and-such.” “TheKing of England,” | “the wife of Socrates,” “the fa- ther of John Stuart Mill,” and so on, all describesome person by means of a one-many relation to agiven term. A person cannot have more than onefather, therefore “the father of John Stuart Mill” de-scribed some one person, even if we did not knowwhom. There is much to say on the subject of de-scriptions, but for the present it is relations thatwe are concerned with, and descriptions are onlyrelevant as exemplifying the uses of one-many rela-tions. It should be observed that all mathematicalfunctions result from one-many relations: the loga-rithm of x, the cosine of x, etc., are, like the fatherof x, terms described by means of a one-many re-

lation (logarithm, cosine, etc.) to a given term (x).The notion of function need not be confined to num-bers, or to the uses to which mathematicians haveaccustomed us; it can be extended to all cases ofone-many relations, and “the father of x” is just aslegitimately a function of which x is the argumentas is “the logarithm of x.” Functions in this senseare descriptive functions. As we shall see later, thereare functions of a still more general and more fun-damental sort, namely, propositional functions; butfor the present we shall confine our attention to de-scriptive functions, i.e. “the term having the relationR to x,” or, for short, “the R of x,” where R is anyone-many relation.

It will be observed that if “the R of x” is to de-scribe a definite term, x must be a term to whichsomething has the relation R, and there must notbe more than one term having the relation R to x,since “the,” correctly used, must imply uniqueness.Thus we may speak of “the father of x” if x is anyhuman being except Adam and Eve; but we cannotspeak of “the father of x” if x is a table or a chair oranything else that does not have a father. We shallsay that the R of x “exists” when there is just oneterm, and no more, having the relation R to x. Thusif R is a one-many relation, the R of x exists when-

ever x belongs to the converse domain of R, and nototherwise. Regarding “the R of x” as a function inthe mathematical | sense, we say that x is the “argu- ment” of the function, and if y is the term which hasthe relation R to x, i.e. if y is the R of x, then y is the“value” of the function for the argument x. If R is aone-many relation, the range of possible argumentsto the function is the converse domain of R, and therange of values is the domain. Thus the range ofpossible arguments to the function “the father ofx” is all who have fathers, i.e. the converse domainof the relation father, while the range of possiblevalues for the function is all fathers, i.e. the domainof the relation.

Many of the most important notions in the logicof relations are descriptive functions, for example:converse, domain, converse domain, field. Other ex-amples will occur as we proceed.

Among one-many relations, one-one relations area specially important class. We have already hadoccasion to speak of one-one relations in connectionwith the definition of number, but it is necessary tobe familiar with them, and not merely to know theirformal definition. Their formal definition may bederived from that of one-many relations: they maybe defined as one-many relations which are also

the converses of one-many relations, i.e. as relationswhich are both one-many and many-one. One-manyrelations may be defined as relations such that, if xhas the relation in question to y, there is no otherterm x′ which also has the relation to y. Or, again,they may be defined as follows: Given two terms xand x′, the terms to which x has the given relationand those to which x′ has it have no member in com-mon. Or, again, they may be defined as relationssuch that the relative product of one of them and itsconverse implies identity, where the “relative prod-uct” of two relations R and S is that relation whichholds between x and z when there is an intermediateterm y, such that x has the relation R to y and y hasthe relation S to z. Thus, for example, if R is therelation of father to son, the relative product of Rand its converse will be the relation which holdsbetween x and a man z when there is a person y,such that x is the father of y and y is the son of z. Itis obvious that x and z must be | the same person. If, on the other hand, we take the relation of parentand child, which is not one-many, we can no longerargue that, if x is a parent of y and y is a child ofz, x and z must be the same person, because onemay be the father of y and the other the mother.This illustrates that it is characteristic of one-many

relations when the relative product of a relation andits converse implies identity. In the case of one-onerelations this happens, and also the relative productof the converse and the relation implies identity.Given a relation R, it is convenient, if x has the rela-tion R to y, to think of y as being reached from x byan “R-step” or an “R-vector.” In the same case x willbe reached from y by a “backward R-step.” Thus wemay state the characteristic of one-many relationswith which we have been dealing by saying that anR-step followed by a backward R-step must bringus back to our starting-point. With other relations,this is by no means the case; for example, if R is therelation of child to parent, the relative product ofR and its converse is the relation “self or brotheror sister,” and if R is the relation of grandchild tograndparent, the relative product of R and its con-verse is “self or brother or sister or first cousin.” Itwill be observed that the relative product of tworelations is not in general commutative, i.e. the rel-ative product of R and S is not in general the samerelation as the relative product of S and R. E.g. therelative product of parent and brother is uncle, butthe relative product of brother and parent is parent.

One-one relations give a correlation of two class-es, term for term, so that each term in either class

has its correlate in the other. Such correlations aresimplest to grasp when the two classes have nomembers in common, like the class of husbandsand the class of wives; for in that case we knowat once whether a term is to be considered as onefrom which the correlating relation R goes, or as oneto which it goes. It is convenient to use the wordreferent for the term from which the relation goes,and the term relatum for the term to which it goes.Thus if x and y are husband and wife, then, withrespect to the relation | “husband,” x is referent and y relatum, but with respect to the relation “wife,” yis referent and x relatum. We say that a relation andits converse have opposite “senses”; thus the “sense”of a relation that goes from x to y is the opposite ofthat of the corresponding relation from y to x. Thefact that a relation has a “sense” is fundamental,and is part of the reason why order can be generatedby suitable relations. It will be observed that theclass of all possible referents to a given relation isits domain, and the class of all possible relata is itsconverse domain.

But it very often happens that the domain andconverse domain of a one-one relation overlap. Take,for example, the first ten integers (excluding ), andadd to each; thus instead of the first ten integers

we now have the integers

, , , , , , , , , .

These are the same as those we had before, exceptthat has been cut off at the beginning and hasbeen joined on at the end. There are still ten inte-gers: they are correlated with the previous ten bythe relation of n to n+, which is a one-one relation.Or, again, instead of adding to each of our originalten integers, we could have doubled each of them,thus obtaining the integers

, , , , , , , , , .

Here we still have five of our previous set of integers,namely, , , , , . The correlating relation inthis case is the relation of a number to its double,which is again a one-one relation. Or we might havereplaced each number by its square, thus obtainingthe set

, , , , , , , , , .

On this occasion only three of our original set areleft, namely, , , . Such processes of correlationmay be varied endlessly.

The most interesting case of the above kind isthe case where our one-one relation has a converse

domain which is part, but | not the whole, of the domain. If, instead of confining the domain to thefirst ten integers, we had considered the whole of theinductive numbers, the above instances would haveillustrated this case. We may place the numbersconcerned in two rows, putting the correlate directlyunder the number whose correlate it is. Thus whenthe correlator is the relation of n to n+ , we havethe two rows:

, , , , , . . . n . . ., , , , , . . . n+ . . .

When the correlator is the relation of a number toits double, we have the two rows:

, , , , , . . . n . . ., , , , , . . . n . . .

When the correlator is the relation of a number toits square, the rows are:

, , , , , . . . n . . ., , , , , . . . n . . .

In all these cases, all inductive numbers occur in thetop row, and only some in the bottom row.

Cases of this sort, where the converse domainis a “proper part” of the domain (i.e. a part not thewhole), will occupy us again when we come to dealwith infinity. For the present, we wish only to notethat they exist and demand consideration.

Another class of correlations which are often im-portant is the class called “permutations,” wherethe domain and converse domain are identical. Con-sider, for example, the six possible arrangements ofthree letters:

a, b, ca, c, bb, c, ab, a, cc, a, bc, b, a |

Each of these can be obtained from any one of the others by means of a correlation. Take, for example,the first and last, (a, b, c) and (c, b, a). Here a iscorrelated with c, b with itself, and c with a. It isobvious that the combination of two permutationsis again a permutation, i.e. the permutations of agiven class form what is called a “group.”

These various kinds of correlations have impor-tance in various connections, some for one purpose,

some for another. The general notion of one-onecorrelations has boundless importance in the phi-losophy of mathematics, as we have partly seen al-ready, but shall see much more fully as we proceed.One of its uses will occupy us in our next chapter.

Chapter VI

Similarity of Relations

We saw in Chapter II. that two classes have the same number of terms when they are “similar,” i.e.when there is a one-one relation whose domain isthe one class and whose converse domain is theother. In such a case we say that there is a “one-onecorrelation” between the two classes.

In the present chapter we have to define a re-lation between relations, which will play the samepart for them that similarity of classes plays forclasses. We will call this relation “similarity of rela-tions,” or “likeness” when it seems desirable to usea different word from that which we use for classes.How is likeness to be defined?

We shall employ still the notion of correlation:we shall assume that the domain of the one rela-tion can be correlated with the domain of the other,

and the converse domain with the converse domain;but that is not enough for the sort of resemblancewhich we desire to have between our two relations.What we desire is that, whenever either relationholds between two terms, the other relation shallhold between the correlates of these two terms. Theeasiest example of the sort of thing we desire is amap. When one place is north of another, the placeon the map corresponding to the one is above theplace on the map corresponding to the other; whenone place is west of another, the place on the mapcorresponding to the one is to the left of the place onthe map corresponding to the other; and so on. Thestructure of the map corresponds with that of | the country of which it is a map. The space-relationsin the map have “likeness” to the space-relations inthe country mapped. It is this kind of connectionbetween relations that we wish to define.

We may, in the first place, profitably introducea certain restriction. We will confine ourselves, indefining likeness, to such relations as have “fields,”i.e. to such as permit of the formation of a singleclass out of the domain and the converse domain.This is not always the case. Take, for example, therelation “domain,” i.e. the relation which the do-main of a relation has to the relation. This relation

has all classes for its domain, since every class isthe domain of some relation; and it has all relationsfor its converse domain, since every relation has adomain. But classes and relations cannot be addedtogether to form a new single class, because they areof different logical “types.” We do not need to enterupon the difficult doctrine of types, but it is well toknow when we are abstaining from entering uponit. We may say, without entering upon the groundsfor the assertion, that a relation only has a “field”when it is what we call “homogeneous,” i.e. when itsdomain and converse domain are of the same logicaltype; and as a rough-and-ready indication of whatwe mean by a “type,” we may say that individuals,classes of individuals, relations between individu-als, relations between classes, relations of classes toindividuals, and so on, are different types. Now thenotion of likeness is not very useful as applied torelations that are not homogeneous; we shall, there-fore, in defining likeness, simplify our problem byspeaking of the “field” of one of the relations con-cerned. This somewhat limits the generality of ourdefinition, but the limitation is not of any practicalimportance. And having been stated, it need nolonger be remembered.

We may define two relations P and Q as “simi-

lar,” or as having “likeness,” when there is a one-onerelation S whose domain is the field of P and whoseconverse domain is the field of Q, and which is suchthat, if one term has the relation P | to another, the correlate of the one has the relation Q to the corre-late of the other, and vice versa. A figure will makethis clearer. Let x and y be two terms having the

z w

x y

Q

P

S S

relation P. Thenthere are to betwo terms z, w,such that x hasthe relation S toz, y has the re-lation S to w,and z has the re-lation Q to w.If this happenswith every pair

of terms such as x and y, and if the converse hap-pens with every pair of terms such as z and w, it isclear that for every instance in which the relation Pholds there is a corresponding instance in which therelation Q holds, and vice versa; and this is what wedesire to secure by our definition. We can eliminatesome redundancies in the above sketch of a defini-tion, by observing that, when the above conditions

are realised, the relation P is the same as the relativeproduct of S and Q and the converse of S, i.e. theP-step from x to y may be replaced by the successionof the S-step from x to z, the Q-step from z to w, andthe backward S-step from w to y. Thus we may setup the following definitions:—

A relation S is said to be a “correlator” or an“ordinal correlator” of two relations P and Q if S isone-one, has the field of Q for its converse domain,and is such that P is the relative product of S and Qand the converse of S.

Two relations P and Q are said to be “similar,”or to have “likeness,” when there is at least onecorrelator of P and Q.

These definitions will be found to yield what weabove decided to be necessary.

It will be found that, when two relations are sim-ilar, they share all properties which do not dependupon the actual terms in their fields. For instance,if one implies diversity, so does the other; if one istransitive, so is the other; if one is connected, sois the other. Hence if one is serial, so is the other.Again, if one is one-many or one-one, the other isone-many | or one-one; and so on, through all the general properties of relations. Even statements in-volving the actual terms of the field of a relation,

though they may not be true as they stand when ap-plied to a similar relation, will always be capable oftranslation into statements that are analogous. Weare led by such considerations to a problem whichhas, in mathematical philosophy, an importanceby no means adequately recognised hitherto. Ourproblem may be stated as follows:—

Given some statement in a language of whichwe know the grammar and the syntax, but not thevocabulary, what are the possible meanings of sucha statement, and what are the meanings of the un-known words that would make it true?

The reason that this question is important is thatit represents, much more nearly than might be sup-posed, the state of our knowledge of nature. Weknow that certain scientific propositions—which, inthe most advanced sciences, are expressed in mathe-matical symbols—are more or less true of the world,but we are very much at sea as to the interpreta-tion to be put upon the terms which occur in thesepropositions. We know much more (to use, for a mo-ment, an old-fashioned pair of terms) about the formof nature than about the matter. Accordingly, whatwe really know when we enunciate a law of natureis only that there is probably some interpretation ofour terms which will make the law approximately

true. Thus great importance attaches to the ques-tion: What are the possible meanings of a law ex-pressed in terms of which we do not know the sub-stantive meaning, but only the grammar and syntax?And this question is the one suggested above.

For the present we will ignore the general ques-tion, which will occupy us again at a later stage;the subject of likeness itself must first be furtherinvestigated.

Owing to the fact that, when two relations aresimilar, their properties are the same except whenthey depend upon the fields being composed of justthe terms of which they are composed, it is desirableto have a nomenclature which collects | together all the relations that are similar to a given relation. Justas we called the set of those classes that are similarto a given class the “number” of that class, so wemay call the set of all those relations that are similarto a given relation the “number” of that relation.But in order to avoid confusion with the numbersappropriate to classes, we will speak, in this case,of a “relation-number.” Thus we have the followingdefinitions:—

The “relation-number” of a given relation is theclass of all those relations that are similar to thegiven relation.

“Relation-numbers” are the set of all those class-es of relations that are relation-numbers of vari-ous relations; or, what comes to the same thing, arelation-number is a class of relations consisting ofall those relations that are similar to one member ofthe class.

When it is necessary to speak of the numbersof classes in a way which makes it impossible toconfuse them with relation-numbers, we shall callthem “cardinal numbers.” Thus cardinal numbersare the numbers appropriate to classes. These in-clude the ordinary integers of daily life, and alsocertain infinite numbers, of which we shall speaklater. When we speak of “numbers” without qualifi-cation, we are to be understood as meaning cardinalnumbers. The definition of a cardinal number, itwill be remembered, is as follows:—

The “cardinal number” of a given class is theset of all those classes that are similar to the givenclass.

The most obvious application of relation-num-bers is to series. Two series may be regarded asequally long when they have the same relation-num-ber. Two finite series will have the same relation-number when their fields have the same cardinalnumber of terms, and only then—i.e. a series of (say)

terms will have the same relation-number as anyother series of fifteen terms, but will not have thesame relation-number as a series of or terms,nor, of course, the same relation-number as a rela-tion which is not serial. Thus, in the quite specialcase of finite series, there is parallelism between car-dinal and relation-numbers. The relation-numbersapplicable to series may be | called “serial numbers” (what are commonly called “ordinal numbers” area sub-class of these); thus a finite serial number isdeterminate when we know the cardinal numberof terms in the field of a series having the serialnumber in question. If n is a finite cardinal number,the relation-number of a series which has n terms iscalled the “ordinal” number n. (There are also infi-nite ordinal numbers, but of them we shall speak ina later chapter.) When the cardinal number of termsin the field of a series is infinite, the relation-numberof the series is not determined merely by the cardi-nal number, indeed an infinite number of relation-numbers exist for one infinite cardinal number, aswe shall see when we come to consider infinite se-ries. When a series is infinite, what we may call its“length,” i.e. its relation-number, may vary withoutchange in the cardinal number; but when a series isfinite, this cannot happen.

We can define addition and multiplication forrelation-numbers as well as for cardinal numbers,and a whole arithmetic of relation-numbers can bedeveloped. The manner in which this is to be doneis easily seen by considering the case of series. Sup-pose, for example, that we wish to define the sumof two non-overlapping series in such a way thatthe relation-number of the sum shall be capable ofbeing defined as the sum of the relation-numbersof the two series. In the first place, it is clear thatthere is an order involved as between the two series:one of them must be placed before the other. Thusif P and Q are the generating relations of the twoseries, in the series which is their sum with P putbefore Q, every member of the field of P will pre-cede every member of the field of Q. Thus the serialrelation which is to be defined as the sum of P andQ is not “P or Q” simply, but “P or Q or the rela-tion of any member of the field of P to any memberof the field of Q.” Assuming that P and Q do notoverlap, this relation is serial, but “P or Q” is notserial, being not connected, since it does not holdbetween a member of the field of P and a memberof the field of Q. Thus the sum of P and Q, as abovedefined, is what we need in order | to define the sum of two relation-numbers. Similar modifications

are needed for products and powers. The resultingarithmetic does not obey the commutative law: thesum or product of two relation-numbers generallydepends upon the order in which they are taken.But it obeys the associative law, one form of the dis-tributive law, and two of the formal laws for powers,not only as applied to serial numbers, but as appliedto relation-numbers generally. Relation-arithmetic,in fact, though recent, is a thoroughly respectablebranch of mathematics.

It must not be supposed, merely because seriesafford the most obvious application of the idea oflikeness, that there are no other applications that areimportant. We have already mentioned maps, andwe might extend our thoughts from this illustrationto geometry generally. If the system of relations bywhich a geometry is applied to a certain set of termscan be brought fully into relations of likeness witha system applying to another set of terms, then thegeometry of the two sets is indistinguishable fromthe mathematical point of view, i.e. all the propo-sitions are the same, except for the fact that theyare applied in one case to one set of terms and inthe other to another. We may illustrate this by therelations of the sort that may be called “between,”which we considered in Chapter IV. We there saw

that, provided a three-term relation has certain for-mal logical properties, it will give rise to series, andmay be called a “between-relation.” Given any twopoints, we can use the between-relation to definethe straight line determined by those two points; itconsists of a and b together with all points x, suchthat the between-relation holds between the threepoints a, b, x in some order or other. It has beenshown by O. Veblen that we may regard our wholespace as the field of a three-term between-relation,and define our geometry by the properties we as-sign to our between-relation. Now likeness is justas easily | definable between three-term relations as between two-term relations. If B and B′ are twobetween-relations, so that “xB(y, z)” means “x is be-tween y and z with respect to B,” we shall call Sa correlator of B and B′ if it has the field of B′ forits converse domain, and is such that the relation Bholds between three terms when B′ holds betweentheir S-correlates, and only then. And we shall saythat B is like B′ when there is at least one correlatorof B with B′ . The reader can easily convince himself

This does not apply to elliptic space, but only to spaces inwhich the straight line is an open series. Modern Mathematics,edited by J. W. A. Young, pp. – (monograph by O. Veblen on“The Foundations of Geometry”).

that, if B is like B′ in this sense, there can be no dif-ference between the geometry generated by B andthat generated by B′ .

It follows from this that the mathematician neednot concern himself with the particular being or in-trinsic nature of his points, lines, and planes, evenwhen he is speculating as an applied mathematician.We may say that there is empirical evidence of theapproximate truth of such parts of geometry as arenot matters of definition. But there is no empiricalevidence as to what a “point” is to be. It has to besomething that as nearly as possible satisfies ouraxioms, but it does not have to be “very small” or“without parts.” Whether or not it is those thingsis a matter of indifference, so long as it satisfiesthe axioms. If we can, out of empirical material,construct a logical structure, no matter how com-plicated, which will satisfy our geometrical axioms,that structure may legitimately be called a “point.”We must not say that there is nothing else that couldlegitimately be called a “point”; we must only say:“This object we have constructed is sufficient forthe geometer; it may be one of many objects, anyof which would be sufficient, but that is no concernof ours, since this object is enough to vindicate theempirical truth of geometry, in so far as geometry

is not a matter of definition.” This is only an illus-tration of the general principle that what matters inmathematics, and to a very great extent in physicalscience, is not the intrinsic nature of our terms, butthe logical nature of their interrelations.

We may say, of two similar relations, that theyhave the same | “structure.” For mathematical pur- poses (though not for those of pure philosophy)the only thing of importance about a relation is thecases in which it holds, not its intrinsic nature. Justas a class may be defined by various different butco-extensive concepts—e.g. “man” and “featherlessbiped”—so two relations which are conceptuallydifferent may hold in the same set of instances. An“instance” in which a relation holds is to be con-ceived as a couple of terms, with an order, so thatone of the terms comes first and the other second;the couple is to be, of course, such that its first termhas the relation in question to its second. Take (say)the relation “father”: we can define what we maycall the “extension” of this relation as the class of allordered couples (x, y) which are such that x is thefather of y. From the mathematical point of view,the only thing of importance about the relation “fa-ther” is that it defines this set of ordered couples.Speaking generally, we say:

The “extension” of a relation is the class of thoseordered couples (x, y) which are such that x has therelation in question to y.

We can now go a step further in the process ofabstraction, and consider what we mean by “struc-ture.” Given any relation, we can, if it is a suffi-ciently simple one, construct a map of it. For thesake of definiteness, let us take a relation of whichthe extension is the following couples: ab, ac, ad, bc,ce, dc, de, where a, b, c, d, e are five terms, no matter

d c

a b

e

what. We may make a“map” of this relationby taking five points ona plane and connectingthem by arrows, as inthe accompanying fig-ure. What is revealedby the map is what wecall the “structure” ofthe relation.

It is clear that the“structure” of the rela-tion does not depend

upon the particular terms that make up the fieldof the relation. The field may be changed withoutchanging the structure, and the structure may be

changed without changing the field—for | example, if we were to add the couple ae in the above illus-tration we should alter the structure but not thefield. Two relations have the same “structure,” weshall say, when the same map will do for both—or, what comes to the same thing, when either canbe a map for the other (since every relation can beits own map). And that, as a moment’s reflectionshows, is the very same thing as what we have called“likeness.” That is to say, two relations have thesame structure when they have likeness, i.e. whenthey have the same relation-number. Thus whatwe defined as the “relation-number” is the verysame thing as is obscurely intended by the word“structure”—a word which, important as it is, isnever (so far as we know) defined in precise termsby those who use it.

There has been a great deal of speculation in tra-ditional philosophy which might have been avoidedif the importance of structure, and the difficulty ofgetting behind it, had been realised. For example,it is often said that space and time are subjective,but they have objective counterparts; or that phe-nomena are subjective, but are caused by things inthemselves, which must have differences inter se cor-responding with the differences in the phenomena

to which they give rise. Where such hypotheses aremade, it is generally supposed that we can knowvery little about the objective counterparts. In ac-tual fact, however, if the hypotheses as stated werecorrect, the objective counterparts would form aworld having the same structure as the phenomenalworld, and allowing us to infer from phenomenathe truth of all propositions that can be stated inabstract terms and are known to be true of phenom-ena. If the phenomenal world has three dimensions,so must the world behind phenomena; if the phe-nomenal world is Euclidean, so must the other be;and so on. In short, every proposition having a com-municable significance must be true of both worldsor of neither: the only difference must lie in justthat essence of individuality which always eludeswords and baffles description, but which, for thatvery reason, is irrelevant to science. Now the onlypurpose that philosophers | have in view in con- demning phenomena is in order to persuade them-selves and others that the real world is very differentfrom the world of appearance. We can all sympa-thise with their wish to prove such a very desirableproposition, but we cannot congratulate them ontheir success. It is true that many of them do notassert objective counterparts to phenomena, and

these escape from the above argument. Those whodo assert counterparts are, as a rule, very reticent onthe subject, probably because they feel instinctivelythat, if pursued, it will bring about too much of arapprochement between the real and the phenom-enal world. If they were to pursue the topic, theycould hardly avoid the conclusions which we havebeen suggesting. In such ways, as well as in manyothers, the notion of structure or relation-numberis important.

Chapter VII

Rational, Real, and Complex Numbers

We have now seen how to define cardinal num- bers, and also relation-numbers, of which what arecommonly called ordinal numbers are a particularspecies. It will be found that each of these kinds ofnumber may be infinite just as well as finite. Butneither is capable, as it stands, of the more famil-iar extensions of the idea of number, namely, theextensions to negative, fractional, irrational, andcomplex numbers. In the present chapter we shallbriefly supply logical definitions of these variousextensions.

One of the mistakes that have delayed the dis-covery of correct definitions in this region is thecommon idea that each extension of number in-cluded the previous sorts as special cases. It wasthought that, in dealing with positive and negative

integers, the positive integers might be identifiedwith the original signless integers. Again it wasthought that a fraction whose denominator is maybe identified with the natural number which is itsnumerator. And the irrational numbers, such as thesquare root of , were supposed to find their placeamong rational fractions, as being greater than someof them and less than the others, so that rational andirrational numbers could be taken together as oneclass, called “real numbers.” And when the idea ofnumber was further extended so as to include “com-plex” numbers, i.e. numbers involving the squareroot of −, it was thought that real numbers couldbe regarded as those among complex numbers inwhich the imaginary part (i.e. the part | which was a multiple of the square root of −) was zero. Allthese suppositions were erroneous, and must be dis-carded, as we shall find, if correct definitions are tobe given.

Let us begin with positive and negative integers.It is obvious on a moment’s consideration that +and −must both be relations, and in fact must beeach other’s converses. The obvious and sufficientdefinition is that + is the relation of n+ to n, and− is the relation of n to n+ . Generally, if m is anyinductive number, +m will be the relation of n+m

to n (for any n), and −m will be the relation of n ton+m. According to this definition, +m is a relationwhich is one-one so long as n is a cardinal number(finite or infinite) and m is an inductive cardinalnumber. But +m is under no circumstances capableof being identified with m, which is not a relation,but a class of classes. Indeed, +m is every bit asdistinct from m as −m is.

Fractions are more interesting than positive ornegative integers. We need fractions for many pur-poses, but perhaps most obviously for purposes ofmeasurement. My friend and collaborator Dr A. N.Whitehead has developed a theory of fractions spe-cially adapted for their application to measurement,which is set forth in Principia Mathematica. Butif all that is needed is to define objects having therequired purely mathematical properties, this pur-pose can be achieved by a simpler method, whichwe shall here adopt. We shall define the fractionm/n as being that relation which holds between twoinductive numbers x, y when xn = ym. This def-inition enables us to prove that m/n is a one-onerelation, provided neither m nor n is zero. And ofcourse n/m is the converse relation to m/n.

From the above definition it is clear that the frac-Vol. iii. ∗ff., especially .

tion m/ is that relation between two integers x andy which consists in the fact that x =my. This rela-tion, like the relation +m, is by no means capable ofbeing identified with the inductive cardinal numberm, because a relation and a class of classes are ob-jects | of utterly different kinds. It will be seen that /n is always the same relation, whatever inductivenumber n may be; it is, in short, the relation of toany other inductive cardinal. We may call this thezero of rational numbers; it is not, of course, iden-tical with the cardinal number . Conversely, therelation m/ is always the same, whatever induc-tive number m may be. There is not any inductivecardinal to correspond to m/. We may call it “theinfinity of rationals.” It is an instance of the sortof infinite that is traditional in mathematics, andthat is represented by “∞.” This is a totally differ-ent sort from the true Cantorian infinite, which weshall consider in our next chapter. The infinity ofrationals does not demand, for its definition or use,any infinite classes or infinite integers. It is not, inactual fact, a very important notion, and we could

Of course in practice we shall continue to speak of a fractionas (say) greater or less than , meaning greater or less than theratio /. So long as it is understood that the ratio / and thecardinal number are different, it is not necessary to be alwayspedantic in emphasising the difference.

dispense with it altogether if there were any objectin doing so. The Cantorian infinite, on the otherhand, is of the greatest and most fundamental im-portance; the understanding of it opens the way towhole new realms of mathematics and philosophy.

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

There is not any difficulty in defining greater andless among ratios (or fractions). Given two ratiosm/n and p/q, we shall say that m/n is less than p/q ifmq is less than pn. There is no difficulty in provingthat the relation “less than,” so defined, is serial, sothat the ratios form a series in order of magnitude.In this series, zero is the smallest term and infin-ity is the largest. If we omit zero and infinity fromour series, there is no longer any smallest or largestratio; it is obvious that if m/n is any ratio otherthan zero and infinity, m/n is smaller and m/nis larger, though neither is zero or infinity, so thatm/n is neither the smallest | nor the largest ratio, and therefore (when zero and infinity are omitted)there is no smallest or largest, since m/n was chosenarbitrarily. In like manner we can prove that how-ever nearly equal two fractions may be, there arealways other fractions between them. For, let m/n

and p/q be two fractions, of which p/q is the greater.Then it is easy to see (or to prove) that (m+p)/(n+q)will be greater than m/n and less than p/q. Thusthe series of ratios is one in which no two terms areconsecutive, but there are always other terms be-tween any two. Since there are other terms betweenthese others, and so on ad infinitum, it is obviousthat there are an infinite number of ratios betweenany two, however nearly equal these two may be.

A series having the property that there are alwaysother terms between any two, so that no two areconsecutive, is called “compact.” Thus the ratios inorder of magnitude form a “compact” series. Suchseries have many important properties, and it is im-portant to observe that ratios afford an instance of acompact series generated purely logically, withoutany appeal to space or time or any other empiricaldatum.

Positive and negative ratios can be defined in away analogous to that in which we defined positiveand negative integers. Having first defined the sumof two ratios m/n and p/q as (mq+pn)/nq, we define+p/q as the relation of m/n+ p/q to m/n, where m/n

Strictly speaking, this statement, as well as those followingto the end of the paragraph, involves what is called the “axiomof infinity,” which will be discussed in a later chapter.

is any ratio; and −p/q is of course the converse of+p/q. This is not the only possible way of definingpositive and negative ratios, but it is a way which,for our purpose, has the merit of being an obviousadaptation of the way we adopted in the case ofintegers.

We come now to a more interesting extension ofthe idea of number, i.e. the extension to what arecalled “real” numbers, which are the kind that em-brace irrationals. In Chapter I. we had occasion tomention “incommensurables” and their | discovery by Pythagoras. It was through them, i.e. throughgeometry, that irrational numbers were first thoughtof. A square of which the side is one inch long willhave a diagonal of which the length is the squareroot of inches. But, as the ancients discovered,there is no fraction of which the square is . Thisproposition is proved in the tenth book of Euclid,which is one of those books that schoolboys sup-posed to be fortunately lost in the days when Euclidwas still used as a text-book. The proof is extraor-dinarily simple. If possible, let m/n be the squareroot of , so that m/n = , i.e. m = n. Thusm is an even number, and therefore m must be aneven number, because the square of an odd num-ber is odd. Now if m is even, m must divide by ,

for if m = p, then m = p. Thus we shall havep = n, where p is half of m. Hence p = n,and therefore n/p will also be the square root of .But then we can repeat the argument: if n = q, p/qwill also be the square root of , and so on, throughan unending series of numbers that are each half ofits predecessor. But this is impossible; if we dividea number by , and then halve the half, and so on,we must reach an odd number after a finite numberof steps. Or we may put the argument even moresimply by assuming that the m/n we start with is inits lowest terms; in that case, m and n cannot both beeven; yet we have seen that, if m/n = , they mustbe. Thus there cannot be any fraction m/n whosesquare is .

Thus no fraction will express exactly the lengthof the diagonal of a square whose side is one inchlong. This seems like a challenge thrown out bynature to arithmetic. However the arithmeticianmay boast (as Pythagoras did) about the power ofnumbers, nature seems able to baffle him by ex-hibiting lengths which no numbers can estimate interms of the unit. But the problem did not remainin this geometrical form. As soon as algebra wasinvented, the same problem arose as regards the so-lution of equations, though here it took on a wider

form, since it also involved complex numbers.It is clear that fractions can be found which ap-

proach nearer | and nearer to having their square equal to . We can form an ascending series of frac-tions all of which have their squares less than ,but differing from in their later members by lessthan any assigned amount. That is to say, supposeI assign some small amount in advance, say one-billionth, it will be found that all the terms of ourseries after a certain one, say the tenth, have squaresthat differ from by less than this amount. And if Ihad assigned a still smaller amount, it might havebeen necessary to go further along the series, but weshould have reached sooner or later a term in the se-ries, say the twentieth, after which all terms wouldhave had squares differing from by less than thisstill smaller amount. If we set to work to extractthe square root of by the usual arithmetical rule,we shall obtain an unending decimal which, takento so-and-so many places, exactly fulfils the aboveconditions. We can equally well form a descendingseries of fractions whose squares are all greater than, but greater by continually smaller amounts aswe come to later terms of the series, and differing,sooner or later, by less than any assigned amount. Inthis way we seem to be drawing a cordon round the

square root of , and it may seem difficult to believethat it can permanently escape us. Nevertheless, itis not by this method that we shall actually reachthe square root of .

If we divide all ratios into two classes, accordingas their squares are less than or not, we find that,among those whose squares are not less than , allhave their squares greater than . There is no maxi-mum to the ratios whose square is less than , andno minimum to those whose square is greater than. There is no lower limit short of zero to the differ-ence between the numbers whose square is a littleless than and the numbers whose square is a littlegreater than . We can, in short, divide all ratiosinto two classes such that all the terms in one classare less than all in the other, there is no maximumto the one class, and there is no minimum to theother. Between these two classes, where

√ ought to

be, there is nothing. Thus our | cordon, though we have drawn it as tight as possible, has been drawnin the wrong place, and has not caught

√.

The above method of dividing all the terms ofa series into two classes, of which the one whollyprecedes the other, was brought into prominenceby Dedekind, and is therefore called a “Dedekind

Stetigkeit und irrationale Zahlen, nd edition, Brunswick,

cut.” With respect to what happens at the point ofsection, there are four possibilities: () there maybe a maximum to the lower section and a minimumto the upper section, () there may be a maximumto the one and no minimum to the other, () theremay be no maximum to the one, but a minimum tothe other, () there may be neither a maximum tothe one nor a minimum to the other. Of these fourcases, the first is illustrated by any series in whichthere are consecutive terms: in the series of integers,for instance, a lower section must end with somenumber n and the upper section must then beginwith n + . The second case will be illustrated inthe series of ratios if we take as our lower sectionall ratios up to and including , and in our uppersection all ratios greater than . The third case isillustrated if we take for our lower section all ratiosless than , and for our upper section all ratios from upward (including itself). The fourth case, aswe have seen, is illustrated if we put in our lowersection all ratios whose square is less than , and inour upper section all ratios whose square is greaterthan .

We may neglect the first of our four cases, sinceit only arises in series where there are consecutive

.

terms. In the second of our four cases, we say thatthe maximum of the lower section is the lower limitof the upper section, or of any set of terms chosenout of the upper section in such a way that no termof the upper section is before all of them. In thethird of our four cases, we say that the minimumof the upper section is the upper limit of the lowersection, or of any set of terms chosen out of thelower section in such a way that no term of thelower section is after all of them. In the fourth case,we say that | there is a “gap”: neither the upper section nor the lower has a limit or a last term. Inthis case, we may also say that we have an “irrationalsection,” since sections of the series of ratios have“gaps” when they correspond to irrationals.

What delayed the true theory of irrationals was amistaken belief that there must be “limits” of seriesof ratios. The notion of “limit” is of the utmostimportance, and before proceeding further it willbe well to define it.

A term x is said to be an “upper limit” of a class αwith respect to a relation P if () α has no maximumin P, () every member of α which belongs to thefield of P precedes x, () every member of the fieldof P which precedes x precedes some member of α.(By “precedes” we mean “has the relation P to.”)

This presupposes the following definition of a“maximum”:—

A term x is said to be a “maximum” of a classα with respect to a relation P if x is a member of αand of the field of P and does not have the relationP to any other member of α.

These definitions do not demand that the termsto which they are applied should be quantitative.For example, given a series of moments of time ar-ranged by earlier and later, their “maximum” (ifany) will be the last of the moments; but if they arearranged by later and earlier, their “maximum” (ifany) will be the first of the moments.

The “minimum” of a class with respect to P is itsmaximum with respect to the converse of P; and the“lower limit” with respect to P is the upper limitwith respect to the converse of P.

The notions of limit and maximum do not essen-tially demand that the relation in respect to whichthey are defined should be serial, but they have fewimportant applications except to cases when the re-lation is serial or quasi-serial. A notion which isoften important is the notion “upper limit or max-imum,” to which we may give the name “upperboundary.” Thus the “upper boundary” of a set ofterms chosen out of a series is their last member

if they have one, but, if not, it is the first term af-ter all of them, if there is such a term. If there isneither | a maximum nor a limit, there is no upper boundary. The “lower boundary” is the lower limitor minimum.

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

A series is called “Dedekindian” when every sec-tion has a boundary, upper or lower as the case maybe.

We have seen that the series of ratios in order ofmagnitude is not Dedekindian.

From the habit of being influenced by spatialimagination, people have supposed that series musthave limits in cases where it seems odd if they donot. Thus, perceiving that there was no rational limitto the ratios whose square is less than , they al-lowed themselves to “postulate” an irrational limit,which was to fill the Dedekind gap. Dedekind, in

the above-mentioned work, set up the axiom thatthe gap must always be filled, i.e. that every sec-tion must have a boundary. It is for this reasonthat series where his axiom is verified are called“Dedekindian.” But there are an infinite number ofseries for which it is not verified.

The method of “postulating” what we want hasmany advantages; they are the same as the advan-tages of theft over honest toil. Let us leave them toothers and proceed with our honest toil.

It is clear that an irrational Dedekind cut in someway “represents” an irrational. In order to make useof this, which to begin with is no more than a vaguefeeling, we must find some way of eliciting fromit a precise definition; and in order to do this, wemust disabuse our minds of the notion that an ir-rational must be the limit of a set of ratios. Justas ratios whose denominator is are not identicalwith integers, so those rational | numbers which can be greater or less than irrationals, or can haveirrationals as their limits, must not be identifiedwith ratios. We have to define a new kind of num-bers called “real numbers,” of which some will berational and some irrational. Those that are ratio-nal “correspond” to ratios, in the same kind of wayin which the ratio n/ corresponds to the integer

n; but they are not the same as ratios. In order todecide what they are to be, let us observe that anirrational is represented by an irrational cut, and acut is represented by its lower section. Let us con-fine ourselves to cuts in which the lower section hasno maximum; in this case we will call the lowersection a “segment.” Then those segments that cor-respond to ratios are those that consist of all ratiosless than the ratio they correspond to, which is theirboundary; while those that represent irrationals arethose that have no boundary. Segments, both thosethat have boundaries and those that do not, are suchthat, of any two pertaining to one series, one mustbe part of the other; hence they can all be arrangedin a series by the relation of whole and part. A seriesin which there are Dedekind gaps, i.e. in which thereare segments that have no boundary, will give riseto more segments than it has terms, since each termwill define a segment having that term for bound-ary, and then the segments without boundaries willbe extra.

We are now in a position to define a real numberand an irrational number.

A “real number” is a segment of the series ofratios in order of magnitude.

An “irrational number” is a segment of the series

of ratios which has no boundary.A “rational real number” is a segment of the

series of ratios which has a boundary.Thus a rational real number consists of all ratios

less than a certain ratio, and it is the rational realnumber corresponding to that ratio. The real num-ber , for instance, is the class of proper fractions. |

In the cases in which we naturally supposed that an irrational must be the limit of a set of ratios, thetruth is that it is the limit of the corresponding setof rational real numbers in the series of segmentsordered by whole and part. For example,

√ is

the upper limit of all those segments of the seriesof ratios that correspond to ratios whose square isless than . More simply still,

√ is the segment

consisting of all those ratios whose square is lessthan .

It is easy to prove that the series of segmentsof any series is Dedekindian. For, given any set ofsegments, their boundary will be their logical sum,i.e. the class of all those terms that belong to at leastone segment of the set.

For a fuller treatment of the subject of segments andDedekindian relations, see Principia Mathematica, vol. ii. ∗–. For a fuller treatment of real numbers, see ibid., vol. iii.∗ff., and Principles of Mathematics, chaps. xxxiii. and xxxiv.

The above definition of real numbers is an ex-ample of “construction” as against “postulation,”of which we had another example in the definitionof cardinal numbers. The great advantage of thismethod is that it requires no new assumptions, butenables us to proceed deductively from the originalapparatus of logic.

There is no difficulty in defining addition andmultiplication for real numbers as above defined.Given two real numbers µ and ν, each being a classof ratios, take any member of µ and any memberof ν and add them together according to the rulefor the addition of ratios. Form the class of all suchsums obtainable by varying the selected membersof µ and ν. This gives a new class of ratios, and itis easy to prove that this new class is a segment ofthe series of ratios. We define it as the sum of µand ν. We may state the definition more shortly asfollows:—

The arithmetical sum of two real numbers is theclass of the arithmetical sums of a member of theone and a member of the other chosen in all possibleways. |

We can define the arithmetical product of two real numbers in exactly the same way, by multiply-ing a member of the one by a member of the other in

all possible ways. The class of ratios thus generatedis defined as the product of the two real numbers.(In all such definitions, the series of ratios is to bedefined as excluding and infinity.)

There is no difficulty in extending our defini-tions to positive and negative real numbers and theiraddition and multiplication.

It remains to give the definition of complex num-bers.

Complex numbers, though capable of a geomet-rical interpretation, are not demanded by geometryin the same imperative way in which irrationals aredemanded. A “complex” number means a num-ber involving the square root of a negative num-ber, whether integral, fractional, or real. Since thesquare of a negative number is positive, a numberwhose square is to be negative has to be a new sort ofnumber. Using the letter i for the square root of −,any number involving the square root of a negativenumber can be expressed in the form x+yi, where xand y are real. The part yi is called the “imaginary”part of this number, x being the “real” part. (Thereason for the phrase “real numbers” is that they arecontrasted with such as are “imaginary.”) Complexnumbers have been for a long time habitually usedby mathematicians, in spite of the absence of any

precise definition. It has been simply assumed thatthey would obey the usual arithmetical rules, and onthis assumption their employment has been foundprofitable. They are required less for geometry thanfor algebra and analysis. We desire, for example,to be able to say that every quadratic equation hastwo roots, and every cubic equation has three, andso on. But if we are confined to real numbers, suchan equation as x + = has no roots, and such anequation as x − = has only one. Every generali-sation of number has first presented itself as neededfor some simple problem: negative numbers wereneeded in order that subtraction might be alwayspossible, since otherwise a−b would be meaninglessif a were less than b; fractions were needed | in order that division might be always possible; and com-plex numbers are needed in order that extractionof roots and solution of equations may be alwayspossible. But extensions of number are not createdby the mere need for them: they are created by thedefinition, and it is to the definition of complexnumbers that we must now turn our attention.

A complex number may be regarded and definedas simply an ordered couple of real numbers. Here,as elsewhere, many definitions are possible. All thatis necessary is that the definitions adopted shall

lead to certain properties. In the case of complexnumbers, if they are defined as ordered couples ofreal numbers, we secure at once some of the prop-erties required, namely, that two real numbers arerequired to determine a complex number, and thatamong these we can distinguish a first and a second,and that two complex numbers are only identicalwhen the first real number involved in the one isequal to the first involved in the other, and the sec-ond to the second. What is needed further can besecured by defining the rules of addition and multi-plication. We are to have

(x+ yi) + (x′ + y′i) = (x+ x′) + (y + y′)i

(x+ yi)(x′ + y′i) = (xx′ − yy′) + (xy′ + x′y)i.

Thus we shall define that, given two ordered cou-ples of real numbers, (x,y) and (x′, y′), their sum isto be the couple (x + x′, y + y′), and their productis to be the couple (xx′ − yy′, xy′ + x′y). By thesedefinitions we shall secure that our ordered couplesshall have the properties we desire. For example,take the product of the two couples (, y) and (, y′).This will, by the above rule, be the couple (−yy′,).Thus the square of the couple (,) will be the cou-ple (−,). Now those couples in which the secondterm is are those which, according to the usual

nomenclature, have their imaginary part zero; in thenotation x + yi, they are x + i, which it is natural towrite simply x. Just as it is natural (but erroneous) |to identify ratios whose denominator is unity with integers, so it is natural (but erroneous) to identifycomplex numbers whose imaginary part is zero withreal numbers. Although this is an error in theory, itis a convenience in practice; “x+i” may be replacedsimply by “x” and “+ yi” by “yi,” provided we re-member that the “x” is not really a real number, buta special case of a complex number. And when yis , “yi” may of course be replaced by “i.” Thusthe couple (,) is represented by i, and the couple(−,) is represented by −. Now our rules of mul-tiplication make the square of (,) equal to (−,),i.e. the square of i is −. This is what we desiredto secure. Thus our definitions serve all necessarypurposes.

It is easy to give a geometrical interpretation ofcomplex numbers in the geometry of the plane. Thissubject was agreeably expounded by W. K. Cliffordin his Common Sense of the Exact Sciences, a book ofgreat merit, but written before the importance ofpurely logical definitions had been realised.

Complex numbers of a higher order, thoughmuch less useful and important than those what

we have been defining, have certain uses that arenot without importance in geometry, as may be seen,for example, in Dr Whitehead’s Universal Algebra.The definition of complex numbers of order n isobtained by an obvious extension of the definitionwe have given. We define a complex number oforder n as a one-many relation whose domain con-sists of certain real numbers and whose conversedomain consists of the integers from to n. This iswhat would ordinarily be indicated by the notation(x, x, x, . . . xn), where the suffixes denote cor-relation with the integers used as suffixes, and thecorrelation is one-many, not necessarily one-one,because xr and xs may be equal when r and s arenot equal. The above definition, with a suitable ruleof multiplication, will serve all purposes for whichcomplex numbers of higher orders are needed.

We have now completed our review of those ex-tensions of number which do not involve infinity.The application of number to infinite collectionsmust be our next topic.

Cf. Principles of Mathematics, §, p. .

Chapter VIII

Infinite Cardinal Numbers

The definition of cardinal numbers which we gave in Chapter II. was applied in Chapter III. to finitenumbers, i.e. to the ordinary natural numbers. Tothese we gave the name “inductive numbers,” be-cause we found that they are to be defined as num-bers which obey mathematical induction startingfrom . But we have not yet considered collectionswhich do not have an inductive number of terms,nor have we inquired whether such collections canbe said to have a number at all. This is an ancientproblem, which has been solved in our own day,chiefly by Georg Cantor. In the present chapter weshall attempt to explain the theory of transfinite orinfinite cardinal numbers as it results from a combi-nation of his discoveries with those of Frege on thelogical theory of numbers.

It cannot be said to be certain that there are infact any infinite collections in the world. The as-sumption that there are is what we call the “axiomof infinity.” Although various ways suggest them-selves by which we might hope to prove this axiom,there is reason to fear that they are all fallacious,and that there is no conclusive logical reason forbelieving it to be true. At the same time, there iscertainly no logical reason against infinite collec-tions, and we are therefore justified, in logic, ininvestigating the hypothesis that there are such col-lections. The practical form of this hypothesis, forour present purposes, is the assumption that, if nis any inductive number, n is not equal to n + .Various subtleties arise in identifying this form ofour assumption with | the form that asserts the exis- tence of infinite collections; but we will leave theseout of account until, in a later chapter, we come toconsider the axiom of infinity on its own account.For the present we shall merely assume that, if n isan inductive number, n is not equal to n+ . Thisis involved in Peano’s assumption that no two in-ductive numbers have the same successor; for, ifn = n+ , then n− and n have the same successor,namely n. Thus we are assuming nothing that wasnot involved in Peano’s primitive propositions.

Let us now consider the collection of the induc-tive numbers themselves. This is a perfectly well-defined class. In the first place, a cardinal numberis a set of classes which are all similar to each otherand are not similar to anything except each other.We then define as the “inductive numbers” thoseamong cardinals which belong to the posterity of with respect to the relation of n to n+ , i.e. thosewhich possess every property possessed by and bythe successors of possessors, meaning by the “suc-cessor” of n the number n + . Thus the class of“inductive numbers” is perfectly definite. By ourgeneral definition of cardinal numbers, the numberof terms in the class of inductive numbers is to bedefined as “all those classes that are similar to theclass of inductive numbers”—i.e. this set of classesis the number of the inductive numbers accordingto our definitions.

Now it is easy to see that this number is notone of the inductive numbers. If n is any induc-tive number, the number of numbers from to n(both included) is n+ ; therefore the total numberof inductive numbers is greater than n, no matterwhich of the inductive numbers n may be. If wearrange the inductive numbers in a series in orderof magnitude, this series has no last term; but if

n is an inductive number, every series whose fieldhas n terms has a last term, as it is easy to prove.Such differences might be multiplied ad lib. Thusthe number of inductive numbers is a new number,different from all of them, not possessing all induc-tive properties. It may happen that has a certain |property, and that if n has it so has n+, and yet that this new number does not have it. The difficultiesthat so long delayed the theory of infinite numberswere largely due to the fact that some, at least, ofthe inductive properties were wrongly judged to besuch as must belong to all numbers; indeed it wasthought that they could not be denied without con-tradiction. The first step in understanding infinitenumbers consists in realising the mistakenness ofthis view.

The most noteworthy and astonishing differencebetween an inductive number and this new numberis that this new number is unchanged by adding or subtracting or doubling or halving or anyof a number of other operations which we think ofas necessarily making a number larger or smaller.The fact of being not altered by the addition of isused by Cantor for the definition of what he calls“transfinite” cardinal numbers; but for various rea-sons, some of which will appear as we proceed, it is

better to define an infinite cardinal number as onewhich does not possess all inductive properties, i.e.simply as one which is not an inductive number.Nevertheless, the property of being unchanged bythe addition of is a very important one, and wemust dwell on it for a time.

To say that a class has a number which is notaltered by the addition of is the same thing as tosay that, if we take a term x which does not belongto the class, we can find a one-one relation whosedomain is the class and whose converse domain isobtained by adding x to the class. For in that case,the class is similar to the sum of itself and the termx, i.e. to a class having one extra term; so that it hasthe same number as a class with one extra term,so that if n is this number, n = n + . In this case,we shall also have n = n − , i.e. there will be one-one relations whose domains consist of the wholeclass and whose converse domains consist of justone term short of the whole class. It can be shownthat the cases in which this happens are the sameas the apparently more general cases in which somepart (short of the whole) can be put into one-onerelation with the whole. When this can be done, |the correlator by which it is done may be said to “reflect” the whole class into a part of itself; for this

reason, such classes will be called “reflexive.” Thus:A “reflexive” class is one which is similar to a

proper part of itself. (A “proper part” is a part shortof the whole.)

A “reflexive” cardinal number is the cardinalnumber of a reflexive class.

We have now to consider this property of reflex-iveness.

One of the most striking instances of a “reflex-ion” is Royce’s illustration of the map: he imaginesit decided to make a map of England upon a part ofthe surface of England. A map, if it is accurate, hasa perfect one-one correspondence with its original;thus our map, which is part, is in one-one relationwith the whole, and must contain the same numberof points as the whole, which must therefore be areflexive number. Royce is interested in the fact thatthe map, if it is correct, must contain a map of themap, which must in turn contain a map of the mapof the map, and so on ad infinitum. This point isinteresting, but need not occupy us at this moment.In fact, we shall do well to pass from picturesqueillustrations to such as are more completely defi-nite, and for this purpose we cannot do better thanconsider the number-series itself.

The relation of n to n+ , confined to inductive

numbers, is one-one, has the whole of the induc-tive numbers for its domain, and all except for itsconverse domain. Thus the whole class of inductivenumbers is similar to what the same class becomeswhen we omit . Consequently it is a “reflexive”class according to the definition, and the numberof its terms is a “reflexive” number. Again, the re-lation of n to n, confined to inductive numbers, isone-one, has the whole of the inductive numbers forits domain, and the even inductive numbers alonefor its converse domain. Hence the total numberof inductive numbers is the same as the number ofeven inductive numbers. This property was usedby Leibniz (and many others) as a proof that infi-nite numbers are impossible; it was thought self-contradictory that | “the part should be equal to the whole.” But this is one of those phrases that dependfor their plausibility upon an unperceived vague-ness: the word “equal” has many meanings, but ifit is taken to mean what we have called “similar,”there is no contradiction, since an infinite collectioncan perfectly well have parts similar to itself. Thosewho regard this as impossible have, unconsciouslyas a rule, attributed to numbers in general prop-erties which can only be proved by mathematicalinduction, and which only their familiarity makes

us regard, mistakenly, as true beyond the region ofthe finite.

Whenever we can “reflect” a class into a part ofitself, the same relation will necessarily reflect thatpart into a smaller part, and so on ad infinitum. Forexample, we can reflect, as we have just seen, all theinductive numbers into the even numbers; we can,by the same relation (that of n to n) reflect the evennumbers into the multiples of , these into the mul-tiples of , and so on. This is an abstract analogue toRoyce’s problem of the map. The even numbers area “map” of all the inductive numbers; the multiplesof are a map of the map; the multiples of are amap of the map of the map; and so on. If we hadapplied the same process to the relation of n to n+,our “map” would have consisted of all the inductivenumbers except ; the map of the map would haveconsisted of all from onward, the map of the mapof the map of all from onward; and so on. Thechief use of such illustrations is in order to becomefamiliar with the idea of reflexive classes, so thatapparently paradoxical arithmetical propositionscan be readily translated into the language of re-flexions and classes, in which the air of paradox ismuch less.

It will be useful to give a definition of the num-

ber which is that of the inductive cardinals. Forthis purpose we will first define the kind of seriesexemplified by the inductive cardinals in order ofmagnitude. The kind of series which is called a“progression” has already been considered in Chap-ter I. It is a series which can be generated by a re-lation of consecutiveness: | every member of the series is to have a successor, but there is to be justone which has no predecessor, and every memberof the series is to be in the posterity of this termwith respect to the relation “immediate predeces-sor.” These characteristics may be summed up inthe following definition:—

A “progression” is a one-one relation such thatthere is just one term belonging to the domain butnot to the converse domain, and the domain is iden-tical with the posterity of this one term.

It is easy to see that a progression, so defined,satisfies Peano’s five axioms. The term belonging tothe domain but not to the converse domain will bewhat he calls “”; the term to which a term has theone-one relation will be the “successor” of the term;and the domain of the one-one relation will be whathe calls “number.” Taking his five axioms in turn,we have the following translations:—

Cf. Principia Mathematica, vol. ii. ∗.

() “ is a number” becomes: “The member ofthe domain which is not a member of the conversedomain is a member of the domain.” This is equiv-alent to the existence of such a member, which isgiven in our definition. We will call this member“the first term.”

() “The successor of any number is a number”becomes: “The term to which a given member ofthe domain has the relation in question is again amember of the domain.” This is proved as follows:By the definition, every member of the domain isa member of the posterity of the first term; hencethe successor of a member of the domain must be amember of the posterity of the first term (becausethe posterity of a term always contains its own suc-cessors, by the general definition of posterity), andtherefore a member of the domain, because by thedefinition the posterity of the first term is the sameas the domain.

() “No two numbers have the same successor.”This is only to say that the relation is one-many,which it is by definition (being one-one). |

() “ is not the successor of any number” be- comes: “The first term is not a member of the con-verse domain,” which is again an immediate resultof the definition.

() This is mathematical induction, and becomes:“Every member of the domain belongs to the pos-terity of the first term,” which was part of our defi-nition.

Thus progressions as we have defined them havethe five formal properties from which Peano de-duces arithmetic. It is easy to show that two progressionsare “similar” in the sense defined for similarity ofrelations in Chapter VI. We can, of course, derivea relation which is serial from the one-one relationby which we define a progression: the method usedis that explained in Chapter IV., and the relation isthat of a term to a member of its proper posteritywith respect to the original one-one relation.

Two transitive asymmetrical relations which gen-erate progressions are similar, for the same reasonsfor which the corresponding one-one relations aresimilar. The class of all such transitive generatorsof progressions is a “serial number” in the sense ofChapter VI.; it is in fact the smallest of infinite serialnumbers, the number to which Cantor has given thename ω, by which he has made it famous.

But we are concerned, for the moment, with car-dinal numbers. Since two progressions are simi-lar relations, it follows that their domains (or theirfields, which are the same as their domains) are

similar classes. The domains of progressions form acardinal number, since every class which is similarto the domain of a progression is easily shown tobe itself the domain of a progression. This cardinalnumber is the smallest of the infinite cardinal num-bers; it is the one to which Cantor has appropriatedthe Hebrew Aleph with the suffix , to distinguishit from larger infinite cardinals, which have othersuffixes. Thus the name of the smallest of infinitecardinals is ℵ.

To say that a class has ℵ terms is the same thingas to say that it is a member of ℵ, and this is thesame thing as to say | that the members of the class can be arranged in a progression. It is obvious thatany progression remains a progression if we omit afinite number of terms from it, or every other term,or all except every tenth term or every hundredthterm. These methods of thinning out a progressiondo not make it cease to be a progression, and there-fore do not diminish the number of its terms, whichremains ℵ. In fact, any selection from a progres-sion is a progression if it has no last term, howeversparsely it may be distributed. Take (say) inductivenumbers of the form nn, or nn

n. Such numbers grow

very rare in the higher parts of the number series,and yet there are just as many of them as there are

inductive numbers altogether, namely, ℵ.Conversely, we can add terms to the inductive

numbers without increasing their number. Take, forexample, ratios. One might be inclined to think thatthere must be many more ratios than integers, sinceratios whose denominator is correspond to theintegers, and seem to be only an infinitesimal pro-portion of ratios. But in actual fact the number ofratios (or fractions) is exactly the same as the num-ber of inductive numbers, namely, ℵ. This is easilyseen by arranging ratios in a series on the followingplan: If the sum of numerator and denominator inone is less than in the other, put the one before theother; if the sum is equal in the two, put first theone with the smaller numerator. This gives us theseries

, , , , ,

, , , ,

, . . .

This series is a progression, and all ratios occur in itsooner or later. Hence we can arrange all ratios in aprogression, and their number is therefore ℵ.

It is not the case, however, that all infinite collec-tions have ℵ terms. The number of real numbers,for example, is greater than ℵ; it is, in fact, ℵ , andit is not hard to prove that n is greater than n evenwhen n is infinite. The easiest way of proving this isto prove, first, that if a class has n members, it con-

tains n sub-classes—in other words, that there aren ways | of selecting some of its members (includ- ing the extreme cases where we select all or none);and secondly, that the number of sub-classes con-tained in a class is always greater than the numberof members of the class. Of these two propositions,the first is familiar in the case of finite numbers, andis not hard to extend to infinite numbers. The proofof the second is so simple and so instructive that weshall give it:

In the first place, it is clear that the number ofsub-classes of a given class (say α) is at least as greatas the number of members, since each member con-stitutes a sub-class, and we thus have a correlationof all the members with some of the sub-classes.Hence it follows that, if the number of sub-classesis not equal to the number of members, it must begreater. Now it is easy to prove that the numberis not equal, by showing that, given any one-onerelation whose domain is the members and whoseconverse domain is contained among the set of sub-classes, there must be at least one sub-class not be-longing to the converse domain. The proof is asfollows: When a one-one correlation R is estab-This proof is taken from Cantor, with some simplifica-

tions: see Jahresbericht der Deutschen Mathematiker-Vereinigung, i.

lished between all the members of α and some ofthe sub-classes, it may happen that a given memberx is correlated with a sub-class of which it is a mem-ber; or, again, it may happen that x is correlated witha sub-class of which it is not a member. Let us formthe whole class, β say, of those members x whichare correlated with sub-classes of which they arenot members. This is a sub-class of α, and it is notcorrelated with any member of α. For, taking firstthe members of β, each of them is (by the definitionof β) correlated with some sub-class of which it isnot a member, and is therefore not correlated withβ. Taking next the terms which are not membersof β, each of them (by the definition of β) is corre-lated with some sub-class of which it is a member,and therefore again is not correlated with β. Thusno member of α is correlated with β. Since R wasany one-one correlation of all members | with some sub-classes, it follows that there is no correlation ofall members with all sub-classes. It does not matterto the proof if β has no members: all that happensin that case is that the sub-class which is shown tobe omitted is the null-class. Hence in any case thenumber of sub-classes is not equal to the number ofmembers, and therefore, by what was said earlier, it

(), p. .

is greater. Combining this with the proposition that,if n is the number of members, n is the number ofsub-classes, we have the theorem that n is alwaysgreater than n, even when n is infinite.

It follows from this proposition that there is nomaximum to the infinite cardinal numbers. How-ever great an infinite number n may be, n will bestill greater. The arithmetic of infinite numbers issomewhat surprising until one becomes accustomedto it. We have, for example,

ℵ + = ℵ,ℵ +n = ℵ, where n is any

inductive number,

ℵ = ℵ.

(This follows from the case of the ratios, for, since aratio is determined by a pair of inductive numbers,it is easy to see that the number of ratios is thesquare of the number of inductive numbers, i.e. it isℵ; but we saw that it is also ℵ.)

ℵn = ℵ, where n is any

inductive number.

(This follows from ℵ = ℵ by induction;

for if ℵn = ℵ,then ℵn+ = ℵ = ℵ.)

But ℵ > ℵ.

In fact, as we shall see later, ℵ is a very importantnumber, namely, the number of terms in a serieswhich has “continuity” in the sense in which thisword is used by Cantor. Assuming space and timeto be continuous in this sense (as we commonly doin analytical geometry and kinematics), this will bethe number of points in space or of instants in time;it will also be the number of points in any finiteportion of space, whether | line, area, or volume. After ℵ, ℵ is the most important and interestingof infinite cardinal numbers.

Although addition and multiplication are alwayspossible with infinite cardinals, subtraction and di-vision no longer give definite results, and cannottherefore be employed as they are employed in ele-mentary arithmetic. Take subtraction to begin with:so long as the number subtracted is finite, all goeswell; if the other number is reflexive, it remainsunchanged. Thus ℵ − n = ℵ, if n is finite; so far,

subtraction gives a perfectly definite result. But it isotherwise when we subtract ℵ from itself; we maythen get any result, from up to ℵ. This is eas-ily seen by examples. From the inductive numbers,take away the following collections of ℵ terms:—

() All the inductive numbers—remainder,zero.

() All the inductive numbers from n onwards—remainder, the numbers from to n−, numberingn terms in all.

() All the odd numbers—remainder, all theeven numbers, numbering ℵ terms.

All these are different ways of subtracting ℵfrom ℵ, and all give different results.

As regards division, very similar results followfrom the fact that ℵ is unchanged when multipliedby or or any finite number n or by ℵ. It followsthat ℵ divided by ℵ may have any value from up to ℵ.

From the ambiguity of subtraction and divisionit results that negative numbers and ratios cannotbe extended to infinite numbers. Addition, multi-plication, and exponentiation proceed quite satis-factorily, but the inverse operations—subtraction,division, and extraction of roots—are ambiguous,and the notions that depend upon them fail when

infinite numbers are concerned.The characteristic by which we defined finitude

was mathematical induction, i.e. we defined a num-ber as finite when it obeys mathematical inductionstarting from , and a class as finite when its num-ber is finite. This definition yields the sort of resultthat a definition ought to yield, namely, that thefinite | numbers are those that occur in the ordinary number-series , , , , . . . But in the present chap-ter, the infinite numbers we have discussed havenot merely been non-inductive: they have also beenreflexive. Cantor used reflexiveness as the definitionof the infinite, and believes that it is equivalent tonon-inductiveness; that is to say, he believes thatevery class and every cardinal is either inductive orreflexive. This may be true, and may very possiblybe capable of proof; but the proofs hitherto offeredby Cantor and others (including the present authorin former days) are fallacious, for reasons whichwill be explained when we come to consider the“multiplicative axiom.” At present, it is not knownwhether there are classes and cardinals which areneither reflexive nor inductive. If n were such a car-dinal, we should not have n = n+, but n would notbe one of the “natural numbers,” and would be lack-ing in some of the inductive properties. All known

infinite classes and cardinals are reflexive; but forthe present it is well to preserve an open mind asto whether there are instances, hitherto unknown,of classes and cardinals which are neither reflexivenor inductive. Meanwhile, we adopt the followingdefinitions:—

A finite class or cardinal is one which is induc-tive.

An infinite class or cardinal is one which is notinductive. All reflexive classes and cardinals are in-finite; but it is not known at present whether allinfinite classes and cardinals are reflexive. We shallreturn to this subject in Chapter XII.

Chapter IX

Infinite Series and Ordinals

An “infinite series” may be defined as a series of which the field is an infinite class. We have alreadyhad occasion to consider one kind of infinite se-ries, namely, progressions. In this chapter we shallconsider the subject more generally.

The most noteworthy characteristic of an infi-nite series is that its serial number can be alteredby merely re-arranging its terms. In this respectthere is a certain oppositeness between cardinal andserial numbers. It is possible to keep the cardinalnumber of a reflexive class unchanged in spite ofadding terms to it; on the other hand, it is possible tochange the serial number of a series without addingor taking away any terms, by mere re-arrangement.At the same time, in the case of any infinite series itis also possible, as with cardinals, to add terms with-

out altering the serial number: everything dependsupon the way in which they are added.

In order to make matters clear, it will be best tobegin with examples. Let us first consider variousdifferent kinds of series which can be made out ofthe inductive numbers arranged on various plans.We start with the series

, , , , . . . n, . . .,

which, as we have already seen, represents the small-est of infinite serial numbers, the sort that Cantorcalls ω. Let us proceed to thin out this series byrepeatedly performing the | operation of removing to the end the first even number that occurs. Wethus obtain in succession the various series:

, , , , . . . n, . . . ,, , , , . . . n+ , . . . , ,, , , , . . . n+ , . . . , , ,

and so on. If we imagine this process carried on aslong as possible, we finally reach the series

, , , , . . . n+ , . . . , , , , . . . n, . . .,

in which we have first all the odd numbers and thenall the even numbers.

The serial numbers of these various series areω+ , ω + , ω + , . . . ω. Each of these numbers is“greater” than any of its predecessors, in the follow-ing sense:—

One serial number is said to be “greater” thananother if any series having the first number con-tains a part having the second number, but no serieshaving the second number contains a part havingthe first number.

If we compare the two series

, , , , . . . n, . . ., , , , . . . n+ , . . . ,

we see that the first is similar to the part of the sec-ond which omits the last term, namely, the number, but the second is not similar to any part of thefirst. (This is obvious, but is easily demonstrated.)Thus the second series has a greater serial num-ber than the first, according to the definition—i.e.ω + is greater than ω. But if we add a term atthe beginning of a progression instead of the end,we still have a progression. Thus +ω = ω. Thus +ω is not equal to ω + . This is characteristicof relation-arithmetic generally: if µ and ν are tworelation-numbers, the general rule is that µ+ν is notequal to ν +µ. The case of finite ordinals, in which

there is equality, is quite exceptional.The series we finally reached just now consisted

of first all the odd numbers and then all the evennumbers, and its serial | number is ω. This number is greater than ω or ω + n, where n is finite. It isto be observed that, in accordance with the generaldefinition of order, each of these arrangements ofintegers is to be regarded as resulting from somedefinite relation. E.g. the one which merely removes to the end will be defined by the following re-lation: “x and y are finite integers, and either y is and x is not , or neither is and x is less thany.” The one which puts first all the odd numbersand then all the even ones will be defined by: “xand y are finite integers, and either x is odd and y iseven or x is less than y and both are odd or both areeven.” We shall not trouble, as a rule, to give theseformulæ in future; but the fact that they could begiven is essential.

The number which we have called ω, namely,the number of a series consisting of two progres-sions, is sometimes called ω . . Multiplication, likeaddition, depends upon the order of the factors: aprogression of couples gives a series such as

x, y, x, y, x, y, . . . xn, yn, . . . ,

which is itself a progression; but a couple of pro-gressions gives a series which is twice as long as aprogression. It is therefore necessary to distinguishbetween ω and ω . . Usage is variable; we shalluse ω for a couple of progressions and ω . for aprogression of couples, and this decision of coursegoverns our general interpretation of “α . β” whenα and β are relation-numbers: “α . β” will have tostand for a suitably constructed sum of α relationseach having β terms.

We can proceed indefinitely with the process ofthinning out the inductive numbers. For example,we can place first the odd numbers, then their dou-bles, then the doubles of these, and so on. We thusobtain the series

, , , , . . .; , , , , . . .; , , , , . . .;, , , , . . .,

of which the number is ω, since it is a progressionof progressions. Any one of the progressions inthis new series can of course be | thinned out as we thinned out our original progression. We canproceed to ω, ω, . . . ωω, and so on; however farwe have gone, we can always go further.

The series of all the ordinals that can be obtainedin this way, i.e. all that can be obtained by thinning

out a progression, is itself longer than any seriesthat can be obtained by re-arranging the terms ofa progression. (This is not difficult to prove.) Thecardinal number of the class of such ordinals canbe shown to be greater than ℵ; it is the numberwhich Cantor calls ℵ. The ordinal number of theseries of all ordinals that can be made out of an ℵ,taken in order of magnitude, is called ω. Thus aseries whose ordinal number is ω has a field whosecardinal number is ℵ.

We can proceed from ω and ℵ to ω and ℵby a process exactly analogous to that by which weadvanced from ω and ℵ to ω and ℵ. And there isnothing to prevent us from advancing indefinitely inthis way to new cardinals and new ordinals. It is notknown whether ℵ is equal to any of the cardinalsin the series of Alephs. It is not even known whetherit is comparable with them in magnitude; for aughtwe know, it may be neither equal to nor greater norless than any one of the Alephs. This question isconnected with the multiplicative axiom, of whichwe shall treat later.

All the series we have been considering so far inthis chapter have been what is called “well-ordered.”A well-ordered series is one which has a beginning,and has consecutive terms, and has a term next after

any selection of its terms, provided there are anyterms after the selection. This excludes, on the onehand, compact series, in which there are terms be-tween any two, and on the other hand series whichhave no beginning, or in which there are subor-dinate parts having no beginning. The series ofnegative integers in order of magnitude, having nobeginning, but ending with −, is not well-ordered;but taken in the reverse order, beginning with −,it is well-ordered, being in fact a progression. Thedefinition is: |

A “well-ordered” series is one in which every sub-class (except, of course, the null-class) has afirst term.

An “ordinal” number means the relation-numberof a well-ordered series. It is thus a species of serialnumber.

Among well-ordered series, a generalised formof mathematical induction applies. A property maybe said to be “transfinitely hereditary” if, when it be-longs to a certain selection of the terms in a series, itbelongs to their immediate successor provided theyhave one. In a well-ordered series, a transfinitelyhereditary property belonging to the first term ofthe series belongs to the whole series. This makesit possible to prove many propositions concerning

well-ordered series which are not true of all series.It is easy to arrange the inductive numbers in

series which are not well-ordered, and even to ar-range them in compact series. For example, we canadopt the following plan: consider the decimalsfrom · (inclusive) to (exclusive), arranged in or-der of magnitude. These form a compact series;between any two there are always an infinite num-ber of others. Now omit the dot at the beginningof each, and we have a compact series consisting ofall finite integers except such as divide by . If wewish to include those that divide by , there is nodifficulty; instead of starting with ·, we will includeall decimals less than , but when we remove thedot, we will transfer to the right any ’s that occur atthe beginning of our decimal. Omitting these, andreturning to the ones that have no ’s at the begin-ning, we can state the rule for the arrangement ofour integers as follows: Of two integers that do notbegin with the same digit, the one that begins withthe smaller digit comes first. Of two that do beginwith the same digit, but differ at the second digit,the one with the smaller second digit comes first,but first of all the one with no second digit; and soon. Generally, if two integers agree as regards thefirst n digits, but not as regards the (n + )th, that

one comes first which has either no (n+ )th digit ora smaller one than the other. This rule of arrange-ment, | as the reader can easily convince himself, gives rise to a compact series containing all the in-tegers not divisible by ; and, as we saw, there isno difficulty about including those that are divisibleby . It follows from this example that it is possi-ble to construct compact series having ℵ terms. Infact, we have already seen that there are ℵ ratios,and ratios in order of magnitude form a compactseries; thus we have here another example. We shallresume this topic in the next chapter.

Of the usual formal laws of addition, multipli-cation, and exponentiation, all are obeyed by trans-finite cardinals, but only some are obeyed by trans-finite ordinals, and those that are obeyed by themare obeyed by all relation-numbers. By the “usualformal laws” we mean the following:—

I. The commutative law:α + β = β +α and α × β = β ×α.

II. The associative law:(α + β) +γ = α + (β +γ) and

(α × β)×γ = α × (β ×γ).III. The distributive law:

α(β +γ) = αβ +αγ .

When the commutative law does not hold, theabove form of the distributive law must be distin-guished from

(β +γ)α = βα +γα.

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

IV. The laws of exponentiation:αβ . αγ = αβ+γ , αγ . βγ = (αβ)γ , (αβ)γ = αβγ .

All these laws hold for cardinals, whether fi-nite or infinite, and for finite ordinals. But whenwe come to infinite ordinals, or indeed to relation-numbers in general, some hold and some do not.The commutative law does not hold; the associativelaw does hold; the distributive law (adopting theconvention | we have adopted above as regards the order of the factors in a product) holds in the form

(β +γ)α = βα +γα,

but not in the form

α(β +γ) = αβ +αγ ;

the exponential laws

αβ . αγ = αβ+γ and (αβ)γ = αβγ

still hold, but not the law

αγ . βγ = (αβ)γ ,

which is obviously connected with the commutativelaw for multiplication.

The definitions of multiplication and exponen-tiation that are assumed in the above propositionsare somewhat complicated. The reader who wishesto know what they are and how the above laws areproved must consult the second volume of PrincipiaMathematica, ∗–.

Ordinal transfinite arithmetic was developed byCantor at an earlier stage than cardinal transfinitearithmetic, because it has various technical mathe-matical uses which led him to it. But from the pointof view of the philosophy of mathematics it is lessimportant and less fundamental than the theory oftransfinite cardinals. Cardinals are essentially sim-pler than ordinals, and it is a curious historical acci-dent that they first appeared as an abstraction fromthe latter, and only gradually came to be studied ontheir own account. This does not apply to Frege’swork, in which cardinals, finite and transfinite, weretreated in complete independence of ordinals; but itwas Cantor’s work that made the world aware of thesubject, while Frege’s remained almost unknown,

probably in the main on account of the difficultyof his symbolism. And mathematicians, like otherpeople, have more difficulty in understanding andusing notions which are comparatively “simple” inthe logical sense than in manipulating more com-plex notions which are |more akin to their ordinary practice. For these reasons, it was only graduallythat the true importance of cardinals in mathemati-cal philosophy was recognised. The importance ofordinals, though by no means small, is distinctly lessthan that of cardinals, and is very largely mergedin that of the more general conception of relation-numbers.

Chapter X

Limits and Continuity

The conception of a “limit” is one of which the im- portance in mathematics has been found continuallygreater than had been thought. The whole of thedifferential and integral calculus, indeed practicallyeverything in higher mathematics, depends uponlimits. Formerly, it was supposed that infinitesimalswere involved in the foundations of these subjects,but Weierstrass showed that this is an error: wher-ever infinitesimals were thought to occur, what re-ally occurs is a set of finite quantities having zero fortheir lower limit. It used to be thought that “limit”was an essentially quantitative notion, namely, thenotion of a quantity to which others approachednearer and nearer, so that among those others therewould be some differing by less than any assignedquantity. But in fact the notion of “limit” is a purely

ordinal notion, not involving quantity at all (exceptby accident when the series concerned happens tobe quantitative). A given point on a line may be thelimit of a set of points on the line, without its beingnecessary to bring in co-ordinates or measurementor anything quantitative. The cardinal number ℵ isthe limit (in the order of magnitude) of the cardinalnumbers , , , . . . n, . . . , although the numeri-cal difference between ℵ and a finite cardinal isconstant and infinite: from a quantitative point ofview, finite numbers get no nearer to ℵ as theygrow larger. What makes ℵ the limit of the finitenumbers is the fact that, in the series, it comes im-mediately after them, which is an ordinal fact, not aquantitative fact. |

There are various forms of the notion of “lim- it,” of increasing complexity. The simplest andmost fundamental form, from which the rest arederived, has been already defined, but we will hererepeat the definitions which lead to it, in a generalform in which they do not demand that the rela-tion concerned shall be serial. The definitions are asfollows:—

The “minima” of a class α with respect to a rela-tion P are those members of α and the field of P (ifany) to which no member of α has the relation P.

The “maxima” with respect to P are the minimawith respect to the converse of P.

The “sequents” of a class α with respect to arelation P are the minima of the “successors” of α,and the “successors” of α are those members of thefield of P to which every member of the commonpart of α and the field of P has the relation P.

The “precedents” with respect to P are the se-quents with respect to the converse of P.

The “upper limits” of α with respect to P are thesequents provided α has no maximum; but if α hasa maximum, it has no upper limits.

The “lower limits” with respect to P are the up-per limits with respect to the converse of P.

Whenever P has connexity, a class can have atmost one maximum, one minimum, one sequent,etc. Thus, in the cases we are concerned with inpractice, we can speak of “the limit” (if any).

When P is a serial relation, we can greatly sim-plify the above definition of a limit. We can, in thatcase, define first the “boundary” of a class α, i.e. itslimit or maximum, and then proceed to distinguishthe case where the boundary is the limit from thecase where it is a maximum. For this purpose it isbest to use the notion of “segment.”

We will speak of the “segment of P defined by a

class α” as all those terms that have the relation Pto some one or more of the members of α. This willbe a segment in the sense defined | in Chapter VII.; indeed, every segment in the sense there defined isthe segment defined by some class α. If P is serial,the segment defined by α consists of all the termsthat precede some term or other of α. If α has amaximum, the segment will be all the predecessorsof the maximum. But if α has no maximum, everymember of α precedes some other member of α, andthe whole of α is therefore included in the segmentdefined by α. Take, for example, the class consistingof the fractions

, , , , . . .,

i.e. of all fractions of the form − /n for differentfinite values of n. This series of fractions has nomaximum, and it is clear that the segment which itdefines (in the whole series of fractions in order ofmagnitude) is the class of all proper fractions. Or,again, consider the prime numbers, considered asa selection from the cardinals (finite and infinite)in order of magnitude. In this case the segmentdefined consists of all finite integers.

Assuming that P is serial, the “boundary” of aclass α will be the term x (if it exists) whose prede-

cessors are the segment defined by α.A “maximum” of α is a boundary which is a

member of α.An “upper limit” of α is a boundary which is not

a member of α.If a class has no boundary, it has neither maxi-

mum nor limit. This is the case of an “irrational”Dedekind cut, or of what is called a “gap.”

Thus the “upper limit” of a set of terms α withrespect to a series P is that term x (if it exists) whichcomes after all the α’s, but is such that every earlierterm comes before some of the α’s.

We may define all the “upper limiting-points”of a set of terms β as all those that are the upperlimits of sets of terms chosen out of β. We shall, ofcourse, have to distinguish upper limiting-pointsfrom lower limiting-points. If we consider, for ex-ample, the series of ordinal numbers:

, , , . . . ω, ω+ , . . . ω, ω+ , . . .ω, . . . ω, . . . ω, . . . , |

the upper limiting-points of the field of this series are those that have no immediate predecessors, i.e.

, ω, ω, ω, . . . ω, ω +ω, . . . ω, . . . ω . . .

The upper limiting-points of the field of this newseries will be

, ω, ω, . . . ω, ω +ω . . .

On the other hand, the series of ordinals—and in-deed every well-ordered series—has no lower limit-ing-points, because there are no terms except thelast that have no immediate successors. But if weconsider such a series as the series of ratios, ev-ery member of this series is both an upper and alower limiting-point for suitably chosen sets. If weconsider the series of real numbers, and select outof it the rational real numbers, this set (the ratio-nals) will have all the real numbers as upper andlower limiting-points. The limiting-points of a setare called its “first derivative,” and the limiting-points of the first derivative are called the secondderivative, and so on.

With regard to limits, we may distinguish var-ious grades of what may be called “continuity” ina series. The word “continuity” had been used fora long time, but had remained without any precisedefinition until the time of Dedekind and Cantor.Each of these two men gave a precise significance tothe term, but Cantor’s definition is narrower thanDedekind’s: a series which has Cantorian continuity

must have Dedekindian continuity, but the conversedoes not hold.

The first definition that would naturally occur toa man seeking a precise meaning for the continuityof series would be to define it as consisting in whatwe have called “compactness,” i.e. in the fact thatbetween any two terms of the series there are others.But this would be an inadequate definition, becauseof the existence of “gaps” in series such as the se-ries of ratios. We saw in Chapter VII. that thereare innumerable ways in which the series of ratioscan be divided into two parts, of which one whollyprecedes the other, and of which the first has no lastterm, | while the second has no first term. Such a state of affairs seems contrary to the vague feelingwe have as to what should characterise “continuity,”and, what is more, it shows that the series of ra-tios is not the sort of series that is needed for manymathematical purposes. Take geometry, for exam-ple: we wish to be able to say that when two straightlines cross each other they have a point in common,but if the series of points on a line were similar tothe series of ratios, the two lines might cross in a“gap” and have no point in common. This is a crudeexample, but many others might be given to showthat compactness is inadequate as a mathematical

definition of continuity.It was the needs of geometry, as much as any-

thing, that led to the definition of “Dedekindian”continuity. It will be remembered that we defined aseries as Dedekindian when every sub-class of thefield has a boundary. (It is sufficient to assume thatthere is always an upper boundary, or that there isalways a lower boundary. If one of these is assumed,the other can be deduced.) That is to say, a seriesis Dedekindian when there are no gaps. The ab-sence of gaps may arise either through terms havingsuccessors, or through the existence of limits in theabsence of maxima. Thus a finite series or a well-ordered series is Dedekindian, and so is the series ofreal numbers. The former sort of Dedekindian seriesis excluded by assuming that our series is compact;in that case our series must have a property whichmay, for many purposes, be fittingly called continu-ity. Thus we are led to the definition:

A series has “Dedekindian continuity” when itis Dedekindian and compact.

But this definition is still too wide for manypurposes. Suppose, for example, that we desireto be able to assign such properties to geometricalspace as shall make it certain that every point canbe specified by means of co-ordinates which are real

numbers: this is not insured by Dedekindian con-tinuity alone. We want to be sure that every pointwhich cannot be specified by rational co-ordinatescan be specified as the limit of a progression of points| whose co-ordinates are rational, and this is a fur- ther property which our definition does not enableus to deduce.

We are thus led to a closer investigation of serieswith respect to limits. This investigation was madeby Cantor and formed the basis of his definition ofcontinuity, although, in its simplest form, this defi-nition somewhat conceals the considerations whichhave given rise to it. We shall, therefore, first travelthrough some of Cantor’s conceptions in this subjectbefore giving his definition of continuity.

Cantor defines a series as “perfect” when allits points are limiting-points and all its limiting-points belong to it. But this definition does notexpress quite accurately what he means. There is nocorrection required so far as concerns the propertythat all its points are to be limiting-points; this isa property belonging to compact series, and to noothers if all points are to be upper limiting- or alllower limiting-points. But if it is only assumed thatthey are limiting-points one way, without specify-ing which, there will be other series that will have

the property in question—for example, the series ofdecimals in which a decimal ending in a recurring is distinguished from the corresponding terminat-ing decimal and placed immediately before it. Sucha series is very nearly compact, but has exceptionalterms which are consecutive, and of which the firsthas no immediate predecessor, while the second hasno immediate successor. Apart from such series, theseries in which every point is a limiting-point arecompact series; and this holds without qualificationif it is specified that every point is to be an upperlimiting-point (or that every point is to be a lowerlimiting-point).

Although Cantor does not explicitly considerthe matter, we must distinguish different kinds oflimiting-points according to the nature of the small-est sub-series by which they can be defined. Cantorassumes that they are to be defined by progressions,or by regressions (which are the converses of pro-gressions). When every member of our series is thelimit of a progression or regression, Cantor calls ourseries “condensed in itself” (insichdicht). |

We come now to the second property by which perfection was to be defined, namely, the propertywhich Cantor calls that of being “closed” (abgeschloss-en). This, as we saw, was first defined as consisting

in the fact that all the limiting-points of a seriesbelong to it. But this only has any effective signifi-cance if our series is given as contained in some otherlarger series (as is the case, e.g., with a selection ofreal numbers), and limiting-points are taken in re-lation to the larger series. Otherwise, if a series isconsidered simply on its own account, it cannot failto contain its limiting-points. What Cantor meansis not exactly what he says; indeed, on other occa-sions he says something rather different, which iswhat he means. What he really means is that everysubordinate series which is of the sort that mightbe expected to have a limit does have a limit withinthe given series; i.e. every subordinate series whichhas no maximum has a limit, i.e. every subordinateseries has a boundary. But Cantor does not statethis for every subordinate series, but only for pro-gressions and regressions. (It is not clear how far herecognises that this is a limitation.) Thus, finally, wefind that the definition we want is the following:—

A series is said to be “closed” (abgeschlossen)when every progression or regression contained inthe series has a limit in the series.

We then have the further definition:—A series is “perfect” when it is condensed in itself

and closed, i.e. when every term is the limit of a

progression or regression, and every progression orregression contained in the series has a limit in theseries.

In seeking a definition of continuity, what Can-tor has in mind is the search for a definition whichshall apply to the series of real numbers and to anyseries similar to that, but to no others. For this pur-pose we have to add a further property. Among thereal numbers some are rational, some are irrational;although the number of irrationals is greater thanthe number of rationals, yet there are rationals be-tween any two real numbers, however | little the two may differ. The number of rationals, as we saw,is ℵ. This gives a further property which sufficesto characterise continuity completely, namely, theproperty of containing a class of ℵ members insuch a way that some of this class occur betweenany two terms of our series, however near together.This property, added to perfection, suffices to definea class of series which are all similar and are in facta serial number. This class Cantor defines as that ofcontinuous series.

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

A “median class” of a series is a sub-class ofthe field such that members of it are to be found

between any two terms of the series.Thus the rationals are a median class in the series

of real numbers. It is obvious that there cannot bemedian classes except in compact series.

We then find that Cantor’s definition is equiva-lent to the following:—

A series is “continuous” when () it is Dedekin-dian, () it contains a median class having ℵ terms.

To avoid confusion, we shall speak of this kindas “Cantorian continuity.” It will be seen that itimplies Dedekindian continuity, but the converse isnot the case. All series having Cantorian continuityare similar, but not all series having Dedekindiancontinuity.

The notions of limit and continuity which wehave been defining must not be confounded withthe notions of the limit of a function for approachesto a given argument, or the continuity of a functionin the neighbourhood of a given argument. Theseare different notions, very important, but deriva-tive from the above and more complicated. Thecontinuity of motion (if motion is continuous) isan instance of the continuity of a function; on theother hand, the continuity of space and time (if theyare continuous) is an instance of the continuity ofseries, or (to speak more cautiously) of a kind of

continuity which can, by sufficient mathematical |manipulation, be reduced to the continuity of series. In view of the fundamental importance of motion inapplied mathematics, as well as for other reasons, itwill be well to deal briefly with the notions of lim-its and continuity as applied to functions; but thissubject will be best reserved for a separate chapter.

The definitions of continuity which we have beenconsidering, namely, those of Dedekind and Cantor,do not correspond very closely to the vague ideawhich is associated with the word in the mind ofthe man in the street or the philosopher. They con-ceive continuity rather as absence of separateness,the sort of general obliteration of distinctions whichcharacterises a thick fog. A fog gives an impressionof vastness without definite multiplicity or division.It is this sort of thing that a metaphysician meansby “continuity,” declaring it, very truly, to be char-acteristic of his mental life and of that of childrenand animals.

The general idea vaguely indicated by the word“continuity” when so employed, or by the word“flux,” is one which is certainly quite different fromthat which we have been defining. Take, for exam-ple, the series of real numbers. Each is what it is,quite definitely and uncompromisingly; it does not

pass over by imperceptible degrees into another; itis a hard, separate unit, and its distance from everyother unit is finite, though it can be made less thanany given finite amount assigned in advance. Thequestion of the relation between the kind of conti-nuity existing among the real numbers and the kindexhibited, e.g. by what we see at a given time, is adifficult and intricate one. It is not to be maintainedthat the two kinds are simply identical, but it may,I think, be very well maintained that the mathe-matical conception which we have been consideringin this chapter gives the abstract logical scheme towhich it must be possible to bring empirical mate-rial by suitable manipulation, if that material is tobe called “continuous” in any precisely definablesense. It would be quite impossible | to justify this thesis within the limits of the present volume. Thereader who is interested may read an attempt tojustify it as regards time in particular by the presentauthor in the Monist for −, as well as in partsof Our Knowledge of the External World. With theseindications, we must leave this problem, interest-ing as it is, in order to return to topics more closelyconnected with mathematics.

Chapter XI

Limits and Continuity of Functions

In this chapter we shall be concerned with the defi- nition of the limit of a function (if any) as the argu-ment approaches a given value, and also with thedefinition of what is meant by a “continuous func-tion.” Both of these ideas are somewhat technical,and would hardly demand treatment in a mere in-troduction to mathematical philosophy but for thefact that, especially through the so-called infinitesi-mal calculus, wrong views upon our present topicshave become so firmly embedded in the minds ofprofessional philosophers that a prolonged and con-siderable effort is required for their uprooting. It hasbeen thought ever since the time of Leibniz that thedifferential and integral calculus required infinites-imal quantities. Mathematicians (especially Weier-strass) proved that this is an error; but errors incor-

porated, e.g. in what Hegel has to say about mathe-matics, die hard, and philosophers have tended toignore the work of such men as Weierstrass.

Limits and continuity of functions, in works onordinary mathematics, are defined in terms involv-ing number. This is not essential, as Dr Whiteheadhas shown. We will, however, begin with the defi-nitions in the text-books, and proceed afterwards toshow how these definitions can be generalised so asto apply to series in general, and not only to such asare numerical or numerically measurable.

Let us consider any ordinary mathematical func-tion fx, where | x and fx are both real numbers, and fx is one-valued—i.e. when x is given, thereis only one value that fx can have. We call x the“argument,” and fx the “value for the argumentx.” When a function is what we call “continuous,”the rough idea for which we are seeking a precisedefinition is that small differences in x shall corre-spond to small differences in fx, and if we makethe differences in x small enough, we can make thedifferences in fx fall below any assigned amount.We do not want, if a function is to be continuous,that there shall be sudden jumps, so that, for somevalue of x, any change, however small, will make

See Principia Mathematica, vol. ii. ∗–.

a change in fx which exceeds some assigned finiteamount. The ordinary simple functions of mathe-matics have this property: it belongs, for example,to x, x, . . . logx, sinx, and so on. But it is not atall difficult to define discontinuous functions. Take,as a non-mathematical example, “the place of birthof the youngest person living at time t.” This is afunction of t; its value is constant from the time ofone person’s birth to the time of the next birth, andthen the value changes suddenly from one birthplaceto the other. An analogous mathematical examplewould be “the integer next below x,” where x is areal number. This function remains constant fromone integer to the next, and then gives a suddenjump. The actual fact is that, though continuousfunctions are more familiar, they are the exceptions:there are infinitely more discontinuous functionsthan continuous ones.

Many functions are discontinuous for one or sev-eral values of the variable, but continuous for allother values. Take as an example sin/x. The func-tion sin θ passes through all values from − to ev-ery time that θ passes from −π/ to π/, or from π/to π/, or generally from (n−)π/ to (n+)π/,where n is any integer. Now if we consider /xwhen x is very small, we see that as x diminishes

/x grows faster and faster, so that it passes moreand more quickly through the cycle of values fromone multiple of π/ to another as x becomes smallerand smaller. Consequently sin/x passes more andmore quickly from − | to and back again, as x grows smaller. In fact, if we take any interval con-taining , say the interval from −ε to +ε where ε issome very small number, sin/x will go through aninfinite number of oscillations in this interval, andwe cannot diminish the oscillations by making theinterval smaller. Thus round about the argument the function is discontinuous. It is easy to manufac-ture functions which are discontinuous in severalplaces, or in ℵ places, or everywhere. Exampleswill be found in any book on the theory of functionsof a real variable.

Proceeding now to seek a precise definition ofwhat is meant by saying that a function is continu-ous for a given argument, when argument and valueare both real numbers, let us first define a “neigh-bourhood” of a number x as all the numbers fromx − ε to x + ε, where ε is some number which, inimportant cases, will be very small. It is clear thatcontinuity at a given point has to do with what hap-pens in any neighbourhood of that point, howeversmall.

What we desire is this: If a is the argument forwhich we wish our function to be continuous, letus first define a neighbourhood (α say) containingthe value fa which the function has for the argu-ment a; we desire that, if we take a sufficiently smallneighbourhood containing a, all values for argu-ments throughout this neighbourhood shall be con-tained in the neighbourhood α, no matter how smallwe may have made α. That is to say, if we decreethat our function is not to differ from fa by morethan some very tiny amount, we can always find astretch of real numbers, having a in the middle of it,such that throughout this stretch fx will not differfrom fa by more than the prescribed tiny amount.And this is to remain true whatever tiny amountwe may select. Hence we are led to the followingdefinition:—

The function f (x) is said to be “continuous” forthe argument a if, for every positive number σ , dif-ferent from , but as small as we please, there existsa positive number ε, different from , such that, forall values of δ which are numerically | less than ε, the difference f (a+δ)− f (a) is numerically less thanσ .A number is said to be “numerically less” than ε when it lies

between −ε and +ε.

In this definition, σ first defines a neighbour-hood of f (a), namely, the neighbourhood from f (a)−σ to f (a) + σ . The definition then proceeds to saythat we can (by means of ε) define a neighbourhood,namely, that from a − ε to a + ε, such that, for allarguments within this neighbourhood, the value ofthe function lies within the neighbourhood fromf (a)− σ to f (a) + σ . If this can be done, however σmay be chosen, the function is “continuous” for theargument a.

So far we have not defined the “limit” of a func-tion for a given argument. If we had done so, wecould have defined the continuity of a function dif-ferently: a function is continuous at a point whereits value is the same as the limit of its values forapproaches either from above or from below. But itis only the exceptionally “tame” function that has adefinite limit as the argument approaches a givenpoint. The general rule is that a function oscillates,and that, given any neighbourhood of a given argu-ment, however small, a whole stretch of values willoccur for arguments within this neighbourhood. Asthis is the general rule, let us consider it first.

Let us consider what may happen as the argu-ment approaches some value a from below. Thatis to say, we wish to consider what happens for ar-

guments contained in the interval from a − ε to a,where ε is some number which, in important cases,will be very small.

The values of the function for arguments froma− ε to a (a excluded) will be a set of real numberswhich will define a certain section of the set of realnumbers, namely, the section consisting of thosenumbers that are not greater than all the valuesfor arguments from a − ε to a. Given any numberin this section, there are values at least as great asthis number for arguments between a − ε and a,i.e. for arguments that fall very little short | of a (if ε is very small). Let us take all possible ε’s andall possible corresponding sections. The commonpart of all these sections we will call the “ultimatesection” as the argument approaches a. To say thata number z belongs to the ultimate section is tosay that, however small we may make ε, there arearguments between a− ε and a for which the valueof the function is not less than z.

We may apply exactly the same process to uppersections, i.e. to sections that go from some point upto the top, instead of from the bottom up to somepoint. Here we take those numbers that are not lessthan all the values for arguments from a − ε to a;this defines an upper section which will vary as ε

varies. Taking the common part of all such sectionsfor all possible ε’s, we obtain the “ultimate uppersection.” To say that a number z belongs to theultimate upper section is to say that, however smallwe make ε, there are arguments between a− ε anda for which the value of the function is not greaterthan z.

If a term z belongs both to the ultimate sectionand to the ultimate upper section, we shall say thatit belongs to the “ultimate oscillation.” We may il-lustrate the matter by considering once more thefunction sin/x as x approaches the value . Weshall assume, in order to fit in with the above defi-nitions, that this value is approached from below.

Let us begin with the “ultimate section.” Be-tween −ε and , whatever ε may be, the functionwill assume the value for certain arguments, butwill never assume any greater value. Hence the ul-timate section consists of all real numbers, positiveand negative, up to and including ; i.e. it consistsof all negative numbers together with , togetherwith the positive numbers up to and including .

Similarly the “ultimate upper section” consistsof all positive numbers together with , togetherwith the negative numbers down to and including−.

Thus the “ultimate oscillation” consists of allreal numbers from − to , both included. |

We may say generally that the “ultimate oscil- lation” of a function as the argument approaches afrom below consists of all those numbers x whichare such that, however near we come to a, we shallstill find values as great as x and values as small asx.

The ultimate oscillation may contain no terms,or one term, or many terms. In the first two casesthe function has a definite limit for approaches frombelow. If the ultimate oscillation has one term, thisis fairly obvious. It is equally true if it has none; forit is not difficult to prove that, if the ultimate oscil-lation is null, the boundary of the ultimate sectionis the same as that of the ultimate upper section,and may be defined as the limit of the function forapproaches from below. But if the ultimate oscil-lation has many terms, there is no definite limit tothe function for approaches from below. In this casewe can take the lower and upper boundaries of theultimate oscillation (i.e. the lower boundary of theultimate upper section and the upper boundary ofthe ultimate section) as the lower and upper limitsof its “ultimate” values for approaches from below.Similarly we obtain lower and upper limits of the

“ultimate” values for approaches from above. Thuswe have, in the general case, four limits to a functionfor approaches to a given argument. The limit fora given argument a only exists when all these fourare equal, and is then their common value. If it isalso the value for the argument a, the function iscontinuous for this argument. This may be taken asdefining continuity: it is equivalent to our formerdefinition.

We can define the limit of a function for a givenargument (if it exists) without passing through theultimate oscillation and the four limits of the gen-eral case. The definition proceeds, in that case, justas the earlier definition of continuity proceeded. Letus define the limit for approaches from below. Ifthere is to be a definite limit for approaches to afrom below, it is necessary and sufficient that, givenany small number σ , two values for arguments suf-ficiently near to a (but both less than a) will differ| by less than σ ; i.e. if ε is sufficiently small, and our arguments both lie between a − ε and a (a ex-cluded), then the difference between the values forthese arguments will be less than σ . This is to holdfor any σ , however small; in that case the functionhas a limit for approaches from below. Similarly wedefine the case when there is a limit for approaches

from above. These two limits, even when both exist,need not be identical; and if they are identical, theystill need not be identical with the value for the ar-gument a. It is only in this last case that we call thefunction continuous for the argument a.

A function is called “continuous” (without qual-ification) when it is continuous for every argument.

Another slightly different method of reachingthe definition of continuity is the following:—

Let us say that a function “ultimately convergesinto a class α” if there is some real number suchthat, for this argument and all arguments greaterthan this, the value of the function is a member ofthe class α. Similarly we shall say that a function“converges into α as the argument approaches xfrom below” if there is some argument y less than xsuch that throughout the interval from y (included)to x (excluded) the function has values which aremembers of α. We may now say that a function iscontinuous for the argument a, for which it has thevalue fa, if it satisfies four conditions, namely:—

() Given any real number less than fa, the func-tion converges into the successors of this number asthe argument approaches a from below;

() Given any real number greater than fa, thefunction converges into the predecessors of this

number as the argument approaches a from below;() and () Similar conditions for approaches to

a from above.The advantage of this form of definition is that

it analyses the conditions of continuity into four,derived from considering arguments and values re-spectively greater or less than the argument andvalue for which continuity is to be defined. |

We may now generalise our definitions so as to apply to series which are not numerical or known tobe numerically measurable. The case of motion is aconvenient one to bear in mind. There is a story byH. G. Wells which will illustrate, from the case ofmotion, the difference between the limit of a func-tion for a given argument and its value for the sameargument. The hero of the story, who possessed,without his knowledge, the power of realising hiswishes, was being attacked by a policeman, but onejaculating “Go to——” he found that the policemandisappeared. If f (t) was the policeman’s positionat time t, and t the moment of the ejaculation, thelimit of the policeman’s positions as t approachedto t from below would be in contact with the hero,whereas the value for the argument t was —. Butsuch occurrences are supposed to be rare in the realworld, and it is assumed, though without adequate

evidence, that all motions are continuous, i.e. that,given any body, if f (t) is its position at time t, f (t)is a continuous function of t. It is the meaning of“continuity” involved in such statements which wenow wish to define as simply as possible.

The definitions given for the case of functionswhere argument and value are real numbers canreadily be adapted for more general use.

Let P and Q be two relations, which it is wellto imagine serial, though it is not necessary to ourdefinitions that they should be so. Let R be a one-many relation whose domain is contained in thefield of P, while its converse domain is contained inthe field of Q. Then R is (in a generalised sense) afunction, whose arguments belong to the field of Q,while its values belong to the field of P. Suppose, forexample, that we are dealing with a particle movingon a line: let Q be the time-series, P the series ofpoints on our line from left to right, R the relationof the position of our particle on the line at timea to the time a, so that “the R of a” is its positionat time a. This illustration may be borne in mindthroughout our definitions.

We shall say that the function R is continuousfor the argument | a if, given any interval α on the P-series containing the value of the function for the

argument a, there is an interval on the Q-seriescontaining a not as an end-point and such that,throughout this interval, the function has valueswhich are members of α. (We mean by an “interval”all the terms between any two; i.e. if x and y are twomembers of the field of P, and x has the relation P toy, we shall mean by the “P-interval x to y” all termsz such that x has the relation P to z and z has therelation P to y—together, when so stated, with x ory themselves.)

We can easily define the “ultimate section” andthe “ultimate oscillation.” To define the “ultimatesection” for approaches to the argument a from be-low, take any argument y which precedes a (i.e. hasthe relation Q to a), take the values of the functionfor all arguments up to and including y, and formthe section of P defined by these values, i.e. thosemembers of the P-series which are earlier than oridentical with some of these values. Form all suchsections for all y’s that precede a, and take theircommon part; this will be the ultimate section. Theultimate upper section and the ultimate oscillationare then defined exactly as in the previous case.

The adaptation of the definition of convergenceand the resulting alternative definition of continuityoffers no difficulty of any kind.

We say that a function R is “ultimately Q-con-vergent into α” if there is a member y of the conversedomain of R and the field of Q such that the valueof the function for the argument y and for any argu-ment to which y has the relation Q is a member of α.We say that R “Q-converges into α as the argumentapproaches a given argument a” if there is a termy having the relation Q to a and belonging to theconverse domain of R and such that the value of thefunction for any argument in the Q-interval from y(inclusive) to a (exclusive) belongs to α.

Of the four conditions that a function must fulfilin order to be continuous for the argument a, thefirst is, putting b for the value for the argument a: |

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

The second condition is obtained by replacingP by its converse; the third and fourth are obtainedfrom the first and second by replacing Q by its con-verse.

There is thus nothing, in the notions of the limitof a function or the continuity of a function, thatessentially involves number. Both can be definedgenerally, and many propositions about them can beproved for any two series (one being the argument-

series and the other the value-series). It will be seenthat the definitions do not involve infinitesimals.They involve infinite classes of intervals, growingsmaller without any limit short of zero, but they donot involve any intervals that are not finite. This isanalogous to the fact that if a line an inch long behalved, then halved again, and so on indefinitely,we never reach infinitesimals in this way: after nbisections, the length of our bit is /n of an inch;and this is finite whatever finite number n may be.The process of successive bisection does not lead todivisions whose ordinal number is infinite, since itis essentially a one-by-one process. Thus infinitesi-mals are not to be reached in this way. Confusionson such topics have had much to do with the diffi-culties which have been found in the discussion ofinfinity and continuity.

Chapter XII

Selections and the Multiplicative Axiom

In this chapter we have to consider an axiom which can be enunciated, but not proved, in terms of logic,and which is convenient, though not indispensable,in certain portions of mathematics. It is convenient,in the sense that many interesting propositions,which it seems natural to suppose true, cannot beproved without its help; but it is not indispens-able, because even without those propositions thesubjects in which they occur still exist, though in asomewhat mutilated form.

Before enunciating the multiplicative axiom, wemust first explain the theory of selections, and thedefinition of multiplication when the number offactors may be infinite.

In defining the arithmetical operations, the onlycorrect procedure is to construct an actual class (or

relation, in the case of relation-numbers) havingthe required number of terms. This sometimes de-mands a certain amount of ingenuity, but it is essen-tial in order to prove the existence of the numberdefined. Take, as the simplest example, the case ofaddition. Suppose we are given a cardinal numberµ, and a class α which has µ terms. How shall we de-fine µ+µ? For this purpose we must have two classeshaving µ terms, and they must not overlap. We canconstruct such classes from α in various ways, ofwhich the following is perhaps the simplest: Formfirst all the ordered couples whose first term is aclass consisting of a single member of α, and whosesecond term is the null-class; then, secondly, formall the ordered couples whose first term is | the null- class and whose second term is a class consisting ofa single member of α. These two classes of coupleshave no member in common, and the logical sumof the two classes will have µ + µ terms. Exactlyanalogously we can define µ+ ν, given that µ is thenumber of some class α and ν is the number of someclass β.

Such definitions, as a rule, are merely a questionof a suitable technical device. But in the case ofmultiplication, where the number of factors may beinfinite, important problems arise out of the defini-

tion.Multiplication when the number of factors is

finite offers no difficulty. Given two classes α andβ, of which the first has µ terms and the second νterms, we can define µ×ν as the number of orderedcouples that can be formed by choosing the firstterm out of α and the second out of β. It will beseen that this definition does not require that αand β should not overlap; it even remains adequatewhen α and β are identical. For example, let α bethe class whose members are x, x, x. Then theclass which is used to define the product µ×µ is theclass of couples:

(x,x), (x,x), (x,x); (x,x), (x,x), (x,x);(x,x), (x,x), (x,x).

This definition remains applicable when µ or ν orboth are infinite, and it can be extended step by stepto three or four or any finite number of factors. Nodifficulty arises as regards this definition, exceptthat it cannot be extended to an infinite number offactors.

The problem of multiplication when the numberof factors may be infinite arises in this way: Supposewe have a class κ consisting of classes; suppose thenumber of terms in each of these classes is given.

How shall we define the product of all these num-bers? If we can frame our definition generally, itwill be applicable whether κ is finite or infinite. Itis to be observed that the problem is to be able todeal with the case when κ is infinite, not with thecase when its members are. If | κ is not infinite, the method defined above is just as applicable when itsmembers are infinite as when they are finite. It isthe case when κ is infinite, even though its membersmay be finite, that we have to find a way of dealingwith.

The following method of defining multiplicationgenerally is due to Dr Whitehead. It is explainedand treated at length in Principia Mathematica, vol.i. ∗ff., and vol. ii. ∗.

Let us suppose to begin with that κ is a classof classes no two of which overlap—say the con-stituencies in a country where there is no plural vot-ing, each constituency being considered as a classof voters. Let us now set to work to choose oneterm out of each class to be its representative, asconstituencies do when they elect members of Par-liament, assuming that by law each constituency hasto elect a man who is a voter in that constituency.We thus arrive at a class of representatives, whomake up our Parliament, one being selected out

of each constituency. How many different possi-ble ways of choosing a Parliament are there? Eachconstituency can select any one of its voters, andtherefore if there are µ voters in a constituency, itcan make µ choices. The choices of the differentconstituencies are independent; thus it is obviousthat, when the total number of constituencies is fi-nite, the number of possible Parliaments is obtainedby multiplying together the numbers of voters inthe various constituencies. When we do not knowwhether the number of constituencies is finite orinfinite, we may take the number of possible Par-liaments as defining the product of the numbers ofthe separate constituencies. This is the method bywhich infinite products are defined. We must nowdrop our illustration, and proceed to exact state-ments.

Let κ be a class of classes, and let us assume tobegin with that no two members of κ overlap, i.e.that if α and β are two different members of κ, thenno member of the one is a member of the other. Weshall call a class a “selection” from κwhen it consistsof just one term from each member of κ; i.e. µ is a“selection” from κ if every member of µ belongsto some member | of κ, and if α be any member of κ, µ and α have exactly one term in common.

The class of all “selections” from κ we shall call the“multiplicative class” of κ. The number of termsin the multiplicative class of κ, i.e. the number ofpossible selections from κ, is defined as the productof the numbers of the members of κ. This definitionis equally applicable whether κ is finite or infinite.

Before we can be wholly satisfied with these def-initions, we must remove the restriction that no twomembers of κ are to overlap. For this purpose, in-stead of defining first a class called a “selection,” wewill define first a relation which we will call a “se-lector.” A relation R will be called a “selector” fromκ if, from every member of κ, it picks out one termas the representative of that member, i.e. if, givenany member α of κ, there is just one term x which isa member of α and has the relation R to α; and thisis to be all that R does. The formal definition is:

A “selector” from a class of classes κ is a one-many relation, having κ for its converse domain,and such that, if x has the relation to α, then x is amember of α.

If R is a selector from κ, and α is a member ofκ, and x is the term which has the relation R to α,we call x the “representative” of α in respect of therelation R.

A “selection” from κ will now be defined as the

domain of a selector; and the multiplicative class,as before, will be the class of selections.

But when the members of κ overlap, there maybe more selectors than selections, since a term xwhich belongs to two classes α and β may be se-lected once to represent α and once to represent β,giving rise to different selectors in the two cases,but to the same selection. For purposes of definingmultiplication, it is the selectors we require ratherthan the selections. Thus we define:

“The product of the numbers of the members ofa class of classes κ” is the number of selectors fromκ.

We can define exponentiation by an adaptationof the above | plan. We might, of course, define µν as the number of selectors from ν classes, eachof which has µ terms. But there are objections tothis definition, derived from the fact that the mul-tiplicative axiom (of which we shall speak shortly)is unnecessarily involved if it is adopted. We adoptinstead the following construction:—

Let α be a class having µ terms, and β a classhaving ν terms. Let y be a member of β, and formthe class of all ordered couples that have y for theirsecond term and a member of α for their first term.There will be µ such couples for a given y, since any

member of α may be chosen for the first term, andα has µ members. If we now form all the classesof this sort that result from varying y, we obtainaltogether ν classes, since y may be any member ofβ, and β has ν members. These ν classes are eachof them a class of couples, namely, all the couplesthat can be formed of a variable member of α and afixed member of β. We define µν as the number ofselectors from the class consisting of these ν classes.Or we may equally well define µν as the numberof selections, for, since our classes of couples aremutually exclusive, the number of selectors is thesame as the number of selections. A selection fromour class of classes will be a set of ordered couples,of which there will be exactly one having any givenmember of β for its second term, and the first termmay be any member of α. Thus µν is defined bythe selectors from a certain set of ν classes eachhaving µ terms, but the set is one having a certainstructure and a more manageable composition thanis the case in general. The relevance of this to themultiplicative axiom will appear shortly.

What applies to exponentiation applies also tothe product of two cardinals. We might define“µ× ν” as the sum of the numbers of ν classes eachhaving µ terms, but we prefer to define it as the

number of ordered couples to be formed consistingof a member of α followed by a member of β, whereα has µ terms and β has ν terms. This definition,also, is designed to evade the necessity of assumingthe multiplicative axiom. |

With our definitions, we can prove the usual for- mal laws of multiplication and exponentiation. Butthere is one thing we cannot prove: we cannot provethat a product is only zero when one of its factors iszero. We can prove this when the number of factorsis finite, but not when it is infinite. In other words,we cannot prove that, given a class of classes noneof which is null, there must be selectors from them;or that, given a class of mutually exclusive classes,there must be at least one class consisting of oneterm out of each of the given classes. These thingscannot be proved; and although, at first sight, theyseem obviously true, yet reflection brings graduallyincreasing doubt, until at last we become content toregister the assumption and its consequences, as weregister the axiom of parallels, without assumingthat we can know whether it is true or false. Theassumption, loosely worded, is that selectors andselections exist when we should expect them. Thereare many equivalent ways of stating it precisely. Wemay begin with the following:—

“Given any class of mutually exclusive classes, ofwhich none is null, there is at least one class whichhas exactly one term in common with each of thegiven classes.”

This proposition we will call the “multiplica-tive axiom.” We will first give various equivalentforms of the proposition, and then consider certainways in which its truth or falsehood is of interest tomathematics.

The multiplicative axiom is equivalent to theproposition that a product is only zero when at leastone of its factors is zero; i.e. that, if any number ofcardinal numbers be multiplied together, the resultcannot be unless one of the numbers concerned is.

The multiplicative axiom is equivalent to theproposition that, if R be any relation, and κ anyclass contained in the converse domain of R, thenthere is at least one one-many relation implying Rand having κ for its converse domain.

The multiplicative axiom is equivalent to theassumption that if α be any class, and κ all the sub-classes of α with the exception | of the null-class, then there is at least one selector from κ. This is theform in which the axiom was first brought to the no-

See Principia Mathematica, vol. i. ∗. Also vol. iii. ∗–.

tice of the learned world by Zermelo, in his “Beweis,dass jede Menge wohlgeordnet werden kann.” Zer-melo regards the axiom as an unquestionable truth.It must be confessed that, until he made it explicit,mathematicians had used it without a qualm; butit would seem that they had done so unconsciously.And the credit due to Zermelo for having madeit explicit is entirely independent of the questionwhether it is true or false.

The multiplicative axiom has been shown byZermelo, in the above-mentioned proof, to be equiv-alent to the proposition that every class can be well-ordered, i.e. can be arranged in a series in whichevery sub-class has a first term (except, of course,the null-class). The full proof of this propositionis difficult, but it is not difficult to see the generalprinciple upon which it proceeds. It uses the formwhich we call “Zermelo’s axiom,” i.e. it assumesthat, given any class α, there is at least one one-many relation R whose converse domain consists ofall existent sub-classes of α and which is such that,if x has the relation R to ξ, then x is a member ofξ. Such a relation picks out a “representative” fromeach sub-class; of course, it will often happen that

Mathematische Annalen, vol. lix. pp. –. In this form weshall speak of it as Zermelo’s axiom.

two sub-classes have the same representative. WhatZermelo does, in effect, is to count off the membersof α, one by one, by means of R and transfinite in-duction. We put first the representative of α; callit x. Then take the representative of the class con-sisting of all of α except x; call it x. It must bedifferent from x, because every representative isa member of its class, and x is shut out from thisclass. Proceed similarly to take away x, and let xbe the representative of what is left. In this way wefirst obtain a progression x, x, . . . xn, . . ., assumingthat α is not finite. We then take away the wholeprogression; let xω be the representative of what isleft of α. In this way we can go on until nothingis left. The successive representatives will form a |well-ordered series containing all the members of α. (The above is, of course, only a hint of the gen-eral lines of the proof.) This proposition is called“Zermelo’s theorem.”

The multiplicative axiom is also equivalent tothe assumption that of any two cardinals which arenot equal, one must be the greater. If the axiom isfalse, there will be cardinals µ and ν such that µ isneither less than, equal to, nor greater than ν. Wehave seen that ℵ and ℵ possibly form an instanceof such a pair.

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

The propositions that depend upon the axiom,without being known to be equivalent to it, are nu-merous and important. Take first the connectionof addition and multiplication. We naturally thinkthat the sum of ν mutually exclusive classes, eachhaving µ terms, must have µ× ν terms. When ν isfinite, this can be proved. But when ν is infinite, itcannot be proved without the multiplicative axiom,except where, owing to some special circumstance,the existence of certain selectors can be proved. Theway the multiplicative axiom enters in is as follows:Suppose we have two sets of ν mutually exclusiveclasses, each having µ terms, and we wish to provethat the sum of one set has as many terms as thesum of the other. In order to prove this, we mustestablish a one-one relation. Now, since there are ineach case ν classes, there is some one-one relationbetween the two sets of classes; but what we want isa one-one relation between their terms. Let us con-sider some one-one relation S between the classes.Then if κ and λ are the two sets of classes, and α is

some member of κ, there will be a member β of λwhich will be the correlate of α with respect to S.Now α and β each have µ terms, and are thereforesimilar. There are, accordingly, one-one correlationsof α and β. The trouble is that there are so many.In order to obtain a one-one correlation of the sumof κ with the sum of λ, we have to pick out onecorrelator of α with β, and similarly for every otherpair. This requires a selection from a set of classes | of correlators, one class of the set being all the one-onecorrelators of α with β. If κ and λ are infinite, wecannot in general know that such a selection exists,unless we can know that the multiplicative axiom istrue. Hence we cannot establish the usual kind ofconnection between addition and multiplication.

This fact has various curious consequences. Tobegin with, we know that ℵ = ℵ × ℵ = ℵ. Itis commonly inferred from this that the sum of ℵclasses each having ℵ members must itself haveℵ members, but this inference is fallacious, sincewe do not know that the number of terms in sucha sum is ℵ × ℵ, nor consequently that it is ℵ.This has a bearing upon the theory of transfiniteordinals. It is easy to prove that an ordinal whichhas ℵ predecessors must be one of what Cantorcalls the “second class,” i.e. such that a series having

this ordinal number will have ℵ terms in its field.It is also easy to see that, if we take any progressionof ordinals of the second class, the predecessors oftheir limit form at most the sum of ℵ classes eachhaving ℵ terms. It is inferred thence—fallaciously,unless the multiplicative axiom is true—that thepredecessors of the limit are ℵ in number, andtherefore that the limit is a number of the “secondclass.” That is to say, it is supposed to be provedthat any progression of ordinals of the second classhas a limit which is again an ordinal of the secondclass. This proposition, with the corollary that ω(the smallest ordinal of the third class) is not thelimit of any progression, is involved in most of therecognised theory of ordinals of the second class. Inview of the way in which the multiplicative axiomis involved, the proposition and its corollary cannotbe regarded as proved. They may be true, or theymay not. All that can be said at present is that wedo not know. Thus the greater part of the theoryof ordinals of the second class must be regarded asunproved.

Another illustration may help to make the pointclearer. We know that ×ℵ = ℵ. Hence we mightsuppose that the sum of ℵ pairs must have ℵterms. But this, though we can prove that it is some-

times the case, cannot be proved to happen always| unless we assume the multiplicative axiom. This is illustrated by the millionaire who bought a pairof socks whenever he bought a pair of boots, andnever at any other time, and who had such a passionfor buying both that at last he had ℵ pairs of bootsand ℵ pairs of socks. The problem is: How manyboots had he, and how many socks? One would nat-urally suppose that he had twice as many boots andtwice as many socks as he had pairs of each, andthat therefore he had ℵ of each, since that numberis not increased by doubling. But this is an instanceof the difficulty, already noted, of connecting thesum of ν classes each having µ terms with µ × ν.Sometimes this can be done, sometimes it cannot.In our case it can be done with the boots, but notwith the socks, except by some very artificial de-vice. The reason for the difference is this: Amongboots we can distinguish right and left, and there-fore we can make a selection of one out of each pair,namely, we can choose all the right boots or all theleft boots; but with socks no such principle of selec-tion suggests itself, and we cannot be sure, unlesswe assume the multiplicative axiom, that there isany class consisting of one sock out of each pair.Hence the problem.

We may put the matter in another way. To provethat a class has ℵ terms, it is necessary and suffi-cient to find some way of arranging its terms in aprogression. There is no difficulty in doing this withthe boots. The pairs are given as forming an ℵ, andtherefore as the field of a progression. Within eachpair, take the left boot first and the right second,keeping the order of the pair unchanged; in this waywe obtain a progression of all the boots. But withthe socks we shall have to choose arbitrarily, witheach pair, which to put first; and an infinite numberof arbitrary choices is an impossibility. Unless wecan find a rule for selecting, i.e. a relation which isa selector, we do not know that a selection is eventheoretically possible. Of course, in the case of ob-jects in space, like socks, we always can find someprinciple of selection. For example, take the centresof mass of the socks: there will be points p in spacesuch that, with any | pair, the centres of mass of the two socks are not both at exactly the same distancefrom p; thus we can choose, from each pair, that sockwhich has its centre of mass nearer to p. But thereis no theoretical reason why a method of selectionsuch as this should always be possible, and the caseof the socks, with a little goodwill on the part of thereader, may serve to show how a selection might be

impossible.It is to be observed that, if it were impossible to

select one out of each pair of socks, it would followthat the socks could not be arranged in a progression,and therefore that there were not ℵ of them. Thiscase illustrates that, if µ is an infinite number, oneset of µ pairs may not contain the same number ofterms as another set of µ pairs; for, given ℵ pairsof boots, there are certainly ℵ boots, but we cannotbe sure of this in the case of the socks unless weassume the multiplicative axiom or fall back uponsome fortuitous geometrical method of selectionsuch as the above.

Another important problem involving the mul-tiplicative axiom is the relation of reflexiveness tonon-inductiveness. It will be remembered that inChapter VIII. we pointed out that a reflexive num-ber must be non-inductive, but that the converse (sofar as is known at present) can only be proved if weassume the multiplicative axiom. The way in whichthis comes about is as follows:—

It is easy to prove that a reflexive class is onewhich contains sub-classes having ℵ terms. (Theclass may, of course, itself have ℵ terms.) Thuswe have to prove, if we can, that, given any non-inductive class, it is possible to choose a progression

out of its terms. Now there is no difficulty in show-ing that a non-inductive class must contain moreterms than any inductive class, or, what comes tothe same thing, that if α is a non-inductive class andν is any inductive number, there are sub-classes ofα that have ν terms. Thus we can form sets of finitesub-classes of α: First one class having no terms,then classes having term (as many as there aremembers of α), then classes having | terms, and so on. We thus get a progression of sets of sub-classes,each set consisting of all those that have a certaingiven finite number of terms. So far we have notused the multiplicative axiom, but we have onlyproved that the number of collections of sub-classesof α is a reflexive number, i.e. that, if µ is the num-ber of members of α, so that µ is the number ofsub-classes of α and

µis the number of collections

of sub-classes, then, provided µ is not inductive, µ

must be reflexive. But this is a long way from whatwe set out to prove.

In order to advance beyond this point, we mustemploy the multiplicative axiom. From each set ofsub-classes let us choose out one, omitting the sub-class consisting of the null-class alone. That is tosay, we select one sub-class containing one term, α,say; one containing two terms, α, say; one contain-

ing three, α, say; and so on. (We can do this if themultiplicative axiom is assumed; otherwise, we donot know whether we can always do it or not.) Wehave now a progression α, α, α, . . . of sub-classesof α, instead of a progression of collections of sub-classes; thus we are one step nearer to our goal. Wenow know that, assuming the multiplicative axiom,if µ is a non-inductive number, µ must be a reflex-ive number.

The next step is to notice that, although we can-not be sure that new members of α come in at anyone specified stage in the progression α, α, α, . . .we can be sure that new members keep on comingin from time to time. Let us illustrate. The class α,which consists of one term, is a new beginning; letthe one term be x. The class α, consisting of twoterms, may or may not contain x; if it does, it intro-duces one new term; and if it does not, it must intro-duce two new terms, say x, x. In this case it is pos-sible that α consists of x, x, x, and so introducesno new terms, but in that case α must introducea new term. The first ν classes α, α, α, . . . ανcontain, at the very most, + + + . . . + ν terms,i.e. ν(ν + )/ terms; thus it would be possible, ifthere were no repetitions in the first ν classes, togo on with only repetitions from the (ν + )th | class

to the ν(ν + )/th class. But by that time the oldterms would no longer be sufficiently numerous toform a next class with the right number of members,i.e. ν(ν + )/+ , therefore new terms must comein at this point if not sooner. It follows that, if weomit from our progression α, α, α, . . . all thoseclasses that are composed entirely of members thathave occurred in previous classes, we shall still havea progression. Let our new progression be calledβ, β, β . . . (We shall have α = β and α = β,because α and α must introduce new terms. Wemay or may not have α = β, but, speaking gener-ally, βµ will be αν , where ν is some number greaterthan µ; i.e. the β’s are some of the α’s.) Now theseβ’s are such that any one of them, say βµ, containsmembers which have not occurred in any of the pre-vious β’s. Let γµ be the part of βµ which consistsof new members. Thus we get a new progressionγ, γ, γ, . . . (Again γ will be identical with βand with α; if α does not contain the one mem-ber of α, we shall have γ = β = α, but if αdoes contain this one member, γ will consist ofthe other member of α.) This new progression ofγ’s consists of mutually exclusive classes. Hence aselection from them will be a progression; i.e. if xis the member of γ, x is a member of γ, x is a

member of γ, and so on; then x, x, x, . . . is aprogression, and is a sub-class of α. Assuming themultiplicative axiom, such a selection can be made.Thus by twice using this axiom we can prove that,if the axiom is true, every non-inductive cardinalmust be reflexive. This could also be deduced fromZermelo’s theorem, that, if the axiom is true, everyclass can be well-ordered; for a well-ordered seriesmust have either a finite or a reflexive number ofterms in its field.

There is one advantage in the above direct argu-ment, as against deduction from Zermelo’s theorem,that the above argument does not demand the uni-versal truth of the multiplicative axiom, but only itstruth as applied to a set of ℵ classes. It may happenthat the axiom holds for ℵ classes, though not forlarger numbers of classes. For this reason it is better,when | it is possible, to content ourselves with the more restricted assumption. The assumption madein the above direct argument is that a product ofℵ factors is never zero unless one of the factors iszero. We may state this assumption in the form: “ℵis a multipliable number,” where a number ν is de-fined as “multipliable” when a product of ν factorsis never zero unless one of the factors is zero. Wecan prove that a finite number is always multipli-

able, but we cannot prove that any infinite numberis so. The multiplicative axiom is equivalent to theassumption that all cardinal numbers are multipli-able. But in order to identify the reflexive with thenon-inductive, or to deal with the problem of theboots and socks, or to show that any progression ofnumbers of the second class is of the second class,we only need the very much smaller assumptionthat ℵ is multipliable.

It is not improbable that there is much to bediscovered in regard to the topics discussed in thepresent chapter. Cases may be found where propo-sitions which seem to involve the multiplicative ax-iom can be proved without it. It is conceivable thatthe multiplicative axiom in its general form maybe shown to be false. From this point of view, Zer-melo’s theorem offers the best hope: the continuumor some still more dense series might be proved to beincapable of having its terms well-ordered, whichwould prove the multiplicative axiom false, in virtueof Zermelo’s theorem. But so far, no method of ob-taining such results has been discovered, and thesubject remains wrapped in obscurity.

Chapter XIII

The Axiom of Infinity and Logical Types

The axiom of infinity is an assumption which may be enunciated as follows:—

“If n be any inductive cardinal number, there isat least one class of individuals having n terms.”

If this is true, it follows, of course, that thereare many classes of individuals having n terms, andthat the total number of individuals in the world isnot an inductive number. For, by the axiom, thereis at least one class having n+ terms, from whichit follows that there are many classes of n termsand that n is not the number of individuals in theworld. Since n is any inductive number, it followsthat the number of individuals in the world must(if our axiom be true) exceed any inductive number.In view of what we found in the preceding chapter,about the possibility of cardinals which are neither

inductive nor reflexive, we cannot infer from ouraxiom that there are at least ℵ individuals, unlesswe assume the multiplicative axiom. But we doknow that there are at least ℵ classes of classes,since the inductive cardinals are classes of classes,and form a progression if our axiom is true.

The way in which the need for this axiom arisesmay be explained as follows. One of Peano’s as-sumptions is that no two inductive cardinals havethe same successor, i.e. that we shall not havem+ =n+ unlessm = n, if m and n are inductive cardinals.In Chapter VIII. we had occasion to use what is vir-tually the same as the above assumption of Peano’s,namely, that, if n is an inductive cardinal, | n is not equal to n+ . It might be thought that this couldbe proved. We can prove that, if α is an inductiveclass, and n is the number of members of α, then n isnot equal to n+ . This proposition is easily provedby induction, and might be thought to imply theother. But in fact it does not, since there might beno such class as α. What it does imply is this: If nis an inductive cardinal such that there is at leastone class having n members, then n is not equal ton+. The axiom of infinity assures us (whether trulyor falsely) that there are classes having n members,and thus enables us to assert that n is not equal to

n+. But without this axiom we should be left withthe possibility that n and n + might both be thenull-class.

Let us illustrate this possibility by an example:Suppose there were exactly nine individuals in theworld. (As to what is meant by the word “individ-ual,” I must ask the reader to be patient.) Thenthe inductive cardinals from up to would besuch as we expect, but (defined as + ) wouldbe the null-class. It will be remembered that n+ may be defined as follows: n + is the collectionof all those classes which have a term x such that,when x is taken away, there remains a class of nterms. Now applying this definition, we see that, inthe case supposed, + is a class consisting of noclasses, i.e. it is the null-class. The same will be trueof + , or generally of +n, unless n is zero. Thus and all subsequent inductive cardinals will allbe identical, since they will all be the null-class. Insuch a case the inductive cardinals will not form aprogression, nor will it be true that no two have thesame successor, for and will both be succeededby the null-class ( being itself the null-class). It isin order to prevent such arithmetical catastrophesthat we require the axiom of infinity.

As a matter of fact, so long as we are content

with the arithmetic of finite integers, and do notintroduce either infinite integers or infinite classesor series of finite integers or ratios, it is possible toobtain all desired results without the axiom of in-finity. That is to say, we can deal with the addition,|multiplication, and exponentiation of finite inte- gers and of ratios, but we cannot deal with infiniteintegers or with irrationals. Thus the theory of thetransfinite and the theory of real numbers fails us.How these various results come about must now beexplained.

Assuming that the number of individuals in theworld is n, the number of classes of individualswill be n. This is in virtue of the general propo-sition mentioned in Chapter VIII. that the numberof classes contained in a class which has n mem-bers is n. Now n is always greater than n. Hencethe number of classes in the world is greater thanthe number of individuals. If, now, we suppose thenumber of individuals to be , as we did just now,the number of classes will be , i.e. . Thus if wetake our numbers as being applied to the countingof classes instead of to the counting of individuals,our arithmetic will be normal until we reach :the first number to be null will be . And if weadvance to classes of classes we shall do still better:

the number of them will be , a number which isso large as to stagger imagination, since it has about digits. And if we advance to classes of classesof classes, we shall obtain a number representedby raised to a power which has about digits;the number of digits in this number will be aboutthree times . In a time of paper shortage itis undesirable to write out this number, and if wewant larger ones we can obtain them by travellingfurther along the logical hierarchy. In this way anyassigned inductive cardinal can be made to find itsplace among numbers which are not null, merelyby travelling along the hierarchy for a sufficient dis-tance.

As regards ratios, we have a very similar state ofaffairs. If a ratio µ/ν is to have the expected prop-erties, there must be enough objects of whateversort is being counted to insure that the null-classdoes not suddenly obtrude itself. But this can be in-sured, for any given ratio µ/ν, without the axiom of |infinity, by merely travelling up the hierarchy a suf- ficient distance. If we cannot succeed by countingindividuals, we can try counting classes of individ-

On this subject see Principia Mathematica, vol. ii. ∗ff. Onthe corresponding problems as regards ratio, see ibid., vol. iii.∗ff.

uals; if we still do not succeed, we can try classesof classes, and so on. Ultimately, however few in-dividuals there may be in the world, we shall reacha stage where there are many more than µ objects,whatever inductive number µ may be. Even if therewere no individuals at all, this would still be true,for there would then be one class, namely, the null-class, classes of classes (namely, the null-class ofclasses and the class whose only member is the null-class of individuals), classes of classes of classes, at the next stage, , at the next stage, andso on. Thus no such assumption as the axiom ofinfinity is required in order to reach any given ratioor any given inductive cardinal.

It is when we wish to deal with the whole classor series of inductive cardinals or of ratios that theaxiom is required. We need the whole class of in-ductive cardinals in order to establish the existenceof ℵ, and the whole series in order to establish theexistence of progressions: for these results, it is nec-essary that we should be able to make a single classor series in which no inductive cardinal is null. Weneed the whole series of ratios in order of magni-tude in order to define real numbers as segments:this definition will not give the desired result unlessthe series of ratios is compact, which it cannot be if

the total number of ratios, at the stage concerned, isfinite.

It would be natural to suppose—as I supposedmyself in former days—that, by means of construc-tions such as we have been considering, the axiomof infinity could be proved. It may be said: Let usassume that the number of individuals is n, where nmay be without spoiling our argument; then if weform the complete set of individuals, classes, classesof classes, etc., all taken together, the number ofterms in our whole set will be

n+ n + n. . . ad inf.,

which is ℵ. Thus taking all kinds of objects to-gether, and not | confining ourselves to objects of any one type, we shall certainly obtain an infiniteclass, and shall therefore not need the axiom of in-finity. So it might be said.

Now, before going into this argument, the firstthing to observe is that there is an air of hocus-pocusabout it: something reminds one of the conjurerwho brings things out of the hat. The man who haslent his hat is quite sure there wasn’t a live rabbitin it before, but he is at a loss to say how the rabbitgot there. So the reader, if he has a robust senseof reality, will feel convinced that it is impossible

to manufacture an infinite collection out of a finitecollection of individuals, though he may be unableto say where the flaw is in the above construction.It would be a mistake to lay too much stress onsuch feelings of hocus-pocus; like other emotions,they may easily lead us astray. But they afford aprima facie ground for scrutinising very closely anyargument which arouses them. And when the aboveargument is scrutinised it will, in my opinion, befound to be fallacious, though the fallacy is a subtleone and by no means easy to avoid consistently.

The fallacy involved is the fallacy which maybe called “confusion of types.” To explain the sub-ject of “types” fully would require a whole volume;moreover, it is the purpose of this book to avoidthose parts of the subjects which are still obscureand controversial, isolating, for the convenience ofbeginners, those parts which can be accepted as em-bodying mathematically ascertained truths. Nowthe theory of types emphatically does not belong tothe finished and certain part of our subject: much ofthis theory is still inchoate, confused, and obscure.But the need of some doctrine of types is less doubt-ful than the precise form the doctrine should take;and in connection with the axiom of infinity it isparticularly easy to see the necessity of some such

doctrine.This necessity results, for example, from the

“contradiction of the greatest cardinal.” We sawin Chapter VIII. that the number of classes con-tained in a given class is always greater than the |number of members of the class, and we inferred that there is no greatest cardinal number. But if wecould, as we suggested a moment ago, add togetherinto one class the individuals, classes of individu-als, classes of classes of individuals, etc., we shouldobtain a class of which its own sub-classes would bemembers. The class consisting of all objects that canbe counted, of whatever sort, must, if there be sucha class, have a cardinal number which is the greatestpossible. Since all its sub-classes will be membersof it, there cannot be more of them than there aremembers. Hence we arrive at a contradiction.

When I first came upon this contradiction, inthe year , I attempted to discover some flawin Cantor’s proof that there is no greatest cardinal,which we gave in Chapter VIII. Applying this proofto the supposed class of all imaginable objects, I wasled to a new and simpler contradiction, namely, thefollowing:—

The comprehensive class we are considering,which is to embrace everything, must embrace it-

self as one of its members. In other words, if thereis such a thing as “everything,” then “everything”is something, and is a member of the class “every-thing.” But normally a class is not a member of it-self. Mankind, for example, is not a man. Form nowthe assemblage of all classes which are not membersof themselves. This is a class: is it a member of itselfor not? If it is, it is one of those classes that are notmembers of themselves, i.e. it is not a member ofitself. If it is not, it is not one of those classes thatare not members of themselves, i.e. it is a memberof itself. Thus of the two hypotheses—that it is, andthat it is not, a member of itself—each implies itscontradictory. This is a contradiction.

There is no difficulty in manufacturing similarcontradictions ad lib. The solution of such contra-dictions by the theory of types is set forth fully inPrincipia Mathematica, and also, more briefly, in ar-ticles by the present author in the American Journal |of Mathematics and in the Revue de Metaphysique et de Morale. For the present an outline of the solutionmust suffice.Vol. i., Introduction, chap. ii., ∗ and ∗; vol. ii., Prefatory

Statement.“Mathematical Logic as based on the Theory of Types,” vol.

xxx., , pp. –.“Les paradoxes de la logique,” , pp. –.

The fallacy consists in the formation of whatwe may call “impure” classes, i.e. classes which arenot pure as to “type.” As we shall see in a laterchapter, classes are logical fictions, and a statementwhich appears to be about a class will only be sig-nificant if it is capable of translation into a formin which no mention is made of the class. Thisplaces a limitation upon the ways in which what arenominally, though not really, names for classes canoccur significantly: a sentence or set of symbols inwhich such pseudo-names occur in wrong ways isnot false, but strictly devoid of meaning. The sup-position that a class is, or that it is not, a memberof itself is meaningless in just this way. And moregenerally, to suppose that one class of individualsis a member, or is not a member, of another classof individuals will be to suppose nonsense; and toconstruct symbolically any class whose membersare not all of the same grade in the logical hierarchyis to use symbols in a way which makes them nolonger symbolise anything.

Thus if there are n individuals in the world, andn classes of individuals, we cannot form a newclass, consisting of both individuals and classes andhaving n + n members. In this way the attemptto escape from the need for the axiom of infinity

breaks down. I do not pretend to have explainedthe doctrine of types, or done more than indicate,in rough outline, why there is need of such a doc-trine. I have aimed only at saying just so muchas was required in order to show that we cannotprove the existence of infinite numbers and classesby such conjurer’s methods as we have been exam-ining. There remain, however, certain other possiblemethods which must be considered.

Various arguments professing to prove the ex-istence of infinite classes are given in the Principlesof Mathematics, § (p. ). | In so far as these arguments assume that, if n is an inductive cardi-nal, n is not equal to n+ , they have been alreadydealt with. There is an argument, suggested by apassage in Plato’s Parmenides, to the effect that, ifthere is such a number as , then has being; but is not identical with being, and therefore andbeing are two, and therefore there is such a numberas , and together with and being gives a class ofthree terms, and so on. This argument is fallacious,partly because “being” is not a term having any def-inite meaning, and still more because, if a definitemeaning were invented for it, it would be found thatnumbers do not have being—they are, in fact, whatare called “logical fictions,” as we shall see when we

come to consider the definition of classes.The argument that the number of numbers from

to n (both inclusive) is n + depends upon theassumption that up to and including n no number isequal to its successor, which, as we have seen, willnot be always true if the axiom of infinity is false.It must be understood that the equation n = n+ ,which might be true for a finite n if n exceeded thetotal number of individuals in the world, is quitedifferent from the same equation as applied to areflexive number. As applied to a reflexive number,it means that, given a class of n terms, this class is“similar” to that obtained by adding another term.But as applied to a number which is too great for theactual world, it merely means that there is no classof n individuals, and no class of n+ individuals;it does not mean that, if we mount the hierarchyof types sufficiently far to secure the existence of aclass of n terms, we shall then find this class “simi-lar” to one of n+ terms, for if n is inductive this willnot be the case, quite independently of the truth orfalsehood of the axiom of infinity.

There is an argument employed by both Bol-zano and Dedekind to prove the existence of re-

Bolzano, Paradoxien des Unendlichen, .Dedekind, Was sind und was sollen die Zahlen? No. .

flexive classes. The argument, in brief, is this: Anobject is not identical with the idea of the | object, but there is (at least in the realm of being) an ideaof any object. The relation of an object to the ideaof it is one-one, and ideas are only some amongobjects. Hence the relation “idea of” constitutes areflexion of the whole class of objects into a partof itself, namely, into that part which consists ofideas. Accordingly, the class of objects and the classof ideas are both infinite. This argument is inter-esting, not only on its own account, but because themistakes in it (or what I judge to be mistakes) areof a kind which it is instructive to note. The mainerror consists in assuming that there is an idea ofevery object. It is, of course, exceedingly difficult todecide what is meant by an “idea”; but let us assumethat we know. We are then to suppose that, starting(say) with Socrates, there is the idea of Socrates, andthen the idea of the idea of Socrates, and so on adinf. Now it is plain that this is not the case in thesense that all these ideas have actual empirical exis-tence in people’s minds. Beyond the third or fourthstage they become mythical. If the argument is to beupheld, the “ideas” intended must be Platonic ideaslaid up in heaven, for certainly they are not on earth.But then it at once becomes doubtful whether there

are such ideas. If we are to know that there are, itmust be on the basis of some logical theory, provingthat it is necessary to a thing that there should bean idea of it. We certainly cannot obtain this resultempirically, or apply it, as Dedekind does, to “meineGedankenwelt”—the world of my thoughts.

If we were concerned to examine fully the rela-tion of idea and object, we should have to enter upona number of psychological and logical inquiries,which are not relevant to our main purpose. But afew further points should be noted. If “idea” is tobe understood logically, it may be identical with theobject, or it may stand for a description (in the senseto be explained in a subsequent chapter). In the for-mer case the argument fails, because it was essentialto the proof of reflexiveness that object and ideashould be distinct. In the second case the argumentalso fails, because the relation of object and descrip-tion is not | one-one: there are innumerable correct descriptions of any given object. Socrates (e.g.) maybe described as “the master of Plato,” or as “thephilosopher who drank the hemlock,” or as “thehusband of Xantippe.” If—to take up the remaininghypothesis—“idea” is to be interpreted psychologi-cally, it must be maintained that there is not any onedefinite psychological entity which could be called

the idea of the object: there are innumerable beliefsand attitudes, each of which could be called an ideaof the object in the sense in which we might say “myidea of Socrates is quite different from yours,” butthere is not any central entity (except Socrates him-self) to bind together various “ideas of Socrates,”and thus there is not any such one-one relation ofidea and object as the argument supposes. Nor, ofcourse, as we have already noted, is it true psycho-logically that there are ideas (in however extendeda sense) of more than a tiny proportion of the thingsin the world. For all these reasons, the above argu-ment in favour of the logical existence of reflexiveclasses must be rejected.

It might be thought that, whatever may be saidof logical arguments, the empirical arguments deriv-able from space and time, the diversity of colours,etc., are quite sufficient to prove the actual existenceof an infinite number of particulars. I do not be-lieve this. We have no reason except prejudice forbelieving in the infinite extent of space and time,at any rate in the sense in which space and timeare physical facts, not mathematical fictions. Wenaturally regard space and time as continuous, or,at least, as compact; but this again is mainly prej-udice. The theory of “quanta” in physics, whether

true or false, illustrates the fact that physics cannever afford proof of continuity, though it mightquite possibly afford disproof. The senses are notsufficiently exact to distinguish between continu-ous motion and rapid discrete succession, as anyonemay discover in a cinema. A world in which allmotion consisted of a series of small finite jerkswould be empirically indistinguishable from onein which motion was continuous. It would take uptoo much space to | defend these theses adequately; for the present I am merely suggesting them for thereader’s consideration. If they are valid, it followsthat there is no empirical reason for believing thenumber of particulars in the world to be infinite,and that there never can be; also that there is atpresent no empirical reason to believe the numberto be finite, though it is theoretically conceivablethat some day there might be evidence pointing,though not conclusively, in that direction.

From the fact that the infinite is not self-contra-dictory, but is also not demonstrable logically, wemust conclude that nothing can be known a priorias to whether the number of things in the world isfinite or infinite. The conclusion is, therefore, toadopt a Leibnizian phraseology, that some of thepossible worlds are finite, some infinite, and we

have no means of knowing to which of these twokinds our actual world belongs. The axiom of infin-ity will be true in some possible worlds and false inothers; whether it is true or false in this world, wecannot tell.

Throughout this chapter the synonyms “indi-vidual” and “particular” have been used withoutexplanation. It would be impossible to explainthem adequately without a longer disquisition onthe theory of types than would be appropriate tothe present work, but a few words before we leavethis topic may do something to diminish the obscu-rity which would otherwise envelop the meaning ofthese words.

In an ordinary statement we can distinguish averb, expressing an attribute or relation, from thesubstantives which express the subject of the at-tribute or the terms of the relation. “Cæsar lived”ascribes an attribute to Cæsar; “Brutus killed Cæsar”expresses a relation between Brutus and Cæsar. Us-ing the word “subject” in a generalised sense, wemay call both Brutus and Cæsar subjects of thisproposition: the fact that Brutus is grammaticallysubject and Cæsar object is logically irrelevant, sincethe same occurrence may be expressed in the words“Cæsar was killed by Brutus,” where Cæsar is the

grammatical subject. | Thus in the simpler sort of proposition we shall have an attribute or relationholding of or between one, two or more “subjects”in the extended sense. (A relation may have morethan two terms: e.g. “A gives B to C” is a relationof three terms.) Now it often happens that, on acloser scrutiny, the apparent subjects are found tobe not really subjects, but to be capable of anal-ysis; the only result of this, however, is that newsubjects take their places. It also happens that theverb may grammatically be made subject: e.g. wemay say, “Killing is a relation which holds betweenBrutus and Cæsar.” But in such cases the grammaris misleading, and in a straightforward statement,following the rules that should guide philosophi-cal grammar, Brutus and Cæsar will appear as thesubjects and killing as the verb.

We are thus led to the conception of terms which,when they occur in propositions, can only occur assubjects, and never in any other way. This is part ofthe old scholastic definition of substance; but persis-tence through time, which belonged to that notion,forms no part of the notion with which we are con-cerned. We shall define “proper names” as thoseterms which can only occur as subjects in propo-sitions (using “subject” in the extended sense just

explained). We shall further define “individuals” or“particulars” as the objects that can be named byproper names. (It would be better to define them di-rectly, rather than by means of the kind of symbolsby which they are symbolised; but in order to do thatwe should have to plunge deeper into metaphysicsthan is desirable here.) It is, of course, possible thatthere is an endless regress: that whatever appearsas a particular is really, on closer scrutiny, a class orsome kind of complex. If this be the case, the axiomof infinity must of course be true. But if it be not thecase, it must be theoretically possible for analysisto reach ultimate subjects, and it is these that givethe meaning of “particulars” or “individuals.” It isto the number of these that the axiom of infinity isassumed to apply. If it is true of them, it is true |of classes of them, and classes of classes of them, and so on; similarly if it is false of them, it is falsethroughout this hierarchy. Hence it is natural toenunciate the axiom concerning them rather thanconcerning any other stage in the hierarchy. Butwhether the axiom is true or false, there seems noknown method of discovering.

Chapter XIV

Incompatibility and the Theory ofDeduction

We have now explored, somewhat hastily it is true, that part of the philosophy of mathematics whichdoes not demand a critical examination of the ideaof class. In the preceding chapter, however, wefound ourselves confronted by problems which makesuch an examination imperative. Before we can un-dertake it, we must consider certain other parts ofthe philosophy of mathematics, which we have hith-erto ignored. In a synthetic treatment, the partswhich we shall now be concerned with come first:they are more fundamental than anything that wehave discussed hitherto. Three topics will concernus before we reach the theory of classes, namely: ()the theory of deduction, () propositional functions,() descriptions. Of these, the third is not logically

presupposed in the theory of classes, but it is a sim-pler example of the kind of theory that is needed indealing with classes. It is the first topic, the theoryof deduction, that will concern us in the presentchapter.

Mathematics is a deductive science: startingfrom certain premisses, it arrives, by a strict processof deduction, at the various theorems which con-stitute it. It is true that, in the past, mathematicaldeductions were often greatly lacking in rigour; it istrue also that perfect rigour is a scarcely attainableideal. Nevertheless, in so far as rigour is lacking ina mathematical proof, the proof is defective; it isno defence to urge that common sense shows theresult to be correct, for if we were to rely upon that,it would be better to dispense with argument al-together, | rather than bring fallacy to the rescue of common sense. No appeal to common sense,or “intuition,” or anything except strict deductivelogic, ought to be needed in mathematics after thepremisses have been laid down.

Kant, having observed that the geometers of hisday could not prove their theorems by unaided ar-gument, but required an appeal to the figure, in-vented a theory of mathematical reasoning accord-ing to which the inference is never strictly logical,

but always requires the support of what is called“intuition.” The whole trend of modern mathemat-ics, with its increased pursuit of rigour, has beenagainst this Kantian theory. The things in the math-ematics of Kant’s day which cannot be proved, can-not be known—for example, the axiom of parallels.What can be known, in mathematics and by math-ematical methods, is what can be deduced frompure logic. What else is to belong to human knowl-edge must be ascertained otherwise—empirically,through the senses or through experience in someform, but not a priori. The positive grounds forthis thesis are to be found in Principia Mathemat-ica, passim; a controversial defence of it is given inthe Principles of Mathematics. We cannot here domore than refer the reader to those works, since thesubject is too vast for hasty treatment. Meanwhile,we shall assume that all mathematics is deductive,and proceed to inquire as to what is involved indeduction.

In deduction, we have one or more propositionscalled premisses, from which we infer a propositioncalled the conclusion. For our purposes, it will beconvenient, when there are originally several pre-misses, to amalgamate them into a single proposi-tion, so as to be able to speak of the premiss as well

as of the conclusion. Thus we may regard deductionas a process by which we pass from knowledge ofa certain proposition, the premiss, to knowledge ofa certain other proposition, the conclusion. But weshall not regard such a process as logical deductionunless it is correct, i.e. unless there is such a rela-tion between premiss and conclusion that we havea right to believe the conclusion | if we know the premiss to be true. It is this relation that is chieflyof interest in the logical theory of deduction.

In order to be able validly to infer the truth of aproposition, we must know that some other propo-sition is true, and that there is between the two arelation of the sort called “implication,” i.e. that (aswe say) the premiss “implies” the conclusion. (Weshall define this relation shortly.) Or we may knowthat a certain other proposition is false, and thatthere is a relation between the two of the sort called“disjunction,” expressed by “p or q,” so that theknowledge that the one is false allows us to inferthat the other is true. Again, what we wish to infermay be the falsehood of some proposition, not itstruth. This may be inferred from the truth of an-other proposition, provided we know that the two

We shall use the letters p, q, r, s, t to denote variable proposi-tions.

are “incompatible,” i.e. that if one is true, the otheris false. It may also be inferred from the falsehoodof another proposition, in just the same circum-stances in which the truth of the other might havebeen inferred from the truth of the one; i.e. fromthe falsehood of p we may infer the falsehood of q,when q implies p. All these four are cases of infer-ence. When our minds are fixed upon inference, itseems natural to take “implication” as the primi-tive fundamental relation, since this is the relationwhich must hold between p and q if we are to beable to infer the truth of q from the truth of p. Butfor technical reasons this is not the best primitiveidea to choose. Before proceeding to primitive ideasand definitions, let us consider further the variousfunctions of propositions suggested by the above-mentioned relations of propositions.

The simplest of such functions is the negative,“not-p.” This is that function of p which is true whenp is false, and false when p is true. It is convenient tospeak of the truth of a proposition, or its falsehood,as its “truth-value”; i.e. truth is the “truth-value”of a true proposition, and falsehood of a false one.Thus not-p has the opposite truth-value to p. |

We may take next disjunction, “p or q.” This is a This term is due to Frege.

function whose truth-value is truth when p is trueand also when q is true, but is falsehood when bothp and q are false.

Next we may take conjunction, “p and q.” Thishas truth for its truth-value when p and q are bothtrue; otherwise it has falsehood for its truth-value.

Take next incompatibility, i.e. “p and q are notboth true.” This is the negation of conjunction; it isalso the disjunction of the negations of p and q, i.e.it is “not-p or not-q.” Its truth-value is truth when pis false and likewise when q is false; its truth-valueis falsehood when p and q are both true.

Last take implication, i.e. “p implies q,” or “if p,then q.” This is to be understood in the widest sensethat will allow us to infer the truth of q if we knowthe truth of p. Thus we interpret it as meaning:“Unless p is false, q is true,” or “either p is false or qis true.” (The fact that “implies” is capable of othermeanings does not concern us; this is the meaningwhich is convenient for us.) That is to say, “p impliesq” is to mean “not-p or q”: its truth-value is to betruth if p is false, likewise if q is true, and is to befalsehood if p is true and q is false.

We have thus five functions: negation, disjunc-tion, conjunction, incompatibility, and implication.We might have added others, for example, joint

falsehood, “not-p and not-q,” but the above five willsuffice. Negation differs from the other four in beinga function of one proposition, whereas the others arefunctions of two. But all five agree in this, that theirtruth-value depends only upon that of the propo-sitions which are their arguments. Given the truthor falsehood of p, or of p and q (as the case may be),we are given the truth or falsehood of the negation,disjunction, conjunction, incompatibility, or impli-cation. A function of propositions which has thisproperty is called a “truth-function.”

The whole meaning of a truth-function is ex-hausted by the statement of the circumstances un-der which it is true or false. “Not-p,” for example,is simply that function of p which is true when p isfalse, and false when p is true: there is no further| meaning to be assigned to it. The same applies to “p or q” and the rest. It follows that two truth-functions which have the same truth-value for allvalues of the argument are indistinguishable. Forexample, “p and q” is the negation of “not-p or not-q” and vice versa; thus either of these may be definedas the negation of the other. There is no furthermeaning in a truth-function over and above theconditions under which it is true or false.

It is clear that the above five truth-functions are

not all independent. We can define some of them interms of others. There is no great difficulty in reduc-ing the number to two; the two chosen in PrincipiaMathematica are negation and disjunction. Implica-tion is then defined as “not-p or q”; incompatibilityas “not-p or not-q”; conjunction as the negation ofincompatibility. But it has been shown by Sheffer

that we can be content with one primitive idea forall five, and by Nicod that this enables us to reducethe primitive propositions required in the theoryof deduction to two non-formal principles and oneformal one. For this purpose, we may take as ourone indefinable either incompatibility or joint false-hood. We will choose the former.

Our primitive idea, now, is a certain truth-func-tion called “incompatibility,” which we will denoteby p/q. Negation can be at once defined as the in-compatibility of a proposition with itself, i.e. “not-p”is defined as “p/p.” Disjunction is the incompati-bility of not-p and not-q, i.e. it is (p/p) | (q/q). Im-plication is the incompatibility of p and not-q, i.e.p | (q/q). Conjunction is the negation of incompati-bility, i.e. it is (p/q) | (p/q). Thus all our four otherfunctions are defined in terms of incompatibility.

Trans. Am. Math. Soc., vol. xiv. pp. –.Proc. Camb. Phil. Soc., vol. xix., i., January .

It is obvious that there is no limit to the man-ufacture of truth-functions, either by introducingmore arguments or by repeating arguments. Whatwe are concerned with is the connection of this sub-ject with inference. |

If we know that p is true and that p implies q, we can proceed to assert q. There is always unavoidablysomething psychological about inference: inferenceis a method by which we arrive at new knowledge,and what is not psychological about it is the relationwhich allows us to infer correctly; but the actualpassage from the assertion of p to the assertion of qis a psychological process, and we must not seek torepresent it in purely logical terms.

In mathematical practice, when we infer, wehave always some expression containing variablepropositions, say p and q, which is known, in virtueof its form, to be true for all values of p and q; wehave also some other expression, part of the former,which is also known to be true for all values of p andq; and in virtue of the principles of inference, we areable to drop this part of our original expression, andassert what is left. This somewhat abstract accountmay be made clearer by a few examples.

Let us assume that we know the five formal prin-ciples of deduction enumerated in Principia Mathe-

matica. (M. Nicod has reduced these to one, but asit is a complicated proposition, we will begin withthe five.) These five propositions are as follows:—

() “p or p” implies p—i.e. if either p is true or pis true, then p is true.

() q implies “p or q”—i.e. the disjunction “p orq” is true when one of its alternatives is true.

() “p or q” implies “q or p.” This would not berequired if we had a theoretically more perfect no-tation, since in the conception of disjunction thereis no order involved, so that “p or q” and “q or p”should be identical. But since our symbols, in anyconvenient form, inevitably introduce an order, weneed suitable assumptions for showing that the or-der is irrelevant.

() If either p is true or “q or r” is true, theneither q is true or “p or r” is true. (The twist in thisproposition serves to increase its deductive power.) |

() If q implies r, then “p or q” implies “p or r.” These are the formal principles of deduction em-

ployed in Principia Mathematica. A formal principleof deduction has a double use, and it is in order tomake this clear that we have cited the above fivepropositions. It has a use as the premiss of an in-ference, and a use as establishing the fact that thepremiss implies the conclusion. In the schema of an

inference we have a proposition p, and a proposition“p implies q,” from which we infer q. Now whenwe are concerned with the principles of deduction,our apparatus of primitive propositions has to yieldboth the p and the “p implies q” of our inferences.That is to say, our rules of deduction are to be used,not only as rules, which is their use for establishing“p implies q,” but also as substantive premisses, i.e.as the p of our schema. Suppose, for example, wewish to prove that if p implies q, then if q implies rit follows that p implies r. We have here a relationof three propositions which state implications. Put

p = p implies q, p = q implies r, p = p implies r.

Then we have to prove that p implies that p im-plies p. Now take the fifth of our above principles,substitute not-p for p, and remember that “not-p orq” is by definition the same as “p implies q.” Thusour fifth principle yields:

“If q implies r, then ‘p implies q’ implies ‘p impliesr,’” i.e. “p implies that p implies p.” Call thisproposition A.

But the fourth of our principles, when we substi-tute not-p, not-q, for p and q, and remember thedefinition of implication, becomes:

“If p implies that q implies r, then q implies thatp implies r.”

Writing p in place of p, p in place of q, and p inplace of r, this becomes:

“If p implies that p implies p, then p impliesthat p implies p.” Call this B. |

Now we proved by means of our fifth principle that

“p implies that p implies p,” which was whatwe called A.

Thus we have here an instance of the schema ofinference, since A represents the p of our scheme,and B represents the “p implies q.” Hence we arriveat q, namely,

“p implies that p implies p,”

which was the proposition to be proved. In thisproof, the adaptation of our fifth principle, whichyields A, occurs as a substantive premiss; while theadaptation of our fourth principle, which yieldsB, is used to give the form of the inference. Theformal and material employments of premisses inthe theory of deduction are closely intertwined, and

it is not very important to keep them separated,provided we realise that they are in theory distinct.

The earliest method of arriving at new resultsfrom a premiss is one which is illustrated in theabove deduction, but which itself can hardly becalled deduction. The primitive propositions, what-ever they may be, are to be regarded as asserted forall possible values of the variable propositions p, q,r which occur in them. We may therefore substitutefor (say) p any expression whose value is always aproposition, e.g. not-p, “s implies t,” and so on. Bymeans of such substitutions we really obtain sets ofspecial cases of our original proposition, but from apractical point of view we obtain what are virtuallynew propositions. The legitimacy of substitutions ofthis kind has to be insured by means of a non-formalprinciple of inference.

We may now state the one formal principle ofinference to which M. Nicod has reduced the fivegiven above. For this purpose we will first showhow certain truth-functions can be defined in termsof incompatibility. We saw already that

p | (q/q) means “p implies q.” |No such principle is enunciated in Principia Mathematica or

in M. Nicod’s article mentioned above. But this would seem tobe an omission.

We now observe that

p | (q/r) means “p implies both q and r.”

For this expression means “p is incompatible withthe incompatibility of q and r,” i.e. “p implies thatq and r are not incompatible,” i.e. “p implies that qand r are both true”—for, as we saw, the conjunctionof q and r is the negation of their incompatibility.

Observe next that t | (t /t) means “t implies it-self.” This is a particular case of p | (q/q).

Let us write p for the negation of p; thus p/swill mean the negation of p/s, i.e. it will mean theconjunction of p and s. It follows that

(s/q) | p/s

expresses the incompatibility of s/q with the con-junction of p and s; in other words, it states that ifp and s are both true, s/q is false, i.e. s and q areboth true; in still simpler words, it states that p ands jointly imply s and q jointly.

Now, put P = p | (q/r),

π = t | (t /t),

Q = (s/q) | p/s.

Then M. Nicod’s sole formal principle of deductionis

P | π/Q,

in other words, P implies both π and Q.He employs in addition one non-formal princi-

ple belonging to the theory of types (which need notconcern us), and one corresponding to the principlethat, given p, and given that p implies q, we canassert q. This principle is:

“If p | (r /q) is true, and p is true, then q is true.”From this apparatus the whole theory of deductionfollows, except in so far as we are concerned withdeduction from or to the existence or the universaltruth of “propositional functions,” which we shallconsider in the next chapter.

There is, if I am not mistaken, a certain confu-sion in the |minds of some authors as to the relation, between propositions, in virtue of which an infer-ence is valid. In order that it may be valid to inferq from p, it is only necessary that p should be trueand that the proposition “not-p or q” should be true.Whenever this is the case, it is clear that q mustbe true. But inference will only in fact take placewhen the proposition “not-p or q” is known other-wise than through knowledge of not-p or knowl-edge of q. Whenever p is false, “not-p or q” is true,

but is useless for inference, which requires that pshould be true. Whenever q is already known to betrue, “not-p or q” is of course also known to be true,but is again useless for inference, since q is alreadyknown, and therefore does not need to be inferred.In fact, inference only arises when “not-p or q” canbe known without our knowing already which ofthe two alternatives it is that makes the disjunc-tion true. Now, the circumstances under which thisoccurs are those in which certain relations of formexist between p and q. For example, we know that ifr implies the negation of s, then s implies the nega-tion of r. Between “r implies not-s” and “s impliesnot-r” there is a formal relation which enables usto know that the first implies the second, withouthaving first to know that the first is false or to knowthat the second is true. It is under such circum-stances that the relation of implication is practicallyuseful for drawing inferences.

But this formal relation is only required in orderthat we may be able to know that either the pre-miss is false or the conclusion is true. It is the truthof “not-p or q” that is required for the validity ofthe inference; what is required further is only re-quired for the practical feasibility of the inference.

Professor C. I. Lewis has especially studied the nar-rower, formal relation which we may call “formaldeducibility.” He urges that the wider relation, thatexpressed by “not-p or q,” should not be called “im-plication.” That is, however, a matter of words. |Provided our use of words is consistent, it matters little how we define them. The essential point ofdifference between the theory which I advocate andthe theory advocated by Professor Lewis is this: Hemaintains that, when one proposition q is “formallydeducible” from another p, the relation which weperceive between them is one which he calls “strictimplication,” which is not the relation expressed by“not-p or q” but a narrower relation, holding onlywhen there are certain formal connections betweenp and q. I maintain that, whether or not there besuch a relation as he speaks of, it is in any caseone that mathematics does not need, and thereforeone that, on general grounds of economy, ought notto be admitted into our apparatus of fundamentalnotions; that, whenever the relation of “formal de-ducibility” holds between two propositions, it is thecase that we can see that either the first is false orthe second true, and that nothing beyond this fact

See Mind, vol. xxi., , pp. –; and vol. xxiii., ,pp. –.

is necessary to be admitted into our premisses; andthat, finally, the reasons of detail which ProfessorLewis adduces against the view which I advocatecan all be met in detail, and depend for their plausi-bility upon a covert and unconscious assumption ofthe point of view which I reject. I conclude, there-fore, that there is no need to admit as a fundamentalnotion any form of implication not expressible as atruth-function.

Chapter XV

Propositional Functions

When, in the preceding chapter, we were discussing propositions, we did not attempt to give a definitionof the word “proposition.” But although the wordcannot be formally defined, it is necessary to saysomething as to its meaning, in order to avoid thevery common confusion with “propositional func-tions,” which are to be the topic of the present chap-ter.

We mean by a “proposition” primarily a form ofwords which expresses what is either true or false.I say “primarily,” because I do not wish to excludeother than verbal symbols, or even mere thoughtsif they have a symbolic character. But I think theword “proposition” should be limited to what may,in some sense, be called “symbols,” and furtherto such symbols as give expression to truth and

falsehood. Thus “two and two are four” and “twoand two are five” will be propositions, and so will“Socrates is a man” and “Socrates is not a man.”The statement: “Whatever numbers a and b maybe, (a + b) = a + ab + b” is a proposition; butthe bare formula “(a+ b) = a + ab + b” alone isnot, since it asserts nothing definite unless we arefurther told, or led to suppose, that a and b are tohave all possible values, or are to have such-and-such values. The former of these is tacitly assumed,as a rule, in the enunciation of mathematical for-mulæ, which thus become propositions; but if nosuch assumption were made, they would be “propo-sitional functions.” A “propositional function,” infact, is an expression containing one or more un-determined constituents, | such that, when values are assigned to these constituents, the expressionbecomes a proposition. In other words, it is a func-tion whose values are propositions. But this latterdefinition must be used with caution. A descrip-tive function, e.g. “the hardest proposition in A’smathematical treatise,” will not be a propositionalfunction, although its values are propositions. Butin such a case the propositions are only described:in a propositional function, the values must actuallyenunciate propositions.

Examples of propositional functions are easy togive: “x is human” is a propositional function; solong as x remains undetermined, it is neither truenor false, but when a value is assigned to x it be-comes a true or false proposition. Any mathemati-cal equation is a propositional function. So long asthe variables have no definite value, the equationis merely an expression awaiting determination inorder to become a true or false proposition. If itis an equation containing one variable, it becomestrue when the variable is made equal to a root of theequation, otherwise it becomes false; but if it is an“identity” it will be true when the variable is anynumber. The equation to a curve in a plane or to asurface in space is a propositional function, true forvalues of the co-ordinates belonging to points on thecurve or surface, false for other values. Expressionsof traditional logic such as “all A is B” are propo-sitional functions: A and B have to be determinedas definite classes before such expressions becometrue or false.

The notion of “cases” or “instances” dependsupon propositional functions. Consider, for exam-ple, the kind of process suggested by what is called“generalisation,” and let us take some very primitiveexample, say, “lightning is followed by thunder.” We

have a number of “instances” of this, i.e. a numberof propositions such as: “this is a flash of lightningand is followed by thunder.” What are these oc-currences “instances” of? They are instances of thepropositional function: “If x is a flash of lightning,x is followed by thunder.” The process of generali-sation (with whose validity we are | fortunately not concerned) consists in passing from a number ofsuch instances to the universal truth of the propo-sitional function: “If x is a flash of lightning, x isfollowed by thunder.” It will be found that, in ananalogous way, propositional functions are alwaysinvolved whenever we talk of instances or cases orexamples.

We do not need to ask, or attempt to answer,the question: “What is a propositional function?”A propositional function standing all alone may betaken to be a mere schema, a mere shell, an emptyreceptacle for meaning, not something already sig-nificant. We are concerned with propositional func-tions, broadly speaking, in two ways: first, as in-volved in the notions “true in all cases” and “truein some cases”; secondly, as involved in the theoryof classes and relations. The second of these topicswe will postpone to a later chapter; the first mustoccupy us now.

When we say that something is “always true” or“true in all cases,” it is clear that the “something”involved cannot be a proposition. A proposition isjust true or false, and there is an end of the matter.There are no instances or cases of “Socrates is a man”or “Napoleon died at St Helena.” These are proposi-tions, and it would be meaningless to speak of theirbeing true “in all cases.” This phrase is only appli-cable to propositional functions. Take, for example,the sort of thing that is often said when causationis being discussed. (We are not concerned with thetruth or falsehood of what is said, but only with itslogical analysis.) We are told that A is, in every in-stance, followed by B. Now if there are “instances”of A, A must be some general concept of which it issignificant to say “x is A,” “x is A,” “x is A,” andso on, where x, x, x are particulars which are notidentical one with another. This applies, e.g., to ourprevious case of lightning. We say that lightning (A)is followed by thunder (B). But the separate flashesare particulars, not identical, but sharing the com-mon property of being lightning. The only way ofexpressing a | common property generally is to say that a common property of a number of objects isa propositional function which becomes true whenany one of these objects is taken as the value of the

variable. In this case all the objects are “instances”of the truth of the propositional function—for apropositional function, though it cannot itself betrue or false, is true in certain instances and false incertain others, unless it is “always true” or “alwaysfalse.” When, to return to our example, we say thatA is in every instance followed by B, we mean that,whatever x may be, if x is an A, it is followed by a B;that is, we are asserting that a certain propositionalfunction is “always true.”

Sentences involving such words as “all,” “every,”“a,” “the,” “some” require propositional functionsfor their interpretation. The way in which proposi-tional functions occur can be explained by means oftwo of the above words, namely, “all” and “some.”

There are, in the last analysis, only two thingsthat can be done with a propositional function: oneis to assert that it is true in all cases, the other toassert that it is true in at least one case, or in somecases (as we shall say, assuming that there is to be nonecessary implication of a plurality of cases). All theother uses of propositional functions can be reducedto these two. When we say that a propositionalfunction is true “in all cases,” or “always” (as weshall also say, without any temporal suggestion),we mean that all its values are true. If “φx” is the

function, and a is the right sort of object to be anargument to “φx,” then φa is to be true, however amay have been chosen. For example, “if a is human,a is mortal” is true whether a is human or not; infact, every proposition of this form is true. Thus thepropositional function “if x is human, x is mortal”is “always true,” or “true in all cases.” Or, again,the statement “there are no unicorns” is the same asthe statement “the propositional function ‘x is not aunicorn’ is true in all cases.” The assertions in thepreceding chapter about propositions, e.g. “‘p or q’implies ‘q or p,’” are really assertions | that certain propositional functions are true in all cases. Wedo not assert the above principle, for example, asbeing true only of this or that particular p or q, butas being true of any p or q concerning which it canbe made significantly. The condition that a functionis to be significant for a given argument is the sameas the condition that it shall have a value for thatargument, either true or false. The study of theconditions of significance belongs to the doctrine oftypes, which we shall not pursue beyond the sketchgiven in the preceding chapter.

Not only the principles of deduction, but all theprimitive propositions of logic, consist of assertionsthat certain propositional functions are always true.

If this were not the case, they would have to men-tion particular things or concepts—Socrates, or red-ness, or east and west, or what not—and clearlyit is not the province of logic to make assertionswhich are true concerning one such thing or con-cept but not concerning another. It is part of thedefinition of logic (but not the whole of its defini-tion) that all its propositions are completely general,i.e. they all consist of the assertion that some propo-sitional function containing no constant terms isalways true. We shall return in our final chapter tothe discussion of propositional functions containingno constant terms. For the present we will proceedto the other thing that is to be done with a propo-sitional function, namely, the assertion that it is“sometimes true,” i.e. true in at least one instance.

When we say “there are men,” that means thatthe propositional function “x is a man” is some-times true. When we say “some men are Greeks,”that means that the propositional function “x is aman and a Greek” is sometimes true. When wesay “cannibals still exist in Africa,” that means thatthe propositional function “x is a cannibal now inAfrica” is sometimes true, i.e. is true for some valuesof x. To say “there are at least n individuals in theworld” is to say that the propositional function “α is

a class of individuals and a member of the cardinalnumber n” is sometimes true, or, as we may say, istrue for certain | values of α. This form of expres- sion is more convenient when it is necessary to indi-cate which is the variable constituent which we aretaking as the argument to our propositional func-tion. For example, the above propositional function,which we may shorten to “α is a class of n individu-als,” contains two variables, α and n. The axiom ofinfinity, in the language of propositional functions,is: “The propositional function ‘if n is an inductivenumber, it is true for some values of α that α is aclass of n individuals’ is true for all possible val-ues of n.” Here there is a subordinate function, “αis a class of n individuals,” which is said to be, inrespect of α, sometimes true; and the assertion thatthis happens if n is an inductive number is said tobe, in respect of n, always true.

The statement that a function φx is always trueis the negation of the statement that not-φx is some-times true, and the statement that φx is sometimestrue is the negation of the statement that not-φx isalways true. Thus the statement “all men are mor-tals” is the negation of the statement that the func-tion “x is an immortal man” is sometimes true. Andthe statement “there are unicorns” is the negation of

the statement that the function “x is not a unicorn”is always true. We say that φx is “never true” or“always false” if not-φx is always true. We can, ifwe choose, take one of the pair “always,” “some-times” as a primitive idea, and define the other bymeans of the one and negation. Thus if we choose“sometimes” as our primitive idea, we can define:“‘φx is always true’ is to mean ‘it is false that not-φx is sometimes true.’” But for reasons connectedwith the theory of types it seems more correct totake both “always” and “sometimes” as primitiveideas, and define by their means the negation ofpropositions in which they occur. That is to say,assuming that we have already | defined (or adopted as a primitive idea) the negation of propositions ofthe type to which φx belongs, we define: “The nega-tion of ‘φx always’ is ‘not-φx sometimes’; and thenegation of ‘φx sometimes’ is ‘not-φx always.’” Inlike manner we can re-define disjunction and theother truth-functions, as applied to propositionscontaining apparent variables, in terms of the defi-nitions and primitive ideas for propositions contain-ing no apparent variables. Propositions containing

For linguistic reasons, to avoid suggesting either the pluralor the singular, it is often convenient to say “φx is not alwaysfalse” rather than “φx sometimes” or “φx is sometimes true.”

no apparent variables are called “elementary propo-sitions.” From these we can mount up step by step,using such methods as have just been indicated, tothe theory of truth-functions as applied to propo-sitions containing one, two, three . . . variables, orany number up to n, where n is any assigned finitenumber.

The forms which are taken as simplest in tra-ditional formal logic are really far from being so,and all involve the assertion of all values or somevalues of a compound propositional function. Take,to begin with, “all S is P.” We will take it that S isdefined by a propositional function φx, and P by apropositional function ψx. E.g., if S is men, φx willbe “x is human”; if P is mortals, ψx will be “thereis a time at which x dies.” Then “all S is P” means:“‘φx implies ψx’ is always true.” It is to be observedthat “all S is P” does not apply only to those termsthat actually are S’s; it says something equally aboutterms which are not S’s. Suppose we come acrossan x of which we do not know whether it is an S ornot; still, our statement “all S is P” tells us some-thing about x, namely, that if x is an S, then x is aP. And this is every bit as true when x is not an S as

The method of deduction is given in Principia Mathematica,vol. i. ∗.

when x is an S. If it were not equally true in bothcases, the reductio ad absurdum would not be a validmethod; for the essence of this method consists inusing implications in cases where (as it afterwardsturns out) the hypothesis is false. We may put thematter another way. In order to understand “all S isP,” it is not necessary to be able to enumerate whatterms are S’s; provided we know what is meant bybeing an S and what by being a P, we can understandcompletely what is actually affirmed | by “all S is P,” however little we may know of actual instances ofeither. This shows that it is not merely the actualterms that are S’s that are relevant in the statement“all S is P,” but all the terms concerning which thesupposition that they are S’s is significant, i.e. allthe terms that are S’s, together with all the termsthat are not S’s—i.e. the whole of the appropriatelogical “type.” What applies to statements about allapplies also to statements about some. “There aremen,” e.g., means that “x is human” is true for somevalues of x. Here all values of x (i.e. all values forwhich “x is human” is significant, whether true orfalse) are relevant, and not only those that in factare human. (This becomes obvious if we considerhow we could prove such a statement to be false.)Every assertion about “all” or “some” thus involves

not only the arguments that make a certain func-tion true, but all that make it significant, i.e. all forwhich it has a value at all, whether true or false.

We may now proceed with our interpretation ofthe traditional forms of the old-fashioned formallogic. We assume that S is those terms x for whichφx is true, and P is those for which ψx is true. (Aswe shall see in a later chapter, all classes are derivedin this way from propositional functions.) Then:

“All S is P” means “‘φx implies ψx’ is alwaystrue.”

“Some S is P” means “‘φx and ψx’ is sometimestrue.”

“No S is P” means “‘φx implies not-ψx’ is alwaystrue.”

“Some S is not P” means “‘φx and not-ψx’ issometimes true.”

It will be observed that the propositional functionswhich are here asserted for all or some values arenot φx and ψx themselves, but truth-functions ofφx and ψx for the same argument x. The easiest wayto conceive of the sort of thing that is intended isto start not from φx and ψx in general, but fromφa and ψa, where a is some constant. Suppose weare considering “all men are mortal”: we will beginwith

“If Socrates is human, Socrates is mortal,” |

and then we will regard “Socrates” as replaced by a variable x wherever “Socrates” occurs. The objectto be secured is that, although x remains a variable,without any definite value, yet it is to have the samevalue in “φx” as in “ψx” when we are assertingthat “φx implies ψx” is always true. This requiresthat we shall start with a function whose valuesare such as “φa implies ψa,” rather than with twoseparate functions φx and ψx; for if we start withtwo separate functions we can never secure that thex, while remaining undetermined, shall have thesame value in both.

For brevity we say “φx always implies ψx” whenwe mean that “φx impliesψx” is always true. Propo-sitions of the form “φx always impliesψx” are called“formal implications”; this name is given equally ifthere are several variables.

The above definitions show how far removedfrom the simplest forms are such propositions as“all S is P,” with which traditional logic begins. Itis typical of the lack of analysis involved that tra-ditional logic treats “all S is P” as a proposition ofthe same form as “x is P”—e.g., it treats “all men aremortal” as of the same form as “Socrates is mortal.”As we have just seen, the first is of the form “φx

always implies ψx,” while the second is of the form“ψx.” The emphatic separation of these two forms,which was effected by Peano and Frege, was a veryvital advance in symbolic logic.

It will be seen that “all S is P” and “no S is P”do not really differ in form, except by the substitu-tion of not-ψx for ψx, and that the same applies to“some S is P” and “some S is not P.” It should alsobe observed that the traditional rules of conversionare faulty, if we adopt the view, which is the onlytechnically tolerable one, that such propositions as“all S is P” do not involve the “existence” of S’s, i.e.do not require that there should be terms which areS’s. The above definitions lead to the result that, ifφx is always false, i.e. if there are no S’s, then “all Sis P” and “no S is P” will both be true, | whatever P may be. For, according to the definition in the lastchapter, “φx implies ψx” means “not-φx or ψx,”which is always true if not-φx is always true. Atthe first moment, this result might lead the readerto desire different definitions, but a little practicalexperience soon shows that any different definitionswould be inconvenient and would conceal the im-portant ideas. The proposition “φx always impliesψx, and φx is sometimes true” is essentially com-posite, and it would be very awkward to give this as

the definition of “all S is P,” for then we should haveno language left for “φx always implies ψx,” whichis needed a hundred times for once that the other isneeded. But, with our definitions, “all S is P” doesnot imply “some S is P,” since the first allows thenon-existence of S and the second does not; thusconversion per accidens becomes invalid, and somemoods of the syllogism are fallacious, e.g. Darapti:“All M is S, all M is P, therefore some S is P,” whichfails if there is no M.

The notion of “existence” has several forms, oneof which will occupy us in the next chapter; but thefundamental form is that which is derived immedi-ately from the notion of “sometimes true.” We saythat an argument a “satisfies” a function φx if φais true; this is the same sense in which the roots ofan equation are said to satisfy the equation. Now ifφx is sometimes true, we may say there are x’s forwhich it is true, or we may say “arguments satis-fying φx exist.” This is the fundamental meaningof the word “existence.” Other meanings are eitherderived from this, or embody mere confusion ofthought. We may correctly say “men exist,” mean-ing that “x is a man” is sometimes true. But if wemake a pseudo-syllogism: “Men exist, Socrates is aman, therefore Socrates exists,” we are talking non-

sense, since “Socrates” is not, like “men,” merelyan undetermined argument to a given propositionalfunction. The fallacy is closely analogous to thatof the argument: “Men are numerous, Socrates isa man, therefore Socrates is numerous.” In thiscase it is obvious that the conclusion is nonsensical,but | in the case of existence it is not obvious, for reasons which will appear more fully in the nextchapter. For the present let us merely note the factthat, though it is correct to say “men exist,” it is in-correct, or rather meaningless, to ascribe existenceto a given particular x who happens to be a man.Generally, “terms satisfying φx exist” means “φx issometimes true”; but “a exists” (where a is a termsatisfying φx) is a mere noise or shape, devoid ofsignificance. It will be found that by bearing inmind this simple fallacy we can solve many ancientphilosophical puzzles concerning the meaning ofexistence.

Another set of notions as to which philosophyhas allowed itself to fall into hopeless confusionsthrough not sufficiently separating propositions andpropositional functions are the notions of “modal-ity”: necessary, possible, and impossible. (Sometimescontingent or assertoric is used instead of possible.)The traditional view was that, among true propo-

sitions, some were necessary, while others weremerely contingent or assertoric; while among falsepropositions some were impossible, namely, thosewhose contradictories were necessary, while othersmerely happened not to be true. In fact, however,there was never any clear account of what was addedto truth by the conception of necessity. In the caseof propositional functions, the threefold divisionis obvious. If “φx” is an undetermined value of acertain propositional function, it will be necessary ifthe function is always true, possible if it is sometimestrue, and impossible if it is never true. This sort ofsituation arises in regard to probability, for exam-ple. Suppose a ball x is drawn from a bag whichcontains a number of balls: if all the balls are white,“x is white” is necessary; if some are white, it ispossible; if none, it is impossible. Here all that isknown about x is that it satisfies a certain proposi-tional function, namely, “x was a ball in the bag.”This is a situation which is general in probabilityproblems and not uncommon in practical life—e.g.when a person calls of whom we know nothing ex-cept that he brings a letter of introduction from ourfriend so-and-so. In all such | cases, as in regard to modality in general, the propositional function isrelevant. For clear thinking, in many very diverse

directions, the habit of keeping propositional func-tions sharply separated from propositions is of theutmost importance, and the failure to do so in thepast has been a disgrace to philosophy.

Chapter XVI

Descriptions

We dealt in the preceding chapter with the words all and some; in this chapter we shall consider theword the in the singular, and in the next chapterwe shall consider the word the in the plural. It maybe thought excessive to devote two chapters to oneword, but to the philosophical mathematician it isa word of very great importance: like Browning’sGrammarian with the enclitic δε, I would give thedoctrine of this word if I were “dead from the waistdown” and not merely in a prison.

We have already had occasion to mention “de-scriptive functions,” i.e. such expressions as “thefather of x” or “the sine of x.” These are to be de-fined by first defining “descriptions.”

A “description” may be of two sorts, definite andindefinite (or ambiguous). An indefinite description

is a phrase of the form “a so-and-so,” and a definitedescription is a phrase of the form “the so-and-so”(in the singular). Let us begin with the former.

“Who did you meet?” “I met a man.” “Thatis a very indefinite description.” We are thereforenot departing from usage in our terminology. Ourquestion is: What do I really assert when I assert “Imet a man”? Let us assume, for the moment, thatmy assertion is true, and that in fact I met Jones. Itis clear that what I assert is not “I met Jones.” I maysay “I met a man, but it was not Jones”; in that case,though I lie, I do not contradict myself, as I shoulddo if when I say I met a | man I really mean that I met Jones. It is clear also that the person to whom Iam speaking can understand what I say, even if heis a foreigner and has never heard of Jones.

But we may go further: not only Jones, but noactual man, enters into my statement. This becomesobvious when the statement is false, since then thereis no more reason why Jones should be supposedto enter into the proposition than why anyone elseshould. Indeed the statement would remain signif-icant, though it could not possibly be true, even ifthere were no man at all. “I met a unicorn” or “Imet a sea-serpent” is a perfectly significant asser-tion, if we know what it would be to be a unicorn

or a sea-serpent, i.e. what is the definition of thesefabulous monsters. Thus it is only what we maycall the concept that enters into the proposition. Inthe case of “unicorn,” for example, there is only theconcept: there is not also, somewhere among theshades, something unreal which may be called “aunicorn.” Therefore, since it is significant (thoughfalse) to say “I met a unicorn,” it is clear that thisproposition, rightly analysed, does not contain aconstituent “a unicorn,” though it does contain theconcept “unicorn.”

The question of “unreality,” which confronts usat this point, is a very important one. Misled bygrammar, the great majority of those logicians whohave dealt with this question have dealt with it onmistaken lines. They have regarded grammaticalform as a surer guide in analysis than, in fact, itis. And they have not known what differences ingrammatical form are important. “I met Jones” and“I met a man” would count traditionally as propo-sitions of the same form, but in actual fact they areof quite different forms: the first names an actualperson, Jones; while the second involves a proposi-tional function, and becomes, when made explicit:“The function ‘I met x and x is human’ is sometimestrue.” (It will be remembered that we adopted the

convention of using “sometimes” as not implyingmore than once.) This proposition is obviously notof the form “I met x,” which accounts | for the exis- tence of the proposition “I met a unicorn” in spite ofthe fact that there is no such thing as “a unicorn.”

For want of the apparatus of propositional func-tions, many logicians have been driven to the con-clusion that there are unreal objects. It is argued, e.g.by Meinong, that we can speak about “the goldenmountain,” “the round square,” and so on; we canmake true propositions of which these are the sub-jects; hence they must have some kind of logicalbeing, since otherwise the propositions in whichthey occur would be meaningless. In such theories,it seems to me, there is a failure of that feeling forreality which ought to be preserved even in the mostabstract studies. Logic, I should maintain, must nomore admit a unicorn than zoology can; for logic isconcerned with the real world just as truly as zo-ology, though with its more abstract and generalfeatures. To say that unicorns have an existence inheraldry, or in literature, or in imagination, is a mostpitiful and paltry evasion. What exists in heraldryis not an animal, made of flesh and blood, mov-ing and breathing of its own initiative. What exists

Untersuchungen zur Gegenstandstheorie und Psychologie, .

is a picture, or a description in words. Similarly,to maintain that Hamlet, for example, exists in hisown world, namely, in the world of Shakespeare’simagination, just as truly as (say) Napoleon existedin the ordinary world, is to say something deliber-ately confusing, or else confused to a degree whichis scarcely credible. There is only one world, the“real” world: Shakespeare’s imagination is part ofit, and the thoughts that he had in writing Hamletare real. So are the thoughts that we have in readingthe play. But it is of the very essence of fiction thatonly the thoughts, feelings, etc., in Shakespeare andhis readers are real, and that there is not, in additionto them, an objective Hamlet. When you have takenaccount of all the feelings roused by Napoleon inwriters and readers of history, you have not touchedthe actual man; but in the case of Hamlet you havecome to the end of him. If no one thought aboutHamlet, there would be nothing | left of him; if no one had thought about Napoleon, he would havesoon seen to it that some one did. The sense of real-ity is vital in logic, and whoever juggles with it bypretending that Hamlet has another kind of realityis doing a disservice to thought. A robust sense ofreality is very necessary in framing a correct analysisof propositions about unicorns, golden mountains,

round squares, and other such pseudo-objects.In obedience to the feeling of reality, we shall

insist that, in the analysis of propositions, nothing“unreal” is to be admitted. But, after all, if thereis nothing unreal, how, it may be asked, could weadmit anything unreal? The reply is that, in deal-ing with propositions, we are dealing in the first in-stance with symbols, and if we attribute significanceto groups of symbols which have no significance,we shall fall into the error of admitting unrealities,in the only sense in which this is possible, namely,as objects described. In the proposition “I met aunicorn,” the whole four words together make a sig-nificant proposition, and the word “unicorn” by it-self is significant, in just the same sense as the word“man.” But the two words “a unicorn” do not forma subordinate group having a meaning of its own.Thus if we falsely attribute meaning to these twowords, we find ourselves saddled with “a unicorn,”and with the problem how there can be such a thingin a world where there are no unicorns. “A unicorn”is an indefinite description which describes nothing.It is not an indefinite description which describessomething unreal. Such a proposition as “x is un-real” only has meaning when “x” is a description,definite or indefinite; in that case the proposition

will be true if “x” is a description which describesnothing. But whether the description “x” describessomething or describes nothing, it is in any case nota constituent of the proposition in which it occurs;like “a unicorn” just now, it is not a subordinategroup having a meaning of its own. All this resultsfrom the fact that, when “x” is a description, “x isunreal” or “x does not exist” is not nonsense, but isalways significant and sometimes true. |

We may now proceed to define generally the meaning of propositions which contain ambiguousdescriptions. Suppose we wish to make some state-ment about “a so-and-so,” where “so-and-so’s” arethose objects that have a certain property φ, i.e.those objects x for which the propositional functionφx is true. (E.g. if we take “a man” as our instance of“a so-and-so,” φx will be “x is human.”) Let us nowwish to assert the property ψ of “a so-and-so,” i.e.we wish to assert that “a so-and-so” has that prop-erty which x has when ψx is true. (E.g. in the caseof “I met a man,” ψx will be “I met x.”) Now theproposition that “a so-and-so” has the property ψis not a proposition of the form “ψx.” If it were, “aso-and-so” would have to be identical with x for asuitable x; and although (in a sense) this may be truein some cases, it is certainly not true in such a case

as “a unicorn.” It is just this fact, that the statementthat a so-and-so has the property ψ is not of theform ψx, which makes it possible for “a so-and-so”to be, in a certain clearly definable sense, “unreal.”The definition is as follows:—

The statement that “an object having the propertyφ has the property ψ”

means:

“The joint assertion of φx and ψx is not alwaysfalse.”

So far as logic goes, this is the same proposi-tion as might be expressed by “some φ’s are ψ’s”;but rhetorically there is a difference, because in theone case there is a suggestion of singularity, andin the other case of plurality. This, however, is notthe important point. The important point is that,when rightly analysed, propositions verbally about“a so-and-so” are found to contain no constituentrepresented by this phrase. And that is why suchpropositions can be significant even when there isno such thing as a so-and-so.

The definition of existence, as applied to am-biguous descriptions, results from what was saidat the end of the preceding chapter. We say that

“men exist” or “a man exists” if the | propositional function “x is human” is sometimes true; and gener-ally “a so-and-so” exists if “x is so-and-so” is some-times true. We may put this in other language. Theproposition “Socrates is a man” is no doubt equiv-alent to “Socrates is human,” but it is not the verysame proposition. The is of “Socrates is human” ex-presses the relation of subject and predicate; the isof “Socrates is a man” expresses identity. It is a dis-grace to the human race that it has chosen to employthe same word “is” for these two entirely differentideas—a disgrace which a symbolic logical languageof course remedies. The identity in “Socrates isa man” is identity between an object named (ac-cepting “Socrates” as a name, subject to qualifica-tions explained later) and an object ambiguouslydescribed. An object ambiguously described will“exist” when at least one such proposition is true,i.e. when there is at least one true proposition of theform “x is a so-and-so,” where “x” is a name. It ischaracteristic of ambiguous (as opposed to definite)descriptions that there may be any number of truepropositions of the above form—Socrates is a man,Plato is a man, etc. Thus “a man exists” follows fromSocrates, or Plato, or anyone else. With definite de-scriptions, on the other hand, the corresponding

form of proposition, namely, “x is the so-and-so”(where “x” is a name), can only be true for one valueof x at most. This brings us to the subject of definitedescriptions, which are to be defined in a way anal-ogous to that employed for ambiguous descriptions,but rather more complicated.

We come now to the main subject of the presentchapter, namely, the definition of the word the (inthe singular). One very important point about thedefinition of “a so-and-so” applies equally to “theso-and-so”; the definition to be sought is a definitionof propositions in which this phrase occurs, not adefinition of the phrase itself in isolation. In the caseof “a so-and-so,” this is fairly obvious: no one couldsuppose that “a man” was a definite object, whichcould be defined by itself. | Socrates is a man, Plato is a man, Aristotle is a man, but we cannot inferthat “a man” means the same as “Socrates” meansand also the same as “Plato” means and also thesame as “Aristotle” means, since these three nameshave different meanings. Nevertheless, when wehave enumerated all the men in the world, there isnothing left of which we can say, “This is a man, andnot only so, but it is the ‘a man,’ the quintessentialentity that is just an indefinite man without beinganybody in particular.” It is of course quite clear

that whatever there is in the world is definite: if itis a man it is one definite man and not any other.Thus there cannot be such an entity as “a man” tobe found in the world, as opposed to specific men.And accordingly it is natural that we do not define“a man” itself, but only the propositions in which itoccurs.

In the case of “the so-and-so” this is equally true,though at first sight less obvious. We may demon-strate that this must be the case, by a considerationof the difference between a name and a definite de-scription. Take the proposition, “Scott is the authorof Waverley.” We have here a name, “Scott,” anda description, “the author of Waverley,” which areasserted to apply to the same person. The distinc-tion between a name and all other symbols may beexplained as follows:—

A name is a simple symbol whose meaning issomething that can only occur as subject, i.e. some-thing of the kind that, in Chapter XIII., we definedas an “individual” or a “particular.” And a “simple”symbol is one which has no parts that are symbols.Thus “Scott” is a simple symbol, because, thoughit has parts (namely, separate letters), these partsare not symbols. On the other hand, “the author ofWaverley” is not a simple symbol, because the sepa-

rate words that compose the phrase are parts whichare symbols. If, as may be the case, whatever seemsto be an “individual” is really capable of furtheranalysis, we shall have to content ourselves withwhat may be called “relative individuals,” whichwill be terms that, throughout the context in ques-tion, are never analysed and never occur | otherwise than as subjects. And in that case we shall havecorrespondingly to content ourselves with “rela-tive names.” From the standpoint of our presentproblem, namely, the definition of descriptions, thisproblem, whether these are absolute names or onlyrelative names, may be ignored, since it concernsdifferent stages in the hierarchy of “types,” whereaswe have to compare such couples as “Scott” and “theauthor of Waverley,” which both apply to the sameobject, and do not raise the problem of types. Wemay, therefore, for the moment, treat names as ca-pable of being absolute; nothing that we shall haveto say will depend upon this assumption, but thewording may be a little shortened by it.

We have, then, two things to compare: () aname, which is a simple symbol, directly designat-ing an individual which is its meaning, and havingthis meaning in its own right, independently of themeanings of all other words; () a description, which

consists of several words, whose meanings are al-ready fixed, and from which results whatever is tobe taken as the “meaning” of the description.

A proposition containing a description is notidentical with what that proposition becomes whena name is substituted, even if the name names thesame object as the description describes. “Scottis the author of Waverley” is obviously a differentproposition from “Scott is Scott”: the first is a factin literary history, the second a trivial truism. Andif we put anyone other than Scott in place of “theauthor of Waverley,” our proposition would becomefalse, and would therefore certainly no longer be thesame proposition. But, it may be said, our proposi-tion is essentially of the same form as (say) “Scott isSir Walter,” in which two names are said to applyto the same person. The reply is that, if “Scott isSir Walter” really means “the person named ‘Scott’is the person named ‘Sir Walter,’” then the namesare being used as descriptions: i.e. the individual,instead of being named, is being described as theperson having that name. This is a way in whichnames are frequently used | in practice, and there will, as a rule, be nothing in the phraseology toshow whether they are being used in this way oras names. When a name is used directly, merely to

indicate what we are speaking about, it is no partof the fact asserted, or of the falsehood if our as-sertion happens to be false: it is merely part of thesymbolism by which we express our thought. Whatwe want to express is something which might (forexample) be translated into a foreign language; itis something for which the actual words are a ve-hicle, but of which they are no part. On the otherhand, when we make a proposition about “the per-son called ‘Scott,’” the actual name “Scott” entersinto what we are asserting, and not merely into thelanguage used in making the assertion. Our propo-sition will now be a different one if we substitute“the person called ‘Sir Walter.’” But so long as weare using names as names, whether we say “Scott”or whether we say “Sir Walter” is as irrelevant towhat we are asserting as whether we speak Englishor French. Thus so long as names are used as names,“Scott is Sir Walter” is the same trivial propositionas “Scott is Scott.” This completes the proof that“Scott is the author of Waverley” is not the sameproposition as results from substituting a name for“the author of Waverley,” no matter what name maybe substituted.

When we use a variable, and speak of a propo-sitional function, φx say, the process of applying

general statements about φx to particular cases willconsist in substituting a name for the letter “x,” as-suming that φ is a function which has individualsfor its arguments. Suppose, for example, that φxis “always true”; let it be, say, the “law of identity,”x = x. Then we may substitute for “x” any namewe choose, and we shall obtain a true proposition.Assuming for the moment that “Socrates,” “Plato,”and “Aristotle” are names (a very rash assumption),we can infer from the law of identity that Socratesis Socrates, Plato is Plato, and Aristotle is Aristotle.But we shall commit a fallacy if we attempt to infer,without further premisses, that the author of Wa-verley is the author of Waverley. This results | from what we have just proved, that, if we substitute aname for “the author of Waverley” in a proposition,the proposition we obtain is a different one. Thatis to say, applying the result to our present case: If“x” is a name, “x = x” is not the same propositionas “the author of Waverley is the author of Waver-ley,” no matter what name “x” may be. Thus fromthe fact that all propositions of the form “x = x”are true we cannot infer, without more ado, thatthe author of Waverley is the author of Waverley.In fact, propositions of the form “the so-and-so isthe so-and-so” are not always true: it is necessary

that the so-and-so should exist (a term which will beexplained shortly). It is false that the present Kingof France is the present King of France, or that theround square is the round square. When we substi-tute a description for a name, propositional func-tions which are “always true” may become false, ifthe description describes nothing. There is no mys-tery in this as soon as we realise (what was provedin the preceding paragraph) that when we substi-tute a description the result is not a value of thepropositional function in question.

We are now in a position to define propositionsin which a definite description occurs. The onlything that distinguishes “the so-and-so” from “aso-and-so” is the implication of uniqueness. Wecannot speak of “the inhabitant of London,” be-cause inhabiting London is an attribute which isnot unique. We cannot speak about “the presentKing of France,” because there is none; but we canspeak about “the present King of England.” Thuspropositions about “the so-and-so” always implythe corresponding propositions about “a so-and-so,” with the addendum that there is not more thanone so-and-so. Such a proposition as “Scott is theauthor of Waverley” could not be true if Waverleyhad never been written, or if several people had

written it; and no more could any other propositionresulting from a propositional function φx by thesubstitution of “the author of Waverley” for “x.” Wemay say that “the author of Waverley” means “thevalue of x for which ‘x wrote | Waverley’ is true.” Thus the proposition “the author of Waverley wasScotch,” for example, involves:

() “x wrote Waverley” is not always false;() “if x and y wrote Waverley, x and y are identi-

cal” is always true;() “if x wrote Waverley, x was Scotch” is always

true.

These three propositions, translated into ordinarylanguage, state:

() at least one person wrote Waverley;() at most one person wrote Waverley;() whoever wrote Waverley was Scotch.

All these three are implied by “the author of Waver-ley was Scotch.” Conversely, the three together (butno two of them) imply that the author of Waverleywas Scotch. Hence the three together may be takenas defining what is meant by the proposition “theauthor of Waverley was Scotch.”

We may somewhat simplify these three proposi-tions. The first and second together are equivalent

to: “There is a term c such that ‘x wrote Waver-ley’ is true when x is c and is false when x is notc.” In other words, “There is a term c such that ‘xwrote Waverley’ is always equivalent to ‘x is c.’” (Twopropositions are “equivalent” when both are trueor both are false.) We have here, to begin with, twofunctions of x, “x wrote Waverley” and “x is c,” andwe form a function of c by considering the equiva-lence of these two functions of x for all values of x;we then proceed to assert that the resulting functionof c is “sometimes true,” i.e. that it is true for at leastone value of c. (It obviously cannot be true for morethan one value of c.) These two conditions togetherare defined as giving the meaning of “the author ofWaverley exists.”

We may now define “the term satisfying thefunction φx exists.” This is the general form ofwhich the above is a particular case. “The authorof Waverley” is “the term satisfying the function ‘xwrote Waverley.’” And “the so-and-so” will | always involve reference to some propositional function,namely, that which defines the property that makesa thing a so-and-so. Our definition is as follows:—

“The term satisfying the function φx exists”means:

“There is a term c such that φx is always equiva-

lent to ‘x is c.’”In order to define “the author of Waverley was

Scotch,” we have still to take account of the third ofour three propositions, namely, “Whoever wroteWaverley was Scotch.” This will be satisfied bymerely adding that the c in question is to be Scotch.Thus “the author of Waverley was Scotch” is:

“There is a term c such that () ‘x wrote Waverley’is always equivalent to ‘x is c,’ () c is Scotch.”

And generally: “the term satisfying φx satisfies ψx”is defined as meaning:

“There is a term c such that ()φx is always equiv-alent to ‘x is c,’ () ψc is true.”

This is the definition of propositions in which de-scriptions occur.

It is possible to have much knowledge concern-ing a term described, i.e. to know many propositionsconcerning “the so-and-so,” without actually know-ing what the so-and-so is, i.e. without knowing anyproposition of the form “x is the so-and-so,” where“x” is a name. In a detective story propositionsabout “the man who did the deed” are accumu-lated, in the hope that ultimately they will suffice todemonstrate that it was A who did the deed. We may

even go so far as to say that, in all such knowledgeas can be expressed in words—with the exception of“this” and “that” and a few other words of which themeaning varies on different occasions—no names,in the strict sense, occur, but what seem like namesare really descriptions. We may inquire significantlywhether Homer existed, which we could not do if“Homer” were a name. The proposition “the so-and-so exists” is significant, whether true or false; but ifa is the so-and-so (where “a” is a name), the words“a exists” are meaningless. It is only of descriptions|—definite or indefinite—that existence can be sig- nificantly asserted; for, if “a” is a name, it must namesomething: what does not name anything is not aname, and therefore, if intended to be a name, is asymbol devoid of meaning, whereas a description,like “the present King of France,” does not becomeincapable of occurring significantly merely on theground that it describes nothing, the reason beingthat it is a complex symbol, of which the meaning isderived from that of its constituent symbols. Andso, when we ask whether Homer existed, we areusing the word “Homer” as an abbreviated descrip-tion: we may replace it by (say) “the author of theIliad and the Odyssey.” The same considerationsapply to almost all uses of what look like proper

names.When descriptions occur in propositions, it is

necessary to distinguish what may be called “pri-mary” and “secondary” occurrences. The abstractdistinction is as follows. A description has a “pri-mary” occurrence when the proposition in whichit occurs results from substituting the descriptionfor “x” in some propositional function φx; a de-scription has a “secondary” occurrence when theresult of substituting the description for x in φxgives only part of the proposition concerned. An in-stance will make this clearer. Consider “the presentKing of France is bald.” Here “the present King ofFrance” has a primary occurrence, and the proposi-tion is false. Every proposition in which a descrip-tion which describes nothing has a primary occur-rence is false. But now consider “the present Kingof France is not bald.” This is ambiguous. If we arefirst to take “x is bald,” then substitute “the presentKing of France” for “x,” and then deny the result,the occurrence of “the present King of France” issecondary and our proposition is true; but if we areto take “x is not bald” and substitute “the presentKing of France” for “x,” then “the present King ofFrance” has a primary occurrence and the proposi-tion is false. Confusion of primary and secondary

occurrences is a ready source of fallacies where de-scriptions are concerned. |

Descriptions occur in mathematics chiefly in the form of descriptive functions, i.e. “the term havingthe relation R to y,” or “the R of y” as we may say, onthe analogy of “the father of y” and similar phrases.To say “the father of y is rich,” for example, is to saythat the following propositional function of c: “c isrich, and ‘x begat y’ is always equivalent to ‘x is c,’”is “sometimes true,” i.e. is true for at least one valueof c. It obviously cannot be true for more than onevalue.

The theory of descriptions, briefly outlined inthe present chapter, is of the utmost importanceboth in logic and in theory of knowledge. But forpurposes of mathematics, the more philosophicalparts of the theory are not essential, and have there-fore been omitted in the above account, which hasconfined itself to the barest mathematical requisites.

Chapter XVII

Classes

In the present chapter we shall be concerned with the in the plural: the inhabitants of London, thesons of rich men, and so on. In other words, weshall be concerned with classes. We saw in ChapterII. that a cardinal number is to be defined as a classof classes, and in Chapter III. that the number isto be defined as the class of all unit classes, i.e. of allthat have just one member, as we should say but forthe vicious circle. Of course, when the number isdefined as the class of all unit classes, “unit classes”must be defined so as not to assume that we knowwhat is meant by “one”; in fact, they are definedin a way closely analogous to that used for descrip-tions, namely: A class α is said to be a “unit” classif the propositional function “‘x is an α’ is alwaysequivalent to ‘x is c’” (regarded as a function of c)

is not always false, i.e., in more ordinary language,if there is a term c such that x will be a memberof α when x is c but not otherwise. This gives us adefinition of a unit class if we already know whata class is in general. Hitherto we have, in dealingwith arithmetic, treated “class” as a primitive idea.But, for the reasons set forth in Chapter XIII., iffor no others, we cannot accept “class” as a prim-itive idea. We must seek a definition on the samelines as the definition of descriptions, i.e. a defini-tion which will assign a meaning to propositionsin whose verbal or symbolic expression words orsymbols apparently representing classes occur, butwhich will assign a meaning that altogether elimi-nates all mention of classes from a right analysis |of such propositions. We shall then be able to say that the symbols for classes are mere conveniences,not representing objects called “classes,” and thatclasses are in fact, like descriptions, logical fictions,or (as we say) “incomplete symbols.”

The theory of classes is less complete than thetheory of descriptions, and there are reasons (whichwe shall give in outline) for regarding the defini-tion of classes that will be suggested as not finallysatisfactory. Some further subtlety appears to berequired; but the reasons for regarding the defini-

tion which will be offered as being approximatelycorrect and on the right lines are overwhelming.

The first thing is to realise why classes cannotbe regarded as part of the ultimate furniture ofthe world. It is difficult to explain precisely whatone means by this statement, but one consequencewhich it implies may be used to elucidate its mean-ing. If we had a complete symbolic language, witha definition for everything definable, and an un-defined symbol for everything indefinable, the un-defined symbols in this language would representsymbolically what I mean by “the ultimate furni-ture of the world.” I am maintaining that no sym-bols either for “class” in general or for particularclasses would be included in this apparatus of unde-fined symbols. On the other hand, all the particularthings there are in the world would have to havenames which would be included among undefinedsymbols. We might try to avoid this conclusion bythe use of descriptions. Take (say) “the last thingCæsar saw before he died.” This is a descriptionof some particular; we might use it as (in one per-fectly legitimate sense) a definition of that particu-lar. But if “a” is a name for the same particular, aproposition in which “a” occurs is not (as we sawin the preceding chapter) identical with what this

proposition becomes when for “a” we substitute“the last thing Cæsar saw before he died.” If ourlanguage does not contain the name “a,” or someother name for the same particular, we shall have nomeans of expressing the proposition which we ex-pressed by means of “a” as opposed to the one that| we expressed by means of the description. Thus descriptions would not enable a perfect languageto dispense with names for all particulars. In thisrespect, we are maintaining, classes differ from par-ticulars, and need not be represented by undefinedsymbols. Our first business is to give the reasons forthis opinion.

We have already seen that classes cannot be re-garded as a species of individuals, on account of thecontradiction about classes which are not membersof themselves (explained in Chapter XIII.), and be-cause we can prove that the number of classes isgreater than the number of individuals.

We cannot take classes in the pure extensionalway as simply heaps or conglomerations. If we wereto attempt to do that, we should find it impossibleto understand how there can be such a class as thenull-class, which has no members at all and cannotbe regarded as a “heap”; we should also find it veryhard to understand how it comes about that a class

which has only one member is not identical withthat one member. I do not mean to assert, or todeny, that there are such entities as “heaps.” As amathematical logician, I am not called upon to havean opinion on this point. All that I am maintainingis that, if there are such things as heaps, we cannotidentify them with the classes composed of theirconstituents.

We shall come much nearer to a satisfactory the-ory if we try to identify classes with propositionalfunctions. Every class, as we explained in Chap-ter II., is defined by some propositional functionwhich is true of the members of the class and falseof other things. But if a class can be defined by onepropositional function, it can equally well be de-fined by any other which is true whenever the firstis true and false whenever the first is false. For thisreason the class cannot be identified with any onesuch propositional function rather than with anyother—and given a propositional function, there arealways many others which are true when it is trueand false when it is false. We say that two propo-sitional functions are “formally equivalent” whenthis happens. Two propositions are | “equivalent” when both are true or both false; two propositionalfunctions φx, ψx are “formally equivalent” when

φx is always equivalent to ψx. It is the fact thatthere are other functions formally equivalent to agiven function that makes it impossible to identify aclass with a function; for we wish classes to be suchthat no two distinct classes have exactly the samemembers, and therefore two formally equivalentfunctions will have to determine the same class.

When we have decided that classes cannot bethings of the same sort as their members, that theycannot be just heaps or aggregates, and also thatthey cannot be identified with propositional func-tions, it becomes very difficult to see what they canbe, if they are to be more than symbolic fictions.And if we can find any way of dealing with them assymbolic fictions, we increase the logical securityof our position, since we avoid the need of assum-ing that there are classes without being compelledto make the opposite assumption that there are noclasses. We merely abstain from both assumptions.This is an example of Occam’s razor, namely, “en-tities are not to be multiplied without necessity.”But when we refuse to assert that there are classes,we must not be supposed to be asserting dogmati-cally that there are none. We are merely agnostic asregards them: like Laplace, we can say, “je n’ai pasbesoin de cette hypothese.”

Let us set forth the conditions that a symbolmust fulfil if it is to serve as a class. I think thefollowing conditions will be found necessary andsufficient:—

() Every propositional function must determinea class, consisting of those arguments for which thefunction is true. Given any proposition (true orfalse), say about Socrates, we can imagine Socratesreplaced by Plato or Aristotle or a gorilla or the manin the moon or any other individual in the world.In general, some of these substitutions will give atrue proposition and some a false one. The classdetermined will consist of all those substitutionsthat give a true one. Of course, we have still todecide what we mean by “all those which, etc.” Allthat | we are observing at present is that a class is rendered determinate by a propositional function,and that every propositional function determinesan appropriate class.

() Two formally equivalent propositional func-tions must determine the same class, and two whichare not formally equivalent must determine differ-ent classes. That is, a class is determined by itsmembership, and no two different classes can havethe same membership. (If a class is determined bya function φx, we say that a is a “member” of the

class if φa is true.)() We must find some way of defining not only

classes, but classes of classes. We saw in Chapter II.that cardinal numbers are to be defined as classes ofclasses. The ordinary phrase of elementary mathe-matics, “The combinations of n things m at a time”represents a class of classes, namely, the class of allclasses ofm terms that can be selected out of a givenclass of n terms. Without some symbolic method ofdealing with classes of classes, mathematical logicwould break down.

() It must under all circumstances be mean-ingless (not false) to suppose a class a member ofitself or not a member of itself. This results from thecontradiction which we discussed in Chapter XIII.

() Lastly—and this is the condition which ismost difficult of fulfilment—it must be possible tomake propositions about all the classes that are com-posed of individuals, or about all the classes thatare composed of objects of any one logical “type.” Ifthis were not the case, many uses of classes wouldgo astray—for example, mathematical induction. Indefining the posterity of a given term, we need to beable to say that a member of the posterity belongsto all hereditary classes to which the given term be-longs, and this requires the sort of totality that is in

question. The reason there is a difficulty about thiscondition is that it can be proved to be impossibleto speak of all the propositional functions that canhave arguments of a given type.

We will, to begin with, ignore this last condi-tion and the problems which it raises. The firsttwo conditions may be | taken together. They state that there is to be one class, no more and no less,for each group of formally equivalent propositionalfunctions; e.g. the class of men is to be the sameas that of featherless bipeds or rational animals orYahoos or whatever other characteristic may be pre-ferred for defining a human being. Now, when wesay that two formally equivalent propositional func-tions may be not identical, although they define thesame class, we may prove the truth of the assertionby pointing out that a statement may be true of theone function and false of the other; e.g. “I believethat all men are mortal” may be true, while “I be-lieve that all rational animals are mortal” may befalse, since I may believe falsely that the Phœnixis an immortal rational animal. Thus we are led toconsider statements about functions, or (more cor-rectly) functions of functions.

Some of the things that may be said about a func-tion may be regarded as said about the class defined

by the function, whereas others cannot. The state-ment “all men are mortal” involves the functions “xis human” and “x is mortal”; or, if we choose, wecan say that it involves the classes men and mortals.We can interpret the statement in either way, be-cause its truth-value is unchanged if we substitutefor “x is human” or for “x is mortal” any formallyequivalent function. But, as we have just seen, thestatement “I believe that all men are mortal” cannotbe regarded as being about the class determinedby either function, because its truth-value may bechanged by the substitution of a formally equiva-lent function (which leaves the class unchanged).We will call a statement involving a function φxan “extensional” function of the function φx, if itis like “all men are mortal,” i.e. if its truth-valueis unchanged by the substitution of any formallyequivalent function; and when a function of a func-tion is not extensional, we will call it “intensional,”so that “I believe that all men are mortal” is an in-tensional function of “x is human” or “x is mortal.”Thus extensional functions of a function φx may, forpractical | purposes, be regarded as functions of the class determined by φx, while intensional functionscannot be so regarded.

It is to be observed that all the specific functions

of functions that we have occasion to introduce inmathematical logic are extensional. Thus, for ex-ample, the two fundamental functions of functionsare: “φx is always true” and “φx is sometimes true.”Each of these has its truth-value unchanged if anyformally equivalent function is substituted for φx.In the language of classes, if α is the class deter-mined by φx, “φx is always true” is equivalent to“everything is a member of α,” and “φx is some-times true” is equivalent to “α has members” or(better) “α has at least one member.” Take, again,the condition, dealt with in the preceding chapter,for the existence of “the term satisfying φx.” Thecondition is that there is a term c such that φx isalways equivalent to “x is c.” This is obviously ex-tensional. It is equivalent to the assertion that theclass defined by the function φx is a unit class, i.e.a class having one member; in other words, a classwhich is a member of .

Given a function of a function which may ormay not be extensional, we can always derive fromit a connected and certainly extensional functionof the same function, by the following plan: Letour original function of a function be one whichattributes to φx the property f ; then consider theassertion “there is a function having the property

f and formally equivalent to φx.” This is an exten-sional function of φx; it is true when our originalstatement is true, and it is formally equivalent to theoriginal function of φx if this original function isextensional; but when the original function is inten-sional, the new one is more often true than the oldone. For example, consider again “I believe that allmen are mortal,” regarded as a function of “x is hu-man.” The derived extensional function is: “Thereis a function formally equivalent to ‘x is human’ andsuch that I believe that whatever satisfies it is mor-tal.” This remains true when we substitute “x is arational animal” | for “x is human,” even if I believe falsely that the Phœnix is rational and immortal.

We give the name of “derived extensional func-tion” to the function constructed as above, namely,to the function: “There is a function having theproperty f and formally equivalent to φx,” wherethe original function was “the function φx has theproperty f.”

We may regard the derived extensional functionas having for its argument the class determined bythe function φx, and as asserting f of this class. Thismay be taken as the definition of a proposition abouta class. I.e. we may define:

To assert that “the class determined by the func-

tion φx has the property f” is to assert that φx sat-isfies the extensional function derived from f.

This gives a meaning to any statement about aclass which can be made significantly about a func-tion; and it will be found that technically it yieldsthe results which are required in order to make atheory symbolically satisfactory.

What we have said just now as regards the def-inition of classes is sufficient to satisfy our firstfour conditions. The way in which it secures thethird and fourth, namely, the possibility of classesof classes, and the impossibility of a class being ornot being a member of itself, is somewhat technical;it is explained in Principia Mathematica, but maybe taken for granted here. It results that, but forour fifth condition, we might regard our task ascompleted. But this condition—at once the mostimportant and the most difficult—is not fulfilled invirtue of anything we have said as yet. The difficultyis connected with the theory of types, and must bebriefly discussed.

We saw in Chapter XIII. that there is a hierarchyof logical types, and that it is a fallacy to allow an

See Principia Mathematica, vol. i. pp. – and ∗.The reader who desires a fuller discussion should consult

Principia Mathematica, Introduction, chap. ii.; also ∗.

object belonging to one of these to be substitutedfor an object belonging to another. | Now it is not difficult to show that the various functions whichcan take a given object a as argument are not all ofone type. Let us call them all a-functions. We maytake first those among them which do not involvereference to any collection of functions; these wewill call “predicative a-functions.” If we now pro-ceed to functions involving reference to the totalityof predicative a-functions, we shall incur a fallacy ifwe regard these as of the same type as the predica-tive a-functions. Take such an every-day statementas “a is a typical Frenchman.” How shall we definea “typical Frenchman”? We may define him as one“possessing all qualities that are possessed by mostFrenchmen.” But unless we confine “all qualities”to such as do not involve a reference to any total-ity of qualities, we shall have to observe that mostFrenchmen are not typical in the above sense, andtherefore the definition shows that to be not typicalis essential to a typical Frenchman. This is not alogical contradiction, since there is no reason whythere should be any typical Frenchmen; but it il-lustrates the need for separating off qualities thatinvolve reference to a totality of qualities from thosethat do not.

Whenever, by statements about “all” or “some”of the values that a variable can significantly take,we generate a new object, this new object must notbe among the values which our previous variablecould take, since, if it were, the totality of valuesover which the variable could range would onlybe definable in terms of itself, and we should beinvolved in a vicious circle. For example, if I say“Napoleon had all the qualities that make a greatgeneral,” I must define “qualities” in such a waythat it will not include what I am now saying, i.e.“having all the qualities that make a great general”must not be itself a quality in the sense supposed.This is fairly obvious, and is the principle whichleads to the theory of types by which vicious-circleparadoxes are avoided. As applied to a-functions,we may suppose that “qualities” is to mean “pred-icative functions.” Then when I say “Napoleon hadall the qualities, etc.,” I mean | “Napoleon satisfied all the predicative functions, etc.” This statementattributes a property to Napoleon, but not a pred-icative property; thus we escape the vicious circle.But wherever “all functions which” occurs, the func-tions in question must be limited to one type if avicious circle is to be avoided; and, as Napoleon andthe typical Frenchman have shown, the type is not

rendered determinate by that of the argument. Itwould require a much fuller discussion to set forththis point fully, but what has been said may sufficeto make it clear that the functions which can take agiven argument are of an infinite series of types. Wecould, by various technical devices, construct a vari-able which would run through the first n of thesetypes, where n is finite, but we cannot construct avariable which will run through them all, and, ifwe could, that mere fact would at once generate anew type of function with the same arguments, andwould set the whole process going again.

We call predicative a-functions the first type ofa-functions; a-functions involving reference to thetotality of the first type we call the second type; andso on. No variable a-function can run through allthese different types: it must stop short at somedefinite one.

These considerations are relevant to our defini-tion of the derived extensional function. We therespoke of “a function formally equivalent to φx.” Itis necessary to decide upon the type of our function.Any decision will do, but some decision is unavoid-able. Let us call the supposed formally equivalentfunction ψ. Then ψ appears as a variable, and mustbe of some determinate type. All that we know

necessarily about the type of φ is that it takes argu-ments of a given type—that it is (say) an a-function.But this, as we have just seen, does not determineits type. If we are to be able (as our fifth requisitedemands) to deal with all classes whose membersare of the same type as a, we must be able to de-fine all such classes by means of functions of someone type; that is to say, there must be some type ofa-function, say the nth, such that any a-function isformally | equivalent to some a-function of the nth type. If this is the case, then any extensional func-tion which holds of all a-functions of the nth typewill hold of any a-function whatever. It is chieflyas a technical means of embodying an assumptionleading to this result that classes are useful. The as-sumption is called the “axiom of reducibility,” andmay be stated as follows:—

“There is a type (τ say) of a-functions such that,given any a-function, it is formally equivalent tosome function of the type in question.”

If this axiom is assumed, we use functions of thistype in defining our associated extensional function.Statements about all a-classes (i.e. all classes definedby a-functions) can be reduced to statements aboutall a-functions of the type τ . So long as only exten-sional functions of functions are involved, this gives

us in practice results which would otherwise haverequired the impossible notion of “all a-functions.”One particular region where this is vital is mathe-matical induction.

The axiom of reducibility involves all that is re-ally essential in the theory of classes. It is thereforeworth while to ask whether there is any reason tosuppose it true.

This axiom, like the multiplicative axiom andthe axiom of infinity, is necessary for certain re-sults, but not for the bare existence of deductivereasoning. The theory of deduction, as explainedin Chapter XIV., and the laws for propositions in-volving “all” and “some,” are of the very texture ofmathematical reasoning: without them, or some-thing like them, we should not merely not obtain thesame results, but we should not obtain any resultsat all. We cannot use them as hypotheses, and de-duce hypothetical consequences, for they are rulesof deduction as well as premisses. They must beabsolutely true, or else what we deduce according tothem does not even follow from the premisses. Onthe other hand, the axiom of reducibility, like ourtwo previous mathematical axioms, could perfectlywell be stated as an hypothesis whenever it is used,instead of being assumed to be actually true. We

can deduce | its consequences hypothetically; we can also deduce the consequences of supposing it false.It is therefore only convenient, not necessary. Andin view of the complication of the theory of types,and of the uncertainty of all except its most gen-eral principles, it is impossible as yet to say whetherthere may not be some way of dispensing with theaxiom of reducibility altogether. However, assum-ing the correctness of the theory outlined above,what can we say as to the truth or falsehood of theaxiom?

The axiom, we may observe, is a generalisedform of Leibniz’s identity of indiscernibles. Leibnizassumed, as a logical principle, that two differentsubjects must differ as to predicates. Now predi-cates are only some among what we called “predica-tive functions,” which will include also relations togiven terms, and various properties not to be reck-oned as predicates. Thus Leibniz’s assumption isa much stricter and narrower one than ours. (Not,of course, according to his logic, which regarded allpropositions as reducible to the subject-predicateform.) But there is no good reason for believing hisform, so far as I can see. There might quite well,as a matter of abstract logical possibility, be twothings which had exactly the same predicates, in

the narrow sense in which we have been using theword “predicate.” How does our axiom look whenwe pass beyond predicates in this narrow sense? Inthe actual world there seems no way of doubtingits empirical truth as regards particulars, owing tospatio-temporal differentiation: no two particularshave exactly the same spatial and temporal relationsto all other particulars. But this is, as it were, anaccident, a fact about the world in which we happento find ourselves. Pure logic, and pure mathematics(which is the same thing), aims at being true, inLeibnizian phraseology, in all possible worlds, notonly in this higgledy-piggledy job-lot of a world inwhich chance has imprisoned us. There is a certainlordliness which the logician should preserve: hemust not condescend to derive arguments from thethings he sees about him. |

Viewed from this strictly logical point of view, I do not see any reason to believe that the axiomof reducibility is logically necessary, which is whatwould be meant by saying that it is true in all pos-sible worlds. The admission of this axiom into asystem of logic is therefore a defect, even if the ax-iom is empirically true. It is for this reason thatthe theory of classes cannot be regarded as beingas complete as the theory of descriptions. There

is need of further work on the theory of types, inthe hope of arriving at a doctrine of classes whichdoes not require such a dubious assumption. Butit is reasonable to regard the theory outlined in thepresent chapter as right in its main lines, i.e. in itsreduction of propositions nominally about classesto propositions about their defining functions. Theavoidance of classes as entities by this method must,it would seem, be sound in principle, however thedetail may still require adjustment. It is becausethis seems indubitable that we have included thetheory of classes, in spite of our desire to exclude,as far as possible, whatever seemed open to seriousdoubt.

The theory of classes, as above outlined, reducesitself to one axiom and one definition. For the sakeof definiteness, we will here repeat them. The axiomis:

There is a type τ such that if φ is a function whichcan take a given object a as argument, then there is afunction ψ of the type τ which is formally equivalentto φ.

The definition is:If φ is a function which can take a given object a

as argument, and τ the type mentioned in the aboveaxiom, then to say that the class determined by φ has

the property f is to say that there is a function of typeτ , formally equivalent to φ, and having the property f.

Chapter XVIII

Mathematics and Logic

Mathematics and logic, historically speaking, have been entirely distinct studies. Mathematics has beenconnected with science, logic with Greek. But bothhave developed in modern times: logic has becomemore mathematical and mathematics has becomemore logical. The consequence is that it has nowbecome wholly impossible to draw a line betweenthe two; in fact, the two are one. They differ as boyand man: logic is the youth of mathematics andmathematics is the manhood of logic. This view isresented by logicians who, having spent their timein the study of classical texts, are incapable of fol-lowing a piece of symbolic reasoning, and by math-ematicians who have learnt a technique withouttroubling to inquire into its meaning or justifica-tion. Both types are now fortunately growing rarer.

So much of modern mathematical work is obviouslyon the border-line of logic, so much of modern logicis symbolic and formal, that the very close relation-ship of logic and mathematics has become obviousto every instructed student. The proof of their iden-tity is, of course, a matter of detail: starting withpremisses which would be universally admitted tobelong to logic, and arriving by deduction at resultswhich as obviously belong to mathematics, we findthat there is no point at which a sharp line can bedrawn, with logic to the left and mathematics to theright. If there are still those who do not admit theidentity of logic and mathematics, we may challengethem to indicate at what point, in the successive def-initions and | deductions of Principia Mathematica, they consider that logic ends and mathematics be-gins. It will then be obvious that any answer mustbe quite arbitrary.

In the earlier chapters of this book, starting fromthe natural numbers, we have first defined “cardinalnumber” and shown how to generalise the concep-tion of number, and have then analysed the con-ceptions involved in the definition, until we foundourselves dealing with the fundamentals of logic.In a synthetic, deductive treatment these funda-mentals come first, and the natural numbers are

only reached after a long journey. Such treatment,though formally more correct than that which wehave adopted, is more difficult for the reader, be-cause the ultimate logical concepts and propositionswith which it starts are remote and unfamiliar ascompared with the natural numbers. Also they rep-resent the present frontier of knowledge, beyondwhich is the still unknown; and the dominion ofknowledge over them is not as yet very secure.

It used to be said that mathematics is the sci-ence of “quantity.” “Quantity” is a vague word, butfor the sake of argument we may replace it by theword “number.” The statement that mathematicsis the science of number would be untrue in twodifferent ways. On the one hand, there are recog-nised branches of mathematics which have nothingto do with number—all geometry that does not useco-ordinates or measurement, for example: projec-tive and descriptive geometry, down to the point atwhich co-ordinates are introduced, does not haveto do with number, or even with quantity in thesense of greater and less. On the other hand, throughthe definition of cardinals, through the theory ofinduction and ancestral relations, through the gen-eral theory of series, and through the definitions ofthe arithmetical operations, it has become possible

to generalise much that used to be proved only inconnection with numbers. The result is that whatwas formerly the single study of Arithmetic has nowbecome divided into a number of separate studies,no one of which is specially concerned with num-bers. The most | elementary properties of numbers are concerned with one-one relations, and similar-ity between classes. Addition is concerned with theconstruction of mutually exclusive classes respec-tively similar to a set of classes which are not knownto be mutually exclusive. Multiplication is mergedin the theory of “selections,” i.e. of a certain kindof one-many relations. Finitude is merged in thegeneral study of ancestral relations, which yieldsthe whole theory of mathematical induction. Theordinal properties of the various kinds of number-series, and the elements of the theory of continu-ity of functions and the limits of functions, can begeneralised so as no longer to involve any essen-tial reference to numbers. It is a principle, in allformal reasoning, to generalise to the utmost, sincewe thereby secure that a given process of deduc-tion shall have more widely applicable results; weare, therefore, in thus generalising the reasoningof arithmetic, merely following a precept which isuniversally admitted in mathematics. And in thus

generalising we have, in effect, created a set of newdeductive systems, in which traditional arithmeticis at once dissolved and enlarged; but whether anyone of these new deductive systems—for example,the theory of selections—is to be said to belong tologic or to arithmetic is entirely arbitrary, and inca-pable of being decided rationally.

We are thus brought face to face with the ques-tion: What is this subject, which may be called in-differently either mathematics or logic? Is there anyway in which we can define it?

Certain characteristics of the subject are clear.To begin with, we do not, in this subject, deal withparticular things or particular properties: we dealformally with what can be said about any thing orany property. We are prepared to say that one andone are two, but not that Socrates and Plato are two,because, in our capacity of logicians or pure math-ematicians, we have never heard of Socrates andPlato. A world in which there were no such individ-uals would still be a world in which one and one aretwo. It is not open to us, as pure mathematicians orlogicians, to mention anything at all, because, if wedo so, | we introduce something irrelevant and not formal. We may make this clear by applying it tothe case of the syllogism. Traditional logic says: “All

men are mortal, Socrates is a man, therefore Socratesis mortal.” Now it is clear that what we mean to as-sert, to begin with, is only that the premisses implythe conclusion, not that premisses and conclusionare actually true; even the most traditional logicpoints out that the actual truth of the premisses isirrelevant to logic. Thus the first change to be madein the above traditional syllogism is to state it in theform: “If all men are mortal and Socrates is a man,then Socrates is mortal.” We may now observe thatit is intended to convey that this argument is validin virtue of its form, not in virtue of the particularterms occurring in it. If we had omitted “Socratesis a man” from our premisses, we should have hada non-formal argument, only admissible becauseSocrates is in fact a man; in that case we could nothave generalised the argument. But when, as above,the argument is formal, nothing depends upon theterms that occur in it. Thus we may substitute α formen, β for mortals, and x for Socrates, where α andβ are any classes whatever, and x is any individual.We then arrive at the statement: “No matter whatpossible values x and α and β may have, if all α’sare β’s and x is an α, then x is a β”; in other words,“the propositional function ‘if all α’s are β’s and xis an α, then x is a β’ is always true.” Here at last

we have a proposition of logic—the one which isonly suggested by the traditional statement aboutSocrates and men and mortals.

It is clear that, if formal reasoning is what we areaiming at, we shall always arrive ultimately at state-ments like the above, in which no actual things orproperties are mentioned; this will happen throughthe mere desire not to waste our time proving ina particular case what can be proved generally. Itwould be ridiculous to go through a long argumentabout Socrates, and then go through precisely thesame argument again about Plato. If our argumentis one (say) which holds of all men, we shall prove itconcerning “x,” with the hypothesis “if x is a man.”With | this hypothesis, the argument will retain its hypothetical validity even when x is not a man. Butnow we shall find that our argument would stillbe valid if, instead of supposing x to be a man, wewere to suppose him to be a monkey or a goose ora Prime Minister. We shall therefore not waste ourtime taking as our premiss “x is a man” but shalltake “x is an α,” where α is any class of individuals,or “φx” where φ is any propositional function ofsome assigned type. Thus the absence of all mentionof particular things or properties in logic or puremathematics is a necessary result of the fact that

this study is, as we say, “purely formal.”At this point we find ourselves faced with a

problem which is easier to state than to solve. Theproblem is: “What are the constituents of a logi-cal proposition?” I do not know the answer, but Ipropose to explain how the problem arises.

Take (say) the proposition “Socrates was beforeAristotle.” Here it seems obvious that we have a re-lation between two terms, and that the constituentsof the proposition (as well as of the correspondingfact) are simply the two terms and the relation, i.e.Socrates, Aristotle, and before. (I ignore the fact thatSocrates and Aristotle are not simple; also the factthat what appear to be their names are really trun-cated descriptions. Neither of these facts is relevantto the present issue.) We may represent the generalform of such propositions by “xRy,” which may beread “x has the relation R to y.” This general formmay occur in logical propositions, but no particularinstance of it can occur. Are we to infer that thegeneral form itself is a constituent of such logicalpropositions?

Given a proposition, such as “Socrates is beforeAristotle,” we have certain constituents and also acertain form. But the form is not itself a new con-stituent; if it were, we should need a new form to

embrace both it and the other constituents. We can,in fact, turn all the constituents of a propositioninto variables, while keeping the form unchanged.This is what we do when we use such a schema as“xRy,” which stands for any | one of a certain class of propositions, namely, those asserting relationsbetween two terms. We can proceed to general as-sertions, such as “xRy is sometimes true”—i.e. thereare cases where dual relations hold. This assertionwill belong to logic (or mathematics) in the sense inwhich we are using the word. But in this assertionwe do not mention any particular things or partic-ular relations; no particular things or relations canever enter into a proposition of pure logic. We areleft with pure forms as the only possible constituentsof logical propositions.

I do not wish to assert positively that pureforms—e.g. the form “xRy”—do actually enter intopropositions of the kind we are considering. Thequestion of the analysis of such propositions is adifficult one, with conflicting considerations on theone side and on the other. We cannot embark uponthis question now, but we may accept, as a first ap-proximation, the view that forms are what enter intological propositions as their constituents. And wemay explain (though not formally define) what we

mean by the “form” of a proposition as follows:—The “form” of a proposition is that, in it, that

remains unchanged when every constituent of theproposition is replaced by another.

Thus “Socrates is earlier than Aristotle” has thesame form as “Napoleon is greater than Welling-ton,” though every constituent of the two proposi-tions is different.

We may thus lay down, as a necessary (thoughnot sufficient) characteristic of logical or mathemat-ical propositions, that they are to be such as can beobtained from a proposition containing no variables(i.e. no such words as all, some, a, the, etc.) by turn-ing every constituent into a variable and assertingthat the result is always true or sometimes true, orthat it is always true in respect of some of the vari-ables that the result is sometimes true in respectof the others, or any variant of these forms. Andanother way of stating the same thing is to say thatlogic (or mathematics) is concerned only with forms,and is concerned with them only in the way of stat-ing that they are always or | sometimes true—with all the permutations of “always” and “sometimes”that may occur.

There are in every language some words whosesole function is to indicate form. These words,

broadly speaking, are commonest in languages hav-ing fewest inflections. Take “Socrates is human.”Here “is” is not a constituent of the proposition, butmerely indicates the subject-predicate form. Sim-ilarly in “Socrates is earlier than Aristotle,” “is”and “than” merely indicate form; the proposition isthe same as “Socrates precedes Aristotle,” in whichthese words have disappeared and the form is oth-erwise indicated. Form, as a rule, can be indicatedotherwise than by specific words: the order of thewords can do most of what is wanted. But thisprinciple must not be pressed. For example, it isdifficult to see how we could conveniently expressmolecular forms of propositions (i.e. what we call“truth-functions”) without any word at all. We sawin Chapter XIV. that one word or symbol is enoughfor this purpose, namely, a word or symbol express-ing incompatibility. But without even one we shouldfind ourselves in difficulties. This, however, is notthe point that is important for our present purpose.What is important for us is to observe that form maybe the one concern of a general proposition, evenwhen no word or symbol in that proposition desig-nates the form. If we wish to speak about the formitself, we must have a word for it; but if, as in math-ematics, we wish to speak about all propositions

that have the form, a word for the form will usuallybe found not indispensable; probably in theory it isnever indispensable.

Assuming—as I think we may—that the forms ofpropositions can be represented by the forms of thepropositions in which they are expressed withoutany special words for forms, we should arrive at alanguage in which everything formal belonged tosyntax and not to vocabulary. In such a languagewe could express all the propositions of mathemat-ics even if we did not know one single word of thelanguage. The language of |mathematical logic, if it were perfected, would be such a language. Weshould have symbols for variables, such as “x” and“R” and “y,” arranged in various ways; and the wayof arrangement would indicate that something wasbeing said to be true of all values or some valuesof the variables. We should not need to know anywords, because they would only be needed for giv-ing values to the variables, which is the business ofthe applied mathematician, not of the pure math-ematician or logician. It is one of the marks of aproposition of logic that, given a suitable language,such a proposition can be asserted in such a lan-guage by a person who knows the syntax withoutknowing a single word of the vocabulary.

But, after all, there are words that express form,such as “is” and “than.” And in every symbolismhitherto invented for mathematical logic there aresymbols having constant formal meanings. We maytake as an example the symbol for incompatibilitywhich is employed in building up truth-functions.Such words or symbols may occur in logic. Thequestion is: How are we to define them?

Such words or symbols express what are called“logical constants.” Logical constants may be de-fined exactly as we defined forms; in fact, they arein essence the same thing. A fundamental logicalconstant will be that which is in common among anumber of propositions, any one of which can re-sult from any other by substitution of terms onefor another. For example, “Napoleon is greaterthan Wellington” results from “Socrates is earlierthan Aristotle” by the substitution of “Napoleon”for “Socrates,” “Wellington” for “Aristotle,” and“greater” for “earlier.” Some propositions can beobtained in this way from the prototype “Socratesis earlier than Aristotle” and some cannot; thosethat can are those that are of the form “xRy,” i.e.express dual relations. We cannot obtain from theabove prototype by term-for-term substitution suchpropositions as “Socrates is human” or “the Atheni-

ans gave the hemlock to Socrates,” because the firstis of the subject- | predicate form and the second expresses a three-term relation. If we are to haveany words in our pure logical language, they mustbe such as express “logical constants,” and “logicalconstants” will always either be, or be derived from,what is in common among a group of propositionsderivable from each other, in the above manner, byterm-for-term substitution. And this which is incommon is what we call “form.”

In this sense all the “constants” that occur inpure mathematics are logical constants. The num-ber , for example, is derivative from propositionsof the form: “There is a term c such that φx is truewhen, and only when, x is c.” This is a function ofφ, and various different propositions result fromgiving different values to φ. We may (with a lit-tle omission of intermediate steps not relevant toour present purpose) take the above function of φas what is meant by “the class determined by φis a unit class” or “the class determined by φ is amember of ” ( being a class of classes). In thisway, propositions in which occurs acquire a mean-ing which is derived from a certain constant logicalform. And the same will be found to be the case withall mathematical constants: all are logical constants,

or symbolic abbreviations whose full use in a propercontext is defined by means of logical constants.

But although all logical (or mathematical) propo-sitions can be expressed wholly in terms of logi-cal constants together with variables, it is not thecase that, conversely, all propositions that can beexpressed in this way are logical. We have foundso far a necessary but not a sufficient criterion ofmathematical propositions. We have sufficiently de-fined the character of the primitive ideas in terms ofwhich all the ideas of mathematics can be defined,but not of the primitive propositions from whichall the propositions of mathematics can be deduced.This is a more difficult matter, as to which it is notyet known what the full answer is.

We may take the axiom of infinity as an exampleof a proposition which, though it can be enunciatedin logical terms, | cannot be asserted by logic to be true. All the propositions of logic have a character-istic which used to be expressed by saying that theywere analytic, or that their contradictories were self-contradictory. This mode of statement, however, isnot satisfactory. The law of contradiction is merelyone among logical propositions; it has no specialpre-eminence; and the proof that the contradictoryof some proposition is self-contradictory is likely

to require other principles of deduction besides thelaw of contradiction. Nevertheless, the characteris-tic of logical propositions that we are in search of isthe one which was felt, and intended to be defined,by those who said that it consisted in deducibilityfrom the law of contradiction. This characteristic,which, for the moment, we may call tautology, ob-viously does not belong to the assertion that thenumber of individuals in the universe is n, whatevernumber n may be. But for the diversity of types, itwould be possible to prove logically that there areclasses of n terms, where n is any finite integer; oreven that there are classes of ℵ terms. But, owingto types, such proofs, as we saw in Chapter XIII.,are fallacious. We are left to empirical observationto determine whether there are as many as n indi-viduals in the world. Among “possible” worlds, inthe Leibnizian sense, there will be worlds havingone, two, three, . . . individuals. There does not evenseem any logical necessity why there should be evenone individual—why, in fact, there should be anyworld at all. The ontological proof of the existenceof God, if it were valid, would establish the logical

The primitive propositions in Principia Mathematica are suchas to allow the inference that at least one individual exists. But Inow view this as a defect in logical purity.

necessity of at least one individual. But it is gen-erally recognised as invalid, and in fact rests upona mistaken view of existence—i.e. it fails to realisethat existence can only be asserted of something de-scribed, not of something named, so that it is mean-ingless to argue from “this is the so-and-so” and “theso-and-so exists” to “this exists.” If we reject the on-tological | argument, we seem driven to conclude that the existence of a world is an accident—i.e. it isnot logically necessary. If that be so, no principle oflogic can assert “existence” except under a hypothe-sis, i.e. none can be of the form “the propositionalfunction so-and-so is sometimes true.” Propositionsof this form, when they occur in logic, will have tooccur as hypotheses or consequences of hypotheses,not as complete asserted propositions. The com-plete asserted propositions of logic will all be suchas affirm that some propositional function is alwaystrue. For example, it is always true that if p implies qand q implies r then p implies r, or that, if all α’s areβ’s and x is an α then x is a β. Such propositions mayoccur in logic, and their truth is independent of theexistence of the universe. We may lay it down that,if there were no universe, all general propositionswould be true; for the contradictory of a generalproposition (as we saw in Chapter XV.) is a proposi-

tion asserting existence, and would therefore alwaysbe false if no universe existed.

Logical propositions are such as can be known apriori, without study of the actual world. We onlyknow from a study of empirical facts that Socratesis a man, but we know the correctness of the syl-logism in its abstract form (i.e. when it is stated interms of variables) without needing any appeal toexperience. This is a characteristic, not of logicalpropositions in themselves, but of the way in whichwe know them. It has, however, a bearing uponthe question what their nature may be, since thereare some kinds of propositions which it would bevery difficult to suppose we could know withoutexperience.

It is clear that the definition of “logic” or “math-ematics” must be sought by trying to give a new def-inition of the old notion of “analytic” propositions.Although we can no longer be satisfied to definelogical propositions as those that follow from thelaw of contradiction, we can and must still admitthat they are a wholly different class of proposi-tions from those that we come to know empirically.They all have the characteristic which, a momentago, we agreed to call “tautology.” This, | combined with the fact that they can be expressed wholly in

terms of variables and logical constants (a logicalconstant being something which remains constantin a proposition even when all its constituents arechanged)—will give the definition of logic or puremathematics. For the moment, I do not know howto define “tautology.” It would be easy to offer adefinition which might seem satisfactory for a while;but I know of none that I feel to be satisfactory, inspite of feeling thoroughly familiar with the char-acteristic of which a definition is wanted. At thispoint, therefore, for the moment, we reach the fron-tier of knowledge on our backward journey into thelogical foundations of mathematics.

We have now come to an end of our somewhatsummary introduction to mathematical philosophy.It is impossible to convey adequately the ideas thatare concerned in this subject so long as we abstainfrom the use of logical symbols. Since ordinarylanguage has no words that naturally express ex-actly what we wish to express, it is necessary, solong as we adhere to ordinary language, to strainwords into unusual meanings; and the reader is

The importance of “tautology” for a definition of mathemat-ics was pointed out to me by my former pupil Ludwig Wittgen-stein, who was working on the problem. I do not know whetherhe has solved it, or even whether he is alive or dead.

sure, after a time if not at first, to lapse into attach-ing the usual meanings to words, thus arriving atwrong notions as to what is intended to be said.Moreover, ordinary grammar and syntax is extraor-dinarily misleading. This is the case, e.g., as regardsnumbers; “ten men” is grammatically the same formas “white men,” so that might be thought to bean adjective qualifying “men.” It is the case, again,wherever propositional functions are involved, andin particular as regards existence and descriptions.Because language is misleading, as well as becauseit is diffuse and inexact when applied to logic (forwhich it was never intended), logical symbolism isabsolutely necessary to any exact or thorough treat-ment of our subject. Those readers, | therefore, who wish to acquire a mastery of the principles of math-ematics, will, it is to be hoped, not shrink from thelabour of mastering the symbols—a labour whichis, in fact, much less than might be thought. As theabove hasty survey must have made evident, thereare innumerable unsolved problems in the subject,and much work needs to be done. If any studentis led into a serious study of mathematical logic bythis little book, it will have served the chief purposefor which it has been written.

Index

[Online edition note: This is a hyperlinked recre- ation of the original index. The page numbers listedare for the original edition, i.e., those marked in themargins of this edition.]

Aggregates, .Alephs, , , ,.

Aliorelatives, .All, ff.Analysis, .Ancestors, , .Argument of a

function, , .Arithmetising of

mathematics, .Associative law, ,.

Axioms, .

Between, ff., .Bolzano, n.Boots and socks, .Boundary, , , .

Cantor, Georg, , ,n., , , , ,.

Classes, , , ff.;reflexive, , ,; similar, , .

Clifford, W. K., .Collections, infinite,.

Commutative law, ,.

Conjunction, .Consecutiveness, ,, .

Constants, .Construction, method

of, .Continuity, , ff.;

Cantorian, ff.;Dedekindian, ; inphilosophy, ; offunctions,ff.

Contradictions,ff.

Convergence, .Converse, , , .Correlators, .Counterparts, objective,.

Counting, , .

Dedekind, , ,n.

Deduction, ff.Definition, ;

extensional andintensional, .

Derivatives, .Descriptions, , ,ff.

Dimensions, .Disjunction, .Distributive law, ,.

Diversity, .Domain, , , .

Equivalence, .Euclid, .Existence, , ,.

Exponentiation, ,.

Extension of a relation,.

Fictions, logical, n.,

, .Field of a relation, ,.

Finite, .Flux, .Form, .Fractions, , .Frege, , [], n.,, , n.

Functions, ;descriptive, , ;intensional andextensional, ;predicative, ;propositional, ,, ff.

Gap, Dedekindian,ff., .

Generalisation, .Geometry, , , ,, , ;analytical, ,.

Greater and less, , .

Hegel, .Hereditary properties,.

Implication, , ;formal, .

Incommensurables, ,.

Incompatibility, ff.,.

Incomplete symbols,.

Indiscernibles, .Individuals, , ,.

Induction,mathematical, ff.,, ,.

Inductive properties,.

Inference, ff.Infinite, ; of rationals,; Cantorian, ; ofcardinals, ff.; andseries and ordinals,

ff.Infinity, axiom of, n.,, ff., .

Instances, .Integers, positive and

negative, .Intervals, .Intuition, .Irrationals, , . |

Kant, .

Leibniz, , , .Lewis, C. I., , .Likeness, .Limit, , ff., ff.; of

functions,ff.

Limiting points, .Logic, , , ff.;

mathematical, v, ,.

Logicising ofmathematics, .

Maps, , ff., .

Mathematics, ff.Maximum, , .Median class, .Meinong, .Method, vi.Minimum, , .Modality, .Multiplication, ff.Multiplicative axiom,, ff.

Names, , .Necessity, .Neighbourhood, .Nicod, , ,n.

Null-class, , .Number, cardinal, ff.,, ff., ; complex,ff.; finite, ff.;inductive, , , ;infinite, ff.;irrational, , ;maximum ? ;multipliable, ;natural, ff., ;

non-inductive, ,; real, , , ;reflexive, , ;relation, , ; serial,.

Occam, .Occurrences, primary

and secondary,.

Ontological proof,.

Order, ff.; cyclic,.

Oscillation, ultimate,.

Parmenides, .Particulars, ff.,.

Peano, ff., , , ,, , .

Peirce, n.Permutations, .Philosophy,

mathematical, v, .

Plato, .Plurality, .Poincare, .Points, .Posterity, ff.; proper,.

Postulates, , .Precedent, .Premisses of arithmetic,.

Primitive ideas andpropositions, ,.

Progressions, , ff.Propositions, ;

analytic, ;elementary, .

Pythagoras, , .

Quantity, , .

Ratios, , , ,.

Reducibility, axiomof, .

Referent, .

Relation-numbers,ff.

Relations,asymmetrical, , ;connected, ;many-one, ;one-many, , ;one-one, , , ;reflexive, ; serial,; similar, ff.;squares of, ;symmetrical, , ;transitive, , .

Relatum, .Representatives, .Rigour, .Royce, .

Section, Dedekindian,ff.; ultimate,.

Segments, , .Selections, ff.Sequent, .Series, ff.; closed,; compact, , ,

; condensed initself, ;Dedekindian, , ,; generation of, ;infinite, ff.; perfect,, ;well-ordered, ,.

Sheffer, .Similarity, of classes,ff.; of relations,ff., .

Some, ff.Space, , , .Structure, ff.Sub-classes, ff.Subjects, .Subtraction, .Successor of a number,, .

Syllogism, .

Tautology, , .The, , ff.Time, , , .Truth-function, .

Truth-value, .Types, logical, ,ff., , .

Unreality, .

Value of a function, ,.

Variables, , ,.

Veblen, .

Verbs, .

Weierstrass, , .Wells, H. G., .Whitehead, , , ,.

Wittgenstein, n.

Zermelo, , .Zero, .

Changes to Online Edition

This Online Corrected Edition was created by KevinC. Klement; this is version . (February , ).It is based on the April so-called “second edi-tion” published by Allen & Unwin, which, by con-temporary standards, was simply a second printingof the original edition but incorporating vari-ous, mostly minor, fixes. This edition incorporatesfixes from later printings as well, and some newfixes, mentioned below. The pagination of the Allen& Unwin edition is given in the margins, with pagebreaks marked with the sign “|”. These are in red,as are other additions to the text not penned byRussell.

Thanks to members of the Russell-l and HEAPS-l mailing lists for help in checking and proofreadingthe version, including Adam Killian, Pierre Grenon,David Blitz, Brandon Young, Rosalind Carey, and,especially, John Ongley. A tremendous debt of

thanks is owed to Kenneth Blackwell of the BertrandRussell Archives/Research Centre, McMaster Uni-versity, for proofreading the bulk of the edition,checking it against Russell’s handwritten manu-script, and providing other valuable advice and as-sistance. Another large debt of gratitude is owed toChristof Graber who compared this version to theprint versions and showed remarkable aptitude inspotting discrepancies. I take full responsibility forany remaining errors. If you discover any, pleaseemail me at klement@philos.umass.edu.

The online edition differs from the Allen& Unwin edition, and reprintings thereof, in cer-tain respects. Some are mere stylistic differences.Others represent corrections based on discrepanciesbetween Russell’s manuscript and the print edition,or fix small grammatical or typographical errors.The stylistic differences are these:

• In the original, footnote numbering beginsanew with each page. Since this version usesdifferent pagination, it was necessary to num-ber footnotes sequentially through each chap-ter. Thus, for example, the footnote listed asnote on page of this edition was listed asnote on page of the original.

• With some exceptions, the Allen & Unwin edi-tion uses linear fractions of the style “x/y”mid-paragraph, but vertical fractions of theform “ xy ” in displays. Contrary to this usualpractice, those in the display on page of theoriginal (page of this edition) were linear,but have been converted to vertical fractionsin this edition. Similarly, the mid-paragraphfractions on pages , , and of theoriginal (pages , , and here) wereprinted vertically in the original, but here arehorizontal.

The following more significant changes and revi-sions are marked in green in this edition. Most ofthese result from Ken Blackwell’s comparison withRussell’s manuscript. A few were originally noted inan early review of the book by G. A. Pfeiffer (Bulletinof the American Mathematical Society : (), pp.–).

. (page n. / original page n.) Russell wrote thewrong publication date () for the secondvolume of Principia Mathematica; this has beenfixed to .

. (page / original page ) “. . . or all that areless than . . . ” is changed to “. . . or all

that are not less than . . . ” to match Rus-sell’s manuscript and the obviously intendedmeaning of the passage. This error was notedby Pfeiffer in but unfixed in Russell’slifetime.

. (page / original page ) “. . . either by lim-iting the domain to males or by limiting theconverse to females” is changed to “. . . eitherby limiting the domain to males or by limit-ing the converse domain to females”, which ishow it read in Russell’s manuscript, and seemsbetter to fit the context.

. (page / original page ) “. . . providedneither m or n is zero.” is fixed to “. . . pro-vided neither m nor n is zero.” Thanks to JohnOngley for spotting this error, which existseven in Russell’s manuscript.

. (page n. / original page n.) The word“deutschen” in the original’s (and the manu-script’s) “Jahresbericht der deutschen Mathemat-iker-Vereinigung” has been capitalized.

. (page / original page ) “. . . of a classα, i.e. its limits or maximum, and then . . . ”is changed to “. . . of a class α, i.e. its limit or

maximum, and then . . . ” to match Russell’smanuscript, and the apparent meaning of thepassage.

. (page / original page ) “. . . the limitof its value for approaches either from . . . ”is changed to “. . . the limit of its values forapproaches either from . . . ”, which matchesRussell’s manuscript, and is more appropriatefor the meaning of the passage.

. (page / original page ) The ungram-matical “. . . advantages of this form of defini-tion is that it analyses . . . ” is changed to “. . .advantage of this form of definition is that itanalyses . . . ” to match Russell’s manuscript.

. (page / original page ) “. . . all termsz such that x has the relation P to x and z hasthe relation P to y . . . ” is fixed to “. . . all termsz such that x has the relation P to z and z hasthe relation P to y . . . ” Russell himself hand-corrected this in his manuscript, but not in aclear way, and at his request, it was changedin the printing.

. (page / original page ) The words “cor-relator of α with β, and similarly for every

other pair. This requires a”, which constituteexactly one line of Russell’s manuscript, wereomitted, thereby amalgamating two sentencesinto one. The missing words are now restored.

. (page / original page ) The passage “. . .if x is the member of y, x is a member of y,x is a member of y, and so on; then . . . ” ischanged to “. . . if x is the member of γ, x isa member of γ, x is a member of γ, and soon; then . . . ” to match Russell’s manuscript,and the obviously intended meaning of thepassage.

. (page / original page ) The words “andthen the idea of the idea of Socrates” althoughpresent in Russell’s manuscript, were left outof previous print editions. Note that Russellmentions “all these ideas” in the next sen-tence.

. (pages – / original page ) The twofootnotes on this page were misplaced. Thesecond, the reference to Principia Mathematica∗, was attached in previous versions to thesentence that now refers to the first footnotein the chapter. That footnote was placed threesentences below. The footnote references have

been returned to where they had been placedin Russell’s manuscript.

. (page / original page ) “. . . the nega-tion of propositions of the type to which xbelongs . . . ” is changed to “. . . the negation ofpropositions of the type to which φx belongs. . . ” to match Russell’s manuscript. This isanother error noted by Pfeiffer.

. (page / original page ) “Suppose weare considering all “men are mortal”: we will. . . ” is changed to “Suppose we are consid-ering “all men are mortal”: we will . . . ” tomatch the obviously intended meaning of thepassage, and the placement of the openingquotation mark in Russell’s manuscript (al-though he here used single quotation marks,as he did sporadically throughout). Thanks toChristof Graber for spotting this error.

. (page / original page ) “. . . as opposedto specific man.” is fixed to “. . . as opposedto specific men.” Russell sent this change toUnwin in , and it was made in the printing.

. (page / original page ) The “φ” in “. . .

the process of applying general statementsabout φx to particular cases . . . ”, present inRussell’s manuscript, was excluded from theAllen & Unwin printings, and has been re-stored.

. (page / original page ) The “φ” in “. . .resulting from a propositional function φx bythe substitution of . . . ” was excluded fromprevious published versions, though it doesappear in Russell’s manuscript, and seemsnecessary for the passage to make sense.Thanks to John Ongley for spotting this er-ror, which had also been noted by Pfeiffer.

. (page / original pages –) The twooccurrences of “φ” in “. . . extensional func-tions of a function φx may, for practical pur-poses, be regarded as functions of the classdetermined by φx, while intensional functionscannot . . . ” were omitted from previous pub-lished versions, but do appear in Russell’smanuscript. Again thanks to John Ongley.

. (page / original page ) The Allen &Unwin printings have the sentence as “Howshall we define a “typical” Frenchman?” Here,the closing quotation mark has been moved

to make it “How shall we define a “typicalFrenchman”?” Although Russell’s manuscriptis not entirely clear here, it appears the latterwas intended, and it also seems to make moresense in context.

. (page / original page ) “There is a type(r say) . . . ” has been changed to “There is atype (τ say) . . . ” to match Russell’s manu-script, and conventions followed elsewhere inthe chapter.

. (page / original page ) “. . . dividedinto numbers of separate studies . . . ” has beenchanged to “. . . divided into a number of sep-arate studies . . . ” Russell’s manuscript justhad “number”, in the singular, without theindefinite article. Some emendation was nec-essary to make the passage grammatical, butthe fix adopted here seems more likely whatwas meant.

. (page / original page ) The passage“the propositional function ‘if all α’s are β andx is an α, then x is a β’ is always true” has beenchanged to “the propositional function ‘if allα’s are β’s and x is an α, then x is a β’ is alwaystrue” to match Russell’s manuscript, as well

as to make it consistent with the other para-phrase given earlier in the sentence. Thanksto Christof Graber for noticing this error.

. (page / original page ) “. . . withoutany special word for forms . . . ” has beenchanged to “. . . without any special words forforms . . . ”, which matches Russell’s manu-script and seems to fit better in the context.

. (page / original page ) The originalindex listed a reference to Frege on page ,but in fact, the discussion of Frege occurs onpage . Here, “” is crossed out, and “[]”inserted.

Some very minor corrections to punctuation havebeen made to the Allen & Unwin printing, butnot marked in green.

a) Ellipses have been regularized to three closeddots throughout.

b) (page / original page ) “We may definetwo relations . . . ” did not start a new para-graph in previous editions, but does in Rus-sell’s manuscript, and is changed to do so.

c) (page / original page ) What appears inthe and later printings as “. . . is the field

of Q. and which is . . . ” is changed to “. . . isthe field of Q, and which is . . . ”

d) (page / original page ) “. . . a relationnumber is a class of . . . ” is changed to “. . .a relation-number is a class of . . . ” to matchthe hyphenation in the rest of the book (andin Russell’s manuscript). A similar change ismade in the index.

e) (page / original page ) “. . . and “feath-erless biped,”—so two . . . ” is changed to “. . .and “featherless biped”—so two . . . ”

f) (pages – / original pages –) Onemisprint of “progession” for “progression”,and one misprint of “progessions” for “pro-gressions”, have been corrected. (Thanks toChristof Graber for noticing these errors inthe original.)

g) (page / original page ) In the Allen& Unwin printing, the “s” in “y’s” in whatappears here as “Form all such sections forall y’s . . . ” was italicized along with the “y”.Nothing in Russell’s manuscript suggests itshould be italicized, however. (Again thanksto Christof Graber.)

h) (page / original page ) In the Allen& Unwin printing, “Let y be a member of β

. . . ” begins a new paragraph, but it does not inRussell’s manuscript, and clearly should not.

i) (page / original pages –) The phrase“well ordered” has twice been changed to “well-ordered” to match Russell’s manuscript (in thefirst case) and the rest of the book (in the sec-ond).

j) (page / original page ) “The way inwhich the need for this axiom arises may beexplained as follows:—One of Peano’s . . . ” ischanged to “The way in which the need forthis axiom arises may be explained as follows.One of Peano’s . . . ” and has been made tostart a new paragraph, as it did in Russell’smanuscript.

k) (page / original page ) The accent on“Metaphysique”, included in Russell’s manu-script but left off in print, has been restored.

l) (page / original page ) “. . . or whatnot,—and clearly . . . ” is changed to “. . . orwhat not—and clearly . . . ”

m) (page / original page ) Italics havebeen added to one occurrence of “Waverley”to make it consistent with the others.

n) (page / original page ) “. . . most diffi-cult of fulfilment,—it must . . . ” is changed to

“. . . most difficult of fulfilment—it must . . . ”o) (page / original page ) In the Allen &

Unwin printings, “Socrates” was not italicizedin “. . . we may substitute α for men, β for mor-tals, and x for Socrates, where . . . ” Russell hadmarked it for italicizing in the manuscript,and it seems natural to do so for the sake ofconsistency, so it has been italicized.

p) (page / original page ) The word “seem”was not italicized in “. . . a definition whichmight seem satisfactory for a while . . . ” in theAllen & Unwin editions, but was marked to bein Russell’s manuscript; it is italicized here.

q) (page / original page ) Under “Rela-tions” in the index, “similar, ff;” has beenchanged to “similar, ff.;” to match the punc-tuation elsewhere.

There are, however, a number of other places wherethe previous print editions differ from Russell’smanuscript in minor ways that were left unchangedin this edition. For a detailed examination of thedifferences between Russell’s manuscript and theprint editions, and between the various printingsthemselves (including the changes from the to the printings not documented here), seeKenneth Blackwell, “Variants, Misprints and a Bib-

liographical Index for Introduction to MathematicalPhilosophy”, Russell n.s. (): –.

p Bertrand Russell’s Introduction to MathematicalPhilosophy is in the Public Domain.See http://creativecommons.org/

cba This typesetting (including LATEX code)and list of changes are licensed under a CreativeCommons Attribution—Share Alike .United StatesLicense.See http://creativecommons.org/licenses/by-sa/./us/